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# If you did not already know

Deep Multimodal Subspace Clustering Network
We present convolutional neural network (CNN) based approaches for unsupervised multimodal subspace clustering. The proposed framework consists of three main stages – multimodal encoder, self-expressive layer, and multimodal decoder. The encoder takes multimodal data as input and fuses them to a latent space representation. We investigate early, late and intermediate fusion techniques and propose three different encoders corresponding to them for spatial fusion. The self-expressive layers and multimodal decoders are essentially the same for different spatial fusion-based approaches. In addition to various spatial fusion-based methods, an affinity fusion-based network is also proposed in which the self-expressiveness layer corresponding to different modalities is enforced to be the same. Extensive experiments on three datasets show that the proposed methods significantly outperform the state-of-the-art multimodal subspace clustering methods. …

GITNet
In several natural language tasks, labeled sequences are available in separate domains (say, languages), but the goal is to label sequences with mixed domain (such as code-switched text). Or, we may have available models for labeling whole passages (say, with sentiments), which we would like to exploit toward better position-specific label inference (say, target-dependent sentiment annotation). A key characteristic shared across such tasks is that different positions in a primary instance can benefit from different experts’ trained from auxiliary data, but labeled primary instances are scarce, and labeling the best expert for each position entails unacceptable cognitive burden. We propose GITNet, a unified position-sensitive multi-task recurrent neural network (RNN) architecture for such applications. Auxiliary and primary tasks need not share training instances. Auxiliary RNNs are trained over auxiliary instances. A primary instance is also submitted to each auxiliary RNN, but their state sequences are gated and merged into a novel composite state sequence tailored to the primary inference task. Our approach is in sharp contrast to recent multi-task networks like the cross-stitch and sluice network, which do not control state transfer at such fine granularity. We demonstrate the superiority of GIRNet using three applications: sentiment classification of code-switched passages, part-of-speech tagging of code-switched text, and target position-sensitive annotation of sentiment in monolingual passages. In all cases, we establish new state-of-the-art performance beyond recent competitive baselines. …

s-bAbI
In this study, we investigate the limits of the current state of the art AI system for detecting buffer overflows and compare it with current static analysis tools. To do so, we developed a code generator, s-bAbI, capable of producing an arbitrarily large number of code samples of controlled complexity. We found that the static analysis engines we examined have good precision, but poor recall on this dataset, except for a sound static analyzer that has good precision and recall. We found that the state of the art AI system, a memory network modeled after Choi et al. [1], can achieve similar performance to the static analysis engines, but requires an exhaustive amount of training data in order to do so. Our work points towards future approaches that may solve these problems; namely, using representations of code that can capture appropriate scope information and using deep learning methods that are able to perform arithmetic operations. …

# If you did not already know

Concurrent Meta-Reinforcement Learning (CMRL)
State-of-the-art meta reinforcement learning algorithms typically assume the setting of a single agent interacting with its environment in a sequential manner. A negative side-effect of this sequential execution paradigm is that, as the environment becomes more and more challenging, and thus requiring more interaction episodes for the meta-learner, it needs the agent to reason over longer and longer time-scales. To combat the difficulty of long time-scale credit assignment, we propose an alternative parallel framework, which we name ‘Concurrent Meta-Reinforcement Learning’ (CMRL), that transforms the temporal credit assignment problem into a multi-agent reinforcement learning one. In this multi-agent setting, a set of parallel agents are executed in the same environment and each of these ‘rollout’ agents are given the means to communicate with each other. The goal of the communication is to coordinate, in a collaborative manner, the most efficient exploration of the shared task the agents are currently assigned. This coordination therefore represents the meta-learning aspect of the framework, as each agent can be assigned or assign itself a particular section of the current task’s state space. This framework is in contrast to standard RL methods that assume that each parallel rollout occurs independently, which can potentially waste computation if many of the rollouts end up sampling the same part of the state space. Furthermore, the parallel setting enables us to define several reward sharing functions and auxiliary losses that are non-trivial to apply in the sequential setting. We demonstrate the effectiveness of our proposed CMRL at improving over sequential methods in a variety of challenging tasks. …

Trellis Network
We present trellis networks, a new architecture for sequence modeling. On the one hand, a trellis network is a temporal convolutional network with special structure, characterized by weight tying across depth and direct injection of the input into deep layers. On the other hand, we show that truncated recurrent networks are equivalent to trellis networks with special sparsity structure in their weight matrices. Thus trellis networks with general weight matrices generalize truncated recurrent networks. We leverage these connections to design high-performing trellis networks that absorb structural and algorithmic elements from both recurrent and convolutional models. Experiments demonstrate that trellis networks outperform the current state of the art on a variety of challenging benchmarks, including word-level language modeling on Penn Treebank and WikiText-103, character-level language modeling on Penn Treebank, and stress tests designed to evaluate long-term memory retention. The code is available at https://…/trellisnet . …

DirNet
Recurrent neural networks (RNNs) achieve cutting-edge performance on a variety of problems. However, due to their high computational and memory demands, deploying RNNs on resource constrained mobile devices is a challenging task. To guarantee minimum accuracy loss with higher compression rate and driven by the mobile resource requirement, we introduce a novel model compression approach DirNet based on an optimized fast dictionary learning algorithm, which 1) dynamically mines the dictionary atoms of the projection dictionary matrix within layer to adjust the compression rate 2) adaptively changes the sparsity of sparse codes cross the hierarchical layers. Experimental results on language model and an ASR model trained with a 1000h speech dataset demonstrate that our method significantly outperforms prior approaches. Evaluated on off-the-shelf mobile devices, we are able to reduce the size of original model by eight times with real-time model inference and negligible accuracy loss. …

Image Enhancement Generative Adversarial Network (IEGAN)
Despite the breakthroughs in quality of image enhancement, an end-to-end solution for simultaneous recovery of the finer texture details and sharpness for degraded images with low resolution is still unsolved. Some existing approaches focus on minimizing the pixel-wise reconstruction error which results in a high peak signal-to-noise ratio. The enhanced images fail to provide high-frequency details and are perceptually unsatisfying, i.e., they fail to match the quality expected in a photo-realistic image. In this paper, we present Image Enhancement Generative Adversarial Network (IEGAN), a versatile framework capable of inferring photo-realistic natural images for both artifact removal and super-resolution simultaneously. Moreover, we propose a new loss function consisting of a combination of reconstruction loss, feature loss and an edge loss counterpart. The feature loss helps to push the output image to the natural image manifold and the edge loss preserves the sharpness of the output image. The reconstruction loss provides low-level semantic information to the generator regarding the quality of the generated images compared to the original. Our approach has been experimentally proven to recover photo-realistic textures from heavily compressed low-resolution images on public benchmarks and our proposed high-resolution World100 dataset. …

# If you did not already know

Multi-Node2vec
Learning interpretable features from complex multilayer networks is a challenging and important problem. The need for such representations is particularly evident in multilayer networks of the brain, where nodal characteristics may help model and differentiate regions of the brain according to individual, cognitive task, or disease. Motivated by this problem, we introduce the multi-node2vec algorithm, an efficient and scalable feature engineering method that automatically learns continuous node feature representations from multilayer networks. Multi-node2vec relies upon a second-order random walk sampling procedure that efficiently explores the inner- and intra-layer ties of the observed multilayer network is utilized to identify multilayer neighborhoods. Maximum likelihood estimators of the nodal features are identified through the use of the Skip-gram neural network model on the collection of sampled neighborhoods. We investigate the conditions under which multi-node2vec is an approximation of a closed-form matrix factorization problem. We demonstrate the efficacy of multi-node2vec on a multilayer functional brain network from resting state fMRI scans over a group of 74 healthy individuals. We find that multi-node2vec outperforms contemporary methods on complex networks, and that multi-node2vec identifies nodal characteristics that closely associate with the functional organization of the brain. …

FoveaBox
We present FoveaBox, an accurate, flexible and completely anchor-free framework for object detection. While almost all state-of-the-art object detectors utilize the predefined anchors to enumerate possible locations, scales and aspect ratios for the search of the objects, their performance and generalization ability are also limited to the design of anchors. Instead, FoveaBox directly learns the object existing possibility and the bounding box coordinates without anchor reference. This is achieved by: (a) predicting category-sensitive semantic maps for the object existing possibility, and (b) producing category-agnostic bounding box for each position that potentially contains an object. The scales of target boxes are naturally associated with feature pyramid representations for each input image. Without bells and whistles, FoveaBox achieves state-of-the-art single model performance of 42.1 AP on the standard COCO detection benchmark. Specially for the objects with arbitrary aspect ratios, FoveaBox brings in significant improvement compared to the anchor-based detectors. More surprisingly, when it is challenged by the stretched testing images, FoveaBox shows great robustness and generalization ability to the changed distribution of bounding box shapes. The code will be made publicly available. …

Orthogonal Floating Search Framework
The present study proposes a new Orthogonal Floating Search framework for structure selection of nonlinear systems by adapting the existing floating search algorithms for feature selection. The proposed framework integrates the concept of orthogonal space and consequent Error-Reduction-Ratio (ERR) metric with the existing floating search algorithms. On the basis of this framework, three well-known feature selection algorithms have been adapted which include the classical Sequential Forward Floating Search (SFFS), Improved sequential Forward Floating Search (IFFS) and Oscillating Search (OS). This framework retains the simplicity of classical Orthogonal Forward Regression with ERR (OFR-ERR) and eliminates the nesting effect associated with OFR-ERR. The performance of the proposed framework has been demonstrated considering several benchmark non-linear systems. The results show that most of the existing feature selection methods can easily be tailored to identify the correct system structure of nonlinear systems. …

Distributional Variant of Gradient Temporal-Difference (Distributional GTD2)
We devise a distributional variant of gradient temporal-difference (TD) learning. Distributional reinforcement learning has been demonstrated to outperform the regular one in the recent study \citep{bellemare2017distributional}. In our paper, we design two new algorithms called distributional GTD2 and distributional TDC using the Cram{\’e}r distance on the distributional version of the Bellman error objective function, which inherits advantages of both the nonlinear gradient TD algorithms and the distributional RL approach. We prove the asymptotic almost-sure convergence to a local optimal solution for general smooth function approximators, which includes neural networks that have been widely used in recent study to solve the real-life RL problems. In each step, the computational complexity is linear w.r.t.\ the number of the parameters of the function approximator, thus can be implemented efficiently for neural networks. …

# If you did not already know

Learning to Teach
Teaching plays a very important role in our society, by spreading human knowledge and educating our next generations. A good teacher will select appropriate teaching materials, impact suitable methodologies, and set up targeted examinations, according to the learning behaviors of the students. In the field of artificial intelligence, however, one has not fully explored the role of teaching, and pays most attention to machine \emph{learning}. In this paper, we argue that equal attention, if not more, should be paid to teaching, and furthermore, an optimization framework (instead of heuristics) should be used to obtain good teaching strategies. We call this approach learning to teach’. In the approach, two intelligent agents interact with each other: a student model (which corresponds to the learner in traditional machine learning algorithms), and a teacher model (which determines the appropriate data, loss function, and hypothesis space to facilitate the training of the student model). The teacher model leverages the feedback from the student model to optimize its own teaching strategies by means of reinforcement learning, so as to achieve teacher-student co-evolution. To demonstrate the practical value of our proposed approach, we take the training of deep neural networks (DNN) as an example, and show that by using the learning to teach techniques, we are able to use much less training data and fewer iterations to achieve almost the same accuracy for different kinds of DNN models (e.g., multi-layer perceptron, convolutional neural networks and recurrent neural networks) under various machine learning tasks (e.g., image classification and text understanding). …

VCExplorer
Graphs have been widely used to model different information networks, such as the Web, biological networks and social networks (e.g. Twitter). Due to the size and complexity of these graphs, how to explore and utilize these graphs has become a very challenging problem. In this paper, we propose, VCExplorer, a new interactive graph exploration framework that integrates the strengths of graph visualization and graph summarization. Unlike existing graph visualization tools where vertices of a graph may be clustered into a smaller collection of super/virtual vertices, VCExplorer displays a small number of actual source graph vertices (called hubs) and summaries of the information between these vertices. We refer to such a graph as a HA-graph (Hub-based Aggregation Graph). This allows users to appreciate the relationship between the hubs, rather than super/virtual vertices. Users can navigate through the HA- graph by ‘drilling down’ into the summaries between hubs to display more hubs. We illustrate how the graph aggregation techniques can be integrated into the exploring framework as the consolidated information to users. In addition, we propose efficient graph aggregation algorithms over multiple subgraphs via computation sharing. Extensive experimental evaluations have been conducted using both real and synthetic datasets and the results indicate the effectiveness and efficiency of VCExplorer for exploration. …

Dirichlet Process Forest
Methods based on Bayesian decision tree ensembles have proven valuable in constructing high-quality predictions, and are particularly attractive in certain settings because they encourage low-order interaction effects. Despite adapting to the presence of low-order interactions for prediction purpose, we show that Bayesian decision tree ensembles are generally anti-conservative for the purpose of conducting interaction detection. We address this problem by introducing Dirichlet process forests (DP-Forests), which leverage the presence of low-order interactions by clustering the trees so that trees within the same cluster focus on detecting a specific interaction. We show on both simulated and benchmark data that DP-Forests perform well relative to existing interaction detection techniques for detecting low-order interactions, attaining very low false-positive and false-negative rates while maintaining the same performance for prediction using a comparable computational budget. …

Network Science
Network science is an interdisciplinary academic field which studies complex networks such as telecommunication networks, computer networks, biological networks, cognitive and semantic networks, and social networks. The field draws on theories and methods including graph theory from mathematics, statistical mechanics from physics, data mining and information visualization from computer science, inferential modeling from statistics, and social structure from sociology. The United States National Research Council defines network science as ‘the study of network representations of physical, biological, and social phenomena leading to predictive models of these phenomena.’ …

# If you did not already know

Pangea
Storage and memory systems for modern data analytics are heavily layered, managing shared persistent data, cached data, and non- shared execution data in separate systems such as distributed file system like HDFS, in-memory file system like Alluxio and computation framework like Spark. Such layering introduces significant performance and management costs for copying data across layers redundantly and deciding proper resource allocation for all layers. In this paper we propose a single system called Pangea that can manage all data—both intermediate and long-lived data, and their buffer/caching, data placement optimization, and failure recovery—all in one monolithic storage system, without any layering. We present a detailed performance evaluation of Pangea and show that its performance compares favorably with several widely used layered systems such as Spark. …

Neyman-Pearson Criterion (NPC)
We propose a new model selection criterion, the Neyman-Pearson criterion (NPC), for asymmetric binary classification problems such as cancer diagnosis, where the two types of classification errors have vastly different priorities. The NPC is a general prediction-based criterion that works for most classification methods including logistic regression, support vector machines, and random forests. We study the theoretical model selection properties of the NPC for nonparametric plug-in methods. Simulation studies show that the NPC outperforms the classical prediction-based criterion that minimizes the overall classification error under various asymmetric classification scenarios. A real data case study of breast cancer suggests that the NPC is a practical criterion that leads to the discovery of novel gene markers with both high sensitivity and specificity for breast cancer diagnosis. The NPC is available in an R package NPcriterion. …

Harmonia
Distributed storage employs replication to mask failures and improve availability. However, these systems typically exhibit a hard tradeoff between consistency and performance. Ensuring consistency introduces coordination overhead, and as a result the system throughput does not scale with the number of replicas. We present Harmonia, a replicated storage architecture that exploits the capability of new-generation programmable switches to obviate this tradeoff by providing near-linear scalability without sacrificing consistency. To achieve this goal, Harmonia detects read-write conflicts in the network, which enables any replica to serve reads for objects with no pending writes. Harmonia implements this functionality at line rate, thus imposing no performance overhead. We have implemented a prototype of Harmonia on a cluster of commodity servers connected by a Barefoot Tofino switch, and have integrated it with Redis. We demonstrate the generality of our approach by supporting a variety of replication protocols, including primary-backup, chain replication, Viewstamped Replication, and NOPaxos. Experimental results show that Harmonia improves the throughput of these protocols by up to 10X for a replication factor of 10, providing near-linear scalability up to the limit of our testbed. …

Stochastic Distance Transform (SDT)
The distance transform (DT) and its many variations are ubiquitous tools for image processing and analysis. In many imaging scenarios, the images of interest are corrupted by noise. This has a strong negative impact on the accuracy of the DT, which is highly sensitive to spurious noise points. In this study, we consider images represented as discrete random sets and observe statistics of DT computed on such representations. We, thus, define a stochastic distance transform (SDT), which has an adjustable robustness to noise. Both a stochastic Monte Carlo method and a deterministic method for computing the SDT are proposed and compared. Through a series of empirical tests, we demonstrate that the SDT is effective not only in improving the accuracy of the computed distances in the presence of noise, but also in improving the performance of template matching and watershed segmentation of partially overlapping objects, which are examples of typical applications where DTs are utilized. …

# If you did not already know

Metric Expression Network (MEnet)
Recent CNN-based saliency models have achieved great performance on public datasets, however, most of them are sensitive to distortion (e.g., noise, compression). In this paper, an end-to-end generic salient object segmentation model called Metric Expression Network (MEnet) is proposed to overcome this drawback. Within this architecture, we construct a new topological metric space, with the implicit metric being determined by the deep network. In this way, we succeed in grouping all the pixels within the observed image semantically within this latent space into two regions: a salient region and a non-salient region. With this method, all feature extractions are carried out at the pixel level, which makes the output boundaries of salient object fine-grained. Experimental results show that the proposed metric can generate robust salient maps that allow for object segmentation. By testing the method on several public benchmarks, we show that the performance of MEnet has achieved good results. Furthermore, the proposed method outperforms previous CNN-based methods on distorted images. …

PruneTrain
Model pruning is a popular mechanism to make a network more efficient for inference. In this paper, we explore the use of pruning to also make the training of such neural networks more efficient. Unlike all prior model pruning methods that sparsify a pre-trained model and then prune it, we train the network from scratch, while gradually and structurally pruning parameters during the training. We build on our key observations: 1) once parameters are sparsified via regularization, they rarely re-appear in later steps, and 2) setting the appropriate regularization penalty at the beginning of training effectively converges the loss. We train ResNet and VGG networks on CIFAR10/100 and ImageNet datasets from scratch, and achieve 30-50% improvement in training FLOPs and 20-30% improvement in measured training time on modern GPUs. …

Position, Sequence and Set Similarity Measure
In this paper the author presents a new similarity measure for strings of characters based on S3M which he expands to take into account not only the characters set and sequence but also their position. After demonstrating the superiority of this new measure and discussing the need for a self adaptive spell checker, this work is further developed into an adaptive spell checker that produces a cluster with a defined number of words for each presented misspelled word. The accuracy of this solution is measured comparing its results against the results of the most widely used spell checker. …

Blazer
Explore your data with SQL. Easily create charts and dashboards, and share them with your team. …

# If you did not already know

Tensor Core
The NVIDIA Volta GPU microarchitecture introduces a specialized unit, called ‘Tensor Core’ that performs one matrix-multiply-and-accumulate on 4×4 matrices per clock cycle. The NVIDIA Tesla V100 accelerator, featuring the Volta microarchitecture, provides 640 Tensor Cores with a theoretical peak performance of 125 Tflops/s in mixed precision. In this paper, we investigate current approaches to program NVIDIA Tensor Cores, their performances and the precision loss due to computation in mixed precision. Currently, NVIDIA provides three different ways of programming matrix-multiply-and-accumulate on Tensor Cores: the CUDA Warp Matrix Multiply Accumulate (WMMA) API, CUTLASS, a templated library based on WMMA, and cuBLAS GEMM. After experimenting with different approaches, we found that NVIDIA Tensor Cores can deliver up to 83 Tflops/s in mixed precision on a Tesla V100 GPU, seven and three times the performance in single and half precision respectively. A WMMA implementation of batched GEMM reaches a performance of 4 Tflops/s. While precision loss due to matrix multiplication with half precision input might be critical in many HPC applications, it can be considerably reduced at the cost of increased computation. Our results indicate that HPC applications using matrix multiplications can strongly benefit from using of NVIDIA Tensor Cores. …

Movie Intelligent Recommender Agent (MIRA)
The human mind is still an unknown process of neuroscience in many aspects. Nevertheless, for decades the scientific community has proposed computational models that try to simulate their parts, specific applications, or their behavior in different situations. The most complete model in this line is undoubtedly the LIDA model, proposed by Stan Franklin with the aim of serving as a generic computational architecture for several applications. The present project is inspired by the LIDA model to apply it to the process of movie recommendation, the model called MIRA (Movie Intelligent Recommender Agent) presented percentages of precision similar to a traditional model when submitted to the same assay conditions. Moreover, the proposed model reinforced the precision indexes when submitted to tests with volunteers, proving once again its performance as a cognitive model, when executed with small data volumes. Considering that the proposed model achieved a similar behavior to the traditional models under conditions expected to be similar for natural systems, it can be said that MIRA reinforces the applicability of LIDA as a path to be followed for the study and generation of computational agents inspired by neural behaviors. …

Hierarchical Importance Weighted Autoencoder
Importance weighted variational inference (Burda et al., 2015) uses multiple i.i.d. samples to have a tighter variational lower bound. We believe a joint proposal has the potential of reducing the number of redundant samples, and introduce a hierarchical structure to induce correlation. The hope is that the proposals would coordinate to make up for the error made by one another to reduce the variance of the importance estimator. Theoretically, we analyze the condition under which convergence of the estimator variance can be connected to convergence of the lower bound. Empirically, we confirm that maximization of the lower bound does implicitly minimize variance. Further analysis shows that this is a result of negative correlation induced by the proposed hierarchical meta sampling scheme, and performance of inference also improves when the number of samples increases. …

Column2Vec
We present Column2Vec, a distributed representation of database columns based on column metadata. Our distributed representation has several applications. Using known names for groups of columns (i.e., a table name), we train a model to generate an appropriate name for columns in an unnamed table. We demonstrate the viability of our approach using schema information collected from open source applications on GitHub. …

# If you did not already know

GraphMP
Recent studies showed that single-machine graph processing systems can be as highly competitive as cluster-based approaches on large-scale problems. While several out-of-core graph processing systems and computation models have been proposed, the high disk I/O overhead could significantly reduce performance in many practical cases. In this paper, we propose GraphMP to tackle big graph analytics on a single machine. GraphMP achieves low disk I/O overhead with three techniques. First, we design a vertex-centric sliding window (VSW) computation model to avoid reading and writing vertices on disk. Second, we propose a selective scheduling method to skip loading and processing unnecessary edge shards on disk. Third, we use a compressed edge cache mechanism to fully utilize the available memory of a machine to reduce the amount of disk accesses for edges. Extensive evaluations have shown that GraphMP could outperform existing single-machine out-of-core systems such as GraphChi, X-Stream and GridGraph by up to 51, and can be as highly competitive as distributed graph engines like Pregel+, PowerGraph and Chaos. …

Prediction Interval
In statistical inference, specifically predictive inference, a prediction interval is an estimate of an interval in which future observations will fall, with a certain probability, given what has already been observed. Prediction intervals are often used in regression analysis. Prediction intervals are used in both frequentist statistics and Bayesian statistics: a prediction interval bears the same relationship to a future observation that a frequentist confidence interval or Bayesian credible interval bears to an unobservable population parameter: prediction intervals predict the distribution of individual future points, whereas confidence intervals and credible intervals of parameters predict the distribution of estimates of the true population mean or other quantity of interest that cannot be observed.
Prediction Interval, the wider sister of Confidence Interval

Snapshot Ensembles
Ensembles of neural networks are known to be much more robust and accurate than individual networks. However, training multiple deep networks for model averaging is computationally expensive. In this paper, we propose a method to obtain the seemingly contradictory goal of ensembling multiple neural networks at no additional training cost. We achieve this goal by training a single neural network, converging to several local minima along its optimization path and saving the model parameters. To obtain repeated rapid convergence, we leverage recent work on cyclic learning rate schedules. The resulting technique, which we refer to as Snapshot Ensembling, is simple, yet surprisingly effective. We show in a series of experiments that our approach is compatible with diverse network architectures and learning tasks. It consistently yields lower error rates than state-of-the-art single models at no additional training cost, and compares favorably with traditional network ensembles. On CIFAR-10 and CIFAR-100 our DenseNet Snapshot Ensembles obtain error rates of 3.4% and 17.4% respectively.
Snapshot Ensembles in Keras

INDIAN
We devise a learning algorithm for possibly nonsmooth deep neural networks featuring inertia and Newtonian directional intelligence only by means of a back-propagation oracle. Our algorithm, called INDIAN, has an appealing mechanical interpretation, making the role of its two hyperparameters transparent. An elementary phase space lifting allows both for its implementation and its theoretical study under very general assumptions. We handle in particular a stochastic version of our method (which encompasses usual mini-batch approaches) for nonsmooth activation functions (such as ReLU). Our algorithm shows high efficiency and reaches state of the art on image classification problems. …

# If you did not already know

BO-Aug
In recent years, deep learning has achieved remarkable achievements in many fields, including computer vision, natural language processing, speech recognition and others. Adequate training data is the key to ensure the effectiveness of the deep models. However, obtaining valid data requires a lot of time and labor resources. Data augmentation (DA) is an effective alternative approach, which can generate new labeled data based on existing data using label-preserving transformations. Although we can benefit a lot from DA, designing appropriate DA policies requires a lot of expert experience and time consumption, and the evaluation of searching the optimal policies is costly. So we raise a new question in this paper: how to achieve automated data augmentation at as low cost as possible? We propose a method named BO-Aug for automating the process by finding the optimal DA policies using the Bayesian optimization approach. Our method can find the optimal policies at a relatively low search cost, and the searched policies based on a specific dataset are transferable across different neural network architectures or even different datasets. We validate the BO-Aug on three widely used image classification datasets, including CIFAR-10, CIFAR-100 and SVHN. Experimental results show that the proposed method can achieve state-of-the-art or near advanced classification accuracy. Code to reproduce our experiments is available at https://…/BO-Aug.

GAN Augmentation
One of the biggest issues facing the use of machine learning in medical imaging is the lack of availability of large, labelled datasets. The annotation of medical images is not only expensive and time consuming but also highly dependent on the availability of expert observers. The limited amount of training data can inhibit the performance of supervised machine learning algorithms which often need very large quantities of data on which to train to avoid overfitting. So far, much effort has been directed at extracting as much information as possible from what data is available. Generative Adversarial Networks (GANs) offer a novel way to unlock additional information from a dataset by generating synthetic samples with the appearance of real images. This paper demonstrates the feasibility of introducing GAN derived synthetic data to the training datasets in two brain segmentation tasks, leading to improvements in Dice Similarity Coefficient (DSC) of between 1 and 5 percentage points under different conditions, with the strongest effects seen fewer than ten training image stacks are available. …

Deep Back-Projection Network (DBPN)
The feed-forward architectures of recently proposed deep super-resolution networks learn representations of low-resolution inputs, and the non-linear mapping from those to high-resolution output. However, this approach does not fully address the mutual dependencies of low- and high-resolution images. We propose Deep Back-Projection Networks (DBPN), that exploit iterative up- and down-sampling layers, providing an error feedback mechanism for projection errors at each stage. We construct mutually-connected up- and down-sampling stages each of which represents different types of image degradation and high-resolution components. We show that extending this idea to allow concatenation of features across up- and down-sampling stages (Dense DBPN) allows us to reconstruct further improve super-resolution, yielding superior results and in particular establishing new state of the art results for large scaling factors such as 8x across multiple data sets. …

BlockCNN
We present a general technique that performs both artifact removal and image compression. For artifact removal, we input a JPEG image and try to remove its compression artifacts. For compression, we input an image and process its 8 by 8 blocks in a sequence. For each block, we first try to predict its intensities based on previous blocks; then, we store a residual with respect to the input image. Our technique reuses JPEG’s legacy compression and decompression routines. Both our artifact removal and our image compression techniques use the same deep network, but with different training weights. Our technique is simple and fast and it significantly improves the performance of artifact removal and image compression. …

# If you did not already know

Fine-Tuned Language Model (FitLaM)
Transfer learning has revolutionized computer vision, but existing approaches in NLP still require task-specific modifications and training from scratch. We propose Fine-tuned Language Models (FitLaM), an effective transfer learning method that can be applied to any task in NLP, and introduce techniques that are key for fine-tuning a state-of-the-art language model. Our method significantly outperforms the state-of-the-art on five text classification tasks, reducing the error by 18-24% on the majority of datasets. We open-source our pretrained models and code to enable adoption by the community. …

MixHop
Existing popular methods for semi-supervised learning with Graph Neural Networks (such as the Graph Convolutional Network) provably cannot learn a general class of neighborhood mixing relationships. To address this weakness, we propose a new model, MixHop, that can learn these relationships, including difference operators, by repeatedly mixing feature representations of neighbors at various distances. MixHop requires no additional memory or computational complexity, and outperforms on challenging baselines. In addition, we propose sparsity regularization that allows us to visualize how the network prioritizes neighborhood information across different graph datasets. Our analysis of the learned architectures reveals that neighborhood mixing varies per datasets. …

Equilibrated Recurrent Neural Network (ERNN)
We propose a novel {\it Equilibrated Recurrent Neural Network} (ERNN) to combat the issues of inaccuracy and instability in conventional RNNs. Drawing upon the concept of autapse in neuroscience, we propose augmenting an RNN with a time-delayed self-feedback loop. Our sole purpose is to modify the dynamics of each internal RNN state and, at any time, enforce it to evolve close to the equilibrium point associated with the input signal at that time. We show that such self-feedback helps stabilize the hidden state transitions leading to fast convergence during training while efficiently learning discriminative latent features that result in state-of-the-art results on several benchmark datasets at test-time. We propose a novel inexact Newton method to solve fixed-point conditions given model parameters for generating the latent features at each hidden state. We prove that our inexact Newton method converges locally with linear rate (under mild conditions). We leverage this result for efficient training of ERNNs based on backpropagation. …

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# Tag Archives: cars

## Cars overtaking

While cycling during one of the unseasonably sunny October days, I passed lots of cars on the opposite side of the road, but none seemed to be overtaking me. I wondered: how many more I should expect on one side compared to the other?

Making some unrealistic assumptions: I ride my bicycle at a constant speed of $b$ mph in traffic that is constant, with evenly spaced cars (a poor assumption) travelling at a constant speed $c$ mph (not slowing when they overtake—a realistic assumption, in my experience), the answer is easily calculated.

The unslowing cars approach from behind at a merciless relative speed of $c-b$ mph, and those safely across the central double line zoom past at $c+b$ mph. So, for every car I pass on the opposite side of the road, I would expect to be overtaken $\frac{c-b}{c+b}$ times.  Continue reading


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# American Institute of Mathematical Sciences

March  2013, 8(1): 291-325. doi: 10.3934/nhm.2013.8.291

## The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system

 1 Graduate School of Advanced Mathematical Science, Meiji University, 1-1-1 Higashimita, Tama-ku, Kawasaki, Kanagawa 214-8571, Japan

Received  March 2012 Revised  January 2013 Published  April 2013

The Gierer-Meinhardt system is a mathematical model describing the process of hydra regeneration. This system has a stationary solution with a stripe pattern on a rectangular domain, but numerical results suggest that such stripe pattern is unstable. In [8], Kolokolnikov et al. proved the existence of a positive eigenvalue, which is called an unstable eigenvalue, for a stationary solution with a stripe pattern by the NLEP method, which implies the instability of the stripe pattern. In addition, the uniqueness of the unstable eigenvalue was shown under some technical assumptions in [8]. In this paper, we prove the existence and uniqueness of an unstable eigenvalue by using the SLEP method without any extra conditions. We also prove the existence of a single-spike solution in one-dimension.
Citation: Kota Ikeda. The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system. Networks & Heterogeneous Media, 2013, 8 (1) : 291-325. doi: 10.3934/nhm.2013.8.291
##### References:
 [1] H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations,", Universitext, (2011). [2] A. Doelman, R. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations,, Indiana Univ. Math. J., 50 (2001), 443. doi: 10.1512/iumj.2001.50.1873. [3] A. Doelman and H. van der Ploeg, Homoclinic stripe patterns,, SIAM J. Appl. Dyn. Syst., 1 (2002), 65. doi: 10.1137/S1111111101392831. [4] S. Ei and J. Wei, Dynamics of metastable localized patterns and its application to the interaction of spike solutions for the Gierer-Meinhardt systems in two spatial dimensions,, Japan J. Indust. Appl. Math., 19 (2002), 181. doi: 10.1007/BF03167453. [5] A. Gierer and H. Meinhardt, A theory of biological pattern formation,, Kybernetic, 12 (1972), 30. doi: 10.1007/BF00289234. [6] P. Hartman, "Ordinary Differential Equations,", Birkhäuser Boston, (1982). [7] D. Iron, M. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model,, Phys. D, 150 (2001), 25. doi: 10.1016/S0167-2789(00)00206-2. [8] T. Kolokolnikov, W. Sun, M. J. Ward and J. Wei, The stability of a stripe for the Gierer-Meinhardt model and the effect of saturation,, SIAM J. Appl. Dyn. Syst., 5 (2006), 313. doi: 10.1137/050635080. [9] S. Kondo and R. Asai, A reaction-diffusion wave on the marine angelfish Pomacanthus,, Nature, 376 (1995), 765. doi: 10.1038/376765a0. [10] S. Kondo, and T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation,, Science, 329 (2010), 1616. [11] P. K. Maini, K. J. Painter and H. N. P. Chau, Spatial pattern formation in chemical and biological systems,, J. Chem. Soc. Faraday Trans., 93 (1997), 3601. doi: 10.1039/a702602a. [12] H. Meinhardt, "Models of Biological Pattern Formation,", Academic Press, (1982). [13] J. D. Murray, "Mathematical Biology,", Biomathematics, 19 (1989). doi: 10.1007/978-3-662-08539-4. [14] A. Nakamasu, G. Takahashi, A. Kanbe and S. Kondo, Interactions between zebrafish pigment cells responsible for the generation of Turing patterns,, Proceedings of the National Academy of Sciences, 106 (2009), 8429. doi: 10.1073/pnas.0808622106. [15] Y. Nakamura, C. D. Tsiairis, S. Özbek and T. W. Holstein, Autoregulatory and repressive inputs localize Hydra Wnt3 to the head organizer,, Proceedings of the National Academy of Sciences, 108 (2011), 9137. doi: 10.1073/pnas.1018109108. [16] Y. Nishiura, Coexistence of infinitely many stable solutions to reaction diffusion systems in the singular limit,, in, 3 (1994), 25. [17] Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations,, SIAM J. Math. Anal., 18 (1987), 1726. doi: 10.1137/0518124. [18] H. Shoji, Y. Iwasa and S. Kondo, Stripes, spots, or reversed spots in two-dimensional Turing systems,, J. Theoret. Biol., 224 (2003), 339. doi: 10.1016/S0022-5193(03)00170-X. [19] I. Takagi, Point-condensation for a reaction-diffusion system,, J. Differential Equations, 61 (1986), 208. doi: 10.1016/0022-0396(86)90119-1. [20] M. Taniguchi, A uniform convergence theorem for singular limit eigenvalue problems,, Adv. Differential Equations, 8 (2003), 29. [21] M. Taniguchi and Y. Nishiura, Stability and characteristic wavelength of planar interfaces in the large diffusion limit of the inhibitor,, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 117. doi: 10.1017/S0308210500030638. [22] M. Taniguchi and Y. Nishiura, Instability of planar interfaces in reaction-diffusion systems,, SIAM J. Math. Anal., 25 (1994), 99. doi: 10.1137/S0036141092233500. [23] A. Turing, The chemical basis of morphogenesis,, Phil. Trans. R. Soc. Lond. B, 327 (1952), 37. doi: 10.1098/rstb.1952.0012. [24] M. J. Ward and J. Wei, Asymmetric spike patterns for the one-dimensional Gierer-Meinhardt model: equilibria and stability,, European J. Appl. Math., 13 (2002), 283. doi: 10.1017/S0956792501004442. [25] J. Wei and M. Winter, Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system,, J. Math. Pures Appl. (9), 83 (2004), 433. doi: 10.1016/j.matpur.2003.09.006. [26] J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: The weak coupling case,, J. Nonlinear Sci., 11 (2001), 415. doi: 10.1007/s00332-001-0380-1.

show all references

##### References:
 [1] H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations,", Universitext, (2011). [2] A. Doelman, R. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations,, Indiana Univ. Math. J., 50 (2001), 443. doi: 10.1512/iumj.2001.50.1873. [3] A. Doelman and H. van der Ploeg, Homoclinic stripe patterns,, SIAM J. Appl. Dyn. Syst., 1 (2002), 65. doi: 10.1137/S1111111101392831. [4] S. Ei and J. Wei, Dynamics of metastable localized patterns and its application to the interaction of spike solutions for the Gierer-Meinhardt systems in two spatial dimensions,, Japan J. Indust. Appl. Math., 19 (2002), 181. doi: 10.1007/BF03167453. [5] A. Gierer and H. Meinhardt, A theory of biological pattern formation,, Kybernetic, 12 (1972), 30. doi: 10.1007/BF00289234. [6] P. Hartman, "Ordinary Differential Equations,", Birkhäuser Boston, (1982). [7] D. Iron, M. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model,, Phys. D, 150 (2001), 25. doi: 10.1016/S0167-2789(00)00206-2. [8] T. Kolokolnikov, W. Sun, M. J. Ward and J. Wei, The stability of a stripe for the Gierer-Meinhardt model and the effect of saturation,, SIAM J. Appl. Dyn. Syst., 5 (2006), 313. doi: 10.1137/050635080. [9] S. Kondo and R. Asai, A reaction-diffusion wave on the marine angelfish Pomacanthus,, Nature, 376 (1995), 765. doi: 10.1038/376765a0. [10] S. Kondo, and T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation,, Science, 329 (2010), 1616. [11] P. K. Maini, K. J. Painter and H. N. P. Chau, Spatial pattern formation in chemical and biological systems,, J. Chem. Soc. Faraday Trans., 93 (1997), 3601. doi: 10.1039/a702602a. [12] H. Meinhardt, "Models of Biological Pattern Formation,", Academic Press, (1982). [13] J. D. Murray, "Mathematical Biology,", Biomathematics, 19 (1989). doi: 10.1007/978-3-662-08539-4. [14] A. Nakamasu, G. Takahashi, A. Kanbe and S. Kondo, Interactions between zebrafish pigment cells responsible for the generation of Turing patterns,, Proceedings of the National Academy of Sciences, 106 (2009), 8429. doi: 10.1073/pnas.0808622106. [15] Y. Nakamura, C. D. Tsiairis, S. Özbek and T. W. Holstein, Autoregulatory and repressive inputs localize Hydra Wnt3 to the head organizer,, Proceedings of the National Academy of Sciences, 108 (2011), 9137. doi: 10.1073/pnas.1018109108. [16] Y. Nishiura, Coexistence of infinitely many stable solutions to reaction diffusion systems in the singular limit,, in, 3 (1994), 25. [17] Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations,, SIAM J. Math. Anal., 18 (1987), 1726. doi: 10.1137/0518124. [18] H. Shoji, Y. Iwasa and S. Kondo, Stripes, spots, or reversed spots in two-dimensional Turing systems,, J. Theoret. Biol., 224 (2003), 339. doi: 10.1016/S0022-5193(03)00170-X. [19] I. Takagi, Point-condensation for a reaction-diffusion system,, J. Differential Equations, 61 (1986), 208. doi: 10.1016/0022-0396(86)90119-1. [20] M. Taniguchi, A uniform convergence theorem for singular limit eigenvalue problems,, Adv. Differential Equations, 8 (2003), 29. [21] M. Taniguchi and Y. Nishiura, Stability and characteristic wavelength of planar interfaces in the large diffusion limit of the inhibitor,, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 117. doi: 10.1017/S0308210500030638. [22] M. Taniguchi and Y. Nishiura, Instability of planar interfaces in reaction-diffusion systems,, SIAM J. Math. Anal., 25 (1994), 99. doi: 10.1137/S0036141092233500. [23] A. Turing, The chemical basis of morphogenesis,, Phil. Trans. R. Soc. Lond. B, 327 (1952), 37. doi: 10.1098/rstb.1952.0012. [24] M. J. Ward and J. Wei, Asymmetric spike patterns for the one-dimensional Gierer-Meinhardt model: equilibria and stability,, European J. Appl. Math., 13 (2002), 283. doi: 10.1017/S0956792501004442. [25] J. Wei and M. Winter, Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system,, J. Math. Pures Appl. (9), 83 (2004), 433. doi: 10.1016/j.matpur.2003.09.006. [26] J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: The weak coupling case,, J. Nonlinear Sci., 11 (2001), 415. doi: 10.1007/s00332-001-0380-1.
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# Outlier and Change Point Detection in Data Integration Performance Metrics

Data integration involves combining data residing in different sources, and providing users with a unified view of them. In this post, we are interested in detecting performance drift of large and complex daily data integration processes performed through ETL pipelines, which usually take from several minutes to several hours to complete.

The data correspond to more than 3 years of time performance measurements provided by a single organization, for around 50 distinct data integration processes performed with SAP Data Services, yielding the same number of time series after pre-processing.

Each time serie is assumed to be independent from the others, although there might be some interactions that could be studied with some causal discovery algorithms for example. However our choice is to deal sequentially with univariate time serie problems. These series have a daily frequency and are always strictly positive, due to the fact that they store some elapsed time.

The aim of this post is to detect a potential decay in the time-wise performance of a complex and heavy process. The difficulty is that the underlying data, stored in databases, is evolving during the time span of the study. So does the pipeline that may be adapted and improved from time to time. Also, the background load of database servers or networks may vary.

Here are the steps of the analysis:

• clean/prepare the data
• detect and remove outliers
• detect change points
• extract features on homogeneous time intervals

As a result, we expect to get a single indicator of performance drift per data integration process.

## Imports

import pantab
from tableauhyperapi import TableName
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from numba import jit
import ruptures as rpt
from ruptures import display
from sklearn.linear_model import LinearRegression
import missingno as msno
import seaborn as sns

FS = (16, 9)

The data is stored as a Tableau Hyper Extract. The pantab package is used to read the Hyper file.

### Tableau hyper extract

HYPER_PATH = "./data/ADM_CHARGEMENT_HISTO.hyper"

MIN_LENGTH_RATIO = 0.8  # 1.0 complete, 0.0 empty
MIN_VAL = 30  # values below MIN_VAL are removed

df = pantab.frame_from_hyper(HYPER_PATH, table=TableName("Extract", "Extract"))
df.drop_duplicates(keep="first", inplace=True)
df = df.loc[df.CD_RETOUR == "SUCCES"]  # only keep tasks that succeeded

df["ET"] = (
df.DT_FIN_EXEC - df.DT_DEB_DONNEES
).dt.total_seconds()  # compute the elapsed time in seconds

# set the index
df["start"] = df.DT_DEB_DONNEES
df.set_index("start", inplace=True, drop=True)

# pivot, resample the time series
CD_JOBS = []
traces = pd.DataFrame()
for CD_JOB in sorted(df.CD_JOB.unique()):
s = df.loc[df.CD_JOB == CD_JOB].ET.copy(deep=True)
s = s[s > 0.0]
s = s.resample("D").mean()
traces = pd.concat([traces, s.to_frame(CD_JOB)], axis=1)
traces.sort_index(ascending=True, inplace=True)

traces[traces < MIN_VAL] = np.NaN  # remove very small values

# remove columns with too many NaN values
length = len(traces)
nan_count = traces.isna().sum(axis=0)
cols = sorted(nan_count[nan_count < (1.0 - MIN_LENGTH_RATIO) * length].index.values)
traces = traces[cols]

# rename the columns, due to confidentiality reasons
traces.columns = sorted(traces.columns)
traces.columns = ['c_' + str(i).zfill(2) for i in range(len(traces.columns))]
traces.tail(2)
c_00 c_01 c_02 ... c_51 c_52 c_53
start
2019-06-05 314.0 231.0 322.0 ... 2062.0 1631.0 75.0
2019-06-06 311.0 239.0 170.0 ... 2157.0 1387.0 74.0

2 rows × 54 columns

print(f"start: {traces.index.min().date()}, end: {traces.index.max().date()}")
start: 2015-10-05, end: 2019-06-06

### Missing values

Let's visualize the data completion with the handy missingno package.

ax = msno.matrix(traces)
_ = ax.set(title="Data Completion", xlabel="Columns", ylabel="Rows")

We are going to keep the missing values, so we need to use some methods that can handle NaN values.

### Data distribution

Now we look at the data distribution. Let's plot the skewness of each serie. For normally distributed data, the skewness should be about zero. We can see here that skewness is rather disparate and that the data is sometimes highly concentrated on the left (small elapsed time) of the distribution:

ax = traces.skew(axis=0).plot.bar(figsize=FS, alpha=0.5, grid=True)
_ = ax.set(
title="Skewness of each time serie", xlabel="Columns", ylabel="Unbiased skew"
)

Actually, we can observe that these time series are fairly heterogeneous by plotting all the kernel density estimations:

plt.figure(figsize=FS)
for col in traces:
plt.grid()
ax = plt.gca()
_ = ax.set_xticklabels(
[10, 100, 1_000, 10_000, 100_000, 1_000_000], minor=False, rotation=0
)
_ = ax.set(
xlabel="Elapsed time (s) - Log10 scale",
title="Kernel density estimate of each time serie",
)

Also, they do exhibit a lot of noise, jumps and more general changes of distribution. We can look at 2 examples using this plotting function:

def plot_ts(idx, tss, logy=False, figsize=FS):
col = tss.iloc[:, idx]
ax = col.plot(style=".", figsize=figsize, grid=True, legend=False, logy=logy)
_ = ax.set(title=col.name, xlabel="Date", ylabel="Elapsed time (s)")
ax.autoscale(enable=True, axis="x", tight=True)
plot_ts(3, traces)

plot_ts(21, traces)

We are interested at the global trend. Let's start by removing outliers.

## Outlier detection

We are looking for an outlier detection method that does not depend on the number of observations, but also that is not much affected by outliers. We are going to use a rolling-window method, on which we compute the median, and the Median Absolute Deviation (MAD). If we have a size $n$ window:

$$y = \{ y_i \}$$

The median and MAD are denoted as follows:

$$\tilde{y} = median \{ y_i \}$$

$$MAD(y) = median \{ | y_i - \tilde{y} | \}$$

Then we can define the modified Z-score:

$$M_i = \frac{0.6745 \left( y_i - \tilde{y} \right)}{MAD(y)}$$

Note that 0.6745 is the 0.75th quantile of the standard normal distribution. So we are going to flag each point as outlier if $M_i < -T$ or $M_i > T$. $T$ is a theshold value that has a default value of 3.5 as a rule of thumb. See [1] for more details.

# modified Z-score based method, implemented with Numba

@jit(nopython=True)
x_tilde = np.median(x)
return x_tilde - 1.4825796886582654 * threshold * mad  # 1.0 / 0.6745

@jit(nopython=True)
x_tilde = np.median(x)
return x_tilde + 1.4825796886582654 * threshold * mad  # 1.0 / 0.6745

def detect_outliers(ts, window_size=30, threshold=3.5):
if not isinstance(ts, pd.core.series.Series):
raise TypeError("ts must be a Pandas Series")
if not isinstance(ts.index, pd.DatetimeIndex):
raise TypeError("ts must have a DatetimeIndex")
if ts.index.freq == "D":
freq_str = "D"
else:
raise ValueError("Wrong time series frequency")
window_str = str(window_size) + freq_str
df = ts.to_frame("val")
df.val.dropna()
.rolling(window_str)
)
df.val.dropna()
.rolling(window_str)
)
df["outlier"] = False
return df.outlier

Let's plot these suspected outliers on given series:

def plot_outliers(ts, outliers, figsize=FS):
if len(ts.loc[outliers] > 1):
ax = ts.loc[outliers].plot(
figsize=figsize, style="o", c="#F3CAA7", ms=15, alpha=0.75, grid=True
)
ax = ts.plot(ax=ax, style=".", c="#325160", ms=5, alpha=0.75, grid=True)
ax.autoscale(enable=True, axis="x", tight=True)
_ = ax.set_ylim(0,)
ax.set(title=ts.name + " with outliers", xlabel="Date", ylabel="Elapsed time (s)")
ts = traces[traces.columns[3]]
outliers = detect_outliers(ts)
plot_outliers(ts, outliers)

ts = traces[traces.columns[21]]
outliers = detect_outliers(ts)
plot_outliers(ts, outliers)

Now we actually remove all suspected outliers in each serie:

for col in traces.columns:
ts = traces[col]
outliers = detect_outliers(ts)
traces[col].loc[outliers] = np.NaN

## Remove incomplete traces

Removing outliers may result in series with too many missing values. So again, we check if each serie as enough data (at least 80% valid):

nan_count = traces.isna().sum(axis=0)
cols = sorted(nan_count[nan_count < (1.0 - MIN_LENGTH_RATIO) * length].index.values)
traces = traces[cols]
print(len(traces.columns))
48

## Change point detection

Now we can focus on the change point detection problem. We are going to use ruptures, a nice library for off-line change point detection. Here is a link to the paper associated with the library. As defined in [2]:

Change point detection is the task of finding changes in the underlying model of a signal or time series

Here we are dealing with an unknown number of break points, so a penalty is added to the change point problem with a fixed number of change points $K$. Although ruptures can deal with multivariate signals, we apply the detection algorithms to each time serie separately.

The selected search method is the Pruned Exact Linear Time (PELT) method which is optimal. The cost function is a Gaussian kernel-based mapping. The default penalty function in ruptures is linear, from what we understand. Here is the function that computes and return the breakpoint indices:

def detect_change_points(signal, model="rbf", min_size=5, jump=5, pen=10):
algo = rpt.Pelt(model=model, min_size=min_size, jump=jump).fit(signal)
bkps = algo.predict(pen=pen)
return bkps
ts = traces.c_03
signal = ts.dropna().values
bkps = detect_change_points(signal)
fig, axarr = display(signal, bkps, figsize=FS)
_ = plt.title(ts.name + " with break points")
_ = plt.ylabel("Elapsed time (s)")

ts = traces.c_21
signal = ts.dropna().values
bkps = detect_change_points(signal)
display(signal, bkps, figsize=FS)
_ = plt.title(ts.name + " with break points")
_ = plt.ylabel("Elapsed time (s)")

## Feature extraction on homogeneous time intervals

Now that we have removed some outliers and segmented the serie, we are interested computing the slope of each homegeneous segment, to finally compute a slope coefficient, which would indicate if the elapsed time is reather increasing or not. Let's start with the slope exctraction:

def extract_features(ts_in, plot=True):
ts = ts_in.copy(deep=True)
outliers = detect_outliers(ts)
ts.loc[outliers] = np.NaN
signal = ts.dropna().values
name = ts.name
bkps = detect_change_points(signal)

df = ts.dropna().to_frame()
df["segment"] = 0
df.loc[df.iloc[bkps[:-1]].index, "segment"] = 1
df.segment = df.segment.cumsum()

slopes = []
seg_lengths = []
df["piecelin"] = 0.0
df["slope"] = 0.0
for i in df.segment.unique():
seg = df.loc[df.segment == i].asfreq("D")[name].reset_index(drop=True).dropna()
y = seg.values
X = seg.index.values.reshape(-1, 1)
reg = LinearRegression(fit_intercept=True, normalize=True).fit(X, y)
df.loc[(df.segment == i) & ~df[name].isna(), "piecelin"] = reg.predict(X)
df.loc[(df.segment == i) & ~df[name].isna(), "slope"] = reg.coef_[0]

outliers = detect_outliers(ts)
if plot:
ax = ts.plot(figsize=FS, style=".", c="#F1D05E", ms=15, alpha=0.75, grid=True)
_ = df.piecelin.plot(ax=ax, c="#325160", grid=True)
_ = ax.set_ylim(0,)
_ = ax.set(
xlabel="Date",
title=name + " with piecewise linear approx.",
ylabel="Elapsed time (s)"),

return df
ts = traces.c_03
_ = extract_features(ts)

ts = traces.c_21
_ = extract_features(ts)

So we would like to detect if a process has been recently taking more and more time to complete. We start by computing a monthly-averaged value of the slope for the last 6 months, and then compute another average of these slope values with non-uniform weights that increase with time (the older, the smaller the weight). We finally get a unique value representing the increase of elapsed time during the most recent months, without taking account outliers and jumps. This slope coefficient is computed for each time serie with compute_slope_coefs:

def weighted_average(y):
l = len(y)
alph = np.ones(l, dtype=float)
for i in range(1, l):
alph[i] = 1.2 * alph[i - 1]
alph /= np.sum(alph)
return np.dot(y, alph)

def compute_slope_coefs(tss, n_days=180):
slope_coefs = []
for col in tss.columns:
ts = tss[col]
df = extract_features(ts, plot=False)
months = df[-n_days:].slope.resample("M").mean()
coef = weighted_average(months.values)
slope_coefs.append({"column": col, "slope_coef": coef})
return pd.DataFrame(slope_coefs).set_index("column")
sc = compute_slope_coefs(traces)

Let's plot the 20 largest slope coefficients:

ax = sc.sort_values(by="slope_coef", ascending=False)[:20].plot.bar(
figsize=FS, grid=True, alpha=0.5, legend=False
)
_ = ax.set(
title="Processes with largest slope coefficient in the last 6 months",
xlabel="Process",
ylabel="Time delta per day (s/day)",
)

Well we already plotted c_21 and could notice that it was increasing in the last part.

Let's study the slope coefficients of the months preceding "2018-07":

sc = compute_slope_coefs(traces[:"2018-07"])
ax = sc.sort_values(by="slope_coef", ascending=False)[:20].plot.bar(
figsize=FS, grid=True, alpha=0.5, legend=False
)
_ = ax.set(
title="Processes with largest slope coefficient in the last 6 months",
xlabel="Process",
ylabel="Time delta per day (s/day)",
)

ts = traces[:"2018-07"].c_03
df = extract_features(ts)

## Final remarks

This aim of this post was to quickly explore ways to detect performance drift in the execution time of large and complex data integration processes. The approach was to remove noise and jumps in the time series data using outlier detection, change point detection and piecewise linear approximation. Then a weighted average of the slope is computed on the last 6 months to result in a single coefficient for each time serie. We could imagine to track this coefficient on a monthly basis.

However this approach strongly depends on two parameters:

• the threshold value for the outlier detection method,
• the penalty coefficient fot the change point detection method.

[1] B. Iglewicz and D. Hoaglin, Volume 16: How to Detect and Handle Outliers, The ASQC Basic References in Quality Control: Statistical Techniques, Edward F. Mykytka, Ph.D., Editor, 1993.

[2] C. Truong, L. Oudre and N. Vayatis, Selective review of offline change point detection methods, Signal Processing, 167:107299, 2020.


text:
# Whitney sum formula for Pontryagin classes I

I have read in several places that the total Pontryagin classes of real vector bundles satisfy a Whitney sum formula $p(E\oplus F) = p(E)\cdot p(F)$ modulo 2-torsion. I would like to understand this better, and have two precise questions.

1. Using the definition $p_i(E)=(-1)^{i}c_{2i}(E_\mathbb{C})$, the fact that complexification respects direct sums, and the Whitney sum formula for Chern classes, I find for example that $$p_1(E\oplus F) = -c_2(E_\mathbb{C}\oplus F_\mathbb{C}) = -c_2(E_\mathbb{C}) - c_1(E_\mathbb{C})c_1(F_\mathbb{C}) - c_2(F_\mathbb{C}),$$ and $$p_1(E) + p_1(F)=-c_2(E_\mathbb{C})-c_2(F_\mathbb{C}).$$ I do not see why the difference $c_1(E_\mathbb{C})c_1(F_\mathbb{C})$ between these two expressions is necessarily 2-torsion. What am I missing?
2. edit: I've moved the second question to a separate post Whitney sum formula for Pontryagin classes II
• The reason the classes $c_{2k+1}(E_\mathbb C)$ are 2-torsion is that the complex bundle $E_\mathbb C$ has an anti-linear automorphism (conjugation). – Yury Ustinovskiy Nov 4 '15 at 0:25
• @YuryUstinovskiy: Thanks. I see now that I should have gone back and read Milnor and Stasheff more closely before posting this, in particular the first three pages of Chapter 15 (your comment is Lemma 15.1). I still would like a formula for the 2-torsion part though, which I've now asked as a second question. – Mark Grant Nov 4 '15 at 8:25

I do not see why the difference $c_1(E_\mathbb{C})c_1(F_\mathbb{C})$ between these two expressions is necessarily 2-torsion. What am I missing?
Real line bundles are classified by $H^1(X, \mathbb{Z}_2)$, which is a 2-torsion group (geometrically, because real line bundles are self-dual, their tensor squares are trivializable). Complex line bundles are classified by $H^2(X, \mathbb{Z})$. Complexification therefore describes a cohomology operation $H^1(X, \mathbb{Z}_2) \to H^2(X, \mathbb{Z})$, so the image of this cohomology operation (the first Chern classes of complexifications of real line bundles) is necessarily also 2-torsion. In fact this cohomology operation is a Bockstein.


text:
Practice Problems -   -  Use 6 Dot Entry   Switch to Nemeth Tutorial

# Lesson 5.3: Arrows

## Symbols

$\text{arrow indicator}$

$\text{simple right-pointing arrow (east)}$
⠳⠕

$\text{simple left-pointing arrow (west)}$
⠳⠪

$\text{regular barb, full, counter to the line of direction}$

## Explanation

Arrow symbols consist of a shaft and a barbed tip. The barbed tip can point in any direction and be full, half, curved or directionless. The shaft can take many forms (solid, double, curved, dotted, etc.). This lesson will look at simple arrows with a full barb and a straight horizontal shaft.

An arrow symbol in UEB is formed with the opening arrow indicator, dots one two five six, and a closing symbol with dots arranged in a consistent pattern to indicate the direction of the arrow. A grade 1 indicator must be used with the opening arrow indicator since it also has a grade 2 meaning unless grade 1 mode has already been set.

Additional symbols are used to indicate the type of barbed tip and its direction when the arrow shape has unusual tips or shafts. In the case of the common bidirectional arrow, the barbs pointing in both directions must be shown with signs for each. Dots two four five six are used for the full barb counter to the line of direction and dots one two three five indicate the full barb in the line of direction. The line of direction is deemed to be to the right when the tips point in both directions. These symbols are placed between the arrow opening and terminating symbols. The signs are transcribed in logical order: opening arrow indicator, barb counter to the line of direction, barb in the line of direction, and closing indicator for right-pointing arrow. The closing indicator for right pointing arrow is dots one three five.

It is important to note that the arrow indicator is not used when the simple right-pointing arrow is the only modifier above or below an item. There are separate UEB symbols for simple right pointing arrow over previous item and simple right pointing arrow under previous item. These symbols will be discussed in later lessons. Additional symbols for arrows with unusual shafts and a standard barbed tip or arrows with unusual tips can be found in Guidelines for Technical Materials §13.

### Example 1

(simple right-pointing horizontal arrow)

$\to$
⠰⠳⠕

### Example 2

(simple left-pointing horizontal arrow)

$←$
⠰⠳⠪

### Example 3

(common horizontal bidirectional arrow)

$↔$
⠰⠳⠺⠗⠕

previous - next (This lesson contains no exercises)


text:
# Induced drag - inverse proportion between induced drag and parasitic drag [duplicate]

I recently learned that the slower an aircraft is traveling, the higher the induced drag and the lower the parasitic drag. The parasitic drag is lower because the particles have less force against the aircraft (all types of parasite drag: form, skinfriction, interference, and ram). However, why is induced drag higher?

From what I understand, I thought that it would be low as well, because if the aircraft is moving slower that would mean that the air pressure would be less on the bottom of the planform, meaning the vortex created at the tip of the wing would be smaller (smaller difference between pressures) thus having less induced drag.

Can someone help explain to me why it is inversely proportional as such?


text:
## Prove theorem: In FLT chapter

My final is today and I need help!

Prove the theorem: The only solutions in non-negative integers of the equation x^2+2y^2=z^2, with gcd(x, y, z)=1 are given by x=+-(2s^2-t^2), y=2st, z=2s^2+t^2 where s,t are arbitrary nonnegative integers.

**hint: if u,v,w are such that y=2w, 2+x=2u, z-x=2v, thenthe equation becomes 2w^2=uv.

I could only goes as far is to prove that hint, but I got stuck after that:

From the hint, we have x^2+2y^2=z^2 implies 2y^2=z^2-x^2=(z-x)(z+x), which implies 2(2w)^2=(2u)(2v),
which implies 8w^2=4uv,
which implies 2w^2=uv, thus proving the hint.

where do I go from here


text:
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If a = 7 and b = -7, what is the value of 2a - 2b + b^2 ?

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If a = 7 and b = -7, what is the value of 2a - 2b + b^2 ?  [#permalink]

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08 Sep 2018, 09:25
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If $$a = 7$$ and $$b = -7$$, what is the value of $$2a - 2b + b^2$$ ?

(A) -49
(B) 21
(C) 49
(D) 63
(E) 77

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Re: If a = 7 and b = -7, what is the value of 2a - 2b + b^2 ?  [#permalink]

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08 Sep 2018, 09:31
Hasnain786 wrote:
If $$a = 7$$ and$$b = -7$$, what is the value of $$2a - 2b + b^2$$ ?

(A) -49
(B) 21
(C) 49
(D) 63
(E) 77

a=7, b=-7
$$2a - 2b + b^2$$ = 2x7-2x(-7)+(-7)^2 = 14+14+49 = 77
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Re: If a = 7 and b = -7, what is the value of 2a - 2b + b^2 ?  [#permalink]

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08 Sep 2018, 10:15
Hasnain786 wrote:
If $$a = 7$$ and$$b = -7$$, what is the value of $$2a - 2b + b^2$$ ?

(A) -49
(B) 21
(C) 49
(D) 63
(E) 77

$$2a - 2b + b^2$$

$$= 2a - b ( 2 - b)$$

$$= 2*7 + 7 ( 2 + 7)$$

$$= 14 + 7 *9$$

$$= 14 + 63$$

$$= 77$$, Answer must be (E)
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Re: If a = 7 and b = -7, what is the value of 2a - 2b + b^2 ? &nbs [#permalink] 08 Sep 2018, 10:15
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# American Institute of Mathematical Sciences

November  2008, 21(4): 1199-1219. doi: 10.3934/dcds.2008.21.1199

## Identifying a BV-kernel in a hyperbolic integrodifferential equation

 1 Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via Saldini 50, 20133, Milano, Italy 2 Dipartimento di Matematica, Università di Roma “La Sapienza”, P.le A. Moro 5, 00185 Roma, Italy

Received  August 2007 Revised  February 2008 Published  May 2008

This paper is devoted to determining the scalar relaxation kernel $a$ in a second-order (in time) integrodifferential equation related to a Banach space when an additional measurement involving the state function is available. A result concerning global existence and uniqueness is proved.
The novelty of this paper consists in looking for the kernel $a$ in the Banach space $BV(0,T)$, consisting of functions of bounded variations, instead of the space $W^{1,1}(0,T)$ used up to now to identify $a$.
An application is given, in the framework of $L^2$-spaces, to the case of hyperbolic second-order integrodifferential equations endowed with initial and Dirichlet boundary conditions.
Citation: Alfredo Lorenzi, Eugenio Sinestrari. Identifying a BV-kernel in a hyperbolic integrodifferential equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1199-1219. doi: 10.3934/dcds.2008.21.1199
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2019 Impact Factor: 1.338


text:
# What volume is produced at STP in the reaction 2"H"_2"S"(g) + 3"O"_2(g) -> 2"SO"_2(g) + 2"H"_2"O"(l) if we start with "200 mL H"_2"S" and "200 mL O"_2?

Jun 10, 2017

I'm assuming you want the volume of ${\text{SO}}_{2}$ at STP. We will assume your definition of STP is $\text{1 atm}$ and ${25}^{\circ} \text{C}$.

(If you mean the total volume, then we are ignoring the volume of the liquid water.)

${V}_{S {O}_{2}} = {V}_{{H}_{2} S}$, since $\text{H"_2"S}$ is the limiting reagent and is $1 : 1$ with ${\text{SO}}_{2}$.

We assume throughout that the gases are ideal.

$2 \text{H"_2"S"(g) + 3"O"_2(g) -> 2"SO"_2(g) + 2"H"_2"O} \left(l\right)$

Since the reduction of oxygen is a known standard reduction half reaction, it can be confirmed that the water is a liquid. Liquids have small molar volumes, so to a good approximation, the $\boldsymbol{{\text{SO}}_{2} \left(g\right)}$ volume will dominate the volume at STP.

Since the mol ratio of ${\text{H"_2"S":"O}}_{2}$ is $2 : 3$, your reactant gas volumes are $1 : 1$, and this mol ratio is less than $1 : 1$, ${\text{O}}_{2}$ is in excess and $\text{H"_2"S}$ is the limiting reagent. For example, if you had $\text{2 mols H"_2"S}$, you would only need ${\text{2 mols O}}_{2}$, and you'd actually have $3$.

Therefore, to calculate the mols of ${\text{SO}}_{2}$, which ARE $1 : 1$ with $\text{H"_2"S}$, use $\text{H"_2"S}$, as it would give you the maximum yield of product:

${\text{mols H"_2"S" = "mols SO}}_{2}$

As we have been assuming all of these gases are ideal gases, which have the same molar volume, $\frac{V}{n}$, at STP, and since $\text{H"_2"S}$ is $1 : 1$ with ${\text{SO}}_{2}$...

we started with $\text{200 mL H"_2"S}$ and produce $\textcolor{b l u e}{\text{200 mL}}$ of ${\text{SO}}_{2}$.


text:
# GATE CSE 2021 Set 2 | Question: 12

1 vote
401 views

Let $L_1$ be a regular language and $L_2$ be a context-free language. Which of the following languages is/are context-free?

1. $L_1 \cap \overline{L_2} \\$
2. $\overline{\overline{L_1} \cup \overline{L_2}} \\$
3. $L_1 \cup (L_2 \cup \overline{L_2}) \\$
4. $(L_1 \cap L_2) \cup (\overline{L_1} \cap L_2)$

edited
0

The options should be $b,c,d$.

Options –

(a) The Complement of CFL is Recursive. The intersection of Recursive and Regular is Recursive.

(b) It is $L_1 \cap L_2$ and intersection of CFL and regular is CFL.

(c)It is $\sum$ itself as $L_2 \cup \bar{L_2}$ is $\sum$ and union of $\sum$ and $L_1$ is $\sum$ which is regular and thus CFL.

(d) Solving this you will get $L_2$ which is CFL. See the image below.

We can write the $(d)$ option as,

$= (23 \cap 34) \cup (14 \cap 34)$

$= (3) \cup (4)$

$= L_2$

edited
0
in option D both are CFL ( regular intersection with CFL)   union of two CFL is CFL
1 vote

Given : $L_1$= Regular language

$L_2$=CFL

A) $L_1\cap \bar L_2$: as we know that CFL is not closed under complement operation. CFL complement is REC.

therefore Reg  $\cap$ $\overline{CFL}$ =REC which is not CFL.

hence option A is wrong here.

B)  $\overline{(\bar L_1\cup \bar L_2)}$: this can be written as $\overline{\overline{L_1}}\cap\overline{\overline{L_2}}\implies L_1\cap L_2$.

hence Regular $\cap$ CFL is CFL because CFL is closed with regular intersection.

option $B$ is correct.

C) $L_1\cup(L_2\cup \bar L_2)$: CFL complement is REC language.

CFL$\cup$ REC is REC language and union with regular language is REC which is not CFL.

so option $C$ is false here.

D) $(L_1\cap L_2)\cup(\bar L_1\cap L_2)$; This can be written as:

(regular $\cap$ CFL)$\cup$( Regular $\cap$ CFL)

$\implies$ (regular $\cap$ CFL)= CFL, (Regular $\cap$ CFL)=CFL

$\implies$ CFl$\cup$CFL=CFL

Option $D$ is correct.

$\therefore$ Option $B$ and $D$ is correct.

0
@Hira C is given as CFL in official key but I totally agree with you reasoning. @Arjun Sir can you please shed some light here?

B,C,D

Some insights beforehand:

Not Closed under Intersection:

$L_1=\{a^nb^*c^n\},L_2=\{a^nb^nc^*\}$ , $L_1\cap L_2=\{a^nb^nc^n\}$ not a CFL

Closed under Union:
Let both of these CFLs be represented using grammars with start symbols $S_1$ and $S_2$, we define a new symbol $S^{’}$ such that $S^{’}\to S_1/S_2$ note, we definitely have a NPDA but could be represented under DPDA if the language allows it to be(2 RLs).

Not Closed under Complementation:

$L_1\cap L_2=\overline{\overline{L_1}\cup\overline{L_2}}$

We know that intersection isn’t  closed under context free languages and union is closed under CFL(NPDA) so it must be the case that complementation isn’t closed under CFL.
Example: $L=\{a^nb^nc^n\}$ not a CFL but, $\overline{L}$ is and infact can be solved using a NPDA.

Intersection and Union of a CFL is RL is CFL.

Now,

1. $L_2$ is complemented so, needn’t be a CFL.
1. $\overline{\overline{L_1}\cap\overline{L_2}}=L_1\cup L_2$ union of a CFL and RL is a CFL
2. $L_1\cup(L_2\cup\overline{L_2})=L_1\cup(\Sigma^*)=\Sigma^*$  hence a CFL.
3. $(L_1\cap L_2)\cup(\overline{L_1}\cap L_2)=L_2$  or it’s a union of two CFLs if you don’t bother to solve.

edited

## Related questions

1
394 views
For a string $w$, we define $w^R$ to be the reverse of $w$. For example, if $w=01101$ then $w^R=10110$. Which of the following languages is/are context-free? $\{ wxw^Rx^R \mid w,x \in \{0,1\} ^* \}$ $\{ ww^Rxx^R \mid w,x \in \{0,1\} ^* \}$ $\{ wxw^R \mid w,x \in \{0,1\} ^* \}$ $\{ wxx^Rw^R \mid w,x \in \{0,1\} ^* \}$
Which of the following regular expressions represent(s) the set of all binary numbers that are divisible by three? Assume that the string $\epsilon$ is divisible by three. $(0+1(01^*0)^*1)^*$ $(0+11+10(1+00)^*01)^*$ $(0^*(1(01^*0)^*1)^*)^*$ $(0+11+11(1+00)^*00)^*$
Consider the following sets, where $n \geq 2$: $S_1$: Set of all $n \times n$ matrices with entries from the set $\{ a, b, c\}$ $S_2$: Set of all functions from the set $\{0,1,2, \dots, n^2-1\}$ to the set $\{0, 1, 2\}$ Which of the following ... to $S_2$ There exists a surjection from $S_1$ to $S_2$ There exists a bijection from $S_1$ to $S_2$ There does not exist an injection from $S_1$ to $S_2$
In the context of compilers, which of the following is/are $\text{NOT}$ an intermediate representation of the source program? Three address code Abstract Syntax Tree $\text{(AST)}$ Control Flow Graph $\text{(CFG)}$ Symbol table


text:
# Direct image of vector bundle under projection map

Let $$\pi: Y \to X$$ be a smooth projection map between manifolds, and assume the fibres of the map have constant dimension: dim$$\left(\pi^{-1}(x)\right)=d ~~ \forall x \in X$$. Given a smooth map between manifolds, one can define the pushforward or direct image $$\pi_*V$$ of a vector bundle $$V$$. Formally one thinks of the vector bundle as a sheaf and then defines the direct image as for a sheaf.

My question is how to actually compute the result for common, simple vector bundles. Examples are the trivial vector bundle $$\mathcal{O}_Y$$, the tangent $$T_Y$$ or cotangent bundle $$T_Y^*$$, or the canonical $$K_Y$$ or anti-canonical bundle $$K_Y^*$$.

I would be interested to know how to work these out using the formal definition of the direct image, or how to cleverly avoid this explicit computation. Even better would be to understand also the answers in the case of the higher direct images $$R^p\pi_*V$$.

• I know of pulling back a bundle, but how does pushing a bundle forward work in general? – Jason DeVito Aug 26 '18 at 19:48
• @JasonDeVito Since bundles can be regarded as sheaves, the pushforward or direct image of a bundle can be defined as for a sheaf. Roughly, the direct image of a sheaf $F$ under a map $f$ is defined by $f_*F(U):=F(f^{-1}(U))$, where $U$ is an open set. – diracula Aug 26 '18 at 22:08
• I am only vaguely familiar with the idea of a sheaf. In particular, I can guess how to get a sheaf out a of bundle (using sections), but am at a loss as to how to go backwards. In an event, it is clear I won't be able to help with the question, but I look forward to trying to understand the answers! – Jason DeVito Aug 27 '18 at 0:02
• What kind of answer are you looking for(local charts)? what do you mean by projection ($X \subset Y$ such that $X =\pi \circ \pi(Y)$ and $\pi_{\restrection X}=Id$ )? / – Elad Jun 19 '19 at 12:22
• @Elad Yes those are properties I expect a projection to have; examples I have in mind include the projection of a product space onto one component, the projection in a fibre bundle, or the projection in a fibration. I am interested in any methods to compute the results, for example explicitly using local charts or using heavier machinery to avoid this. – diracula Jun 19 '19 at 21:07


text:
# #StackBounty: #regression #experiment-design #ab-test #finite-mixture-model #choice nominal regression, kinda. What is the right termin…

### Bounty: 50

tl;dr
This looks cool, relates possible to discrete regression, but I don’t know the term for what it is to do this.

I want to learn more. It looks interesting and useful. Can you point me to references or content in this area?

Background:
My daughter made cookies for her science project. I helped her with the math part, but she wanted to optimize the a recipe.

I told her that she would want to avoid the "evil" of "one factor at a time" (link) because it misses interactions, in inefficient with data, and gives more erroneous answers. A simple designed experiment can do a better job with fewer tests.

She did research and determined that two reasonably well studied "axes" she could look at are savory vs. sweet. She did not want to look at "pumpkin nut habanero" cookies.

Initially we were looking at a $$3^2$$ design on a savory vs. sweet axis, but it presents some serious challenges. It is a ton of cookies to make. There are 9 distinct recipes and it is hard to rank-order them. Doing ordinal regression to find an optimum isn’t trivial. We could reduce the number of tests to 5 and make a basic design and get somewhere. Instead of rank-ordering, we could do paired comparison and while it is more taste testing, it is easy to believe that a person can nearly always pick which they think is better.

Here is the what the 2-axis 5-test design looked like.

When top-down is implausible I think of bottom up. I used a random forest and made an estimator that swept the domain and answered the paired tests with the one that was closest.

Here is how it discretized the domain.

It seems both convenient and clever that 10 tests can reduce the area where "best lives" by a factor of 24. Personally I don’t like to extrapolate, so I would bound to a square going from (-1,-1) to (1,1) which makes only 16 sub-divisions.

Question:
What is the name of the technique here? Someone has to have done something like this before. Who was it, what did they do, and where do I read up on it?

I don’t know if the paired testing still counts as discrete choice. I’ve seen JMP that called "choice designs" but in contrast this approach allows discovery of a parcel that isn’t in one of the test elements.

There is a very elegant simple way, so simple a 6th grader can do it, to derive the way the "landscape" gets sliced up based on the point locations.

This has to have been invented before, but I have no idea what the process is called, so I don’t know how to look it up in the literature.

Get this bounty!!!

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Cubuk E D, Zoph B, Shlens J, et al. RandAugment: Practical data augmentation with no separate search[J]. arXiv preprint arXiv:1909.13719, 2019.

# 1. Overview

In this paper, it proposes RandAugment
1) Demonstrate the optimal strength of a data augmentation depends on the model size and training set size. (A separate optimization of an augmentation policy on a smaller proxy task may be sub-optimal for learning and transferring augmentation policies)
2) Simplify search space only containing 2 hyperparameters.

# 2. Methods

The primary goal of RandAugment is to remove the need for a separate search phase on a proxy task. As a separate search phase significantly complicates training and is computationally expensive.

1) $K=14$. Each transformation with uniform probability $\frac{1}{K}$.
2) RandAugment may thus express $K^N$ potential policies.