arXiv subjects

Yury Polyanskiy

Publications and source records attributed to Yury Polyanskiy.

At least 55 records · Page 3Linked to original sources

arXiv2025

Residual connections provably mitigate oversmoothing in graph neural networks

Graph neural networks (GNNs) have achieved remarkable empirical success in processing and representing graph-structured data across various domains. However, a significant challenge known as "oversmoothing" persists, where vertex features become nearly indistinguishable in deep GNNs, severely restricting their expressive power and practical utility. In this work, we analyze the asymptotic oversmoothing rates of deep GNNs with and without residual connections by deriving explicit convergence rates for a normalized vertex similarity measure. Our analytical framework is grounded in the multiplicative ergodic theorem. Furthermore, we demonstrate that adding residual connections effectively mitigates or prevents oversmoothing across several broad families of parameter distributions. The theoretical findings are strongly supported by numerical experiments.

cs.LG↗

arXiv2024

Clustering in Causal Attention Masking

This work presents a modification of the self-attention dynamics proposed by Geshkovski et al. (arXiv:2312.10794) to better reflect the practically relevant, causally masked attention used in transformer architectures for generative AI. This modification translates into an interacting particle system that cannot be interpreted as a mean-field gradient flow. Despite this loss of structure, we significantly strengthen the results of Geshkovski et al. (arXiv:2312.10794) in this context: While previous rigorous results focused on cases where all three matrices (Key, Query, and Value) were scaled identities, we prove asymptotic convergence to a single cluster for arbitrary key-query matrices and a value matrix equal to the identity. Additionally, we establish a connection to the classical Rényi parking problem from combinatorial geometry to make initial theoretical steps towards demonstrating the existence of meta-stable states.

cs.LG↗

arXiv2024

Dynamic metastability in the self-attention model

We consider the self-attention model - an interacting particle system on the unit sphere, which serves as a toy model for Transformers, the deep neural network architecture behind the recent successes of large language models. We prove the appearance of dynamic metastability conjectured in [GLPR23] - although particles collapse to a single cluster in infinite time, they remain trapped near a configuration of several clusters for an exponentially long period of time. By leveraging a gradient flow interpretation of the system, we also connect our result to an overarching framework of slow motion of gradient flows proposed by Otto and Reznikoff [OR07] in the context of coarsening and the Allen-Cahn equation. We finally probe the dynamics beyond the exponentially long period of metastability, and illustrate that, under an appropriate time-rescaling, the energy reaches its global maximum in finite time and has a staircase profile, with trajectories manifesting saddle-to-saddle-like behavior, reminiscent of recent works in the analysis of training dynamics via gradient descent for two-layer neural networks.

cs.LG↗

arXiv2024

Entropy Contractions in Markov Chains: Half-Step, Full-Step and Continuous-Time

This paper considers the speed of convergence (mixing) of a finite Markov kernel $P$ with respect to the Kullback-Leibler divergence (entropy). Given a Markov kernel one defines either a discrete-time Markov chain (with the $n$-step transition kernel given by the matrix power $P^n$) or a continuous-time Markov process (with the time-$t$ transition kernel given by $e^{t(P-\mathrm{Id})}$). The contraction of entropy for $n=1$ or $t=0+$ are characterized by the famous functional inequalities, the strong data processing inequality (SDPI) and the modified log-Sobolev inequality (MLSI), respectively. When $P=KK^*$ is written as the product of a kernel and its adjoint, one could also consider the ``half-step'' contraction, which is the SDPI for $K$, while the ``full-step'' contraction refers to the SDPI for $P$. The work [DMLM03] claimed that these contraction coefficients (half-step, full-step, and continuous-time) are generally within a constant factor of each other. We disprove this and related conjectures by working out a number of different counterexamples. In particular, we construct (a) a continuous-time Markov process that contracts arbitrarily faster than its discrete-time counterpart; and (b) a kernel $P$ such that $P^{m+1}$ contracts arbitrarily better than $P^m$. Hence, our main conclusion is that the four standard inequalities comparing five common notions of entropy and variance contraction are generally not improvable. In the process of analyzing the counterexamples, we survey and sharpen the tools for bounding the contraction coefficients and characterize properties of extremizers of the respective functional inequalities. As our examples range from Bernoulli-Laplace model, random walks on graphs, to birth-death chains, the paper is also intended as a tutorial on computing MLSI, SDPI and other constants for these types of commonly occurring Markov chains.

math.PR↗

arXiv2024

Unsourced Multiple Access: A Coding Paradigm for Massive Random Access

This paper is a tutorial introduction to the field of unsourced multiple access (UMAC) protocols. We first provide a historical survey of the evolution of random access protocols, focusing specifically on the case in which uncoordinated users share a wireless broadcasting medium. Next, we highlight the change of perspective originated by the UMAC model, in which the physical and medium access layer's protocols cooperate, thus reframing random access as a novel coding-theoretic problem. By now, a large variety of UMAC protocols (codes) emerged, necessitating a certain classification that we indeed propose here. Although some random access schemes require a radical change of the physical layer, others can be implemented with minimal changes to existing industry standards. As an example, we discuss a simple modification to the 5GNR Release 16 random access channel that builds on the UMAC theory and that dramatically improves energy efficiency for systems with even moderate number of simultaneous users (e.g., $5-10$ dB gain for $10-50$ users), and also enables handling of high number of users, something completely out of reach of the state-of-the-art.

cs.IT↗

arXiv2024

Evolution of the 5G New Radio Two-Step Random Access towards 6G Unsourced MAC

This report summarizes some considerations on possible evolutions of grant-free random access in the next generation of the 3GPP wireless cellular standard. The analysis is carried out by mapping the problem to the recently-introduced unsourced multiple access channel (UMAC) setup. By doing so, the performance of existing solutions can be benchmarked with information-theoretic bounds, assessing the potential gains that can be achieved over legacy 3GPP schemes. The study focuses on the two-step random access (2SRA) protocol introduced by Release 16 of the 5G New Radio standard, investigating its applicability to support large MTC / IoT terminal populations in a grant-free fashion. The analysis shows that the existing 2SRA scheme may not succeed in providing energy-efficient support to large user populations. Modifications to the protocol are proposed that enable remarkable gains in both energy and spectral efficiency while retaining a strong resemblance to the legacy protocol.

cs.IT↗

arXiv2024

Likelihood-free hypothesis testing

Consider the problem of binary hypothesis testing. Given $Z$ coming from either $\mathbb P^{\otimes m}$ or $\mathbb Q^{\otimes m}$, to decide between the two with small probability of error it is sufficient, and in many cases necessary, to have $m\asymp1/ε^2$, where $ε$ measures the separation between $\mathbb P$ and $\mathbb Q$ in total variation ($\mathsf{TV}$). Achieving this, however, requires complete knowledge of the distributions and can be done, for example, using the Neyman-Pearson test. In this paper we consider a variation of the problem which we call likelihood-free hypothesis testing, where access to $\mathbb P$ and $\mathbb Q$ is given through $n$ i.i.d. observations from each. In the case when $\mathbb P$ and $\mathbb Q$ are assumed to belong to a non-parametric family, we demonstrate the existence of a fundamental trade-off between $n$ and $m$ given by $nm\asymp n_\sf{GoF}^2(ε)$, where $n_\sf{GoF}(ε)$ is the minimax sample complexity of testing between the hypotheses $H_0:\, \mathbb P=\mathbb Q$ vs $H_1:\, \mathsf{TV}(\mathbb P,\mathbb Q)\geqε$. We show this for three families of distributions, in addition to the family of all discrete distributions for which we obtain a more complicated trade-off exhibiting an additional phase-transition. Our results demonstrate the possibility of testing without fully estimating $\mathbb P$ and $\mathbb Q$, provided $m \gg 1/ε^2$.

math.ST↗

arXiv2024

The Sample Complexity of Approximate Rejection Sampling with Applications to Smoothed Online Learning

Suppose we are given access to $n$ independent samples from distribution $μ$ and we wish to output one of them with the goal of making the output distributed as close as possible to a target distribution $ν$. In this work we show that the optimal total variation distance as a function of $n$ is given by $\tildeΘ(\frac{D}{f'(n)})$ over the class of all pairs $ν,μ$ with a bounded $f$-divergence $D_f(ν\|μ)\leq D$. Previously, this question was studied only for the case when the Radon-Nikodym derivative of $ν$ with respect to $μ$ is uniformly bounded. We then consider an application in the seemingly very different field of smoothed online learning, where we show that recent results on the minimax regret and the regret of oracle-efficient algorithms still hold even under relaxed constraints on the adversary (to have bounded $f$-divergence, as opposed to bounded Radon-Nikodym derivative). Finally, we also study efficacy of importance sampling for mean estimates uniform over a function class and compare importance sampling with rejection sampling.

stat.ML↗

arXiv2024

The emergence of clusters in self-attention dynamics

Viewing Transformers as interacting particle systems, we describe the geometry of learned representations when the weights are not time dependent. We show that particles, representing tokens, tend to cluster toward particular limiting objects as time tends to infinity. Cluster locations are determined by the initial tokens, confirming context-awareness of representations learned by Transformers. Using techniques from dynamical systems and partial differential equations, we show that the type of limiting object that emerges depends on the spectrum of the value matrix. Additionally, in the one-dimensional case we prove that the self-attention matrix converges to a low-rank Boolean matrix. The combination of these results mathematically confirms the empirical observation made by Vaswani et al. [VSP'17] that leaders appear in a sequence of tokens when processed by Transformers.

cs.LG↗

arXiv2024

Score-based Source Separation with Applications to Digital Communication Signals

We propose a new method for separating superimposed sources using diffusion-based generative models. Our method relies only on separately trained statistical priors of independent sources to establish a new objective function guided by maximum a posteriori estimation with an $α$-posterior, across multiple levels of Gaussian smoothing. Motivated by applications in radio-frequency (RF) systems, we are interested in sources with underlying discrete nature and the recovery of encoded bits from a signal of interest, as measured by the bit error rate (BER). Experimental results with RF mixtures demonstrate that our method results in a BER reduction of 95% over classical and existing learning-based methods. Our analysis demonstrates that our proposed method yields solutions that asymptotically approach the modes of an underlying discrete distribution. Furthermore, our method can be viewed as a multi-source extension to the recently proposed score distillation sampling scheme, shedding additional light on its use beyond conditional sampling. The project webpage is available at https://alpha-rgs.github.io

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arXiv2023

Kernel-Based Tests for Likelihood-Free Hypothesis Testing

Given $n$ observations from two balanced classes, consider the task of labeling an additional $m$ inputs that are known to all belong to \emph{one} of the two classes. Special cases of this problem are well-known: with complete knowledge of class distributions ($n=\infty$) the problem is solved optimally by the likelihood-ratio test; when $m=1$ it corresponds to binary classification; and when $m\approx n$ it is equivalent to two-sample testing. The intermediate settings occur in the field of likelihood-free inference, where labeled samples are obtained by running forward simulations and the unlabeled sample is collected experimentally. In recent work it was discovered that there is a fundamental trade-off between $m$ and $n$: increasing the data sample $m$ reduces the amount $n$ of training/simulation data needed. In this work we (a) introduce a generalization where unlabeled samples come from a mixture of the two classes -- a case often encountered in practice; (b) study the minimax sample complexity for non-parametric classes of densities under \textit{maximum mean discrepancy} (MMD) separation; and (c) investigate the empirical performance of kernels parameterized by neural networks on two tasks: detection of the Higgs boson and detection of planted DDPM generated images amidst CIFAR-10 images. For both problems we confirm the existence of the theoretically predicted asymmetric $m$ vs $n$ trade-off.

stat.ML↗

arXiv2023

Efficient Representation of Large-Alphabet Probability Distributions

A number of engineering and scientific problems require representing and manipulating probability distributions over large alphabets, which we may think of as long vectors of reals summing to $1$. In some cases it is required to represent such a vector with only $b$ bits per entry. A natural choice is to partition the interval $[0,1]$ into $2^b$ uniform bins and quantize entries to each bin independently. We show that a minor modification of this procedure -- applying an entrywise non-linear function (compander) $f(x)$ prior to quantization -- yields an extremely effective quantization method. For example, for $b=8 (16)$ and $10^5$-sized alphabets, the quality of representation improves from a loss (under KL divergence) of $0.5 (0.1)$ bits/entry to $10^{-4} (10^{-9})$ bits/entry. Compared to floating point representations, our compander method improves the loss from $10^{-1}(10^{-6})$ to $10^{-4}(10^{-9})$ bits/entry. These numbers hold for both real-world data (word frequencies in books and DNA $k$-mer counts) and for synthetic randomly generated distributions. Theoretically, we set up a minimax optimality criterion and show that the compander $f(x) ~\propto~ \mathrm{ArcSinh}(\sqrt{(1/2) (K \log K) x})$ achieves near-optimal performance, attaining a KL-quantization loss of $\asymp 2^{-2b} \log^2 K$ for a $K$-letter alphabet and $b\to \infty$. Interestingly, a similar minimax criterion for the quadratic loss on the hypercube shows optimality of the standard uniform quantizer. This suggests that the $\mathrm{ArcSinh}$ quantizer is as fundamental for KL-distortion as the uniform quantizer for quadratic distortion.

cs.IT↗

arXiv2023

Capacity of Noisy Permutation Channels

We establish the capacity of a class of communication channels introduced in [1]. The $n$-letter input from a finite alphabet is passed through a discrete memoryless channel $P_{Z|X}$ and then the output $n$-letter sequence is uniformly permuted. We show that the maximal communication rate (normalized by $\log n$) equals $1/2 (rank(P_{Z|X})-1)$ whenever $P_{Z|X}$ is strictly positive. This is done by establishing a converse bound matching the achievability of [1]. The two main ingredients of our proof are (1) a sharp bound on the entropy of a uniformly sampled vector from a type class and observed through a DMC; and (2) the covering $ε$-net of a probability simplex with Kullback-Leibler divergence as a metric. In addition to strictly positive DMC we also find the noisy permutation capacity for $q$-ary erasure channels, the Z-channel and others.

cs.IT↗

arXiv2023

Ising Model on Locally Tree-like Graphs: Uniqueness of Solutions to Cavity Equations

In the study of Ising models on large locally tree-like graphs, in both rigorous and non-rigorous methods one is often led to understanding the so-called belief propagation distributional recursions and its fixed points. We prove that there is at most one non-trivial fixed point for Ising models with zero or certain random external fields. Previously this was only known for sufficiently ``low-temperature'' models. Our main innovation is in applying information-theoretic ideas of channel comparison leading to a new metric (degradation index) between binary-input-symmetric (BMS) channels under which the Belief Propagation (BP) operator is a strict contraction (albeit non-multiplicative). A key ingredient of our proof is a strengthening of the classical stringy tree lemma of (Evans-Kenyon-Peres-Schulman'00). Our result simultaneously closes the following 6 conjectures in the literature: 1) independence of robust reconstruction accuracy to leaf noise in broadcasting on trees (Mossel-Neeman-Sly'16); 2) uselessness of global information for a labeled 2-community stochastic block model, or 2-SBM (Kanade-Mossel-Schramm'16); 3) optimality of local algorithms for 2-SBM under noisy side information (Mossel-Xu'16); 4) uniqueness of BP fixed point in broadcasting on trees in the Gaussian (large degree) limit (ibid); 5) boundary irrelevance in broadcasting on trees (Abbe-Cornacchia-Gu-Polyanskiy'21); 6) characterization of entropy (and mutual information) of community labels given the graph in 2-SBM (ibid).

cs.IT↗

arXiv2023

Empirical Bayes via ERM and Rademacher complexities: the Poisson model

We consider the problem of empirical Bayes estimation for (multivariate) Poisson means. Existing solutions that have been shown theoretically optimal for minimizing the regret (excess risk over the Bayesian oracle that knows the prior) have several shortcomings. For example, the classical Robbins estimator does not retain the monotonicity property of the Bayes estimator and performs poorly under moderate sample size. Estimators based on the minimum distance and non-parametric maximum likelihood (NPMLE) methods correct these issues, but are computationally expensive with complexity growing exponentially with dimension. Extending the approach of Barbehenn and Zhao (2022), in this work we construct monotone estimators based on empirical risk minimization (ERM) that retain similar theoretical guarantees and can be computed much more efficiently. Adapting the idea of offset Rademacher complexity Liang et al. (2015) to the non-standard loss and function class in empirical Bayes, we show that the shape-constrained ERM estimator attains the minimax regret within constant factors in one dimension and within logarithmic factors in multiple dimensions.

math.ST↗

arXiv2023

Coded Orthogonal Modulation for the Multi-Antenna Multiple-Access Channel

This study focuses on (traditional and unsourced) multiple-access communication over a single transmit and multiple ($M$) receive antennas. We assume full or partial channel state information (CSI) at the receiver. It is known that to fully achieve the fundamental limits (even asymptotically) the decoder needs to jointly estimate all user codewords, doing which directly is computationally infeasible. We propose a low-complexity solution, termed coded orthogonal modulation multiple-access (COMMA), in which users first encode their messages via a long (multi-user interference aware) outer code operating over a $q$-ary alphabet. These symbols are modulated onto $q$ orthogonal waveforms. At the decoder a multiple-measurement vector approximate message passing (MMV-AMP) algorithm estimates several candidates (out of $q$) for each user, with the remaining uncertainty resolved by the single-user outer decoders. Numerically, we show that COMMA outperforms a standard solution based on linear multiuser detection (MUD) with Gaussian signaling. Theoretically, we derive bounds and scaling laws for $M$, the number of users $K_a$, SNR, and $q$, allowing to quantify the trade-off between receive antennas and spectral efficiency. The orthogonal signaling scheme is applicable to unsourced random access and, with chirp sequences as basis, allows for low-complexity fast Fourier transform (FFT) based receivers that are resilient to frequency and timing offsets.

cs.IT↗

arXiv2023

Comparing Poisson and Gaussian channels (extended)

Consider a pair of input distributions which after passing through a Poisson channel become $ε$-close in total variation. We show that they must necessarily then be $ε^{0.5+o(1)}$-close after passing through a Gaussian channel as well. In the opposite direction, we show that distributions inducing $ε$-close outputs over the Gaussian channel must induce $ε^{1+o(1)}$-close outputs over the Poisson. This quantifies a well-known intuition that ''smoothing'' induced by Poissonization and Gaussian convolution are similar. As an application, we improve a recent upper bound of Han-Miao-Shen'2021 for estimating mixing distribution of a Poisson mixture in Gaussian optimal transport distance from $n^{-0.1 + o(1)}$ to $n^{-0.25 + o(1)}$.

cs.IT↗

arXiv2023

Uniqueness of BP fixed point for the Potts model and applications to community detection

In the study of sparse stochastic block models (SBMs) one often needs to analyze a distributional recursion, known as the belief propagation (BP) recursion. Uniqueness of the fixed point of this recursion implies several results about the SBM, including optimal recovery algorithms for SBM (Mossel et al. (2016)) and SBM with side information (Mossel and Xu (2016)), and a formula for SBM mutual information (Abbe et al. (2021)). The 2-community case corresponds to an Ising model, for which Yu and Polyanskiy (2022) established uniqueness for all cases. In this paper we analyze the $q$-ary Potts model, i.e., broadcasting of $q$-ary spins on a Galton-Watson tree with expected offspring degree $d$ through Potts channels with second-largest eigenvalue $λ$. We allow the intermediate vertices to be observed through noisy channels (side information). We prove that BP uniqueness holds with and without side information when $dλ^2 \ge 1 + C \max\{λ, q^{-1}\}\log q$ for some absolute constant $C>0$ independent of $q,λ,d$. For large $q$ and $λ= o(1/\log q)$, this is asymptotically achieving the Kesten-Stigum threshold $dλ^2=1$. These results imply mutual information formulas and optimal recovery algorithms for the $q$-community SBM in the corresponding ranges. For $q\ge 4$, Sly (2011); Mossel et al. (2022) showed that there exist choices of $q,λ,d$ below Kesten-Stigum (i.e. $dλ^2 < 1$) but reconstruction is possible. Somewhat surprisingly, we show that in such regimes BP uniqueness does not hold at least in the presence of weak side information. Our technical tool is a theory of $q$-ary symmetric channels, that we initiate here, generalizing the classical and widely-utilized information-theoretic characterization of BMS (binary memoryless symmetric) channels.

math.PR↗