CCSP SeminarThe online CCSP Seminar, on recent advances in communication, control and signal processing at large, is a teaching seminar. Each invited speaker is requested to present a lecture (of duration 60 - 90 minutes) that describes just one or two mathematical techniques and just as many key results. The lecture will be given at a Zoom whiteboard in classroom fashion, at classroom pace, and will be videotaped for open access if the speaker so desires.
http://localhost:4000/
Online Learning of Structured Matrices in Recommendation Systems<p>We consider an online model for recommendation systems, with each user being recommended an item at each time-step and providing ‘like’ or ‘dislike’ feedback. A latent variable model specifies the user preferences: both users and items are clustered into types. The model captures structure in both the item and user spaces, and our focus is on the simultaneous use of both structures. We analyze the situation in which the type preference matrix has i.i.d. entries. Our analysis elucidates the system operating regimes in which existing algorithms are nearly optimal, as well as highlighting the sub-optimality of using only one of item or user structure (as is done in commonly used item-item and user-user collaborative filtering). This prompts a new algorithm that is optimal in essentially all parameter regimes.</p>
<p><a href="https://arxiv.org/abs/1711.02198">https://arxiv.org/abs/1711.02198</a></p>
<h3 id="recorded-talk">Recorded Talk</h3>
<div class="video-container">
<iframe src="https://www.youtube.com/embed/wJEOwsn0I8w" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen=""></iframe>
</div>
Thu, 06 May 2021 07:21:29 -0400
http://localhost:4000/2021/05/06/karzand-online-learning-structured-matrices/
http://localhost:4000/2021/05/06/karzand-online-learning-structured-matrices/Function Correcting Codes<p>Motivated by applications in machine learning and archival data storage, we introduce function-correcting codes, a new class of codes designed to protect a function evaluation on the data against errors. We show that function-correcting codes are equivalent to irregular distance codes, i.e., codes that obey some given distance requirement between each pair of codewords. Using these connections, we study irregular distance codes and derive general upper and lower bounds on their optimal redundancy. Since these bounds heavily depend on the specific function, we provide simplified, suboptimal bounds that are easier to evaluate. We further employ our general results to specific functions of interest and we show that function-correcting codes can achieve significantly less redundancy than standard error-correcting codes which protect the whole data.</p>
<p>The talk is based on joint work with Andreas Lenz, Rawad Bitar and Eitan Yaakobi.</p>
<h3 id="recorded-talk">Recorded Talk</h3>
<p>Coming soon!
<!--<div class="video-container">
<iframe src="https://www.youtube.com/embed/0OTczuUDWnw" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
</div>--></p>
Thu, 29 Apr 2021 07:21:29 -0400
http://localhost:4000/2021/04/29/wachter-zeh-function-correcting-codes/
http://localhost:4000/2021/04/29/wachter-zeh-function-correcting-codes/Adversarial Robustness for Non-Parametric Methods<p>There has been much recent interest in adversarially robust learning – where the goal is to learn a classifier which can accurately classify, not just data from the underlying distribution, but also small perturbations thereof. In this talk, we will look at this phenomenon from the point of view of non-parametric methods, such as nearest neighbors and decision trees.</p>
<p>For non-parametric methods, Stone in 1977 proved a consistency theorem that shows that when the training data size goes to infinity, the accuracy of many methods approach that of the Bayes optimal classifier. In this talk, we will establish a robustness analogue of the Bayes optimal, called the r-optimal, and show an analogue of Stone’s theorem for robustness for the case when data from different classes is well-separated. We will then briefly discuss what happens when this is not the case.</p>
<p>Talk based on joint work with Robi Bhattacharjee.</p>
<h3 id="recorded-talk">Recorded Talk</h3>
<div class="video-container">
<iframe src="https://www.youtube.com/embed/1MSiBV1c8Wg" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen=""></iframe>
</div>
Thu, 22 Apr 2021 07:21:29 -0400
http://localhost:4000/2021/04/22/chaudhuri-adversarial-robustness-nonparametric/
http://localhost:4000/2021/04/22/chaudhuri-adversarial-robustness-nonparametric/Mitigating Coherent Noise in Quantum Computing using the Classical MacWilliams Identities<p>Noise in quantum systems can be coherent or stochastic (incoherent). The former is more damaging since such noise can accumulate in one direction and grow quadratically in the number of qubits. While standard quantum error correction (QEC) addresses noise actively by measuring syndromes and applying corrections, we develop conditions for a stabilizer code to passively tackle coherent noise. When the noise introduces a coherent Z-rotation by an angle theta on all qubits, our codes remain unaffected and act as a decoherence free subspace (DFS). Given any [[n,k,d]] stabilizer code and even M, we can produce a [[Mn,k,>= d]] code that is a DFS for this noise.</p>
<p>In this talk, we will begin by revisiting the classical result of MacWilliams that relates the weight enumerator of a code to the weight enumerator of the dual. Then, after reviewing the essentials of stabilizer codes, we will discuss conditions for a transversal Z-rotation exp(i<em>theta</em>Z) to fix the code space of a stabilizer code. Subsequently, we consider the case where we impose that this transversal rotation fixes the code for all theta, and show that this necessitates the presence of a large amount of weight-2 Z-stabilizers. By organizing these suitably, we will develop DFSs for the aforesaid form of coherent noise. If time permits, we will briefly discuss how we analyze the case where we want only theta <= pi/2^l to preserve the code, in order to induce non-trivial logical gates.</p>
<h3 id="recorded-talk">Recorded Talk</h3>
<div class="video-container">
<iframe src="https://www.youtube.com/embed/PFr6Ux1GMbg" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen=""></iframe>
</div>
Thu, 08 Apr 2021 07:21:29 -0400
http://localhost:4000/2021/04/08/rengaswamy-coherent-noise-in-quantum-computing/
http://localhost:4000/2021/04/08/rengaswamy-coherent-noise-in-quantum-computing/Self-regularizing Property of Nonparametric Maximum Likelihood Estimator in Mixture Models<p>Introduced by Kiefer and Wolfowitz 1956, the nonparametric maximum likelihood estimator (NPMLE) is a widely used methodology for learning mixture models and empirical Bayes estimation. Sidestepping the non-convexity in mixture likelihood, the NPMLE estimates the mixing distribution by maximizing the total likelihood over the space of probability measures, which can be viewed as an extreme form of overparameterization.</p>
<p>In this work, we discover a surprising property of the NPMLE solution. Consider, for example, a Gaussian mixture model on the real line with a subgaussian mixing distribution. Leveraging complex-analytic techniques, we show that with high probability the NPMLE based on a sample of size n has O(\log n) atoms (mass points), significantly improving the deterministic upper bound of n due to Lindsay 1983. Notably, any such Gaussian mixture is statistically indistinguishable from a finite
one with O(\log n) components (and this is tight for certain mixtures). Thus, absent any explicit form of model selection, NPMLE automatically chooses the right model complexity, a property we term self-regularization. Statistical applications and extensions to other exponential families will be given. Connections to rate-distortion functions will be briefly discussed.</p>
<p>This is based on joint work with Yury Polyanskiy (MIT): <a href="https://arxiv.org/abs/2008.08244">https://arxiv.org/abs/2008.08244</a></p>
<h3 id="recorded-talk">Recorded Talk</h3>
<div class="video-container">
<iframe src="https://www.youtube.com/embed/glCfe1Saq2s" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen=""></iframe>
</div>
Thu, 01 Apr 2021 07:21:29 -0400
http://localhost:4000/2021/04/01/wu-self-regularising-property-of-npmles/
http://localhost:4000/2021/04/01/wu-self-regularising-property-of-npmles/A Single-Letter Upper Bound on the Mismatch Capacity<p>The question of finding a single-letter formula for the mismatch capacity, which is the supremum of achievable rates of reliable communication when the receiver uses a sub-optimal decoding rule, has been a long-standing open problem. This question has many applications in communications, Information Theory and Computer Science. For example, the zero-error capacity of a channel is a special case of mismatch capacity.</p>
<p>In this talk, I will give a brief overview of the problem, and introduce a new bounding technique called the “multicasting approach,” which straightforwardly yields single-letter upper bounds on the mismatch capacity of stationary memoryless channels. I will also present equivalence classes of isomorphic channel-metric pairs that share the same mismatch capacity, and a sufficient condition for the tightness of the bound for the entire equivalence class.</p>
<h3 id="recorded-talk">Recorded Talk</h3>
<p>Coming soon!
<!--<div class="video-container">
<iframe src="https://www.youtube.com/embed/0OTczuUDWnw" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
</div>--></p>
Thu, 25 Mar 2021 07:21:29 -0400
http://localhost:4000/2021/03/25/somekh-baruch-mismatched-capacity/
http://localhost:4000/2021/03/25/somekh-baruch-mismatched-capacity/Codes for Adversaries - Between Worst-Case and Average-Case Jamming<p>Over the last 70 years, information theory and coding have enabled communication technologies that have had an astounding impact on our lives. This is possible due to the match between encoding/decoding strategies and corresponding channel models. Traditional studies of channels have mostly taken one of two extremes: Shannon-theoretic models are inherently average-case in which channel noise is governed by a memoryless stochastic process whereas coding-theoretic (referred to as “Hamming”) models take a worst-case, adversarial, view of the noise. However, for several existing and emerging communication systems, the Shannon/average-case view may be too optimistic, whereas the Hamming/worst-case view may be too pessimistic. In this talk, I will survey a collection of results on the study of channel models that fall between the Shannon and Hamming perspectives.</p>
<p>The talk is based on joint works with Z. Chen, A. Budkuley, B. K. Dey, I. Haviv, S. Jaggi, A. D. Sarwate, C. Wang, and Y. Zhang.</p>
<h3 id="recorded-talk">Recorded Talk</h3>
<div class="video-container">
<iframe src="https://www.youtube.com/embed/aSYpsqH-B1M" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen=""></iframe>
</div>
Thu, 11 Mar 2021 06:21:29 -0500
http://localhost:4000/2021/03/11/langberg-codes-for-adversaries/
http://localhost:4000/2021/03/11/langberg-codes-for-adversaries/The zero-error list decoding capacity of the q/(q-1) channel<p>We will start by reviewing the arguments of Krichevskii, Hansel and Pippenger on covering graphs using bipartite graphs, and using them motivate Korner’s graph entropy. We will combine the graph covering argument with some counting of increasing complexity to derive the following:</p>
<ol>
<li>The Fredman-Komlos lower bound on the size of a family of perfect hash functions;</li>
<li>A bound on the zero-error list decoding capacity of the <em>4/3</em> channel;</li>
<li>A bound on the zero-error list decoding capacity of the <em>q(q-1)</em> channel.</li>
</ol>
<p>Handwritten notes for the talk can be found <a href="/files/jaikumar-notes.pdf">here</a>.</p>
<h3 id="references">References</h3>
<ol>
<li>M. Dalai, V. Guruswami and J. Radhakrishnan, “An Improved Bound on the Zero-Error List-Decoding Capacity of the 4/3 Channel,” in <em>IEEE Transactions on Information Theory</em>, vol. 66, no. 2, pp. 749-756, Feb. 2020, doi: 10.1109/TIT.2019.2933424. <a href="https://ieeexplore.ieee.org/document/8788642">Link</a></li>
<li>S. Bhandari and J. Radhakrishnan, “Bounds on the Zero-Error List-Decoding Capacity of the q/(q-1) Channel,” <em>2018 IEEE International Symposium on Information Theory (ISIT)</em>, Vail, CO, 2018, pp. 906-910, doi: 10.1109/ISIT.2018.8437609. <a href="https://ieeexplore.ieee.org/document/8437609">Link</a></li>
</ol>
<h3 id="recorded-talk">Recorded Talk</h3>
<div class="video-container">
<iframe src="https://www.youtube.com/embed/qxaiDfJq4h8" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen=""></iframe>
</div>
Thu, 11 Feb 2021 06:21:29 -0500
http://localhost:4000/2021/02/11/radhakrishnan-zero-error-list-decoding-capacity-q-q-1-channel/
http://localhost:4000/2021/02/11/radhakrishnan-zero-error-list-decoding-capacity-q-q-1-channel/Sharp Thresholds for Random Subspaces, and Applications to LDPC Codes<p>What combinatorial properties are likely to be satisfied by a random subspace over a finite field? For example, is it likely that not too many points lie in any Hamming ball? What about any cube? We show that there is a sharp threshold on the dimension of the subspace at which the answers to these questions change from “extremely likely” to “extremely unlikely,” and moreover we give a simple characterization of this threshold for different properties. Our motivation comes from error correcting codes, and we use this characterization to make progress on the questions of list-decoding and list-recovery for random linear codes, and also to establish the list-decodability of random Low Density Parity-Check (LDPC) codes.</p>
<p>This talk is based on joint works with Venkat Guruswami, Ray Li, Jonathan Mosheiff, Nicolas Resch, Noga Ron-Zewi, and Shashwat Silas.</p>
<h3 id="references">References</h3>
<ul>
<li><a href="https://arxiv.org/abs/1909.06430">LDPC Codes Achieve List Decoding Capacity</a> [FOCS2020]</li>
<li><a href="https://arxiv.org/abs/2004.13247">Bounds for list-decoding and list-recovery of random linear codes</a> [RANDOM2020]</li>
<li><a href="https://arxiv.org/abs/2009.04553">Sharp threshold rates for random codes</a> [ITCS 2021]</li>
</ul>
<h3 id="recorded-talk">Recorded Talk</h3>
<p>Thanks to Mary for allowing us to record the talk!</p>
<div class="video-container">
<iframe src="https://www.youtube.com/embed/W4CqtwKpIX4" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen=""></iframe>
</div>
Thu, 03 Dec 2020 06:21:29 -0500
http://localhost:4000/2020/12/03/wootters-sharp-thresholds-for-random-subspaces/
http://localhost:4000/2020/12/03/wootters-sharp-thresholds-for-random-subspaces/Rényi information inequalities and their mathematical ramifications<p>Rényi entropies are a natural one-parameter generalization of Shannon entropy that were first introduced over half a century ago, but about which fundamental questions remain incompletely answered. After a (very) brief introduction to why Rényi information functionals (entropies, divergences, etc.) are of interest from an information-theoretic viewpoint, we will attempt to expose the relevance of Rényi information inequalities for several areas of mathematics. For example, they allow for the unification of several interesting inequalities — including the entropy power inequality (which plays a fundamental role in information theory), the Brunn-Minkowski inequality (which plays a fundamental role in convex geometry), and Rogozin’s convolution inequality (which is fundamental to the area of “small ball” estimates in probability theory). They also allow for the quantification of uncertainty principles in harmonic analysis. In another direction, they are relevant to the field of additive combinatorics, which has seen burgeoning activity over the last two decades due to applications in theoretical computer science as well as other parts of mathematics.</p>
<h3 id="recorded-talk">Recorded Talk</h3>
<p>Thanks to Mokshay for allowing us to record the talk!</p>
<div class="video-container">
<iframe src="https://www.youtube.com/embed/2256Hd7WSKo" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen=""></iframe>
</div>
Thu, 12 Nov 2020 06:21:29 -0500
http://localhost:4000/2020/11/12/madiman-renyi-information-inequalities/
http://localhost:4000/2020/11/12/madiman-renyi-information-inequalities/