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## AI ALIGNMENT FORUMAF

Rob Bensinger

Communications lead at MIRI. Unless otherwise indicated, my posts and comments here reflect my own views, and not necessarily my employer's.

# Sequences

Late 2021 MIRI Conversations

# Wiki Contributions

Visible Thoughts Project and Bounty Announcement

We have now received the first partial run that meets our quality bar. The run was submitted by LessWrong user Vanilla_cabs. Vanilla's team is still expanding the run (and will probably fix some typos, etc. later), but I'm providing a copy of it here with Vanilla's permission, to give others an example of the kind of thing we're looking for:

Vanilla's run is currently 266 steps long. Per the Visible Thoughts Project FAQ, we're willing to pay authors $20 / step for partial runs that meet our quality bar (up to at least the first 5,000 total steps we're sent), so the partial run here will receive$5320 from the prize pool (though the final version will presumably be much longer and receive more; we expect a completed run to be about 1000 steps).

Vanilla_cabs is open to doing paid consultation for anyone who's working on this project. So if you want feedback from someone who understands our quality bar and can demonstrably pass it, contact Vanilla_cabs via their LessWrong profile.

Visible Thoughts Project and Bounty Announcement

In case you missed it: we now have an FAQ for this project, last updated Jan. 7.

Soares, Tallinn, and Yudkowsky discuss AGI cognition

how do you get some substance into every human's body within the same 1 second period? Aren't a bunch of people e.g. in the middle of some national park, away from convenient air vents? Is the substance somehow everywhere in the atmosphere all at once?

I think the intended visualization is simply that you create a very small self-replicating machine, and have it replicate enough times in the atmosphere that every human-sized organism on the planet will on average contain many copies of it.

One of my co-workers at MIRI comments:

(further conjunctive detail for visualizer-plausibility: most of your replication time is in all the doublings before the last doubling, and in particular you can make a shitload in a pretty small space before launching it into the jet stream to disperse. the jet stream can be used to disperse stuff throughout the atmosphere (and it can use solar radiation, at least, to keep reproducing). it could in principle be powered and do minor amounts of steering.

example things the [AGI] who has no better plan than this paltry human-conceivable plan has to think about are "how does the time-cost of making sure [I hit the people] at the south pole base and [on] all the cruise liners and in all the nuclear submarines, trade off against the risk-cost of leaving that fragment of humanity alive", etc.)

Regarding the idea of diamondoid nanotechnology, Drexler's Nanosystems and http://www.molecularassembler.com/Nanofactory/index.htm talk about the general concept.

Biology-Inspired AGI Timelines: The Trick That Never Works

Making a map of your map is another one of those techniques that seem to provide more grounding but do not actually.

Sounds to me like one of the things Eliezer is pointing at in Hero Licensing:

Look, thinking things like that is just not how the inside of my head is organized. There’s just the book I have in my head and the question of whether I can translate that image into reality. My mental world is about the book, not about me.

You do want to train your brain, and you want to understand your strengths and weaknesses. But dwelling on your biases at the expense of the object level isn't actually usually the best way to give your brain training data and tweak its performance.

I think there's a lesson here that, e.g., Scott Alexander hadn't fully internalized as of his 2017 Inadequate Equilibria review. There's a temptation to "go meta" and find some cleaner, more principled, more objective-sounding algorithm to follow than just "learn lots and lots of object-level facts so you can keep refining your model, learn some facts about your brain too so you can know how much to trust it in different domains, and just keep doing that".

But in fact there's no a priori reason to expect there to be a shortcut that lets you skip the messy unprincipled your-own-perspective-privileging Bayesian Updating thing. Going meta is just a tool in the toolbox, and it's risky to privilege it on 'sounds more objective/principled' grounds when there's neither a theoretical argument nor an empirical-track-record argument for expecting that approach to actually work.

Teaching the low-description-length principles of probability to your actual map-updating system is much more feasible (or at least more cost-effective) than emitting your actual map into a computationally realizable statistical model.

I think this is a good distillation of Eliezer's view (though I know you're just espousing your own view here). And of mine, for that matter. Quoting Hero Licensing again:

STRANGER:  I believe the technical term for the methodology is “pulling numbers out of your ass.” It’s important to practice calibrating your ass numbers on cases where you’ll learn the correct answer shortly afterward. It’s also important that you learn the limits of ass numbers, and don’t make unrealistic demands on them by assigning multiple ass numbers to complicated conditional events.

ELIEZER:  I’d say I reached the estimate… by thinking about the object-level problem? By using my domain knowledge? By having already thought a lot about the problem so as to load many relevant aspects into my mind, then consulting my mind’s native-format probability judgment—with some prior practice at betting having already taught me a little about how to translate those native representations of uncertainty into 9:1 betting odds.

One framing I use is that there are two basic perspectives on rationality:

• Prosthesis: Human brains are naturally bad at rationality, so we can identify external tools (and cognitive tech that's too simple and straightforward for us to misuse) and try to offload as much of our reasoning as possible onto those tools, so as to not have to put weight down (beyond the bare minimum necessary) on our own fallible judgment.
• Strength training: There's a sense in which every human has a small AGI (or a bunch of AGIs) inside their brain. If we didn't have access to such capabilities, we wouldn't be able to do complicated 'planning and steering of the world into future states' at all.

It's true that humans often behave 'irrationally', in the sense that we output actions based on simpler algorithms (e.g., reinforced habits and reflex behavior) that aren't doing the world-modeling or future-steering thing. But if we want to do better, we mostly shouldn't be leaning on weak reasoning tools like pocket calculators; we should be focusing our efforts on more reliably using (and providing better training data) the AGI inside our brains. Nearly all of the action (especially in hard foresight-demanding domains like AI alignment) is in improving your inner AGI's judgment, intuitions, etc., not in outsourcing to things that are way less smart than an AGI.

In practice, of course, you should do some combination of the two. But I think a lot of the disagreements MIRI folks have with other people in the existential risk ecosystem are related to us falling on different parts of the prosthesis-to-strength-training spectrum.

Techniques that give the illusion of objectivity are usually not useless. But to use them effectively, you have to see through the illusion of objectivity, and treat their outputs as observations of what those techniques output, rather than as glimpses at the light of objective reasonableness.

Strong agreement. I think this is very well-put.

Conversation on technology forecasting and gradualism

Is this 5 years of engineering effort and then humans leaving it alone with infinite compute?

Maybe something like '5 years of engineering effort to start automating work that qualitatively (but incredibly slowly and inefficiently) is helping with AI research, and then a few decades of throwing more compute at that for the AI to reach superintelligence'?

With infinite compute you could just recapitulate evolution, so I doubt Paul thinks there's a crux like that? But there could be a crux that's about whether GPT-3.5 plus a few decades of hardware progress achieves superintelligence, or about whether that's approximately the fastest way to get to superintelligence, or something.

Biology-Inspired AGI Timelines: The Trick That Never Works

When I try to mentally simulate negative reader-reactions to the dialogue, I usually get a complicated feeling that's some combination of:

• Some amount of conflict aversion: Harsh language feels conflict-y, which is inherently unpleasant.
• Empathy for, or identification with, the people or views Eliezer was criticizing. It feels bad to be criticized, and it feels doubly bad to be told 'you are making basic mistakes'.
• Something status-regulation-y: My reader-model here finds the implied threat to the status hierarchy salient (whether or not Eliezer is just trying to honestly state his beliefs), and has some version of an 'anti-cheater' or 'anti-rising-above-your-station' impulse.

How right/wrong do you think this is, as a model of what makes the dialogue harder or less pleasant to read from your perspective?

(I feel a little wary of stating my model above, since (a) maybe it's totally off, and (b) it can be rude to guess at other people's mental states. But so far this conversation has felt very abstract to me, so maybe this can at least serve as a prompt to go more concrete. E.g., 'I find it hard to read condescending things' is very vague about which parts of the dialogue we're talking about, about what makes them feel condescending, and about how the feeling-of-condescension affects the sentence-parsing-and-evaluating experience.)

Shulman and Yudkowsky on AI progress

Note: I've written up short summaries of each entry in this sequence so far on https://intelligence.org/late-2021-miri-conversations/,  and included links to audio recordings of most of the posts.

Biology-Inspired AGI Timelines: The Trick That Never Works

I've gotten one private message expressing more or less the same thing about this post, so I don't think this is a super unusual reaction.

Soares, Tallinn, and Yudkowsky discuss AGI cognition

I don't know Eliezer's view on this — presumably he either disagrees that the example he gave is "mundane AI safety stuff", or he disagrees that "mundane AI safety stuff" is widespread? I'll note that you're a MIRI research associate, so I wouldn't have auto-assumed your stuff is representative of the stuff Eliezer is criticizing.

Safety Interruptible Agents is an example Eliezer's given in the past of work that isn't "real" (back in 2017):

[...]

It seems to me that I've watched organizations like OpenPhil try to sponsor academics to work on AI alignment, and it seems to me that they just can't produce what I'd consider to be real work. The journal paper that Stuart Armstrong coauthored on "interruptibility" is a far step down from Armstrong's other work on corrigibility. It had to be dumbed way down (I'm counting obscuration with fancy equations and math results as "dumbing down") to be published in a mainstream journal. It had to be stripped of all the caveats and any mention of explicit incompleteness, which is necessary meta-information for any ongoing incremental progress, not to mention important from a safety standpoint. The root cause can be debated but the observable seems plain. If you want to get real work done, the obvious strategy would be to not subject yourself to any academic incentives or bureaucratic processes. Particularly including peer review by non-"hobbyists" (peer commentary by fellow "hobbyists" still being potentially very valuable), or review by grant committees staffed by the sort of people who are still impressed by academic sage-costuming and will want you to compete against pointlessly obscured but terribly serious-looking equations.

[...]

The rest of Intellectual Progress Inside and Outside Academia may be useful context. Or maybe this is also not a representative example of the stuff EY has in mind in the OP conversation?


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# How do you simplify 7^sqrt2*7^(3sqrt2)?

${7}^{4 \setminus \sqrt{2}}$

#### Explanation:

Powers on the same base, are added in the multiplication as follows

${7}^{\setminus \sqrt{2}} \setminus \cdot {7}^{3 \setminus \sqrt{2}}$

$= {7}^{\setminus \sqrt{2} + 3 \setminus \sqrt{2}}$

$= {7}^{4 \setminus \sqrt{2}}$

format: plain


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Model Evaluation¶

This tutorial shows how to use an existing approximation model: given a model file, evaluate model responses for values of variables from a test sample and save the results.

Note

This tutorial requires the approximation model trained in the Sample-Based Approximation tutorial.

Before You Begin¶

This tutorial requires an existing Tutorials project. If you have not created this project yet, see Tutorial Project first. You will also need the approximation model trained in the Sample-Based Approximation tutorial (model.gtapprox should be found in the Tutorials project directory).

• Open the Tutorials project.
• Create a new empty workflow. If you have never created a workflow before, see the Simple Workflow tutorial.
• Save the workflow as ApproxEvaluation.
• Switch to Workspace and verify that the workflow file (ApproxEvaluation.p7wf) is added to the project. You can see it in the Project pane.
• While in Workspace, verify that a file named model.gtapprox exists in the project.
• Switch to Edit and select the workflow you have just created to continue with the tutorial.

The tutorial is aimed at using an existing approximation model to predict function values for the values of variables from a test sample (evaluate model responses).

The test sample can be prepared beforehand (stored in a file) or generated by the workflow itself. This tutorial uses a randomly generated sample, since loading sample files is already described in the Sample-Based Approximation tutorial.

The end result is three samples — two model inputs and model responses — which are stored to the project database and can be used to study the model.

Solution¶

Evaluating a pSeven approximation model, which is stored in a file, can be divided into the following general steps:

1. Configure an Approximation model block to read the file and load the model.
2. Prepare a test sample — load it from a file or configure a block which will generate it when the workflow runs.
3. Send the test sample to the Approximation model block to get responses.
4. Save the results for analysis.

Model¶

Approximation models are loaded to and evaluated by the Approximation model block. In this tutorial, the block will load the model from a file. Alternatively, you can also send the model to the Model.Model file port, if it comes from an ApproxBuilder block in the same workflow.

• Add an Approximation model block. Name it Model.

• Double-click Model to open its configuration.

• Click or the path field in the Approximation model pane to bring up the file selection dialog.

• In the Select file dialog, select model.gtapprox from the current project directory and click .

After selecting the model file you can see general model information in the Model summary pane and model inputs and outputs in the Variables and Responses panes.

The model has two inputs named x1 and x2 and one output named f. Note that these names were read from the training sample file and saved to the model in the Sample-Based Approximation tutorial.

In the Model summary pane, note the Input bounds section. It lists the minimum and maximum values of model inputs (variables) which were found in the training sample. These values specify the bounding box of the model and will be required when you configure the block which generates the input sample.

• Valid range for the x1 input is $$(-4.9614, 9.9396)$$.
• Valid range for the x2 input is $$(0.1293, 14.8778)$$.

Values of variables which you send to the model for evaluation should be within these ranges. The model can evaluate any input, but it has no information about the input-output dependency outside the bounding box, because the training data did not contain this information. Due to this, evaluation results for inputs outside the bounding box will be very inaccurate, because the model will tend to linear extrapolation for such inputs.

To save model input and output data for analysis, you have to monitor the Model.x1, Model.x2, and Model.f ports. Monitored data will be stored to the project database when the workflow runs, allowing you to process the results in Analyze.

• Switch to the Ports tab in the Model block configuration dialog.
• In the Inputs pane, enable monitoring for the x1 and x2 ports.
• In the Outputs pane, enable monitoring for the f port.
• Click to save changes and close the configuration dialog.

Test Sample Generation¶

To generate a random test sample, you can use a Design space exploration block.

• Add a Design space exploration block. Name it Test Sample.
• Open Test Sample configuration.

Generating a sample requires you to add sample variables and set their lower and upper bounds.

• Click the button in the Variables pane or select Add variables… from the pane’s menu to bring up the Add variables dialog.

• Specify names of variables: x1, x2. Leave other settings in this dialog default and click .

After adding variables, edit their lower and upper bounds in the Variables pane. Note that the bounds should be set in such a way that the sample’s bounding box is equal or less than the model’s bounding box. For simplicity, set the same bounds as shown in the model details in the Model block.

• For x1: set lower bound to -4.9614 and upper bound to 9.9396.
• For x2: set lower bound to 0.1293 and upper bound to 14.8778.

Note that when you add variables, the block automatically selects the design space exploration technique to use and the number of sample points to generate (the Number of designs setting). For this tutorial, use the automatically selected technique (Latin hypercube sampling) but increase the number of designs to get enough data for plotting in Analyze.

• Set Number of designs to 1000.
• Click in the block configuration dialog to save settings and close it.

The Test Sample block will generate a sample in 2 parts — for x1 and x2. Each of them is a RealMatrix which is output to the Test Sample.x1.Optimal designs and Test Sample.x2.Optimal designs ports, respectively. This data has to be sent to Model for evaluation.

• Link Test Sample.x1.Optimal designs to Model.x1.
• Link Test Sample.x2.Optimal designs to Model.x2.

Monitoring Settings¶

The Test Sample block enables monitoring for some of its ports by default. These monitors are not needed in the current tutorial, so it is recommended to remove them for convenience. In addition, it is recommended to assign monitoring aliases for the Model ports. These aliases that will become the names of the respective records in the project database which store the model’s input and output data.

• Open workflow configuration and switch to the Monitoring tab.
• Select all Test Sample ports and click in the toolbar to remove them.
• Specify monitoring aliases for the Model block ports:
• Input x1 for Model.x1,
• Input x2 for Model.x2, and
• Response for Model.f.
• Verify your monitoring settings and click to save workflow configuration.

Monitored data (model input and response) will be stored to the project database when the workflow runs, allowing you to process the results in Analyze.

Workflow¶

Finished workflow is very simple: it contains two blocks only. Test Sample generates a data sample with specified number of points and bounds of variables. Model loads the model from the model.gtapprox file in the project directory and evaluates model outputs for inputs form the generated sample.

Note that results of this workflow are saved to the project database because Model input and output ports are monitored.

• Verify that monitoring is enabled for the Model.x1, for Model.x2, and Model.f ports.
• Verify that monitoring is disabled for the Test Sample block ports. This block enables monitoring for some of its ports by default, but currently this data is not needed.
• Save the workflow and run it.

When the workflow finishes, switch to Analyze to view the results.

Results¶

The data from Model.x1, Model.x2 and Model.f ports is saved to the project database — the Input x1, Input x2 and Response records, respectively. Note that the records in Monitoring are named the same as the monitoring aliases in workflow configuration.

To visualize the results you can, for example, plot the model’s response surface in 3D.

• In Analyze, create a new report and open the Data series pane to add report data.
• Select the Input x1, Input x2 and Response records in the project database and drag them to the Data series pane. This creates new data series containing model input and response data.

Add a 3D response surface plot.

• Dataset: model inputs as D0 and D1, model response as D2. See 3D Plot for details on using this plot.

Additionally, you can use a 2D plot to show the sample generated by the Test Sample block.

• Dataset: Input x1 as D0, Input x2 as D1. See 2D Plot for details on using this plot.

results_viewers_2dplot

For a guide on how to add and configure plots, see also section Plotting in the Results and Reports tutorial.

Conclusion¶

This tutorial explains the basics of evaluating an approximation model in pSeven in batch mode — that is, the input contains a number of sample points, and all responses are evaluated at once. Approximation model, in fact, is not limited to batch calculations and can process sequential input. For example, you can remove the Test Sample block and uplink Model.x1, Model.x2 and Model.f instead to study the model manually, or use the Model block in an iterative workflow, such as an optimization cycle.


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# Chapter 2 What the hell is R?

🚧 Under Construction 🚧

This chapter contains instruction

## 2.2 RStudio’s interface

RStudio looks like this:

As you can see from the above figure, RStudio have four main windows.

### 2.2.1 console window

Arguably the most important window in RStudio. We write R command in this window.

### 2.2.2 script window

Forget what we said about the previous window. This is the most important window in RStudio. We write our sets of command in this window

### 2.2.3 environment window

You can observe your environment here, such as the data and variables / series you created. This window also have other useful tab such as “history” which can show you the history of what happens in your environment, but we will skip them for now.

### 2.2.4 files window

Although we call it ‘files window’, this window consists of 5 tabs. The tab file shows your working directory. When you run a code that generates file such as graphics, it will be stored in this window. Plots tab shows you your latest plot.

Packages tab show you all the package that you have. You can also add new package from this tab instead of using the install.packages() command. In this tab, you can also activate package that you have instead of using library() command.

“help” tab will show you some information about things that you ask help for when you use ?'command' command. Will show you how it works in a minute.

Lastly, “viewer” tab. I actually forgot why it exists so whatever.

## 2.6 Update

You need to keep your R, RStudio and your installed packages updated as much as you can.


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Review question

# How many normals to a parabola pass through $P$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R5053

## Question

Find the equation of the normal to the parabola $y^2=4ax$ at the point $(at^2,2at)$.

Prove that, if $p^2>8$, two chords can be drawn through the point $(ap^2,2ap)$ which are normal to the parabola at their second points of intersection, and that the line joining these points of intersection meets the axis of the parabola in a fixed point, independent of $p$.


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# How to prove this determinant is positive-II?

Question: Given an arbitrary number of real matrices of the form $A_i= \biggl(\begin{matrix} C_i+E_i & B_i \\ B_i^T & D_i-F_i \end{matrix} \biggr)$, where $B_i$ is an arbitrary $n\times n$ real matrix, $C_i$ and $D_i$ are $n\times n$ real anti-symmetric matrices, $E_i$ and $F_i$ are $n\times n$ real symmetric and positive semidefinite matrices, how to prove the following $$\det \biggl(I_{2n} + \prod_i e^{A_i}\biggl)\ge 0 \,?$$

Background: This is a stronger version of an earlier MO question How to prove this determinant is positive? which was solved by GH from MO and Terry Tao. Their proof addressed the case of $E_i=F_i=0$, where $e^{A_i}$ belongs to the split orthogonal group.

The paper arXiv:1601.01994v2 in fact contains a rigorous proof of the above statement using tools familiar to physicists, e.g. Majorana fermion and reflection positivity (cf Eq.(10)). It would be nice to see an alternate mathematical proof (perhaps of a similar nature to the split orthogonal group proof by GH from MO and Terry Tao).

Addendum: The paper arXiv:1601.01994v2 contains further results on complex matrices, which I do not know how to formulate in simple math language yet.

• I find it interesting such a many-body problem hinges on such a basic result – john mangual Feb 1 '16 at 12:35
• Maybe one can start with a warm up question with $B_i=C_i=D_i=0$ ? Then the statement becomes $det(I_n +\prod_i e^{E_i}) * det(I_n +\prod_i e^{-F_i}) \ge 0$, where $E_i$ and $F_i$ are real symmetric and positive semidefinite matrices. – Lei Wang Feb 1 '16 at 14:34
• to the warm up : $det(I_n +\prod_i e^{-F_i}) \ge 0$ because $\Vert\prod_i e^{-F_i}\Vert \le 1$ and $F_i$ real . Then also $det(I_n +\prod_i e^{E_i}) = det(\prod_i e^{E_i}) det(I_n +(\prod_i e^{E_i})^{-1}) \ge 0$ . – jjcale Feb 3 '16 at 19:49
• The $e^{A_i}$ increase the quadratic form that the split orthogonal group leaves invariant . So maybe this is the generalization of Part I : The split orthogonal group is replaced by the semi group that increases the quadratic form. – jjcale Feb 4 '16 at 18:50
• @jjcale Thanks! Could you elaborate a bit more about "increase the quadratic from" ? – Lei Wang Feb 5 '16 at 7:57

Let $q(x,y) = x^H J y$ for $x,y \in \mathbb{C}^{2n}$ where $J = diag(I_n,-I_n)$ and let $S = \{A \in M_{2n}(\mathbb{R}) : q(Ax, Ax) \ge q(x,x)$ $\forall x \in \mathbb{C}^{2n}\}$. Obviously $S$ is a semi group . Furthermore the $e^{t A_i}$ are in $S$ since $$\frac{d}{dt} q(e^{t A_i} x,e^{t A_i} x) = 2 (e^{t A_i} x)^H diag (E_i,F_i) (e^{t A_i} x) \ge 0$$ for all $x \in \mathbb{C}^{2n}$.

Now let $T : [0,1] \rightarrow S$ analytic where $det(I_{2n}+T(0)) > 0$ and where $T(0)$ has no degenerate eigenvalues.

Let $E_{\lambda}(t)$ be the generalized eigenspace of $T(t)$ to the eigenvalue $\lambda$ . For $G \subset \mathbb{C}$ define $$E_G(t) = \bigoplus_{\lambda \in G}{E_{\lambda}(t)}$$ .

Let $t_0 \in [0,1]$ such that $det(I_{2n}+T(t_0)) = 0$ .

Now we want to show that $dim\,E_{(-1,\infty)}(t)$ can only change by an even number near $t_0$ . Therefore $det(I_{2n}+T(t))$ can't change the sign.

Lemma 1 : Let $U \in S$, $x \in \mathbb{C}^{2n}$ with $0 = q(x,x) = q(Ux,Ux)$ . Then $q(x,y) = q(Ux,Uy)$ for all $y \in \mathbb{C}^{2n}$ .

Proof : We have $q(ax+y,ax+y) \le q(a Ux + Uy,a Ux + Uy)$ for all $a \in \mathbb{C}$ and therefore $0 \le 2 Re\,a (q(Uy,Ux) - q(y,x)) + q(Uy,Uy) - q(y,y)$ . But the right hand side can be made negative for appropriate a if $q(Uy,Ux) \neq q(y,x)$ .

Lemma 2 : Let $p$ a polynomial and $z \in \mathbb{C}$ . If $p(n) z^n$ is constant for all large enough $n \in \mathbb{N}$ then $p$ is constant and $z = 1$ or $p = 0$ or $z = 0$.

Lemma 3 : Let $U \in S$, $x$ a generalized eigenvector of $U$ to the eigenvalue $\lambda$, $y$ a generalized eigenvector of $U$ to the eigenvalue $\mu$ and $q(U^l x,U^l x) = 0$ for all $l \in \mathbb{N}_0$. Then holds $\lambda \bar{\mu} = 1$ or $q(x,y) = 0$ .

Proof : By Lemma 1 we have $q(y,x) = q(U^l y, U^l x)$ for all $l \in \mathbb{N}$ . But $q(U^l y, U^l x)$ has the form $p(l) (\lambda \bar{\mu})^l$ for a polynomial $p$ for all large enough $l \in \mathbb{N}$ . From Lemma 2 then follows Lemma 3 .

Lemma 4 : Let $U \in S$, $x$ a generalized eigenvector of $U$ to the eigenvalue $-1$ and $q(x,U^k x) = 0$ for all $k \in \mathbb{N}_0$ . Then $q(U^k x,U^l x) = 0$ for all $k,l \in \mathbb{N}_0$ .

Proof : Let $x_k = (I_{2n} + U)^k x$ and $m$ minimal such that $q(x_j,x_k) = 0$ for all $j,k \geq m$ . First we want to show that $q(x_j,x_k) = 0$ for $j \geq m$ and $k \geq 0$. If this is not the case then let $j \geq m$ and $k$ be maximal such that $q(x_j,x_k) \neq 0$ . Then $k > 0$ and by Lemma 1 $q(x_j,x_{k-1}) = q(U^l x_j,U^l x_{k-1}) = q(x_j,x_{k-1}) - l q(x_j,x_k)$ for all $l \in \mathbb{N}$ . Contradiction ! Now we get for $m > 1$ $q(x_{m-2},x_{m-2}) \leq q(U^l x_{m-2},U^l x_{m-2}) = l^2 q(x_{m-1},x_{m-1}) + O(l)$ and $q(x_{m-2},x_{m-2}) \geq q(U^{-l} x_{m-2},U^{-l} x_{m-2}) = l^2 q(x_{m-1},x_{m-1}) + O(l)$ . Contradiction to $q(x_{m-1},x_{m-1}) \neq 0$ ! Since $m = 1$ is impossible since $q(x_0,x_k) = 0$ for all $k \geq 0$ we are done .

Lemma 5 : The restriction of $q$ to $E_{-1}(t_0)$ is non degenerate .

Proof : Let $x \in E_{-1}(t_0)$ and $U = T(t_0)$ . We want to show that there exists $y \in E_{-1}(t_0)$ such that $q(x,y) \neq 0$ . If there exists $n \in \mathbb{N}_0$ such that $q(x,U^n x) \neq 0$ we are done. Otherwise ist follows from Lemma 4 and Lemma 3 that x is orthogonal w.r.t. q to all other generalized eigenspaces . But since q is non degenerate there exists $y \in E_{-1}(t_0)$ such that $q(x,y) \neq 0$ .

Lemma 6 : T(t) has degenerated eigenvalues only at isolated points.

Proof : The discriminant of T(t) is analytic in t and nonzero at t = 0 .

Now we can choose $\epsilon > 0$ and $r$ with $0 < r < 1$ such that for $\vert t-t_0\vert < \epsilon$ holds :

i) $det(I_{2n}+T(t)) \neq 0$ for $t \neq t_0$ ,

ii) $E_{\{z: \vert z+1 \vert \leq r\}}(t_0) = E_{-1}(t_0)$ ,

iii) $\sigma (T(t)) \cap \{z: \vert z+1 \vert = r\} = \emptyset$ ,

iv) the signature of the restriction of $q$ to $V(t)$ is constant where $V(t) = E_{\{z: \vert z+1 \vert \leq r\}}(t)$ ,

v) T(t) has no degenerate eigenvalues for $t \neq t_0$ .

Let $D = \{z: \vert z+1 \vert \leq r\}$ . For each eigenvalue $\lambda \in D$ of $T(t)$ with $\vert \lambda \vert = 1$ we can write $E_{\lambda}(t) = E_{\lambda}^+(t) \oplus E_{\lambda}^-(t)$ such that the restriction of $q$ to $E_{\lambda}^+(t)$ is positive semidefinite and the restriction of $q$ to $E_{\lambda}^-(t)$ is negative definite and such that $E_{\bar{\lambda}}^+(t) = \overline{E_{\lambda}^+(t)}$ and $E_{\bar{\lambda}}^-(t) = \overline{E_{\lambda}^-(t)}$ .

Now we can write $V(t) = V^+(t) \oplus V^-(t)$ where $$V^+(t) = E_{D \cap \{z : \vert z \vert > 1\}}(t) \oplus \bigoplus_{\lambda \in D , \vert \lambda \vert = 1} E_{\lambda}^+(t)$$ and $$V^-(t) = E_{D \cap \{z : \vert z \vert < 1\}}(t) \oplus \bigoplus_{\lambda \in D , \vert \lambda \vert = 1} E_{\lambda}^-(t)$$ .

Now we want to show that for $\vert t-t_0\vert < \epsilon$ and $t \neq t_0$ the restriction of $q$ to $V^+(t)$ is positive semidefinite and the restriction to $V^-(t)$ is negative semidefinite :

For $x \in V^+(t)$ we get $$q(x,x) \geq \lim_{m \rightarrow \infty} \frac{1}{m} \sum_{l=1}^m q(U^{-l} x,U^{-l} x) = \sum_{\lambda \in D, \vert \lambda \vert = 1} q(x_{\lambda}^+,x_{\lambda}^+) \geq 0$$ where $x_{\lambda}^+$ is the component of x in $E_{\lambda}^+(t)$ . For $x \in V^-(t)$ we get $$q(x,x) \leq \lim_{m \rightarrow \infty} \frac{1}{m} \sum_{l=1}^m q(U^l x,U^l x) = \sum_{\lambda \in D, \vert \lambda \vert = 1} q(x_{\lambda}^-,x_{\lambda}^-) \leq 0$$ where $x_{\lambda}^-$ is the component of x in $E_{\lambda}^-(t)$ .

Let $n_+$ the number of positive eigenvalues of the restriction of q to $V(t)$ and $n_-$ the number of negative eigenvalues. We have shown that $n_+ \geq dim\, V^+(t)$ and $n_- \geq dim\, V^-(t)$ and therefore $n_+ = dim\, V^+(t)$ . Since $dim\, V^+(t) - dim\, E_{(-1,-1-r)}$ is even $dim\, E_{(-1,-1-r)}$ can only change by an even number. And $dim\, E_{[-1-r,-\infty)}$ can only change if a pair of complex conjugate eigenvalues gets real or vice versa and therefore also only by an even number.

So we have shown that $dim\,E_{(-1,\infty)}(t)$ can only differ by an even number on different points in $\vert t-t_0\vert < \epsilon, t\neq t_0$ .

To finish the proof, choose $W$ such that $e^{t W} \in S$ for $t \in [0,1]$ and such that $e^W$ fullfills the requirements on $T(0)$ . Then choose $$T(t) = e^{(1-t) W} \prod_i e^{t A_i}$$ .

• Thanks a lot! I still need some time to fully digest your proof. Anyway, let me ask the following question first: according to your proof, do you think the statement is already the strongest one can make? In other words, do you see any chance to further generalize it ? Thanks! – Lei Wang Feb 26 '16 at 10:10
• What one needs is $A_i^T J + J A_i \ge 0$ but I haven't checked whether this is a generalization. – jjcale Mar 2 '16 at 18:01
• Oh I see. This definitely generalized the split orthogonal group condition $A_i^TJ + J A_i=0$. – Lei Wang Mar 4 '16 at 0:53

Even though this paper [1] is in the strongly correlated electrons section of the cond-mat (and it's about Quantum Monte Carlo) we can learn a bit about Hermitian matrices. In particular, I argue that reflection-positivity is a natural matrix notion if you're reading a paper like this.

In Supplement I we get general form of the density matrix. Though I am not sure about the size of this matrix:

$$\rho_P = \mathrm{Tr} \prod_{k=1}^M e^{-\tau H_0} e^{-\eta H_I(\eta_k)} = \det \left( I + \prod_{k=1}^M e^{-\tau h^0} e^{-\eta h^I (\eta_k)} \right)$$

This strikes me as ambiguous, since this trace is also a determinant, and in fact $\rho_P$ is a matrix. I suspect this has to do with the "Fermions" which are a way of arranging Hilbert space into certain tensor products.

Eq (5) and (6) say either of $H_0$ or $H_I$ could take the "reflection symmetric" form:

$$H = \gamma^T \left( \begin{array}{cc} A & iB \\ -iB^T & A^* \end{array}\right) \gamma$$

Although $A,B$ are $2N \times 2N$ matrices, it's unclear about the shape of $H$. We have $\gamma = (\gamma_i^{(1)}, \gamma_i^{(2)})$ which lives in a $2\times N$ dimensional space and I am still not sure how to write the tensor product.

Then in supplement 3A we make the choices:

$$A = \frac{1}{2} \left( \begin{array}{cc} C & - iF \\ i F^T & D \end{array}\right) \hspace{0.25in} B = \frac{1}{2} \left( \begin{array}{cc} B_1 & 0\\ 0 & B_2 \end{array}\right)$$

And I have still have not addressed the change of basis between Dirac and Majorana fermions.

If we set $\eta = 0$ and $\tau = -1$, theorem 1 says if $H_0$ is majorana reflection positive" then:

$$\rho_P = \mathrm{Tr} \prod_{k=1}^M e^{-\tau H_0} = \det \left( I + \prod_{k=1}^M e^{-\tau h^0} \right)$$

However, it's ambiguous what to put for $H_0$ or $h^0$, possibly eq (10).

Certainly, your result is much stronger that $\rho_P \geq 0$ is a positive semi-definite matrix.


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My Math Forum The irrational number between irrationals

 Number Theory Number Theory Math Forum

 November 2nd, 2018, 06:04 AM #1 Senior Member   Joined: Nov 2011 Posts: 250 Thanks: 3 The irrational number between irrationals How I can prove that between every two irrational number there is irrational number?
November 2nd, 2018, 06:12 AM   #2
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Quote:
 Originally Posted by shaharhada How I can prove that between every two irrational number there is irrational number?
If $p$ and $q$ are irrational numbers and (without loss of generality) $p > q$, you should be able to show there will be some other irrational number $x$ such that $x \in (q, p)$. There are a number of ways to do this. Take a crack at finding a solution and post what you come up with. If you get stuck, I'm sure someone will assist.

 November 2nd, 2018, 08:50 AM #3 Senior Member   Joined: Oct 2009 Posts: 867 Thanks: 330 Hint: prove first that between two rationals is an irrational, and that between any two irrationals is a rational.
 November 2nd, 2018, 09:26 AM #4 Senior Member   Joined: Dec 2015 From: somewhere Posts: 647 Thanks: 94 I know one way to show it, but it is rare and not as advanced as other methods. Can anyone post the solution? I'm sure there are a lot of ways to show that. Last edited by skipjack; November 3rd, 2018 at 03:21 PM.
 November 2nd, 2018, 09:27 AM #5 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,685 Thanks: 2665 Math Focus: Mainly analysis and algebra Suppose that $p > q$ with both irrational. $p-q > 0$ and so there exists a natural number $n$ such that $n(p-q) > 1 > 0$. Thus $p-q > \frac1n > 0$ and therefore $$p > q + \frac1n > q$$ All that you need now is to prove that $(q + \frac1n )$ is irrational. (Hint: assume that it is rational and show that $q$ must then also be rational - a contradiction - because the rationals are closed under subtraction.) Thanks from SenatorArmstrong and idontknow Last edited by skipjack; November 2nd, 2018 at 09:48 AM.
 November 2nd, 2018, 10:49 AM #6 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,685 Thanks: 2665 Math Focus: Mainly analysis and algebra Given a number $b$, there is a greatest integer $m$ such that $m \le b$ and $m+1 > b$, (because $(m+1)$ is, by definition an integer greater than $m$). Suppose that $a-b > 1$, then $$b + (a-b) = a \gt m+1 \gt b$$ With $a=np$ and $b=nq$ we then have that $np > m+1 > nq$ and so $$p > \frac{m+1}{n} > q$$ and $\frac{m+1}{n}$ is thus a rational between $p$ and $q$.
November 3rd, 2018, 01:52 PM   #7
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Math Focus: Mainly analysis and algebra
Quote:
 Originally Posted by idontknow I know one way to show it

 November 7th, 2018, 12:11 PM #8 Senior Member   Joined: Dec 2015 From: somewhere Posts: 647 Thanks: 94 Each irrational number can equal to a sum of two irrational numbers $\displaystyle i=i_1+i_2$ and $\displaystyle i_2=i_3 + i_4$ $\displaystyle i=i_1 + i_3 + i_4$ such that $\displaystyle i_1 , i_3 , i_4$ do not equal to each other , so there is a relation between them , one possible relation as $\displaystyle i_1 < i_3 < i_4$ so $\displaystyle i_3$ is an irrational number between two irrational numbers Thanks from v8archie
 November 7th, 2018, 04:08 PM #9 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,685 Thanks: 2665 Math Focus: Mainly analysis and algebra I can't say that I can see how that guarantees that two very close irrationals must have a third between them. Your irrationals also have a condition that is not met by every set of three irrationals. I'm not saying it's definitely wrong, but I certainly don't find it very satisfying.
November 7th, 2018, 04:31 PM   #10
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From: USA

Posts: 647
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Math Focus: Dynamical systems, analytic function theory, numerics
Quote:
 Originally Posted by idontknow Each irrational number can equal to a sum of two irrational numbers $\displaystyle i=i_1+i_2$ and $\displaystyle i_2=i_3 + i_4$ $\displaystyle i=i_1 + i_3 + i_4$ such that $\displaystyle i_1 , i_3 , i_4$ do not equal to each other , so there is a relation between them , one possible relation as $\displaystyle i_1 < i_3 < i_4$ so $\displaystyle i_3$ is an irrational number between two irrational numbers
This has several problems.

1. You haven't proved any of your claims and they are far from obvious. For example, prove that every irrational can be written as the sum of two other irrationals.

2. Even if the above claim was proved (it is true after all, just unproved), it doesn't prove what you set out to show. This would show that every irrational can be inserted between two others. It does NOT show that given two irrationals, you can find one between them. The difference is you don't get to pick the two irrationals, they are given to you.

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# Prove property of floor function [duplicate]

Prove that, for any $a, b, c \in \mathbb{N}$, $\bigl\lfloor \frac{a}{bc} \bigr\rfloor = \Bigl\lfloor \frac{\bigl\lfloor \frac{a}{b} \bigr\rfloor}{c} \Bigr\rfloor$.

I have no idea how to prove this. Can you help me, please? Thanks!

## marked as duplicate by dxiv, Namaste, MPW, Shailesh, ClaytonJun 24 '18 at 1:07

• It may be useful to note this would also be $$\bigg\lfloor\frac{\lfloor\frac ac\rfloor}{b}\bigg\rfloor$$ – Rhys Hughes Jun 24 '18 at 0:41
If either $b$ or $c$ divides $a$, this is trivial based off the transposability highlighted in my comment.
Suppose $\frac ab=I+F$, where $I$ is the integer part, $F$ is the fractional part. Thus, $\lfloor I+F\rfloor=I$, and so this problem summarises to proving: $$\bigg\lfloor\frac{I+F}{c}\bigg\rfloor=\bigg\lfloor\frac{I}{c}\bigg\rfloor$$
For $0\le F<1$


text:
Power Set of Group under Induced Operation is Monoid

Theorem

Let $\left({G, \circ}\right)$ be a group with identity $e$.

Let $\left({\mathcal P \left({G}\right), \circ_\mathcal P}\right)$ be the algebraic structure consisting of the power set of $G$ and the operation induced on $\mathcal P \left({G}\right)$ by $\circ$.

Then $\left({\mathcal P \left({G}\right), \circ_\mathcal P}\right)$ is a monoid with identity $\left\{{e}\right\}$.

Proof

By the definition of a group, $\left({G, \circ}\right)$ is a monoid.

The result follows from Power Set of Monoid under Induced Operation is Monoid.

$\blacksquare$


text:
# Heat and Electron question

1. Mar 4, 2005

### misogynisticfeminist

Here's the question,

A beam of electrons, travelling at $$9x10^6ms^{-1}$$ In every second, $$1x10^{14}$$ electrons hit a metal target and all their energy is converted into heat. How long would it take to heat up the metal from 18 degrees celsius to 80 degrees celsius if the heat capacity of the metal is $$0.15JK^{-1}$$. The mass of the electron is $$9x10^{-31} kg$$.

How i solved this problem was to find the momentum of each individual electron using the formula $$p=mv$$. Then i found the energy of each electron using the relation $$E^2=(mc^2)^2+(pc)^2$$. Then afterwards i found the heat energy in joules required to heat up the metal to 80 degrees celsius.

The thing is, i got 10.2 for the energy required to heat the metal object up. And the energy per second gained is $$8.08886$$. Therefore the answer i got is 1.26 seconds. The problem is, the answer is $$1.44x10^5 s$$.

Where did I go wrong? I've got a feeling the momentum-energy relation in SR is a little outta place but I can't think of any other relation which relates momentum and energy. Thanks alot.

2. Mar 4, 2005

### vincentchan

the energy you were using is the kinetic energy + the rest mass energy....... do you see what is your problem?

3. Mar 4, 2005

### xanthym

{Kinetic Energy Each Electron} = (1/2)mv2 = (1/2){9.0e(-31) kg}{9.0e(6) m/s}2 =
= (3.645e(-17) Joules)
{Power Dissipated In Metal Target} = {Kinetic Energy Ea Electron}*{# Electrons/Sec Absorbed} =
= {3.645e(-17)}*{1.0e(14)} = (3.645e(-3) Watts)
{Rate of Temperature Change} = {Power Dissipated In Metal Target}/{0.15 J/oK} =
= (3.645e(-3) Watts)/{0.15 J/oK} = (0.0243 oK/sec)

{Time To Heat Metal 18 to 80 oC} = (80 - 18 oC)/(0.0243 oK/sec) = (2551 sec)

~~

Last edited: Mar 4, 2005
4. Mar 4, 2005

### misogynisticfeminist

ohhhhh yes ! that solved the problem, thanks alot....

5. Mar 4, 2005

6. Mar 4, 2005

### misogynisticfeminist

The method i used is different from Xanthym's one as i did not use the formula $$KE=\frac{1}{2}mv^2$$. What i did was first to find momentum, then relate it with energy using the relativistic momentum-energy relation.

The kinetic energy per electron I got is $$6.0479974_E -19$$ and the kinetic energy hitting the target in one second is $$6.0479974_E -5$$. To find the energy required to heat the metal object, i got $$9.3 J$$.

So, $$\frac{9.3}{60479974_E -5}$$, i get answer $$143,519.0944$$ which is approximately equal to $$1.44_E 5$$, which is the textbook answer.

7. Mar 4, 2005

### dextercioby

The approximation using KE=1/2mv^{2} is excellent.No need to apply relativistic formulas..

Daniel.


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# One-shot DP Top-k mechanisms

In the last blog post, we showed that the exponential mechanism enjoys improved composition bounds over general pure DP mechanisms due to a property called bounded range. For this post, we will present another useful, and somewhat surprising, property of the exponential mechanism in its application of top-$$k$$ selection.

# A Better Privacy Analysis of the Exponential Mechanism

A basic and frequent task in data analysis is selection – given a set of options $$\mathcal{Y}$$, output the (approximately) best one, where “best” is defined by some loss function $$\ell : \mathcal{Y} \times \mathcal{X}^n \to \mathbb{R}$$ and a dataset $$x \in \mathcal{X}^n$$. That is, we want to output some $$y \in \mathcal{Y}$$ that approximately minimizes $$\ell(y,x)$$. Naturally, we are interested in private selection – i.e., the output should be differentially private in terms of the dataset $$x$$. This post discusses algorithms for private selection – in particular, we give an improved privacy analysis of the popular exponential mechanism.

# Open Problem - Optimal Query Release for Pure Differential Privacy

Releasing large sets of statistical queries is a centerpiece of the theory of differential privacy. Here, we are given a dataset $$x = (x_1,\dots,x_n) \in [T]^n$$, and a set of statistical queries $$f_1,\dots,f_k$$, where each query is defined by some bounded function $$f_j : [T] \to [-1,1]$$, and (abusing notation) is defined as $f_j(x) = \frac{1}{n} \sum_{i=1}^{n} f_j(x_i).$ We use $$f(x) = (f_1(x),\dots,f_k(x))$$ to denote the vector consisting of the true answers to all these queries. Our goal is to design an $$(\varepsilon, \delta)$$-differentially private algorithm $$M$$ that takes a dataset $$x\in [T]^n$$ and outputs a random vector $$M(x)\in \mathbb{R}^k$$ such that $$\| M(x) - f(x) \|$$ is small in expectation for some norm $$\|\cdot\|$$. Usually algorithms for this problem also give high probability bounds on the error, but we focus on expected error for simplicity.

# Conference Digest - ICML 2021

ICML 2021, one of the biggest conferences in machine learning, naturally has a ton of interesting sounding papers on the topic of differential privacy. We went through this year’s accepted papers and aggregated all the relevant papers we could find. In addition, this year features three workshops on the topic of privacy, as well as a tutorial. As always, please inform us if we overlooked any papers on differential privacy.

# Statistical Inference is Not a Privacy Violation

On April 28, 2021, the US Census Bureau released a new demonstration of its differentially private Disclosure Avoidance System (DAS) for the 2020 US Census. The public were given a month to submit feedback before the system is finalized. This demonstration data and the feedback has generated a lot of discussion, including media coverage on National Public Radio, in the Washington Post, and via the Associated Press. The DAS is also the subject of an ongoing lawsuit.

# Call for Papers - Workshop on the Theory and Practice of Differential Privacy (TPDP 2021)

Work on differential privacy spans a number of different research communities, including theoretical computer science, machine learning, statistics, security, law, databases, cryptography, programming languages, social sciences, and more. Each of these communities may choose to publish their work in their own community’s venues, which could result in small groups of differential privacy researchers becoming isolated. To alleviate these issues, we have the Workshop on the Theory and Practice of Differential Privacy (TPDP), which is intended to bring these subcommunities together under one roof (well, a virtual one at least for 2020 and 2021).

# ALT Highlights - An Equivalence between Private Learning and Online Learning (ALT '21 Tutorial)

Welcome to ALT Highlights, a series of blog posts spotlighting various happenings at the recent conference ALT 2021, including plenary talks, tutorials, trends in learning theory, and more! To reach a broad audience, the series will be disseminated as guest posts on different blogs in machine learning and theoretical computer science. Given the topic of this post, we felt DifferentialPrivacy.org was a great fit. This initiative is organized by the Learning Theory Alliance, and overseen by Gautam Kamath. All posts in ALT Highlights are indexed on the official Learning Theory Alliance blog.

# What is δ, and what δifference does it make?

There are many variants or flavours of differential privacy (DP) some weaker than others: often, a given variant comes with own guarantees and “conversion theorems” to the others. As an example, “pure” DP has a single parameter $$\varepsilon$$, and corresponds to a very stringent notion of DP:

# Conference Digest - TPDP 2020

TPDP 2020 is a workshop focused on differential privacy. As such, it’s a great place to learn about recent developments in the DP research community. It will be held on 13 November and is co-located with CCS, but, of course, it’s virtual this year. Registration is only US\$35 if you register by Friday, 30 October. Check out the 8 excellent talks and 71 posters below – wow, the workshop has grown!

# Reconstruction Attacks in Practice

This is the second of two posts describing the theory and practice of reconstruction attacks. To read the first post, which covers the theoretical basis of such attacks, [click here].

# The Theory of Reconstruction Attacks

We often see people asking whether or not differential privacy might be overkill. Why do we need strong privacy protections like differential privacy when we’re only releasing approximate, aggregate statistical information about a dataset? Is it really possible to extract information about specific users from releasing these statistics? The answer turns out to be a resounding yes! The textbook by Dwork and Roth [DR14] calls this phenomenon the Fundamental Law of Information Recovery:

Giving overly accurate answers to too many questions will inevitably destroy privacy.

# Conference Digest - NeurIPS 2020

NeurIPS 2020 is the biggest conference on machine learning, with tons of content on differential privacy in many different forms. We were able to find two workshops, a competition, and 31 papers. This was just going off the preliminary accepted papers list, so it’s possible that we might have missed some papers on differential privacy – please let us know! We will update this post later, once all the conference material (papers and videos) are publicly available.

# Open Problem - Avoiding the Union Bound for Multiple Queries

Background: Perhaps the best-studied problem in differential privacy is answering multiple counting queries. The standard approach is to add independent, appropriately-calibrated (Laplace or Gaussian) noise to each query result and apply a composition theorem. To bound the maximum error over the query answers, one takes a union bound over the independent noise samples. However, this is not optimal. The problem is to identify the optimal method (up to constant factors).

# Differentially Private PAC Learning

The study of differentially private PAC learning runs all the way from its introduction in 2008 [KLNRS08] to a best paper award at the Symposium on Foundations of Computer Science (FOCS) this year [BLM20]. In this post, we’ll recap the history of this line of work, aiming for enough detail for a rough understanding of the results and methods.

# Conference Digest - ICML 2020

ICML 2020 is one of the premiere venues in machine learning, and generally features a lot of great work in differentially private machine learning. This year is no exception: the relevant papers are listed below to the best of our ability, including links to the full versions of papers, as well as the conference pages (which contain slides and 15 minute videos for each paper). As always, please inform us if we overlooked any papers on differential privacy.

# Conference Digest - COLT 2020

COLT 2020 was held online in July, and featured nine papers on differential privacy, as well as a keynote talk by Salil Vadhan. While differential privacy has always had a home in the COLT community, it seems like this year was truly exceptional in terms of the number of results. We link all the content below, including pointers to the papers, videos on Youtube, and the page on the conference website. Please let us know if we missed any papers on differential privacy, either in the comments below or by email.

# Why Privacy Needs Composition

We’re back! In our last post we discussed some of the subtle pitfalls of formulating the assumptions underlying average-case relaxations of differential privacy. This time we’re going to look at the composition property of differential privacy—that is, the fact that running two independent differentially private algorithms on your data and combining their outputs is still differentially private. This is a key property of differential privacy and is actually closely related to the worst-case nature of differential privacy.

# Open Problem - Private All-Pairs Distances

Background: Suppose we are interested in computing the distance between two vertices in a graph. Under edge or node differential privacy, this problem is not promising because the removal of a single edge can make distances change from 1 to $$n − 1$$ or can even disconnect the graph. However, a different setting that makes sense to consider is that of a weighted graph $$(G, w)$$ whose topology $$G = (V, E)$$ is publicly known but edge weight function $$w : E \to \mathbb{R}^+$$ must be kept private. (For instance, consider transit times on a road network. The topology of the road network may be publicly available as a map, but the edge weights corresponding to transit times may be based on private GPS locations of individual cars.)

# The Pitfalls of Average-Case Differential Privacy

Differential privacy protects against extremely strong adversaries—even ones who know the entire dataset except for one bit of information about one individual. Since its inception, people have considered ways to relax the definition to assume a more realistic adversary. A natural way to do so is to incorporate some distributional assumptions. That is, rather than considering a worst-case dataset, assume the dataset is drawn from some distribution and provide some form of “average-case” or “Bayesian” privacy guarantee with respect to this distribution. This is especially tempting as it is common for statistical analysis to work under distributional assumptions.

# Conference Digest - STOC 2020

STOC 2020 was recently held online, as one of the first major theory conferences during the COVID-19 era. It featured four papers on differential privacy, which we list and link below. Each one is accompanied by a video from the conference, as well as a longer video if available. Please let us know if we missed any papers on differential privacy, either in the comments below or by email.


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# When two smaller atoms combine into a larger atom what has occurred?

I think you describe $\text{fusion}$...........


text:
# Spectral indices with Landsat 8 imagery

We will use here the same data cube that was introduced in the images example.

A very popular spectral indice is the NDVI which is deffined as the a normalized difference between the Near Infra Red (NIR) and the Red bands. Namely NDVI = (NIR - Red) / (NIR + Red). An heuristic says that NDVI values ~greater than 0.4 indicate green vegetation as higher the indice (maximum = 1) larger is the green vegetation content.

Since the data cube holds in it the information about each band, computing the NDVI is a trivial opration because we known where each band is in the data cube. Let's see it.

using RemoteS, GMT
N = ndvi("c:/v/LC08_L1TP_20210525_02_cube.tiff");
imshow(N, colorbar=true)

The spectral indices functions all have a $threshold$ value that will NaNify all values $< threshold$. Below we wipe out all values < 0.4 to show only the Green stuff

N = ndvi("c:/v/LC08_L1TP_20210525_02_cube.tiff", threshold=0.4);
imshow(N, dpi=150, colorbar=1)

Cool but I would like to check that the result is correct.

The spectral indices functions have another option to obtain just a mask where the values are $> threshold$, so we can use it to mask out the true color image and see what we get.

# Compute a mask based on the condition that threshold >= 0.4
mask = ndvi("c:/v/LC08_L1TP_20210525_02_cube.tiff", threshold=0.4, mask=true);

To mask out the true color image the best way is to use the mask as the alpha band. To make it easier we will recalculate the true color image here.

# Recalculate the true color image
Irgb = truecolor("c:/v/LC08_L1TP_20210525_02_cube.tiff");

# Apply the mask

# And save it to disk so that we can visualize the result

imshow("c:/v/rgb_masked.tiff")

While we can see that only the green color is present in the above image, it's not easy to see the details. So let's zoom in but do also another thing. Let us see also what parts of the RGB image were not selected. To see that we will calculate the $inverse mask$, that is, the mask that retains all values < threshold. To achieve that we use the $mask$ option with a negative number.

# Compute a mask based on the condition that threshold >= 0.4
gmtwrite("c:/v/rgb_inv_masked.tiff", Irgb)
grdimage("c:/v/rgb_masked.tiff", region=(502380,514200,4311630,4321420), figsize=8, frame=:bare)
grdimage!("c:/v/rgb_inv_masked.tiff", figsize=8, projection=:linear, xshift=8, frame=:bare, show=true)
As we can see only the greenest part, the one that is probably more irrigated considering that all the river and cannals margins were retained, was extracted with the condition NDVI > 0.4. Off course, using different threshold values would lead to slightly different results.


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# Stubborn Charachters! Need help converting CHR variable to Numeric (in R)

#### elilieberman

##### New Member
I read in a csv as follows:
My issue is the dollar "amount" field was assigned "character" type, as follows:
str (defense)
'data.frame': 5024 obs. of 11 variables:
$amount: chr "$548,277.00 " "$1,063,790.35 " "$2,988.72 "

I wish to change the "amount" field to numeric, so I can run meaningful aggregate functions i.e. the MEAN for the vector. I failed using the "as.numeric" and "type.convert", see following;
as.numeric ("defense$amount") type.convert(defense$amount, na.strings = "NA", as.is = FALSE, dec = ".")
The str function still tells me that "amount" type is still "chr"
Any ideas? Thank you. Eli

#### Dason

The problem is that you do have characters in there. "," and "$" aren't numeric and aren't the decimal point so it doesn't know how to deal with those. The easiest way to deal with it is to remove those characters and then convert to numeric. Code: > x <- c("$1,000.2", "$3", "2.54") > x [1] "$1,000.2" "$3" "2.54" > gsub("[$,]", "", x)
[1] "1000.2" "3"      "2.54"
> as.numeric(gsub("[$,]", "", x)) [1] 1000.20 3.00 2.54 #### elilieberman ##### New Member Dason, thank you, the first line of code you recommended gsub... did as you indicated, [5021] "279383.90 " "529923.40 " "32912.20 " "8008.80 " However when I tried the as.numeric... I got the following response [5021] 2.793839e+05 5.299234e+05 3.291220e+04 8.008800e+03 Warning message: NAs introduced by coercion And then when I checked the filed with the str command I seem to be back to the where I started with the$ and comma's
> str (defense2$amount) chr [1:5024] "$548,277.00 " "$1,063,790.35 " "$2,988.72 " ...

Any ideas, much thanks. Eli

#### Dason

I don't know why you are getting NAs for some values - I would need to see the data for that. But the way it is displayed is perfectly understandable - it uses scientific notation for large numeric values.

I think the second issue is that my code does the conversion but it doesn't write over the previous data. It's the difference between

Code:
> x <- 3 # original data
> x
[1] 3
> # conversion
> x + 2
[1] 5
> x # x is unchanged though
[1] 3
> # write the changes back to the variable
> x <- x + 2
> x
[1] 5
Hopefully the change you need to make becomes obvious now.

#### elilieberman

##### New Member
I've been pluggin along searching Google and I you got me a lot closer quickly, I'll work on it and play with the data, thanks again, eli


text:
# Is there any symmetry I can use to find this Fourier sine series?

1. Mar 3, 2014

### richyw

1. The problem statement, all variables and given/known data

I am going over a practice exam, and I need to find the FSS of $$f(x)=x(\pi^2-x^2)$$

2. Relevant equations
$$f(x) \sim \sum^\infty_{n=1}a_n sin\left(\frac{n \pi x}{L}\right)$$
$$a_n=\frac{2}{L}\int^L_0 f(x)sin\left(\frac{n\pi x}{L}\right)dx$$
3. The attempt at a solution
I think I need to integrate$$\frac{2}{\pi}\int^\pi_0 x(\pi^2-x^2)sin(n\pi)dx$$which is two integrals, the first one would need me to use IBP once, and the second one would need me to use IBP three times. This is on a practice exam (and my exam is in an hour), so I am guessing that this integral is easier if I can find some symmetry in it. Is this true?

2. Mar 3, 2014

### asdf12312

You can indeed use symmetry. Graph f(x) on your calc: what do you notice? remember if x(t)=x(-t) then there is even symmetry, x(t)=-x(-t) means it has odd symmetry.

3. Mar 3, 2014

### LCKurtz

The fact that the function is odd is the reason you can use the half range sine expansion in the first place. That is why there are no cosine terms. You just need to bite the bullet and do the integration by parts. Don't do two separate integrals though. Write your integrand as$$(\pi^2x-x^3)sin(n\color{red}x)$$and do it all at once. Note your typo correction.


text:
# five month countdown timer circuit?

I'm a beginner in electronics, so I just have an idea in my head and don't know how to actually make it. I want to make a countdown circuit, for an event in next 5 month. The display format is like this:

DDD HH MM SS

Days have 3 digits, hours, minutes and seconds have 2 digits and I want to show the numbers on 7-segment displays. I just want to show remained time to a specific time, some day in October. I want to make it with a PIC or anything that you suggest.
I googled it, but just found a MM:SS countdown circuit. I can't expand it for solving my question. How can I do that?

• Do you want it on 7 segment displays, an LCD, ...? Anyway, it would look pretty similar to a HH:MM circuit - could you show us what you found for that? – user17592 May 18 '13 at 15:52
• +1 to counter the fly-by downvote. The question could do with some improvement, sure, but helping the OP understand what those improvements could be, is way more useful than a silent down-vote. – Anindo Ghosh May 18 '13 at 16:19
• To Netmoon: What exactly do you mean by "make it with ... TTL"? The PIC is a microcontroller, whereas TTL i.e. Transistor-Transistor-Logic, is a set of logic signaling standards used by various electronic components, including some microcontrollers. On the question itself, please edit the question to describe, with relevant links, your findings so far, to enable other members to understand how far you are along the path to your goal, and what level of answer would help you best. – Anindo Ghosh May 18 '13 at 16:23
• @anindo & camil : edited... i hope i can explain right about my question. – Netmoon May 18 '13 at 17:20
• I would do it using a microcontroller with a RTC chip (or the built-in RTC on your microcontroller, like some microcontrollers have). Read from the RTC every second and then calculate the difference between your target date/time and the current date/time. There probably are better ways of doing that. – Renan May 18 '13 at 17:27

In order to expand the display to displaying DDD MM HH, you need five more I/O lines for the DDD and SS. The PIC16F84A microntroller shown in the linked circuit has only 13 I/O lines and there are all used. (If you don't need the buzzer and relay outputs then potentially you have two spare outputs, but as I mentioned above, you need five.)

So If you want to stay with the PIC16F series of microcontrollers in your linked circuit, I would expand it to meet your needs using a PIC16F883 which has 24 I/O pins. (They make PIC16F microcontrollers with 18 and 20 I/O pins, but only in SMD packages, and I assume you want through-hole). Just add five more outputs going to additional transistors like T1-T4 driving the five new seven segment displays, and the a-g inputs of the displays to the parallel bus coming out of resistor pack R5-R11.

For a crystal, you need to choose one that has a tight tolerance as possible to keep the drift to a minimum. Timer1 of the PIC16F883 can run off of a 32KHz 30 ppm watch crystal.

Software wise, set up Timer1 to interrupt once a second. Use the Set and Select buttons as in the previous application to set up the initial DDD:HH:MM:SS values. On each interrupt, update the display -- decrement seconds, when they roll over from 00 to 59, decrement minutes etc.

• as i explained above i'm beginner, but when read your answer i understood that i should start reading about PICs, right? i just have 20 days to create this circuit, i want to present it to my friend and just make her happy. but i think 20 days is too small for reading and understanding PICs. can you lead me to best reference or book or website that help me to create this circuit as soon as possible? – Netmoon May 18 '13 at 19:57
• This may be too ambitious to accomplish in 20 days, unless you can devote a lot of time each day to it. The earlier projects were both written in PIC16 assembly language, not C, so you would need to learn that first -- or you could write the program from scratch in C if you know that language. You will need to get the datasheet for the PICF883 and download a development system from Microchip. (continued)... – tcrosley May 18 '13 at 23:09
• You will need to get a book that covers either assembly such as The PIC Microcontroller: Your Personal Introductory Course, Third Edition or C -- PIC Microcontrollers - Programming in C. Finally you will need to get a programmer such as the PICKit 3 to program the PIC. – tcrosley May 18 '13 at 23:11
• thanks a lot. it's complete answer that i search it :) . i know but i try hard to understand and accomplish this mission in 20 days ;) . i vote this answer, and start reading this books, and some days check this page for any other suggestions. thanks :) – Netmoon May 20 '13 at 7:08

I highly recommend you to use a LCD instead of bunch of 7-segments. You can configure the digits easily. PCB will be much easier to build. Much more less soldering and easier to control in software.

LCD can be used with direct wiring of 8 pins. I recommend you 2x16 LCD with a HD44780 chip in it. You can find in net for HD44780 examples a lot.

As for timer, i recommend a RTC (real time clock). Exp. DS1302. These chips are for real time clocking. So you don't have to use any PIC's timer interrupts and a counter for this job.

• if you look at the circuit the OP linked to, you will see that the 7 wires going to the segments a-g are connected in parallel to all the of the displays, and the segments multiplexed. – tcrosley May 19 '13 at 1:12
• Perfect link. Thank you. Didn't know this 4543 chip. But i think segments are not multiplexed. I think T1-T4 transistors are for deciding working segment. – Hammers May 19 '13 at 1:24

The fastest way to do this is with an Arduino based board. Get a board, a clock module like this one https://www.sparkfun.com/products/99 and connect them together. You will probably need a Mega to have enough IO pins to driver the displays. Ask any coding questions on StackOverflow (the programming Q&A) site.

Or - here is a project with an Arduino type board called the "Teensy" to build a display like the time machine Delorean in "Back To The Future". http://learn.adafruit.com/delorean-time-circuit/background He has a parts list and sample code that could be adapted.


text:
# Angular diameter distance

1. Mar 26, 2014

### kuecken

So the angular diameter distance is given by the angle it extends in the sky and its actual size x
$d_A= \frac{x}{\theta}$
This definition is okay. It means that if I consider the universe to be homogeneous and isotropic, I will have a conformal expansion. Therefore the angle should remain the same and only x expands.
But if I now use the following definition, I am lost.
$d_A = \frac{r(\chi)}{1+z}$

with $\chi$ being the comoving distance.
$r(\chi) = \begin{cases} \sin \left( \sqrt{-\Omega_k} H_0 \chi \right)/\left(H_0\sqrt{|\Omega_k|}\right) & \Omega_k < 0\\ \chi & \Omega_k=0 \\ \sinh \left( \sqrt{\Omega_k} H_0 \chi \right)/\left(H_0\sqrt{|\Omega_k|}\right) & \Omega_k >0 \end{cases}$

I know $r(\chi)$ is called proper motion distance or transverse comoving distance.

1) But what does it represent in a non flat universe?

If I had a light ray travelling along some path. Then this quantity would not include the expansion of the universe. So it cannot represent the physical distance the lightray has travelled.

2) What is then a physical distance in a curved universe?

Well if I use the definition anyway together with
${1+z}=\frac{a_{now}}{a_{past}}$
then for an expanding universe $d_A = \frac{r(\chi)}{1+z}$, the angular diameter distance would be decreased. So why that? 3) If the angle in the sky remains the same, why should it be necessary to decrease this value?
4) Does the angle maybe chance for a curved universe?

I would be very happy about any help.


text:
How to find the sum? [closed]

The problem:

How to find this sum?

$$\sum_{a=0}^{\infty}\frac{1}{(\frac{(a(p-1)+b)!}{p^{q_a}} \mod p) \times p^q}$$

where:

$p \in Primes$

$b \in \mathbb{N}\quad$, $0 \leq b \leq p-2$, but not defined.

$q_a$ is the greatest power of $p$ who divides the term $(a(p-1)+b)!$

Details:

$b$ refers to the congruence modulus $p-1$, so $0 \leq b \leq p-2$.

$a(p-1)+b$ for differents $a$'s and $b$'s we can express all natural numbers,

so the summation looks like the sequence of $e$ but a bit modified.

$q_a$ can also be expressed as: $q_a=\lfloor{\frac{(a(p-1)+b)}{p}}\rfloor+\lfloor{\frac{(a(p-1)+b)}{p^2}}\rfloor+\lfloor{\frac{(a(p-1)+b)}{p^3}}\rfloor+\ldots$

$\frac{(a(p-1)+b)!}{p^{q_a}} \bmod p$ is just the remainder of the division by $p$.

Expandind the sum we have:

$\frac{1}{(\frac{b!}{p^{q_0}} \mod p)p^{q_0}}+\frac{1}{(\frac{(p-1+b)!}{p^{q_1}} \mod p)p^{q_1}}+\frac{1}{(\frac{(2p-2+b)!}{p^{q_2}} \mod p)p^{q_2}}+\frac{1}{(\frac{(3p-3+b)!}{p^{q_3}} \mod p)p^{q_3}}+\ldots$

just to clarify, to $p=2$ and $b=0$ we have:

$\frac{1}{(0! \mod 2)2^0}+\frac{1}{(1! \mod 2)2^0}+\frac{1}{(\frac{2!}{2^1} \mod 2)2^1}+\frac{1}{(\frac{3!}{2^1} \mod 2)2^1}++\frac{1}{(\frac{4!}{2^3} \mod 2)2^3}+\frac{1}{(\frac{5!}{2^3} \mod 2)2^3}+\frac{1}{(\frac{6!}{2^4} \mod 2)2^4}\ldots$

The motivation to find this sum is analyze some properties of the prime numbers using the expansion of $e$.

-
This was posted to math.stackexchange.com/questions/82204/… a few minutes before you posted it here. – Yemon Choi Nov 15 2011 at 1:41
To explicit, cross-posting is bad form, because answers are not linked to questions at the other site, and this is unhelpful to other users. Also, some of the detail you provided is not of research-level (e.g. people here don't need telling what 'mod p' means), and 'analyze some properties of the prime numbers using the expansion of e.' is not exactly detailed motivation. People study analytic number theory/additive combinatorics/arakelov geometry/etc/ for this reason. – David Roberts Nov 15 2011 at 1:57
Although I think this question could be a lot better, would the closers please give a reason? I've started a meta thread meta.mathoverflow.net/discussion/1207/… – David Roberts Nov 15 2011 at 2:07
Do you have a reason to believe this sum is meaningful? When we ask for motivation, we mean that you should describe how such a sum would help you in your study of properties of primes. At any rate, anyone here who knows an answer to your question is welcome to give a response at math.stackexchange.com, because there is a corresponding question there. – S. Carnahan Nov 15 2011 at 2:24
I think close isn't the better way to solve this. The question isn't "too localizable" (even in Math.SE) and have mathematician interest as in FAQ. About the motivation, the truth is I didn't have a theory yet and I'm asking some help (solving this question) to go ahead. Otherwise, where in FAQ says put questions in two different sites aren't allowed? – GarouDan Nov 15 2011 at 19:19


text:
Translator Disclaimer
VOL. 9 | 2013 Efficient estimation in the semiparametric normal regression-copula model with a focus on QTL mapping
Bojan Basrak, Chris A. J. Klaassen

Editor(s) M. Banerjee, F. Bunea, J. Huang, V. Koltchinskii, M. H. Maathuis

## Abstract

The semiparametric normal copula model is studied with a correlation matrix that depends on a covariate. The bivariate version of this regression-copula model has been proposed for statistical analysis of Quantitative Trait Loci (QTL) via twin data. Appropriate linear combinations of Van der Waerden’s normal scores rank correlation coefficients yield $\sqrt{n}$-consistent estimators of the coefficients in the correlation function, i.e. of the regression parameters. They are used to construct semiparametrically efficient estimators of the regression parameters.

## Information

Published: 1 January 2013
First available in Project Euclid: 8 March 2013

zbMATH: 1327.62192
MathSciNet: MR3186746

Digital Object Identifier: 10.1214/12-IMSCOLL903

Subjects:
Primary: 62G05 , 62G20
Secondary: 62P10

Keywords: $\sqrt{n}$-consistency , linkage analysis , QTL mapping , semiparametric inference , van der Waerden


text:
# zbMATH — the first resource for mathematics

Dynamical eigenfunction decomposition of turbulent pipe flow. (English) Zbl 1273.76192
Summary: The results of an analysis of turbulent pipe flow based on a Karhunen-Loève decomposition are presented. The turbulent flow is generated by a direct numerical simulation of the Navier-Stokes equations using a spectral element algorithm at a Reynolds number $$Re = 150$$. This simulation yields a set of basis functions that captures 90% of the energy after 2763 modes. The eigenfunctions are categorized into two classes and six subclasses based on their wavenumber and coherent vorticity structure. Of the total energy, 81% is in the propagating class, characterized by constant phase speeds; the remaining energy is found in the non-propagating subclasses, the shear and roll modes. The four subclasses of the propagating modes are the wall, lift, asymmetric and ring modes. The wall modes display coherent vorticity structures near the wall, the lift modes display coherent vorticity structures that lift away from the wall, the asymmetric modes break the symmetry about the axis, and the ring modes display rings of coherent vorticity. Together, the propagating modes form a wave packet, as found from a circular normal speed locus. The energy transfer mechanism in the flow is a four-step process. The process begins with energy being transferred from mean flow to the shear modes, then to the roll modes. Energy is then transferred from the roll modes to the wall modes, and then eventually to the lift modes. The ring and asymmetric modes act as catalysts that aid in this four-step energy transfer. Physically, this mechanism shows how the energy in the flow starts at the wall and then propagates into the outer layer.

##### MSC:
 76F65 Direct numerical and large eddy simulation of turbulence
Full Text:
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