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# Keras Image Regression Example

even my model gives very less. To begin with, we will define the model. Linear regression is a linear model, e. Dillon, and the TensorFlow Probability team BackgroundAt the 2019 TensorFlow Dev Summit, we announced Probabilistic Layers in TensorFlow Probability (TFP). 0 and decided to revisit my three basic examples — Iris (multi-class classification), Banknote (binary classification), and Boston (regression). Step 2 - Loading the data and performing basic data checks. Model Training with VGG16. Logistic regression with Keras. image import ImageDataGenerator. ImageNet VGG16 Model with Keras. models import Sequential. Each image is a matrix: with shape (28, 28). We'll create sample regression dataset, build the model, train it, and predict the input data. This is a slightly polished and packaged version of the Keras CRNN implementation and the published CRAFT text detection model. As of version 2. models import load_model #Define a model_regression. Next Page. One Variable. You'll use both TensorFlow core and Keras to implement this logistic regression algorithm. Sep 09, 2021 · I do not see any much work on the image regression so I write this for you guys import ModelCheckpoint from tensorflow. 19 th November, 2020: Initial release; 20 th November, 2020: Removed two images. The following are 6 code examples for showing how to use keras. We have build a model using Keras library (Python) and trained it to make predictions. It uses VBScript language for automation. org, and download the Anaconda set up. Evaluation is a process during development of the model to check whether the model is best fit for the given problem and corresponding data. Keras is a deep learning library that wraps the efficient numerical libraries Theano and TensorFlow. Let's see code. Use the global keras. Regression is a process where a model learns to predict a continuous value output for a given input data, e. Import modules and sample image. [ ] !pip install autokeras. models import Sequential. To begin with, we'll need a dataset to train on. We will generate some (mostly) random data and then fit a line to it using stochastic gradient descent. Logistic Regression (LR) is a simple yet quite effective method for carrying out binary classification tasks. Keras: Image Forecasting Keras: Tabular Regression Please add keras. create_mlp(trainX. Technically, it is possible to gather training and test data independently to build the classifier. Keras takes data in a different format and so, you must first reformat the data using datasetslib: x_train_im = mnist. 3% accuracy. Linear Regression with Keras. The first step involves creating a Keras model with the Sequential() constructor. preprocessing. Use hyperparameter optimization to squeeze more performance out of your model. In this fashion, the variational autoencoders can be used as generative models in order to generate fake data. Another fancy name for that kind of problem is regression problem. append(image). Replacing the original batch of images with a new randomly transformed batch. A Generative Adversarial Networks tutorial applied to Image Deblurring with the Keras library. You'll use both TensorFlow core and Keras to implement this logistic regression algorithm. We then build our model using keras. get_file( "mountains. The Fashion MNIST dataset is a part of the available datasets present in the tf. imread ( image_path, cv2. from keras import backend as K. fit (X_train, y_train, batch_size=1, epochs=30, shuffle=False) Computation is done in batches with each size = 1. This tutorials covers: Generating sample dataset Building the model. preprocessing. Example #4: Image Captioning with Attention In this example, we train our model to predict a caption for an image. Keras Convolution Neural Network Layers and Working. Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow, 2nd Edition. Image Classification: image classification using the Fashing MNIST dataset. In keras we define model either as Sequential, or by using the functional API which allows us to build more complicated architectures. CS229: Machine Learning (Details for Fall quarter will be updated soon) Course Description This course provides a broad introduction to machine learning and statistical pattern recognition. The model is defined in two steps. Now we resize the image to the model input size and reshape it adding another axis to the image making it 1 x h x w x 3, and pass that to the model. Introduction. Inception Inceptionresnet 1d 2d Tensorflow Keras ⭐ 2. Regression data can be easily fitted with a Keras Deep Learning API. In this post, Keras CNN used for image classification uses the Kaggle Fashion MNIST dataset. from tensorflow. """ from keras_preprocessing. First and foremost, we will need to get the image data for training the model. AutoKeras also accepts images of three dimensions with the channel: dimension at last, e. Whether you're developing a Keras model from the ground-up or you're bringing an existing model into the cloud, Azure Machine Learning can help you build production-ready models. Quick Test Professional (QTP): HP Quick Test Professional is automated software designed to automate functional and regression test cases. get_file( "mountains. Data preparation For Financial Time Series Forecasting. com/drive/1zjh0tUPYJYgJJunpLC9fW5uf--O0LKeZ?usp=sharingConv1D in Keras playlist: https://youtube. In this post we'll use Keras to build the hello world of machine learning, classify a number in an image from the MNIST database of handwritten digits, and achieve ~99% classification accuracy using a convolutional neural network. Work your way from a bag-of-words model with logistic regression to more advanced methods leading to convolutional neural networks. 2f percent dog. In this section, you will learn about Keras code which will be used to train the neural network for predicting Boston housing price. 1 Solve a linear regression problem with an example. Basic Regression. Image object, regression reducers can be used with reduceRegion or reduceRegions to perform linear regression on the pixels in the region(s. predict price, length, width, etc. Keras Convolution Neural Network Layers and Working. Start with a single-variable linear regression, to predict MPG from …. Image classification of rust via Transfer-Learning Image classification flow. model script: Replace tf. Part 4 - Prediction using Keras. For image classification we use a Dense layer with the number of output neurons equal to the number of classes, NUM_CLASSES. com/watch?v=EF7LBSCfW7cIn this vi. Note that, the value of image_data_format is Now, let's build a simple example to implement linear regression using Keras Sequential model. Sep 09, 2021 · I do not see any much work on the image regression so I write this for you guys import ModelCheckpoint from tensorflow. they regressed to the mean of the parents). predict (img_array) score = predictions [0] print ("This image is %. This guide demonstrates a step-by-step implementation of a Normalized X-Corr model using Keras, which is a modification of a Siamese network 2. models import Sequential from keras. Explore a preview version of Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow, 2nd Edition right now. See full list on analyticsvidhya. This tutorials covers: Generating sample dataset Building the model. Use hyperparameter optimization to squeeze more performance out of your model. Decoder part of autoencoder will try to reverse the. In this article, we will learn step by step, how to tune a Keras deep learning regression model and identify the best set of hyperparameters. keras for Oxford IIIT Pet Image dataset containing 7,390 images of pets spanning across 37 different classes. This example shows how to fine-tune the recognizer using an existing dataset. Dataset that yields batches of images from the subdirectories class_a and class_b, together with labels 0 and 1 (0 corresponding to class_a and 1 corresponding to class_b). With a set of fonts, backgrounds, and alphabet, we now build our data generators. Selenium: This is an open source tool used for automating web applications. After two introductory tutorials, its time to build our first neural network! The network we are building solves a simple regression problem. Part 3: Combining categorical, numerical, and …. validation_split: Float between 0 and 1. We will also see how data augmentation helps in improving the performance of the network. add this line to get the input image from cam. Just notice that it tries to detect one object. My introduction to Neural Networks covers everything you need to know (and. 1) is using GPU: from keras import backend as K K. probabilities that a certain object is present in the image, then we can use ELI5 to check what is it in the image that made the model predict a certain class score. First and foremost, we will need to get the image data for training the model. This article discusses the concepts behind image generation and the code implementation of Variational Autoencoder with a practical example using TensorFlow Keras. The difference between the two is mostly due to the regularization term being added to the loss during training (worth about 0. how much a particular person will spend on buying a car) for a customer based on the following attributes:. It may sound quite complicated, but the available libraries, including Keras, Tensorflow, Theano and scikit-learn. 11), for the green one to the value of one (0. Finally, we ask the model to compute the 'accuracy' metric, which is the percentage of correctly classified images. model_selection import train_test_split X_train, X_test, y_train, y_test = train_test_split (X, output_category, test_size=0. Lets define network which takes two inputs and gives three outputs. The first line of code below calls for the Sequential …. More specifically, that y can be calculated from a linear combination of the input variables (x). This tutorials covers: Generating sample dataset Building the model. image_regression - Colaboratory. tl;dr: keras-pandas allows users to rapidly build and iterate on deep learning models. preprocessing import StandardScaler. Text Classification: text classification using the IMDB dataset. Doing this is the same process as we've needed to do to train the model, so we'll be recycling quite a bit of code. Keras Installation. In that tutorial, we neglected a step which for real-life problems is very vital. This is a slightly polished and packaged version of the Keras CRNN implementation and the published CRAFT text detection model. Classify structured data using Keras Preprocessing Layers. Replacing the original batch of images with a new randomly transformed batch. Image classification is used to solve several Computer Vision problems; right from medical diagnoses, to surveillance systems, on to monitoring agricultural farms. add () method:. resize ( cv2. keras and how to use them, in many situations you need to define your own custom metric because the […]. Daniel Hnyk. Keras metrics are functions that are used to evaluate the performance of your deep learning model. Convert output of the model to usable bounding box tensors. Input is. Use hyperparameter optimization to squeeze more performance out of your model. The sklearn train_test_split() function is able to split the data by giving it the test ratio and the rest is, of course, the training size. To do that, we have the following, which includes support for an augmenter to. Maybe you can use this for your purposes. Linear regression model is initialized with weights w: 0. models import Sequential. It means we will treat prediction targets of MNIST dataset, which are integers ranging from 0 to. My first step was to reproduce hardmaru's work with CPPN. Batch size refers to the number of training examples utilized in one iteration. add this line to get the input image from cam. We created a CNN model and trained it to classify Covid-19 chest X-ray scans and normal chest X-ray scans. In this episode, we'll introduce MobileNets, a class of light weight deep convolutional neural networks that are vastly smaller in size and faster in performance than many other popular models. linear_model provides you Logistic Regression class; you can also use it to make the model. The first line of code below calls for the Sequential …. tasks for other examples): A ready-to-use TCN model can be used that way (cf. Model Evaluation. which are scaled to 28 by 28. The image generator generates (image, lines) tuples where image is a HxWx3 image and lines is a list of lines of text in the. Keras has Scikit-learn API. #Load trained model from keras. 1) is using GPU: from keras import backend as K K. This comment has been minimized. You just trained your very first logistic regression model using TensorFlow for classifying handwritten digit images and got 74. TensorFlow's new 2. Now, see how writing the same model in Keras makes this. Our example goes like this - The first step is to import the necessary libraries and load the. _get_available_gpus() You need to a d d the following block after importing keras if you are working on a machine, for example, which have 56 core cpu, and a gpu. But here, we see the implementation of Logistic Regression using Keras. By the end of this article, you should be able to:. a model that assumes a linear relationship between the input variables (x) and the single output variable (y). layers import Conv2D, MaxPooling2D. MNIST consists of 28 x 28 grayscale images of handwritten digits like these: The dataset also includes labels for each image, telling us which digit it is. linear_model provides you Logistic Regression class; you can also use it to make the model. Note: Creating 5 outputs/targets/labels for this example, but the method easily extends to any number or outputs. I introduced Keras in mishimasyk#9. Work your way from a bag-of-words model with logistic regression to more advanced methods leading to convolutional neural networks. Before going ahead and looking at the Python / Keras code examples and related concepts, you may want to check my post on Convolution Neural Network – Simply Explained in order to get a good understanding of CNN concepts. YOLOv3 has relatively speedy inference times with it taking roughly 30ms per inference. It was created with TensorFlow 2. There are two main types of models available in Keras: the Sequential model, and the Model class used with the functional API. We applied data augmentation to increase the size of our dataset. But here, we see the implementation of Logistic Regression using Keras. In this page, we will discuss how to interpret a regression model when some variables in the model have been log transformed. One of the most popular examples regarding multiclass classification is to label the images of handwritten digits. Lets define network which takes two inputs and gives three outputs. Work your way from a bag-of-words model with logistic regression to more advanced methods leading to convolutional neural networks. load_img('dog. Choosing a good metric for your problem is usually a difficult task. Output the dimension for conv2d_2 will be [ (28-3+2*0)/1] + 1 which is 26. On mobile: the mobile phones the inference with it own processor, using Tensorflow. shuffle () Replace tf. preprocessing import image import requests from skimage. datasets import mnist. Non Linear Regression Example with Keras and Tensorflow Backend. For example, the model focuses near the surfboard in the image when it predicts the word "surfboard". CV_LOAD_IMAGE_COLOR ), ( image_size … › Posted at 1 day ago. model = tf. VGG16 (weights='imagenet', include_top=True) # If you are only. model_selection import train_test_split X_train, X_test = train_test_split(X, test_size= 0. Posted on June 12, 2017 by charleshsliao. Multi label Image Classification. We'll also see how we can work with MobileNets in code using TensorFlow's Keras API. No need to install any software to run code. This Embedding() layer takes the size of the vocabulary. These examples are extracted from open source projects. Part 6 (Section 15-18) - End-to-End Image Recognition project in Python In this section we build a complete image recognition project on colored images. Part 3: Combining categorical, numerical, and …. 1, random_state= 42). Getting data formatted and into keras can be tedious, time consuming, and difficult, whether your a veteran or new to Keras. TensorFlow is one of the top preferred frameworks for deep learning processes. For this project, I have imported numpy and Keras packages only. Here's a comprehensive developer's guide for implementing an image classification and prediction system build with Keras. This type of problem comes under multi label image classification where an instance can be classified into multiple classes among the predefined classes. hdf5" save. You can even do regression tasks that takes images as inputs and target values as outputs and it also supports multiple numerical target columns, so now you can create multi output neural networks easily. For a particular problem, the baseline helps model developers quantify the minimal expected performance that a new model must achieve for the new model to be useful. keras datasets API. Simple Linear Regression model in Keras. image import. Getting Started With Semantic Segmentation Using TensorFlow Keras. We'll use the Boston housing price regression dataset which comes with Keras by default - that'll make the example easier to follow. from keras. Image classification using Convolutional Neural Network (CNN) in tensorflow. This API specifies how software components should interact. This tutorial focuses more on using this model with AI Platform than on the design of the model itself. [ ] ↳ 1 cell hidden. You can find a complete example of this strategy on applied on a specific example on GitHub where codes of data generation as well as the Keras script are available. layers import Activation, Dropout, Flatten, Dense. VGG16 is a built-in neural network in Keras that is pre-trained for image recognition. These heads are shared between all the feature maps of the feature pyramid. It has three main arguments, Test data. 17 which is quite close to the actual median price of \$21,600. You just trained your very first logistic regression model using TensorFlow for classifying handwritten digit images and got 74. Source: opengraph. image import ImageDataGenerator, array_to_img, img_to_array, load_img datagen = ImageDataGenerator (rotation_range = 40, width_shift_range = 0. For example, I really like the implementation of keras to build image analogies. Transfer learning means we use a pretrained model and fine tune the model on new data. SavviHub overview. from sklearn. load_images(x_train). I was trying to implement a regression model in Keras, but am unable to figure out how to calculate the score of my model, i. The first layer in the network, as per the architecture diagram shown previously, is a word embedding layer. The dataset can be downloaded here. So, if we run the preprocess_input() function from Keras on input images will normalize each pixel to a standard range. In this python Colab tutorial you will learn: How to train a Keras model using the ImageDataGenerator class. segmentation. predict() , i get a constant value for all input. vgg_model = applications. model_selection import train_test_split X_train, X_test = train_test_split(X, test_size= 0. Sep 09, 2021 · I do not see any much work on the image regression so I write this for you guys import ModelCheckpoint from tensorflow. Let's see code. import numpy as np import pandas as pd from keras. keras-ocr has a simple method for this for English, but anything that generates strings of characters in your selected alphabet will do!. siamese network with triplet loss in keras 99 across the various network architectures. We resize all images into 128*128. New! - Google Colab version of this code is available in this link. The output of yolo_model is a (m, 19, 19, 5, 85) tensor that needs to pass through non-trivial processing and conversion. Step 2 - Loading the data and performing basic data checks. you need to understand which metrics are already available in Keras and tf. Today I'm going to write about a kaggle competition I started working on recently. Compile Keras Models¶. hdf5" save. import autokeras as ak. Regression data can be easily fitted with a Keras Deep Learning API. Logistic regression with Keras. resize ( cv2. For the tutorial. Use computer vision, TensorFlow, and Keras for image classification and processing. By me, I assume most TF developers had a little hard time with TF 2. In this page, we will discuss how to interpret a regression model when some variables in the model have been log transformed. This article discusses the concepts behind image generation and the code implementation of Variational Autoencoder with a practical example using TensorFlow Keras. Keras is a high-level library that is available as part of TensorFlow. Step3: Click on the next button to move forward. X = data["image"]. Keras offers the very nice model. In our case, we’re going to create a simple, one-dimensional linear regression model to test TensorFlow and Keras. from keras. Keras Fit Function (R): Train Regression Model with multiple Labels Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern) 2019 Moderator Election Q&A - Questionnaire 2019 Community Moderator Election ResultsHow to calculate accuracy on keras model with. Logistic regression with Keras. model = tf. By using Kaggle, you agree to our use of cookies. In this example, we will be using the famous CIFAR-10 dataset. models import Sequential. Keras is a high-level library that is available as part of TensorFlow. For image classification we use a Dense layer with the number of output neurons equal to the number of classes, NUM_CLASSES. Advertisements. Semantic Segmentation laid down the fundamental path to advanced Computer Vision tasks such as object detection, shape recognition, autonomous driving, robotics, and virtual reality. Replacing the original batch of images with a new randomly transformed batch. models import sequential from keras. My first step was to reproduce hardmaru's work with CPPN. After the training of the model, we'll test the model that how accurate it will be on the new dataset i. Another fancy name for that kind of problem is regression problem. i am trying to use a end to end nvidia model for self driving car in keras. py script to convert the. models import Sequential from keras. Logistic regression with Keras. Evaluate model on test data. Let’s create the model for face images. Image Captioning is the process of generating a textual description of an image based on the objects and actions in it. models import load_model #Define a model_regression. The training dataset consists of a single folder containing images, each of which has a single word in it. It outputs a matrix of shape (m,128) that encodes each input face image into a 128-dimensional vector. Part 4 - Prediction using Keras. The random_state, which you are going to see a lot in machine. But here, we see the implementation of Logistic Regression using Keras. img = keras. import autokeras as ak. The dependent variable would have two classes, or we can say that it is binary coded as either 1 or 0, where 1 stands for the Yes and 0 stands for No. slogix offers a best python project for How to build regression model using keras for predicting students weights in python #5, First Floor, 4th Street , Dr. Example #4: Image Captioning with Attention In this example, we train our model to predict a caption for an image. # Note: by specifying the shape of top layers, input tensor shape is forced # to be (224, 224, 3), therefore you can use it only on 224x224 images. Work your way from a bag-of-words model with logistic regression to more advanced methods leading to convolutional neural networks. NET and I want to share this, because there are not that many examples of Keras. First and foremost, we will need to get the image data for training the model. We resize all images into 128*128. , (32, 32, 3), (28, 28, 1). Take a batch of images used for training. See full list on machinelearningmindset. models import load_model #Define a model_regression. Choosing a good metric for your problem is usually a difficult task. This type of problem comes under multi label image classification where an instance can be classified into multiple classes among the predefined classes. For example, the model focuses near the surfboard in the image when it predicts the word "surfboard". Note: Creating 5 outputs/targets/labels for this example, but the method easily extends to any number or outputs. TensorFlow's new 2. Advertisements. As we can see, the spread of latent encodings is in between [-3 to 3 on the x-axis, and also -3 to 3 on the y-axis]. The random_state, which you are going to see a lot in machine. One of the most popular examples regarding multiclass classification is to label the images of handwritten digits. Keras - Regression Prediction using MPL. Squared Hinge. convolutional import Convolution2D, MaxPooling2D. segmentation. U-Net for segmenting seismic images with keras. Active 11 months ago. Selenium can be used for browser-based regression testing. fit(x, y) # Keras model. Sep 09, 2021 · I do not see any much work on the image regression so I write this for you guys import ModelCheckpoint from tensorflow. Image Classification is one of the fundamental supervised tasks in the world of machine learning. pb file to a model XML and bin file. Image Classification: image classification using the Fashing MNIST dataset. How to use this to build a deep learning model? Keras: It is a tensor flow deep learning library to create a deep learning model for both regression and classification problems. [ ] import tensorflow as tf. y = W*x + b. You may also like. org, and download the Anaconda set up. Our example goes like this - The first step is to import the necessary libraries and load the. Import modules and sample image. One part of the model is responsible for extracting the key features from images, like edges etc. The model should run each epoch a bit fast than the larger …. MobileBERT is strangely not compatible with Tensorflow 2. In this post, we learn how to fit and predict regression data through the neural networks model with Keras in R. For example, the initial (Python) compile() function is called keras_compile(); The same holds for other functions, such as for instance fit(), which becomes keras_fit(), or predict(), which is keras_predict when you make use of the kerasR package. After two introductory tutorials, its time to build our first neural network! The network we are building solves a simple regression problem. AutoKeras accepts numpy. Keras is a simple-to-use but powerful deep learning library for Python. With Keras 2. Logistic regression with Keras. Creating the Image Classification Model. In this post, we learn how to fit and predict regression data through the neural networks model with Keras in R. Image classification is used to solve several Computer Vision problems; right from medical diagnoses, to surveillance systems, on to monitoring agricultural farms. Keras, a user-friendly API standard for machine learning, will be the central high-level API used to build and train models. Logistic regression with TensorFlow. First and foremost, we will need to get the image data for training the model. models import load_model #Define a model_regression. This guide demonstrates a step-by-step implementation of a Normalized X-Corr model using Keras, which is a modification of a Siamese network 2. This tutorials covers: Generating sample dataset Building the model. Let's generate a bunch of digits with random latent encodings belonging to this range only. Evaluation is a process during development of the model to check whether the model is best fit for the given problem and corresponding data. Keras example image regression, extract texture height › Search The Best education at www. Here, we will use a CNN network called ResNet-50. Basic Regression. resize(image, (32, 32)) inputImages. MobileNet image classification with TensorFlow's Keras API. ImageNet VGG16 Model with Keras. However, it is widely in use, for example there. In this post you will discover how to develop and evaluate neural …. Keras example image regression, extract texture height › Search The Best education at www. The problem is, that I don't know how to build the Model using Keras. Keras - Deep Learning | Image Classification - A simple Example from keras. 2, horizontal_flip = True, fill_mode = 'nearest') img = load_img ('data/train/cats/cat. Evaluation is a process during development of the model to check whether the model is best fit for the given problem and corresponding data. In this post you will discover how to develop and evaluate neural …. jpg", "https://storage. An example of an image classification problem is to identify a photograph of an animal as a "dog" or "cat" or "monkey. Even though it had only been a few months since I last did the Boston example, I was surprised at how much Keras had changed, and how much my preferred techniques had changed. Keras Convolution Neural Network Layers and Working. Roboflow provides implementations in both Pytorch and Keras. ( Number of samples per gradient update. For example, I really like the implementation of keras to build image analogies. Input to Sequential model. Using these two images you want to do an image classification. Till now, we have only done the classification based prediction. python3 keras_script. The output could be a non-discrete value: for example, given an image, output the rotation value. Using the TensorFlow Image Summary API, you can easily view them in TensorBoard. In this regard, it is much easier to. Batch size refers to the number of training examples utilized in one iteration. Example code: Multilayer Perceptron for regression with TensorFlow 2. its a regression problem to predict the angle of steering by providing image of …. layers import Conv2D, MaxPooling2D. Step 5 - Define, compile, and fit the Keras classification model. A Generative Adversarial Networks tutorial applied to Image Deblurring with the Keras library. Sequential ( [ # Note the input shape is the desired size of the image 300x300 with 3 bytes color tf. The output of yolo_model is a (m, 19, 19, 5, 85) tensor that needs to pass through non-trivial processing and conversion. Note that the further from the separating line, the more sure the classifier is. Where W is a vector called Weights and b is a scalar called Bias. The goal of our Linear …. convolutional import Convolution2D, MaxPooling2D. Introduction. , (32, 32, 3), (28, 28, 1). segmentation. You can find a complete example of this strategy on applied on a specific example on GitHub where codes of data generation as well as the Keras script are available. Minimal example for predicting rgb from coordinates. The difference between the two is mostly due to the regularization term being added to the loss during training (worth about 0. Image classification using Convolutional Neural Network (CNN) in tensorflow. its a regression problem to predict the angle of steering by providing image of camera installed front side of car. Simple Neural Network Model using Keras and Grid Search HyperParametersTuning Meena Vyas In this blog, I have explored using Keras and GridSearch and how we can automatically run different Neural Network models by tuning hyperparameters (like epoch, batch sizes etc. tensorflow_backend. I introduced Keras in mishimasyk#9. For example, the Trauma and Injury Severity Score (), which is widely used to predict mortality in injured patients, was originally developed by Boyd et al. Part 2: Regression with Keras and CNNs — training a CNN to predict house prices from image data (today's tutorial). Test data label. *FREE* shipping on qualifying offers. tensorflow_backend. load_images(x_train). Nov 27, 2017 · This network uses 96×96 dimensional RGB images as its input. Image classification using Convolutional Neural Network (CNN) in tensorflow. applications import resnet50. [ ] import tensorflow as tf. Examples of image augmentation transformations supplied by Keras. 4 you can use ". MNIST consists of 28 x 28 grayscale images of handwritten digits like these: The dataset also includes labels for each image, telling us which digit it is. For example, the initial (Python) compile() function is called keras_compile(); The same holds for other functions, such as for instance fit(), which becomes keras_fit(), or predict(), which is keras_predict when you make use of the kerasR package. These examples are extracted from open source projects. Input to Sequential model. import numpy as np from keras. images in the MNIST dataset do not have the channel dimension. See full list on rosenfelder. 101 \times 101 101× 101 pixels. My first step was to reproduce hardmaru's work with CPPN. Importantly, Keras provides several model-building APIs (Sequential, Functional, and Subclassing), so you can choose the right level of abstraction for your. Classify structured data using Keras Preprocessing Layers. Trains a memory network on the bAbI dataset for reading comprehension. Education Keras example image regression, extract texture height param. Mar 12, 2018 · Keras has this ImageDataGenerator class which allows the users to perform image augmentation on the fly in a very easy way. Released September 2019. from keras. No need to install any software to run code. It means we will treat prediction targets of MNIST dataset, which are integers ranging from 0 to. The following are 6 code examples for showing how to use keras. Whereas there are many steps involved in training a model, the focus will be on those six steps specific to transfer learning. 2, zoom_range = 0. There is still much more you can do. convolutional import Convolution2D, MaxPooling2D. [2]: def get_dataset (): # Create sample data with sklearn make_regression function X, y = make_regression (n_samples=1000, n_features=10, n_informative=7, n. tabular data in a CSV). layers import Dense from keras. by Aurélien Géron. Logistic regression with TensorFlow. The Weights and Bias are called the parameters of the model. 15/05/2021. Daniel Hnyk. Before going ahead and looking at the Python / Keras code examples and related concepts, you may want to check my post on Convolution Neural Network – Simply Explained in order to get a good understanding of CNN concepts. from keras. You can find a complete example of this strategy on applied on a specific example on GitHub where codes of data generation as well as the Keras script are available. For image classification we use a Dense layer with the number of output neurons equal to the number of classes, NUM_CLASSES. Keras offers the very nice model. then, Flatten is used to flatten the dimensions of the image obtained after convolving it. layers import Conv2D, MaxPooling2D. Where W is a vector called Weights and b is a scalar called Bias. Image Classification: image classification using the Fashing MNIST dataset. No need to install any software to run code. Figure 4: The House Prices dataset consists of both numerical/categorical data and image data. Inception Inceptionresnet 1d 2d Tensorflow Keras ⭐ 2. I have a Transportation Engineering (Civil Engineering Domain) background. Squared Hinge. pb file to a model XML and bin file. Keras provides us with a built-in loader that splits it into 50,000 training images and 10,000 test images. Creating the Image Classification Model. In this post, we'll see how easy it is to build a feedforward neural network and train it to solve a real problem with Keras. Regression Loss functions in Keras. keras and how to use them, in many situations you need to define your own custom metric because the […]. I was trying to implement a regression model in Keras, but am unable to figure out how to calculate the score of my model, i. I'm trying to build a CNN similar to this: For this purpose I chose to use Keras since I worked with it before (simple RNN and FFNN only). Model Training with VGG16. img_preprocessed = preprocess_input(img_batch) Select a pre-trained model. Credit: www. you need to understand which metrics are already available in Keras and tf. # Regression Example With Boston Dataset: Standardized and Wider import pandas as pd from keras. , how well it performed on my dataset. This will convert our words (referenced by integers in the data) into meaningful embedding vectors. jpg", "https://storage. Here, we will use a CNN network called ResNet-50. If your data is time series, then you can use LSTM model. Step3: Click on the next button to move forward. Using Keras, we’ll build a model supporting the multiple inputs and mixed data types. from tensorflow. which are scaled to 28 by 28. The MNIST dataset consists of 60,000 hand-drawn numbers, 0 to 9. [ ] !pip install autokeras. [ ] ↳ 1 cell hidden. h5') #make labels according to. py script to convert the. applications. img = keras. com/watch?v=EF7LBSCfW7cIn this vi. MNIST consists of 28 x 28 grayscale images of handwritten digits like these: The dataset also includes labels for each image, telling us which digit it is. 19 th November, 2020: Initial release; 20 th November, 2020: Removed two images. These heads are shared between all the feature maps of the feature pyramid. Another fancy name for that kind of problem is regression problem. 2, height_shift_range = 0. For example, the Trauma and Injury Severity Score (), which is widely used to predict mortality in injured patients, was originally developed by Boyd et al. See why word embeddings are useful and how you can use pretrained word embeddings. The usual way is to import the TCN layer and use it inside a Keras model. The following are 6 code examples for showing how to use keras. hdf5" save. model = tf. The first layer in the network, as per the architecture diagram shown previously, is a word embedding layer. In this tutorial, we will learn the basics of Convolutional Neural Networks ( CNNs ) and how to use them for an Image Classification task. The code will be described using the following sub-topics: Loading the Sklearn Bosting pricing dataset; Training the Keras neural network. CNN Model of Image Detection in Keras (TensorFlow) in Python3. Model Training with VGG16. We have build a model using Keras library (Python) and trained it to make predictions. from keras_preprocessing. Linear regression is a linear model, e. You can compute your gradient on just one example image and update the weights and biases immediately, but doing so on a batch of, for example, 128 images gives a gradient that better represents the constraints imposed by different example images and is therefore likely to converge towards the solution faster. Create a custom function that generates the multi-output regression data. Keras: Image Forecasting Keras: Tabular Regression Please add `keras. Non Linear Regression Example with Keras and Tensorflow Backend. CNN is a type of neural network model which allows working with the images and videos, CNN takes the image's raw pixel data, trains the model, then extracts the features automatically for better classification. Designed to enable fast experimentation with deep neural. Keras Fit Function (R): Train Regression Model with multiple Labels Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern) 2019 Moderator Election Q&A - Questionnaire 2019 Community Moderator Election ResultsHow to calculate accuracy on keras model with. In this section, you will rebuild the same model built earlier with TensorFlow core with Keras: 1. Now, see how writing the same model in Keras makes this. The Weights and Bias are called the parameters of the model. Note that, the value of image_data_format is Now, let's build a simple example to implement linear regression using Keras Sequential model. Keras takes data in a different format and so, you must first reformat the data using datasetslib: x_train_im = mnist. The goal is to classify whether the truck cargo area is up or down. The basic idea of using Autoencoders for Image denoising is as follows: Encoder part of autoencoder will learn how noise is added to original images. But here, we see the implementation of Logistic Regression using Keras. In this section, we will rebuild the same model we built earlier with TensorFlow core with Keras: In the preceding code, we are loading the training images in memory before both the training and test images are scaled, which we do by dividing. Then we read the training data images. In this project, the authors train a neural network to understand an image, and recreate learnt attributes to another image. Jan 30, 2019 · Use computer vision, TensorFlow, and Keras for image classification and processing. The following are 6 code examples for showing how to use keras. Logistic regression with TensorFlow. O'Reilly members get unlimited access to. With a pH of 7, pure water is neutral. Minimal example for predicting rgb from coordinates. 1) is using GPU: from keras import backend as K K. For the tutorial. The Keras machine learning framework provides flexibility to architect custom neural networks, loss functions, optimizers, and also runs on GPU so it trains complex networks much faster than sklearn. See full list on machinelearningmindset. You can find a complete example of this strategy on applied on a specific example on GitHub where codes of data generation as well as the Keras script are available. Using the pygad. preprocessing. After the training of the model, we'll test the model that how accurate it will be on the new dataset i. Most wines have a pH between 2. layers import Dense from keras. then, Flatten is used to flatten the dimensions of the image obtained after convolving it. Doing this is the same process as we've needed to do to train the model, so we'll be recycling quite a bit of code. you need to understand which metrics are already available in Keras and tf. 49% improvement over baseline (as measured on V1) in MRR and [email protected]. Regression data can be easily fitted with a Keras Deep Learning API. placeholder that we can't imagine TensorFlow without. # create our MLP and then compile the model using mean absolute # percentage error as our loss, implying that we seek to minimize # the absolute percentage difference between our price *predictions* # and the *actual prices* model = models. MobileNetV2 model is available with tf. Work your way from a bag-of-words model with logistic regression to more advanced methods leading to convolutional neural networks. We have build a model using Keras library (Python) and trained it to make predictions. In this section, we will rebuild the same model we built earlier with TensorFlow core with Keras: In the preceding code, we are loading the training images in memory before both the training and test images are scaled, which we do by dividing. In this article, we will learn step by step, how to tune a Keras deep learning regression model and identify the best set of hyperparameters. If your data is time series, then you can use LSTM model. For this we use an image from the cifar10 dataset which comes with keras and features similar classes to ImageNet. We take a Kaggle image recognition competition and build CNN model to solve it. Model Training with VGG16. resize(img, (64, 64)) img2=np. Training the model with the data history = model. For classification, the images are captured with 80x80 resolution. Logistic Regression - classification. 0 and Keras, and runs on the Chennai Water Management Dataset. Then we create model we user 3 layers with activation function ReLU and in the last layer add a "softmax" layer. Semantic Segmentation laid down the fundamental path to advanced Computer Vision tasks such as object detection, shape recognition, autonomous driving, robotics, and virtual reality. Essentially, we are trying to predict the value of a potential car sale (i. Inception Inceptionresnet 1d 2d Tensorflow Keras ⭐ 2. Neural Regression using Keras Demo Run This article assumes you have intermediate or better programming skill with a C-family language and a basic familiarity with machine learning. applications import resnet50.


text:
# Writing Parameterized Tests in JUnit 5

Improving code quality using parameterized tests of JUnit 5! This article explains the motivation, the basic syntax, different annotations, and IDE-related actions about parameterized tests. Also, ...

## Introduction

Writing tests is an important part of software development, it’s a good way to ensure that your code works as expected. However, it is difficult to write tests that cover all the scenarios. It’s also difficult to make them easy to understand when the implementation is complex. In this article, we are going to see how parameterized tests of JUnit 5 can help.

• The motivation of using parameterized tests
• How to use parameterized tests
• Different sources of arguments
• The argument conversion
• Interactions with IDE
• When to use or not to use parameterized tests?
• How to go further from here?

## Motivation

Why should we consider using parameterized tests in our code?

Improve test coverage. Using parameterized testing can improve the code coverage. Thanks to parameterized test, you can easily add more parameters, or add more values for the same parameter to an existing test. This is done without heavily changing the logic of the test code. By doing so, your assertions cover more lines or more branches.

Reuse test code. Using parameterized testing reuses the same test case for multiple inputs. Therefore, it simplifies the code set up, improves the maintainability of these tests, and therefore save your time to develop other features or to write tests for other scenarios.

Clarify inputs and outputs. Sometimes the test code is not as readable as you may think about. Without putting additional effort into those tests, it can become a nightmare to understand the purpose of this test, such as what are the real input or output parameters of these tests. By using parameterized testing, we extract those values as external parameters. Therefore, it’s easier to understand. It’s easier because 1) we can see those parameters as method input parameters; 2) we can find the reasoning of the inputs by comparing their common points and differences.

Delegate the complexity to JUnit. But why not just adding a for-loop inside the test? Why using the parameterized testing feature in JUnit 5? Well, this is because by doing so, we delegate the complexity to JUnit. More precisely, JUnit ensures that each execution contains their set up and tear downs; the failure of one execution won’t impact the other, they are isolated; we can have a nice display name in IDE or build to explain the current execution; it can handle different sources and type conversions, etc.

Validate multiple implementations. Parameterized testing is also useful for validating multiple implementations. These implementations may need to respect the same specification. Therefore, we can express the expectations of this specification as tests and require all implementations to pass these tests. This is a good way to prevent regressions.

Testing the orthogonality. Sometimes, one method accepts multiple input parameters, they should be orthogonal. That is, the output of one scenario will only depend on one input parameter, not others. So when changing the value of any other parameters, the test result shouldn’t be changed. We can ensure that the orthogonality is respected through parameterized testing.

These are some reasons that I found during my daily development. I hope they are enough to convince you to try parameterized testing. Now we understood the motivation, we are going to check how to use it in the following section.

## Prerequisite

Before using parameterized testing in JUnit 5, you have to declare 3 dependencies: the specification (API) and the implementation (engine) of the JUnit 5, and also an additional module to support parameterized testing. Here is what do they look like in the Maven POM file (pom.xml).

The specification (API) and the implementation (engine) are required for running JUnit 5. You probably have them in your project already.

<dependency>
<groupId>org.junit.jupiter</groupId>
<artifactId>junit-jupiter-api</artifactId>
<version>5.7.0</version>
<scope>test</scope>
</dependency>
<dependency>
<groupId>org.junit.jupiter</groupId>
<artifactId>junit-jupiter-engine</artifactId>
<version>5.7.0</version>
<scope>test</scope>
</dependency>

Then declare the additional Maven dependency to support parameterized testing:

<dependency>
<groupId>org.junit.jupiter</groupId>
<artifactId>junit-jupiter-params</artifactId>
<version>5.7.0</version>
<scope>test</scope>
</dependency>

## Syntax

The basic syntax of a parameterized test is as follows:

@ParameterizedTest // 1
@ValueSource(strings = { // 2
"racecar",
"able was I ere I saw elba"
})
void palindromes(String candidate) { // 3
assertTrue(StringUtils.isPalindrome(candidate));
}

where you can see that: 1) you need to use annotation @ParameterizedTest as the replacement of the regular @Test; 2) an annotation containing a list of sources to be used for the parameterized test; 3) one input parameter to represent the source value provided by the annotation, it can also be multiple input parameters: it depends on your sources.

## Source Providers

@ValueSource annotation can be used to provide a list of values, which each item contains one single argument. It is supported for types: short, byte, int, long, float, double, char, boolean, java.lang.String, java.lang.Class. However, you may notice that value source can only provide one argument at a time, so you have to keep your test simple to ensure that all these arguments will fit the test.

@ValueSource(ints = { 1, 2, 3 })

@ValueSource(booleans = {true, false})

@ArgumentsSource annotation can be used to link to an arguments-provider, which provides a stream of argument arrays (Arguments) for the parameterized tests. I found it useful for two cases: 1) providing multiple implementations of the same interface, so that we can validate all of them using the same test, as you can see in the code block below; 2) providing values that are computed from another field in the codebase. This is not suitable for @ValueSource because the line will be long and unreadable. Using @ArgumentsSource makes the metadata of the test method more declarative.

@ParameterizedTest
@ArgumentsSource(ChatBotProvider.class)
void sayHello(ChatBot bot) {
assertThat(bot.sayHello("Foo")).isEqualTo("Hello, Foo");
assertThat(bot.sayHello("Bar")).isEqualTo("Hello, Bar");
}

public static class ChatBotProvider implements ArgumentsProvider {

@Override
public Stream<? extends Arguments> provideArguments(ExtensionContext context) {
return Stream.of(new StringFormatChatBot(), new StringConcatenationChatBot())
.map(Arguments::of);
}
}

@CsvSource annotation can be used to declare multiple argument arrays, each array representing one argument array. This is my favorite annotation for providing sources because I can provide multiple arguments for each test case, which is usually the case for enterprise development. You may notice that the string value in the CSV source is converted to primitive type int by JUnit. This makes your tests concise. We will talk more about conversions in the next section.

@ParameterizedTest
@CsvSource({
"1,  2, 2",
"1, -1, 1",
"1,  1, 1",
})
void testMax(int a, int b, int max) {
assertThat(Math.max(a, b)).isEqualTo(max);
}

There are other source providers, such as @EnumSource, @NullSource, @EmptySource, @MethodSource. They provide different types of sources and they are easy to use. You can find the complete documentation in JUnit 5 User Guide - 2.15.3 Sources of Arguments.

## Conversion

From my experience, the CSV source seems to be the most frequently used provider. But inside a CSV file, every row and every cell is considered as a string. In this section, I want to share some tricks about type conversion. It makes your code more concise: just let JUnit handle the work for you ;)

String-to-Primitive. When providing a primitive type as an input parameter for the test case, JUnit knows how to convert it. For example, given an input parameter as int, then JUnit will convert your string into an integer automatically. The same logic applies to other primitive types: long, double, float, boolean, etc.

@ParameterizedTest
@CsvSource({
"1,  2, 2",
"1, -1, 1",
"1,  1, 1",
})
void testMax(int a, int b, int max) {
assertThat(Math.max(a, b)).isEqualTo(max);
}

String-to-Enum. When providing an enumeration type as an input parameter for the test case, JUnit knows how to convert it as well (probably using the valueOf(String) function of enumeration). Therefore, you don’t have to handle the conversion yourself. Here is a concrete example about how to converting a string into enum type FilePermission, and then test the executability for different permission.

@ParameterizedTest
@CsvSource({
"R, false",
"W, false",
"RW, false",
"X, true",
"RX, true",
"WX, true",
"RWX, true",
})
void isExecutable(FilePermission perm, boolean isExecutable) {
assertThat(perm.isExecutable()).isEqualTo(isExecutable);
}

## IDE

Another important part of testing is to understand how to use IDE so that it can help you to be even more productive! In this section, I want to share two things about IDE with you: how to check which parameterized test is running and how to check the test coverage. Here I am using IntelliJ IDEA but I think the methodology is still valid for other IDEs.

### IDE: Display Name of Tests

When running the parameterized in JUnit 5, you can see each test execution with the input parameters. For example, the test case isWritable(FilePermission, boolean) is executed 7 times with different file permission each time:

It is also possible to customize the display names to print something more human-readable. Personally, I prefer to avoid spending time on it. But if you were interested, you can find it in the JUnit 5 User Guide §2.15.6. Customizing Display Names.

### IDE: Test Coverage

The main motivation of using parameterized testing is to increase the test coverage and improve the robustness of the implementation. To review the test coverage, you can run the tests with coverage and then check the coverage. Here is how to trigger a “run tests with coverage” at package level or class level in IntelliJ IDEA:

Right-click on a Java package, such as io.mincong.junit5 and select the option “Run ‘Test in ‘io.mincong.junit5’’ with Coverage” to trigger the tests with coverage enabled:

Left-click on the test icon of the class name or the test name of a test file, such as FilePermissionTest, then select option “Run ‘FilePermissionTest’ with Coverage” to trigger the tests with coverage enabled:

Once the test execution is done, you can find the coverage report in different views, such as the project view, the class file itself, or the dedicated test coverage view:

In the project, we can see the test coverage of each package and each class inside the package. It gives a good overview of the current situation of a given package.

Inside a class, it’s also possible to see the code coverage right next to the line numbers. The color green is covered and the color red is not covered. This is useful when you are modifying some logic, especially for some critical path, because you want them to be covered by at least one test. This information helps you to improve your tests.

The last view is the coverage view, where you can see a list of packages and their coverage. You can also zoom into one package as the screenshot below and inspect the coverage of each file inside the package.

## Pros and Cons

When to use and not to use parameterized tests?

Now we understood how to use parameterized tests and get familiar with IDE, the remaining question is: should we use it for our tests? I think it depends on your needs. It’s not always the right choice. Here are some cases that I summarized during my daily work. I hope that they will give you some inspiration.

As mentioned in the motivation section, you should use parameterized tests to:

• Improve the test coverage
• Reuse test code
• Clarify inputs and outputs of your test
• Delegate the complexity to JUnit
• Validate multiple implementations
• Testing the orthogonality of different parameters

But you shouldn’t use parameterized tests:

• if it makes the logic of the test difficult the understand. Typically, if it introduces if-statements or other branches in the code.
• if different arguments don’t share the same purpose. If they are different, it is probably worth creating multiple test cases. A good way to know this is to summarize the purpose in one sentence, if you cannot, it’s probably worth writing different tests.
• if there are too many input arguments. This can make the tests difficult to maintain.

## Going Further

How to go further from here?

If you want to see the source code of this article, they are available on GitHub under module junit5 of project mincong-h/java-examples.

## Conclusion

In this article, we saw the motivation of using parameterized testing in JUnit 5, how to declared dependencies, the annotations of different source providers (value source, argument source, CSV source, …), the conversation from string to primitive types and enum types, running tests and inspecting coverage in IntelliJ IDEA, the pros and cons of using parameterized tests, and finally how to go further from here. Interested to know more? You can subscribe to the feed of my blog, follow me on Twitter or GitHub. Hope you enjoy this article, see you the next time!

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text:
# Best Way to Get 5V, 2A from a 2-Cell LiPo Battery

Part of a circuit I'm designing needs to control servo motors rated for 5V. I've measured that they can draw over 1A when active.

The issue is that my circuit is powered by a 2-cell LiPo battery, which I've gathered has a 6V minimum, 8.4V maximum output.

Another part of the circuit uses 3.3V, so I used a switching regulator from 6-to-28V down to 3.3V, and it works perfectly. I thought that a similar component to regulate down to 5V would be about as expensive ($2-$3 ish), but I'm finding on DigiKey that I'm looking at at least $16 per regulator for even a 1A-output 5V regulator, and significantly more for a 2A output. This is really way too expensive for me, especially since I will have 4+ servos to power. Is there a better way to power servos from a 2-cell LiPo? I would really like to power everything off of one battery, but it seems like I may just need to power the servos with a separate 5V battery (NiCd or other, not sure). Edit: My search parameters earlier somehow caused me to miss the LM2596, for$2.5. Will this work how I want, if I adjust it to have a 5V output? It seems a little too cheap to be true. :)

• You can find cheap regulators on ebay, aliexpress, dx, etc.
– kva
Jun 15 '17 at 21:03
• @kva Can I trust components from those sources? I've only ever ordered from DigiKey before. (When I see titles on AliExpress with "100% Genuine" it makes me very suspicious, to say the least...) Jun 15 '17 at 21:06
• Depends on what your application is. Definitely not suited for medical applications, but usually good enough for hobby projects. High sellers' reputability score and good user feedback are good indicators of a sellers trustworthiness. Edit: Do not buy things that are too good to be true. They generally arent.
– kva
Jun 15 '17 at 21:14
• "I've measured that they can draw over 1A when active." - each? Note that peak current draw can be much higher than average bhabbott.net.nz/Servo.html Jun 16 '17 at 1:07

The cost of the regulator seems about right, and is definitely not too good to be true. But what makes you think that the regulator is all you need? Look at the data sheet you linked to, and look closely at the applications section. You will need several more components, particularly including an inductor which cannot be just any old inductor. Even worse, you'll need a way to mount all your components, and you cannot just solder them together. Switching supplies are susceptible to layout errors, so a printed circuit board is a really, really good idea.

Between the components and the pcb, for small quantities of finished product (like, let's say 100) you'll almost certainly end up spending more than 16 bucks a unit.

So eBay is probably your best bet.

• Don't worry, this is all going on an existing PCB design, so PCB cost is not an issue. (In fact, I just finished soldering and testing v1 of the board, and it works perfectly!) I did look at the applications section, and I suppose I will need to see how much all the components are worth. The only thing I'm concerned about now is Figure 4, on page 5 of the datasheet. I'm trying to get the full 3A output, so what I'm seeing is that at 25 degrees celcius, the dropout voltage is around 1.6V, which means that I need a minimum of 6.6V battery voltage. (continued...) Jun 15 '17 at 22:38
• However, it could be important that I have a lower dropout so that I can use more of the battery life. I'll have to see if I can even find low-dropout regulators that work at these voltages... Jun 15 '17 at 22:39
• Driving cheap asian electronics to their maximum (e.g. 3A) most often requires a good heatsink. Jun 16 '17 at 1:59

The standard regulator used with 'hobby' servos is called a 'UBEC' (BEC = Battery Eliminator Circuit). These are usually based around an SO8 package switch-mode buck regulator IC, with a small inductor, Schottky diode, and a few capacitors and resistors completing the circuit.

You could get ready made units such as this and wire them onto your board, or buy the individual components and make you own. I used a Sanken SI8005Q, but several other chips are available which have similar specs. If you are not experienced at designing switch-mode supplies then follow the example layout in the datasheet.


text:
## Inner Product

Sure, this could also appropriately be placed under linear alg. this being said, can anyone give me an intuitive explanation for the real inner product? i realize it as:

$$<f(x),g(x)>\doteqdot\int_a^bf(x)g(x)dx$$

where i can think of this as an "infinite" dot product along a to b. geometrically (and intuitively) what does this represent about two functions, say in $$R^3$$ for ease of visualization.

for what its worth this has shown up in a fourier analysis course

 PhysOrg.com science news on PhysOrg.com >> Hong Kong launches first electric taxis>> Morocco to harness the wind in energy hunt>> Galaxy's Ring of Fire
 Recognitions: Homework Help It is the same as a dot product for discrete-valued vectors only for continuous ones. http://en.wikipedia.org/wiki/Dot_product#Inner_product It may help if you examined specific vectors for f and g ...
 Recognitions: Gold Member Science Advisor Staff Emeritus A little more generally, we define the dot product specifically on the vector space Rn by $\sum_{i= 1}^n u_iv_i$, the sum of the products of corresponding components (or on Cn as $\sum_{i=1}^n u_iv_i^*$ where "*" indicate the complex conjugate). In any vector space, V, we say that an "inner product" is function from VxV to the underlying field (real numbers or complex numbers) satisfying 1) is a positive real number. 2) = a+ b 3) = if the underlying field is the real numbers, is the complex conjugate of if the underlying field is the complex numbers. It is easy to show that the dot products in Rn and Cn do satisfy those and are "inner products". It is also fairly easy to show that any n dimensional vector space, of dimension n, is isomorphic to Rn, the specific isomorphism given by a selected ordered basis. And so any inner product is equivalent to a dot product, given a specific basis. But, as I said, that requires a finite dimension which is why Simon Bridge said "It is the same as a dot product for discrete-valued vectors only for continuous ones." The "infinite dimensional vector spaces" typically used are the "function spaces".

## Inner Product

You might want to review the Riemann Integral and see the analog between that infinite sum and the infinite sum of a vector in l^2 (little l^2 as opposed to big L^2).

 Well, vaguely, you could say it's sort of measuring how much the two functions are "in synch" with each other, much as the ordinary dot product measures how much two vectors are "in synch" with each other. The dot product of vectors can be interpreted as projecting one vector onto the other and then multiplying the length of the the projection with the length of the vector being projected to. So, in some sense, it's measuring how much one vector points in the direction of the other, with longer vectors being given more weight. What you really need to get an intuition for is not so much the inner product, but the function space--the idea that functions can be thought of as some sort of vectors because an inner product is something you do to vectors. One way to say it is that functions can be added and multiplied by scalars. Roughly, speaking vectors are things that can be added together and multiplied by scalars. That's probably a bit unsatisfying, though. One thing that might be a little more satisfying is the realization that if you look at a finite vector subspace of the space of functions, you get the vector subspaces that we know and love, namely R^n. Remembering that we are thinking of functions as vectors, we could take the the span, that is all linear combinations, of the functions, 1, x, and x^2. We get all the second degree (or lower) polynomials. There are 3 parameters, so it is a 3-dimensional space. So, it's R^3, a vector space we know and love. As another example, you could take the interval from 0 to 1, divide it into n pieces, and consider all the functions that are constant on each piece. We get R^n. Well, I'm getting tired, but if you keep thinking along those lines, you'll see how the inner product of functions is related to the one in R^n. Just think of doing a discrete approximation.

Recognitions:
Homework Help
 vaguely, you could say it's sort of measuring how much the two functions are "in synch" with each other
Oooh - that's good: I'll have to use it.
Agree with comments on vector spaces.


text:
# Learning Data Manifolds with a Cutting Plane Method

We consider the problem of classifying data manifolds where each manifold represents invariances that are parameterized by continuous degrees of freedom. Conventional data augmentation methods rely upon sampling large numbers of training examples from these manifolds; instead, we propose an iterative algorithm called M_CP based upon a cutting-plane approach that efficiently solves a quadratic semi-infinite programming problem to find the maximum margin solution. We provide a proof of convergence as well as a polynomial bound on the number of iterations required for a desired tolerance in the objective function. The efficiency and performance of M_CP are demonstrated in high-dimensional simulations and on image manifolds generated from the ImageNet dataset. Our results indicate that M_CP is able to rapidly learn good classifiers and shows superior generalization performance compared with conventional maximum margin methods using data augmentation methods.

## Authors

• 12 publications
• 1 publication
• 8 publications
• 29 publications
• ### Isometric Multi-Manifolds Learning

Isometric feature mapping (Isomap) is a promising manifold learning meth...
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• ### ResizeMix: Mixing Data with Preserved Object Information and True Labels

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## 1 Introduction

Handling object variability is a major challenge for machine learning systems. For example, in visual recognition tasks, changes in pose, lighting, identity or background can result in large variability in the appearance of objects

hinton1997modeling

. Techniques to deal with this variability has been the focus of much recent work, especially with convolutional neural networks consisting of many layers. The manifold hypothesis states that natural data variability can be modeled as lower-dimensional manifolds embedded in higher dimensional feature representations

bengio2013representation . A deep neural network can then be understood as disentangling or flattening the data manifolds so that they can be more easily read out in the final layer brahma2016deep . Manifold representations of stimuli have also been utilized in neuroscience, where different brain areas are believed to untangle and reformat their representations riesenhuber1999hierarchical ; serre2005object ; hung2005fast ; dicarlo2007untangling ; pagan2013signals .

This paper addresses the problem of classifying data manifolds that contain invariances with a number of continuous degress of freedom. These invariances may be modeled using prior knowledge, manifold learning algorithms tenenbaum1998mapping ; roweis2000nonlinear ; tenenbaum2000global ; belkin2003laplacian ; belkin2006manifold ; canas2012learning or as generative neural networks via adversarial training goodfellow2014generative . Based upon knowledge of these structures, other work has considered building group-theoretic invariant representations anselmi2013unsupervised or constructing invariant metrics simard1994memory . On the other hand, most approaches today rely upon data augmentation by explicitly generating “virtual” examples from these manifolds niyogi1998incorporating scholkopf1996incorporating . Unfortunately, the number of samples needed to successfully learn the underlying manifolds may increase the original training set by more than a thousand-fold krizhevsky2012imagenet .

We propose a new method, called the Manifold Cutting Plane algorithm or , that uses knowledge of the manifolds to efficiently learn a maximum margin classifier. Figure 1 illustrates the problem in its simplest form, binary classification of manifolds with a linear hyperplane with extensions to this basic model discussed later. Given a number of manifolds embedded in a feature space, the algorithm learns a weight vector that separates positively labeled manifolds from negatively labeled manifolds with the maximum margin. Although the manifolds consist of uncountable sets of points, the algorithm is able to find a good solution in a provably finite number of iterations and training examples.

Support vector machines (SVM) can learn a maximum margin classifier given a finite set of training examples vapnik1998statistical ; however, with conventional data augmentation methods, the number of training examples increase exponentially rendering the standard SVM algorithm intractable. Methods such as shrinkage and chunking to reduce the complexity of SVM have been studied before in the context of dealing with large-scale datasets smola1998learning , but the resultant kernel matrix may still be very large. Other methods which subsample the kernel matrix or reduce the number of training samples wang2005training ,smola1998learning may result in suboptimal solutions that do not generalize well.

Our algorithm directly handles the uncountable set of points in the manifolds by solving a quadratic semi-infinite programming problem (QSIP). is based upon a cutting-plane method which iteratively refines a finite set of training examples to solve the underlying QSIP fang2001solving ; kortanek1993central ; liu2004new . The cutting-plane method was also previously shown to efficiently handle learning problems with a finite number of examples but an exponentially large number of constraints joachims2006training . We provide a novel analysis of the convergence of with both hard and soft margins. When the problem is realizable, the convergence bound explicitly depends upon the margin value whereas with a soft margin and slack variables, the bound depends linearly on the number of manifolds.

The paper is organized as follows. We first consider the hard margin problem and analyze the simplest form of the algorithm. Next, we introduce slack variables in , one for each manifold, and analyze its convergence with additional auxiliary variables. We then demonstrate the application of to both high-dimensional synthetic data manifolds and to feature representations of images undergoing a variety of warpings. We compare its performance, both in efficiency and generalization error, with conventional SVMs using data augmentation techniques. Finally, we discuss some natural extensions and potential future work on and its applications.

## 2 Manifolds Cutting Plane Algorithm with Hard Margin

In this section, we first consider the problem of classifying a set of manifolds when they are linearly separable. This allows us to introduce the simplest version of the algorithm along with the appropriate definitions and QSIP formulation. We analyze the convergence of the simple algorithm and prove an upper bound on the number of errors the algorithm can make in this setting.

### 2.1 Hard Margin QSIP

Formally, we are given a set of manifolds , with binary labels (all points in the same manifold share the same label). Each manifold is defined by ) where , is a compact, convex subset of representing the parameterization of the invariances and is a continuous function of so that the manifolds are bounded: by some . We would like to solve the following semi-infinite quadratic programming problem for the weight vector :

 (1)

This is the primal formulation of the problem, where maximizing the margin is equivalent to minimizing the squared norm We denote the maximum margin attainable by , and the optimal solution as . For simplicity, we do not explicitly include the bias term here. A non-zero bias can be modeled by adding an additional feature of constant value as a component to all the . Note that the dual formulation of this QSIP is more complicated, involving optimization of non-negative measures over the manifolds. In order to solve the hard margin QSIP, we propose the following simple algorithm.

### 2.2 MsimpleCP Algorithm

The algorithm is an iterative algorithm to find the optimal in (1), based upon a cutting plane method for solving the QSIP. The general idea behind is to start with a finite number of training examples, find the maximum margin solution for that training set, augment the training set by finding a point on the manifolds that violates the constraints, and iterating this process until a tolerance criterion is reached.

At each stage of the algorithm there is a finite set of training points and associated labels. The training set at the -th iteration is denoted by the set: with examples. For the -th pattern in , is the index of the manifold, and is its associated label.

On this set of examples, we solve the following finite quadratic programming problem:

 SVMTk:argmin→w12∥→w∥2 s.t.∀→xi∈Tk:ypi⟨→w,→xi⟩≥1 (2)

to obtain the optimal weights on the training set . We then find a constraint-violating point from one of the manifolds such that

 (3)

with a required tolerance If there is no such point, the algorithm terminates. If such a point exists, it is added to the training set, defining the new set . The algorithm then proceeds at the next iteration to solve to obtain . For , the set is initialized with at least one point from each manifold. The pseudocode for is shown in Alg. 1.

In step 3 of the algorithm, a point among the manifolds that violates the margin constraint needs to be found. The use of a “separation oracle” is common in other cutting plane algorithms such as those used for structural SVM’s joachims2006training or linear mixed-integer programming marchand2002cutting . In our case, this requires determining the feasibility of over the -dimensional convex parameter set . When the manifold mapping is convex, feasibility can be determined sometimes analytically or more generally by a variety of modern convex optimization techniques. For non-convex mappings, a feasible separating point can be found with search techniques in the -dimensional parameter set using gradient information or finite differences or approximately via convex relaxation techniques. In our experiments below, we provide some specific examples of how separating points can be efficiently found.

### 2.3 Convergence of MsimpleCP

The algorithm will converge asymptotically to an optimal solution when it exists. Here we show that the asymptotically converges to an optimal . Denote the change in the weight vector in the -th iteration as

. We present a set of lemmas and theorems leading up to the bounds on the number of iterations for convergence, and the estimation of the objective function. More detailed proofs can be found in Supplementary Materials (SM).

###### Lemma 1.

The change in the weights satisfies .

###### Proof.

Define . Then for all , satisfies the constraints on the point set : for all . However, if , there exists a such that , contradicting the fact that is a solution to .

Next, we show that the norm must monotonically increase by a finite amount at each iteration.

###### Theorem 2.

In the iteration of algorithm, the increase in the norm of is lower bounded by , where and .

###### Proof.

First, note that , otherwise the algorithm stops. We have: (Lemma 1). Consider the point added to set . At this point, , , hence . Then, from the Cauchy-Schwartz inequality,

 ∥∥Δ→w(k)∥∥2≥δ2k∥→xk+1∥2>δ2kL2>δ2L2 (4)

Since the solution satisfies the constraints for , . Thus, the sequence of iterations monotonically increase norms and are upper bounded by . Due to convexity, there is a single global optimum and the algorithm is guaranteed to converge.

As a corollary, we see that this procedure is guaranteed to find a realizable solution if it exists in a finite number of steps.

###### Corollary 3.

The algorithm converges to a zero error classifier in less than iterations, where is the optimal margin and bounds the norm of the points on the manifolds.

###### Proof.

When there is an error, we have , and (See (4)). This implies the total number of possible errors is upper bounded by .

With a finite tolerance , we obtain a bound on the number of iterations required for convergence:

###### Corollary 4.

The algorithm for a given tolerance terminates in less than iterations where is the optimal margin and bounds the norm of the points on the manifolds.

###### Proof.

Again, and each iteration increases the squared norm by at least .

We can also bracket the error in the objective function after terminates:

###### Corollary 5.

With tolerance , after terminates with solution , the optimal value of is bracketed by: .

###### Proof.

The lower bound on is as before. Since has terminated, setting would make feasible for , resulting in the upper bound on .

## 3 Mcp with Slack Variables

In many classification problems, the manifolds may not be linearly separable due to their dimensionality, size, or correlation structure. In these situations, will not be able to find a feasible solution. To handle these problems, the classic approach is to introduce slack variables on each point (). Unfortunately, this approach requires integrations over entire manifolds with an appropriate measure defined by the infinite set of slack variables. Thus, we formulate an alternative version of the QSIP with slack variables below.

### 3.1 QSIP with Manifold Slacks

In this work, we propose using only one slack variable per manifold for classification problems with non-separable manifolds. This formulation demands that all the points on each manifold obey an inequality constraint with one manifold slack variable, . As we see below, solving for this constraint is tractable and the algorithm has good convergence guarantees.

However, a single slack requirement for each manifold by itself may not be sufficient for good generalization performance. Our empirical studies show that generalization performance can be improved if we additionally demand that some representative points on each manifold also obey the margin constraint: . In this work, we implement this intuition by specifying appropriate center points for each manifold . This center point could be the center of mass of the manifold, a representative point, or an exemplar used to generate the manifolds krizhevsky2012imagenet . Additional slack variables for these constraints could potentially be introduced; in the present work, we simply demand that these points strictly obey the margin inequalities corresponding to their manifold label. Formally, the QSIP optimization problem is summarized below, where the objective function is minimized over the weight vector and slack variables :

### 3.2 MslackCP Algorithm

With these definitions, we introduce our algorithm with slack variables below.

The proposed algorithm modifies the cutting plane approach to solve a semi-infinite, semi-definite quadratic program. Each iteration involves a finite set: with examples that is used to define the following soft margin SVM:

to obtain the weights and slacks at each iteration. We then find a point from one of the manifolds so that:

 ypk+1⟨→w(k),→xk+1⟩+ξ(k)pk+1=1−δk (5)

where . If there is no such a point, the algorithm terminates. Otherwise, the point is added as a training example to the set and the algorithm proceeds to solve to obtain and .

### 3.3 Convergence of MslackCP

Here we show that the objective function is guaranteed to increase by a finite amount with each iteration. This result is similar to tsochantaridis2005large , but here we present statements in the primal domain over an infinite number of examples. More detailed proofs can be found in SM.

###### Lemma 6.

The change in the weights and slacks satisfy:

 ⟨Δ→w(k),→w(k)⟩+C∑pΔ→ξ(k)p≥0 (6)

where and .

###### Proof.

Define and . Then for all , and satisfy the constraints for . The resulting change in the objective function is given by:

 (7)

If (6) is not satisfied, then there is some such that , which contradicts the fact that and are a solution to .

We derive that the added point at each iteration must be a support vector for the next weight:

###### Lemma 7.

In each iteration of algorithm, the added point must be a support vector for the new weights and slacks, s.t.

 ypk+1⟨→w(k+1),→xk+1⟩+ξ(k+1)pk+1=1 (8)
 ypk+1⟨Δ→w(k),→xk+1⟩+Δξ(k)pk+1=δk (9)
###### Proof.

Suppose for some . Then we can choose so that and satisfy the constraints for . But, from Lemma 6, we have which contradicts the fact that and are a solution to . Thus, and the point must be a support vector for (9) results from subtracting (5) from (8).

We also derive a bound on the following quadratic function over nonnegative values:

###### Lemma 8.

Given ,, then

 12(x−δ)2+Kx≥min(12δ2,12Kδ) (10)
###### Proof.

The minimum value occurs when . When , then and the minimum is . When , the minimum occurs at . Thus, the lower bound is the smaller of these two values.

Using the lemmas above, the lower bound on the change in the objective function can be found:

###### Theorem 9.

In each iteration of algorithm, the increase in the objective function for , defined as , is lower bounded by

 ΔF(k)≥min(18δ2kL2,12Cδk) (11)
###### Proof.

We first note that the change in objective function is strictly increasing:

 F(→w(k+1),→ξ(k+1))−F(→w(k),→ξ(k))=[⟨Δ→w(k),→w(k)⟩+C∑pΔξ(k)p]+12∥∥Δ→w(k)∥∥2>0 (12)

This can be seen immediately from Lemma 6 when . On the other hand, if , we know that from Lemma 7 and since is the solution for . So for , . To compute a general lower bound on the increase in the objective function, we proceed as follows.

The added point comes from a particular manifold . If , from Lemma 7 we have . Then by the Cauchy-Schwarz inequality, which yields .

We next analyze and consider the finite set of points that come from the manifold. There must be at least one such point in by initialization of the algorithm. Each of these points obeys the constraints:

 ypk⟨→w(k),→xν⟩+ξ(k)pk =1+ϵ(k)ν (13) ypk⟨→w(k+1),→xν⟩+ξ(k+1)pk =1+ϵ(k+1)ν (14) ϵ(k)ν,ϵ(k+1)ν ≥0 (15)

We consider the minimum value of the thresholds: . We have two possibilities: is positive so that none of the points are support vectors for , or so that at least one support vector lies in .

Case :

In this case, we define a linear set of slack variables:

 ξp(λ) = {ξ(k)p+λΔξ(k)pp≠pkξ(k)pkp=pk (16)

and weights . Then for , and will satisfy the constraints for . Following similar reasoning in Lemma 6, this implies

 ⟨Δ→w(k),→w(k)⟩+C∑p≠pkΔ→ξ(k)p≥0 (17)

Then in this case, we have

 F(→w(k+1),→ξ(k+1))−F(→w(k),→ξ(k)) =[⟨Δ→w(k),→w(k)⟩+C∑pΔξ(k)p]+12∥∥Δ→w(k)∥∥2 ≥CΔξ(k)pk+12∥∥Δ→w(k)∥∥2 (18) ≥12(δk−Δξ(k)pk)2L2+CΔξ(k)pk (19) ≥min(12L2δ2k,12Cδk) (20)

by applying (17) in (18), Lemma 7 and Cauchy-Schwarz in (19) and Lemma 8 in (20) .

Case :

In this case, we consider , i.e. the minimal increase among the support vectors. We then define

 ξp(λ) = ⎧⎨⎩ξ(k)p+λΔξ(k)pp≠pkξ(k)pk+λ(Δξ(k)p−ε)p=pk (21)

and weights . There will then be a finite range of for which and satisfy the constraints for so that

 ⟨Δ→w(k),→w(k)⟩+C∑p≠pkΔ→ξ(k)p+C(Δξ(k)pk−ε) ≥0 (22) ⟨Δ→w(k),→w(k)⟩+C∑pΔ→ξ(k)p ≥Cε (23)

We also have a support vector so that

 ypk⟨→w(k+1),→xν⟩+ξ(k+1)pk =1+ε (24) =ε (25)

Using Lemma 7 and Cauchy-Schwarz, we get:

 ypk⟨Δ→w(k),→xk−→xν⟩ =δk−ε (26) ∥∥Δ→w(k)∥∥2 ≥14L2(δk−ε)2 (27)

Thus, we have:

 F(→w(k+1),→ξ(k+1))−F(→w(k),→ξ(k)) (28) =[⟨Δ→w(k),→w(k)⟩+C∑pΔξ(k)p]+12∥∥Δ→w(k)∥∥2 (29) ≥Cε+18L2(δk−ε)2 (30) ≥min(18L2δ2k,12Cδk) (31)

by applying (23) and (27) to obtain the final bound.

Since the algorithm is guaranteed to increase the objective by a finite amount, it will terminate in a finite number of iterations if we require for some positive .

###### Corollary 10.

The algorithm for a given will terminate after at most iterations where is the number of manifolds, L bounds the norm of the points on the manifolds.

###### Proof.

and is a feasible solution for . Therefore, the optimal objective function is upper-bounded by . The upper bound on the number of iterations is then provided by Theorem (9).

We can also bound the error in the objective function after terminates:

###### Corollary 11.

With , after terminates with solution , slack and value , then the optimal value of is bracketed by:

 FMCP≤F⋆≤FMCP+PCδ. (32)
###### Proof.

The lower bound on is apparent since includes constraints for all . Setting the slacks will make the solution feasible for resulting in the upper bound.

## 4 Experiments

### 4.1 Lq Ellipsoidal Manifolds

As an illustration of our method, we have generated -dimensional -norm ellipsoids with random radii, centers, and directions. The points on each manifold are parameterized as where the center and basis vectors are random Gaussian and , the ellipsoidal radii, are sampled from with mean .

We compare the performance of to the conventional Point SVM (, ) with samples drawn from the surface of the ellipsoids for training and test examples. Performance is measured by generalization error on the task of classifying points from positively labeled manifolds from negatively labeled ones, as a function of the number of training samples used during learning.

For these manifolds, the separation oracle of returns points that minimize


text:
# Measurement of the top quark mass in the dileptonic $t\overline{t}$ decay channel using the mass observables $M_{b\ell}$, $M_{T2}$, and $M_{b\ellν}$ in pp collisions at $\sqrt{s}=8$ TeV

CMS Collaboration; Canelli, Florencia; Kilminster, Benjamin; Aarestad, Thea; Caminada, Lea; De Cosa, Annapaoloa; Del Burgo, Riccardo; Donato, Silvio; Galloni, Camilla; Hinzmann, Andreas; Hreus, Tomas; Ngadiuba, Jennifer; Pinna, Deborah; Rauco, Giorgia; Robmann, Peter; Salerno, Daniel; Schweiger, Korbinian; Seitz, Claudia; Takahashi, Yuta; Zucchetta, Alberto; et al (2017). Measurement of the top quark mass in the dileptonic $t\overline{t}$ decay channel using the mass observables $M_{b\ell}$, $M_{T2}$, and $M_{b\ellν}$ in pp collisions at $\sqrt{s}=8$ TeV. Physical Review. D, Particles, fields, gravitation and cosmology, D96(3):032002.

## Abstract

A measurement of the top quark mass ($M_t$) in the dileptonic $t\overline{t}$ decay channel is performed using data from proton-proton collisions at a center-of-mass energy of 8 TeV. The data was recorded by the CMS experiment at the LHC and corresponds to an integrated luminosity of 19.7 $\pm$ 0.5 fb$^{−1}$. Events are selected with two oppositely charged leptons ($\ell = e, \mu$) and two jets identified as originating from $b$ quarks. The analysis is based on three kinematic observables whose distributions are sensitive to the value of $M_t$. An invariant mass observable, $M_{b\ell}$, and a “stransverse mass” observable, MT2, are employed in a simultaneous fit to determine the value of Mt and an overall jet energy scale factor (JSF). A complementary approach is used to construct an invariant mass observable, $M_{b\ellν}$, that is combined with $M_{T2}$ to measure $M_t$. The shapes of the observables, along with their evolutions in $M_t$ and JSF, are modeled by a nonparametric Gaussian process regression technique. The sensitivity of the observables to the value of $M_t$ is investigated using a Fisher information density method. The top quark mass is measured to be 172.22 $\pm$ 0.18(stat) $^{+0.89}_{−0.93}$(syst)  GeV.

## Abstract

A measurement of the top quark mass ($M_t$) in the dileptonic $t\overline{t}$ decay channel is performed using data from proton-proton collisions at a center-of-mass energy of 8 TeV. The data was recorded by the CMS experiment at the LHC and corresponds to an integrated luminosity of 19.7 $\pm$ 0.5 fb$^{−1}$. Events are selected with two oppositely charged leptons ($\ell = e, \mu$) and two jets identified as originating from $b$ quarks. The analysis is based on three kinematic observables whose distributions are sensitive to the value of $M_t$. An invariant mass observable, $M_{b\ell}$, and a “stransverse mass” observable, MT2, are employed in a simultaneous fit to determine the value of Mt and an overall jet energy scale factor (JSF). A complementary approach is used to construct an invariant mass observable, $M_{b\ellν}$, that is combined with $M_{T2}$ to measure $M_t$. The shapes of the observables, along with their evolutions in $M_t$ and JSF, are modeled by a nonparametric Gaussian process regression technique. The sensitivity of the observables to the value of $M_t$ is investigated using a Fisher information density method. The top quark mass is measured to be 172.22 $\pm$ 0.18(stat) $^{+0.89}_{−0.93}$(syst)  GeV.

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# Thread: Maximum Modulus of Complex Function

1. ## Maximum Modulus of Complex Function

How do we find the maximum value of $|z^2+2z-3|$ in $|z|=1$?
Can we just simply use the triangle inequality?

2. ## Re: Maximum Modulus of Complex Function

You can use the triangle inequality to get an upper bound for this, but I am unsure whether that is the maximum value though...

3. ## Re: Maximum Modulus of Complex Function

$|z|=1 \Rightarrow z=\cos(\theta) + i \sin(\theta)$

Find a formula for $|z^2 + 2z - 3|^2$ and plug the above $z$ into that.

That gives you a formula in $\theta$ for the square of the modulus.

You can maximize this and then double check that the answers correspond to a maximum for the modulus itself.

Spoiler:
Letting software do all the grubby work

$|z^2 + 2z - 3|^2 = x^4+4 x^3+2 x^2 \left(y^2-1\right)+4 x \left(y^2-3\right)+y^4+10 y^2+9$

with $z = \cos(\theta + i \sin(\theta)$ this becomes (after a ton of computer aided simpilfication)

$|z^2 + 2z - 3|^2 = 8 \sin ^2\left(\frac{\theta }{2}\right) (3 \cos (\theta )+5)$

$\dfrac{d}{d\theta} \left( 8 \sin ^2\left(\frac{\theta }{2}\right) (3 \cos (\theta )+5) \right)=8 \sin \left(\frac{\theta }{2}\right) \left(5 \cos \left(\frac{\theta }{2}\right)+3 \cos \left(\frac{3 \theta }{2}\right)\right)$

setting this equal to 0 and solving for $\theta$ we get

$\theta = 0, ~\pi, ~\pi - \tan^{-1}(2\sqrt{2}), -\pi - \tan^{-1}(2\sqrt{2})$

I leave it to you to determine which of these are maxima and which are local minima using the 2nd derivative.

4. ## Re: Maximum Modulus of Complex Function

Originally Posted by wps
How do we find the maximum value of $|z^2+2z-3|$ in $|z|=1$?
Can we just simply use the triangle inequality?
$|z^2|=1~,~|2z|=2~\&,~|-3|=3$ so $|z^2+2z-3|\le 6$ is true.

Originally Posted by Prove It
You can use the triangle inequality to get an upper bound for this, but I am unsure whether that is the maximum value though...
Good point. If $z=\bf{i}$ then
$|\bf{i}^2+2\bf{i}-3|=|-4+2\bf{i}|=\sqrt{16+4}=2\sqrt5<4.5$

On the other hand, $z=-1$ then $\left|(-1)^2+2(-1)-3\right|=4$

If $z=a+b\bf{i}$ such that |a^2+b^2=1| then $\left|z^2+2z-3\right|^2=\left(a^2-b^2+2a-3\right)^2+\left(2ab+2a\right)^2$.
Can we maximize this?

5. ## Re: Maximum Modulus of Complex Function

Originally Posted by Prove It
You can use the triangle inequality to get an upper bound for this, but I am unsure whether that is the maximum value though...
Originally Posted by Plato
$|z^2|=1~,~|2z|=2~\&,~|-3|=3$ so $|z^2+2z-3|\le 6$ is true.

Good point. If $z=\bf{i}$ then
$|\bf{i}^2+2\bf{i}-3|=|-4+2\bf{i}|=\sqrt{16+4}=2\sqrt5<4.5$

On the other hand, $z=-1$ then $\left|(-1)^2+2(-1)-3\right|=4$

If $z=a+b\bf{i}$ such that |a^2+b^2=1| then $\left|z^2+2z-3\right|^2=\left(a^2-b^2+2a-3\right)^2+\left(2ab+2a\right)^2$ $\rightarrow$ $\mathbf{\left(2ab+2b\right)^2$}.
Can we maximize this?
Yes,got it.
$\left|z^2+2z-3\right|^2=\left( a^2-(1-a^2)+2a-3\right)^2+(1-a^2)\left(2a+2\right)^2$
So we need to maximize a one variable function.
Thanks everyone