arXiv subjects

Dmitriy Kunisky

Publications and source records attributed to Dmitriy Kunisky.

At least 37 records · Page 2Linked to original sources

arXiv2025

On the Structure of Bad Science Matrices

The bad science matrix problem consists in finding, among all matrices $A \in \mathbb{R}^{n \times n}$ with rows having unit $\ell^2$ norm, one that maximizes $β(A) = \frac{1}{2^n} \sum_{x \in \{-1, 1\}^n} \|Ax\|_\infty$. Our main contribution is an explicit construction of an $n \times n$ matrix $A$ showing that $β(A) \geq \sqrt{\log_2(n+1)}$, which is only 18% smaller than the asymptotic rate. We prove that every entry of any optimal matrix is a square root of a rational number, and we find provably optimal matrices for $n \leq 4$.

math.FA↗

arXiv2024

Low coordinate degree algorithms II: Categorical signals and generalized stochastic block models

We study when low coordinate degree functions (LCDF) -- linear combinations of functions depending on small subsets of entries of a vector -- can test for the presence of categorical structure, including community structure and generalizations thereof, in high-dimensional data. This complements the first paper of this series, which studied the power of LCDF in testing for continuous structure like real-valued signals perturbed by additive noise. We apply the tools developed there to a general form of stochastic block model (SBM), where a population is assigned random labels and every $p$-tuple of the population generates an observation according to an arbitrary probability measure associated to the $p$ labels of its members. We show that the performance of LCDF admits a unified analysis for this class of models. As applications, we prove tight lower bounds against LCDF (and therefore also against low degree polynomials) for nearly arbitrary graph and regular hypergraph SBMs, always matching suitable generalizations of the Kesten-Stigum threshold. We also prove tight lower bounds for group synchronization and abelian group sumset problems under the "truth-or-Haar" noise model, and use our technical results to give an improved analysis of Gaussian multi-frequency group synchronization. In most of these models, for some parameter settings our lower bounds give new evidence for conjectural statistical-to-computational gaps. Finally, interpreting some of our findings, we propose a precise analogy between categorical and continuous signals: a general SBM as above behaves, in terms of the tradeoff between subexponential runtime cost of testing algorithms and the signal strength needed for a testing algorithm to succeed, like a spiked $p_*$-tensor model of a certain order $p_*$ that may be computed from the parameters of the SBM.

math.ST↗

arXiv2024

Statistical inference of a ranked community in a directed graph

We study the problem of detecting or recovering a planted ranked subgraph from a directed graph, an analog for directed graphs of the well-studied planted dense subgraph model. We suppose that, among a set of $n$ items, there is a subset $S$ of $k$ items having a latent ranking in the form of a permutation $π$ of $S$, and that we observe a fraction $p$ of pairwise orderings between elements of $\{1, \dots, n\}$ which agree with $π$ with probability $\frac{1}{2} + q$ between elements of $S$ and otherwise are uniformly random. Unlike in the planted dense subgraph and planted clique problems where the community $S$ is distinguished by its unusual density of edges, here the community is only distinguished by the unusual consistency of its pairwise orderings. We establish computational and statistical thresholds for both detecting and recovering such a ranked community. In the log-density setting where $k$, $p$, and $q$ all scale as powers of $n$, we establish the exact thresholds in the associated exponents at which detection and recovery become statistically and computationally feasible. These regimes include a rich variety of behaviors, exhibiting both statistical-computational and detection-recovery gaps. We also give finer-grained results for two extreme cases: (1) $p = 1$, $k = n$, and $q$ small, where a full tournament is observed that is weakly correlated with a global ranking, and (2) $p = 1$, $q = \frac{1}{2}$, and $k$ small, where a small "ordered clique" (totally ordered directed subgraph) is planted in a random tournament.

math.ST↗

arXiv2024

Inference of rankings planted in random tournaments

We consider the problem of inferring an unknown ranking of $n$ items from a random tournament on $n$ vertices whose edge directions are correlated with the ranking. We establish, in terms of the strength of these correlations, the computational and statistical thresholds for detection (deciding whether an observed tournament is purely random or drawn correlated with a hidden ranking) and recovery (estimating the hidden ranking with small error in Spearman's footrule or Kendall's tau metric on permutations). Notably, we find that this problem provides a new instance of a detection-recovery gap: solving the detection problem requires much weaker correlations than solving the recovery problem. In establishing these thresholds, we also identify simple algorithms for detection (thresholding a degree 2 polynomial) and recovery (outputting a ranking by the number of "wins" of a tournament vertex, i.e., the out-degree) that achieve optimal performance up to constants in the correlation strength. For detection, we find that the above low-degree polynomial algorithm is superior to a natural spectral algorithm. We also find that, whenever it is possible to achieve strong recovery (i.e., to estimate with vanishing error in the above metrics) of the hidden ranking, then the above "Ranking By Wins" algorithm not only does so, but also outputs a close approximation of the maximum likelihood estimator, a task that is NP-hard in the worst case.

math.ST↗

arXiv2024

Average-Case Matrix Discrepancy: Asymptotics and Online Algorithms

We study the operator norm discrepancy of i.i.d. random matrices, initiating the matrix-valued analog of a long line of work on the $\ell^{\infty}$ norm discrepancy of i.i.d. random vectors. First, using repurposed results on vector discrepancy and new first moment method calculations, we give upper and lower bounds on the discrepancy of random matrices. We treat i.i.d. matrices drawn from the Gaussian orthogonal ensemble (GOE) and low-rank Gaussian Wishart distributions. In both cases, for what turns out to be the "critical" number of $Θ(n^2)$ matrices of dimension $n \times n$, we identify the discrepancy up to constant factors. Second, we give a new analysis of the matrix hyperbolic cosine algorithm of Zouzias (2011), a matrix version of an online vector discrepancy algorithm of Spencer (1977) studied for average-case inputs by Bansal and Spencer (2020), for the case of i.i.d. random matrix inputs. We both give a general analysis and extract concrete bounds on the discrepancy achieved by this algorithm for matrices with independent entries (including GOE matrices) and Gaussian Wishart matrices.

cs.DS↗

arXiv2024

Fitting an ellipsoid to random points: predictions using the replica method

We consider the problem of fitting a centered ellipsoid to $n$ standard Gaussian random vectors in $\mathbb{R}^d$, as $n, d \to \infty$ with $n/d^2 \to α> 0$. It has been conjectured that this problem is, with high probability, satisfiable (SAT; that is, there exists an ellipsoid passing through all $n$ points) for $α< 1/4$, and unsatisfiable (UNSAT) for $α> 1/4$. In this work we give a precise analytical argument, based on the non-rigorous replica method of statistical physics, that indeed predicts a SAT/UNSAT transition at $α= 1/4$, as well as the shape of a typical fitting ellipsoid in the SAT phase (i.e., the lengths of its principal axes). Besides the replica method, our main tool is the dilute limit of extensive-rank "HCIZ integrals" of random matrix theory. We further study different explicit algorithmic constructions of the matrix characterizing the ellipsoid. In particular, we show that a procedure based on minimizing its nuclear norm yields a solution in the whole SAT phase. Finally, we characterize the SAT/UNSAT transition for ellipsoid fitting of a large class of rotationally-invariant random vectors. Our work suggests mathematically rigorous ways to analyze fitting ellipsoids to random vectors, which is the topic of a companion work.

cond-mat.dis-nn↗

arXiv2024

Computational lower bounds for multi-frequency group synchronization

We consider a group synchronization problem with multiple frequencies which involves observing pairwise relative measurements of group elements on multiple frequency channels, corrupted by Gaussian noise. We study the computational phase transition in the problem of detecting whether a structured signal is present in such observations by analyzing low-degree polynomial algorithms. We show that, assuming the low-degree conjecture, in synchronization models over arbitrary finite groups as well as over the circle group $SO(2)$, a simple spectral algorithm is optimal among algorithms of runtime $\exp(\tildeΩ(n^{1/3}))$ for detection from an observation including a constant number of frequencies. Combined with an upper bound for the statistical threshold shown in Perry et al., our results indicate the presence of a statistical-to-computational gap in such models with a sufficiently large number of frequencies.

math.ST↗

arXiv2024

Tensor cumulants for statistical inference on invariant distributions

Many problems in high-dimensional statistics appear to have a statistical-computational gap: a range of values of the signal-to-noise ratio where inference is information-theoretically possible, but (conjecturally) computationally intractable. A canonical such problem is Tensor PCA, where we observe a tensor $Y$ consisting of a rank-one signal plus Gaussian noise. Multiple lines of work suggest that Tensor PCA becomes computationally hard at a critical value of the signal's magnitude. In particular, below this transition, no low-degree polynomial algorithm can detect the signal with high probability; conversely, various spectral algorithms are known to succeed above this transition. We unify and extend this work by considering tensor networks, orthogonally invariant polynomials where multiple copies of $Y$ are "contracted" to produce scalars, vectors, matrices, or other tensors. We define a new set of objects, tensor cumulants, which provide an explicit, near-orthogonal basis for invariant polynomials of a given degree. This basis lets us unify and strengthen previous results on low-degree hardness, giving a combinatorial explanation of the hardness transition and of a continuum of subexponential-time algorithms that work below it, and proving tight lower bounds against low-degree polynomials for recovering rather than just detecting the signal. It also lets us analyze a new problem of distinguishing between different tensor ensembles, such as Wigner and Wishart tensors, establishing a sharp computational threshold and giving evidence of a new statistical-computational gap in the Central Limit Theorem for random tensors. Finally, we believe these cumulants are valuable mathematical objects in their own right: they generalize the free cumulants of free probability theory from matrices to tensors, and share many of their properties, including additivity under additive free convolution.

math.ST↗

arXiv2024

A degree 4 sum-of-squares lower bound for the clique number of the Paley graph

We prove that the degree 4 sum-of-squares (SOS) relaxation of the clique number of the Paley graph on a prime number $p$ of vertices has value at least $Ω(p^{1/3})$. This is in contrast to the widely believed conjecture that the actual clique number of the Paley graph is $O(\mathrm{polylog}(p))$. Our result may be viewed as a derandomization of that of Deshpande and Montanari (2015), who showed the same lower bound (up to $\mathrm{polylog}(p)$ terms) with high probability for the Erdős-Rényi random graph on $p$ vertices, whose clique number is with high probability $O(\log(p))$. We also show that our lower bound is optimal for the Feige-Krauthgamer construction of pseudomoments, derandomizing an argument of Kelner. Finally, we present numerical experiments indicating that the value of the degree 4 SOS relaxation of the Paley graph may scale as $O(p^{1/2 - ε})$ for some $ε> 0$, and give a matrix norm calculation indicating that the pseudocalibration proof strategy for SOS lower bounds for random graphs will not immediately transfer to the Paley graph. Taken together, our results suggest that degree 4 SOS may break the "$\sqrt{p}$ barrier" for upper bounds on the clique number of Paley graphs, but prove that it can at best improve the exponent from $1/2$ to $1/3$.

cs.DS↗

arXiv2024

Computational hardness of detecting graph lifts and certifying lift-monotone properties of random regular graphs

We introduce a new conjecture on the computational hardness of detecting random lifts of graphs: we claim that there is no polynomial-time algorithm that can distinguish between a large random $d$-regular graph and a large random lift of a Ramanujan $d$-regular base graph (provided that the lift is corrupted by a small amount of extra noise), and likewise for bipartite random graphs and lifts of bipartite Ramanujan graphs. We give evidence for this conjecture by proving lower bounds against the local statistics hierarchy of hypothesis testing semidefinite programs. We then explore the consequences of this conjecture for the hardness of certifying bounds on numerous functions of random regular graphs, expanding on a direction initiated by Bandeira, Banks, Kunisky, Moore, and Wein (2021). Conditional on this conjecture, we show that no polynomial-time algorithm can certify tight bounds on the maximum cut of random 3- or 4-regular graphs, the maximum independent set of random 3- or 4-regular graphs, or the chromatic number of random 7-regular graphs. We show similar gaps asymptotically for large degree for the maximum independent set and for any degree for the minimum dominating set, finding that naive spectral and combinatorial bounds are optimal among all polynomial-time certificates. Likewise, for small-set vertex and edge expansion in the limit of very small sets, we show that the spectral bounds of Kahale (1995) are optimal among all polynomial-time certificates.

cs.CC↗

arXiv2024

Low coordinate degree algorithms I: Universality of computational thresholds for hypothesis testing

We study when low coordinate degree functions (LCDF) -- linear combinations of functions depending on small subsets of entries of a vector -- can hypothesis test between high-dimensional probability measures. These functions are a generalization, proposed in Hopkins' 2018 thesis but seldom studied since, of low degree polynomials (LDP), a class widely used in recent literature as a proxy for all efficient algorithms for tasks in statistics and optimization. Instead of the orthogonal polynomial decompositions used in LDP calculations, our analysis of LCDF is based on the Efron-Stein or ANOVA decomposition, making it much more broadly applicable. By way of illustration, we prove channel universality for the success of LCDF in testing for the presence of sufficiently "dilute" random signals through noisy channels: the efficacy of LCDF depends on the channel only through the scalar Fisher information for a class of channels including nearly arbitrary additive i.i.d. noise and nearly arbitrary exponential families. As applications, we extend lower bounds against LDP for spiked matrix and tensor models under additive Gaussian noise to lower bounds against LCDF under general noisy channels. We also give a simple and unified treatment of the effect of censoring models by erasing observations at random and of quantizing models by taking the sign of the observations. These results are the first computational lower bounds against any large class of algorithms for all of these models when the channel is not one of a few special cases, and thereby give the first substantial evidence for the universality of several statistical-to-computational gaps.

math.ST↗

arXiv2023

Optimality of Glauber dynamics for general-purpose Ising model sampling and free energy approximation

Recently, Eldan, Koehler, and Zeitouni (2020) showed that Glauber dynamics mixes rapidly for general Ising models so long as the difference between the largest and smallest eigenvalues of the coupling matrix is at most $1 - ε$ for any fixed $ε> 0$. We give evidence that Glauber dynamics is in fact optimal for this "general-purpose sampling" task. Namely, we give an average-case reduction from hypothesis testing in a Wishart negatively-spiked matrix model to approximately sampling from the Gibbs measure of a general Ising model for which the difference between the largest and smallest eigenvalues of the coupling matrix is at most $1 + ε$ for any fixed $ε> 0$. Combined with results of Bandeira, Kunisky, and Wein (2019) that analyze low-degree polynomial algorithms to give evidence for the hardness of the former spiked matrix problem, our results in turn give evidence for the hardness of general-purpose sampling improving on Glauber dynamics. We also give a similar reduction to approximating the free energy of general Ising models, and again infer evidence that simulated annealing algorithms based on Glauber dynamics are optimal in the general-purpose setting.

cs.DS↗

arXiv2023

Spectral pseudorandomness and the road to improved clique number bounds for Paley graphs

We study subgraphs of Paley graphs of prime order $p$ induced on the sets of vertices extending a given independent set of size $a$ to a larger independent set. Using a sufficient condition proved in the author's recent companion work, we show that a family of character sum estimates would imply that, as $p \to \infty$, the empirical spectral distributions of the adjacency matrices of any sequence of such subgraphs have the same weak limit (after rescaling) as those of subgraphs induced on a random set including each vertex independently with probability $2^{-a}$, namely, a Kesten-McKay law with parameter $2^a$. We prove the necessary estimates for $a = 1$, obtaining in the process an alternate proof of a character sum equidistribution result of Xi (2022), and provide numerical evidence for this weak convergence for $a \geq 2$. We also conjecture that the minimum eigenvalue of any such sequence converges (after rescaling) to the left edge of the corresponding Kesten-McKay law, and provide numerical evidence for this convergence. Finally, we show that, once $a \geq 3$, this (conjectural) convergence of the minimum eigenvalue would imply bounds on the clique number of the Paley graph improving on the current state of the art due to Hanson and Petridis (2021), and that this convergence for all $a \geq 1$ would imply that the clique number is $o(\sqrt{p})$.

math.CO↗

arXiv2023

Generic MANOVA limit theorems for products of projections

We study the convergence of the empirical spectral distribution of $\mathbf{A} \mathbf{B} \mathbf{A}$ for $N \times N$ orthogonal projection matrices $\mathbf{A}$ and $\mathbf{B}$, where $\frac{1}{N}\mathrm{Tr}(\mathbf{A})$ and $\frac{1}{N}\mathrm{Tr}(\mathbf{B})$ converge as $N \to \infty$, to Wachter's MANOVA law. Using free probability, we show mild sufficient conditions for convergence in moments and in probability, and use this to prove a conjecture of Haikin, Zamir, and Gavish (2017) on random subsets of unit-norm tight frames. This result generalizes previous ones of Farrell (2011) and Magsino, Mixon, and Parshall (2021). We also derive an explicit recursion for the difference between the empirical moments $\frac{1}{N}\mathrm{Tr}((\mathbf{A} \mathbf{B} \mathbf{A})^k)$ and the limiting MANOVA moments, and use this to prove a sufficient condition for convergence in probability of the largest eigenvalue of $\mathbf{A} \mathbf{B} \mathbf{A}$ to the right edge of the support of the limiting law in the special case where that law belongs to the Kesten-McKay family. As an application, we give a new proof of convergence in probability of the largest eigenvalue when $\mathbf{B}$ is unitarily invariant; equivalently, this determines the limiting operator norm of a rectangular submatrix of size $\frac{1}{2}N \times αN$ of a Haar-distributed $N \times N$ unitary matrix for any $α\in (0, 1)$. Unlike previous proofs, we use only moment calculations and non-asymptotic bounds on the unitary Weingarten function, which we believe should pave the way to analyzing the largest eigenvalue for products of random projections having other distributions.

math.PR↗

arXiv2022

On the concentration of Gaussian Cayley matrices

Given a finite group, we study the Gaussian series of the matrices in the image of its left regular representation. We propose such random matrices as a benchmark for improvements to the noncommutative Khintchine inequality, and we highlight an application to the matrix Spencer conjecture.

math.FA↗

arXiv2022

Subexponential-Time Algorithms for Sparse PCA

We study the computational cost of recovering a unit-norm sparse principal component $x \in \mathbb{R}^n$ planted in a random matrix, in either the Wigner or Wishart spiked model (observing either $W + λxx^\top$ with $W$ drawn from the Gaussian orthogonal ensemble, or $N$ independent samples from $\mathcal{N}(0, I_n + βxx^\top)$, respectively). Prior work has shown that when the signal-to-noise ratio ($λ$ or $β\sqrt{N/n}$, respectively) is a small constant and the fraction of nonzero entries in the planted vector is $\|x\|_0 / n = ρ$, it is possible to recover $x$ in polynomial time if $ρ\lesssim 1/\sqrt{n}$. While it is possible to recover $x$ in exponential time under the weaker condition $ρ\ll 1$, it is believed that polynomial-time recovery is impossible unless $ρ\lesssim 1/\sqrt{n}$. We investigate the precise amount of time required for recovery in the "possible but hard" regime $1/\sqrt{n} \ll ρ\ll 1$ by exploring the power of subexponential-time algorithms, i.e., algorithms running in time $\exp(n^δ)$ for some constant $δ\in (0,1)$. For any $1/\sqrt{n} \ll ρ\ll 1$, we give a recovery algorithm with runtime roughly $\exp(ρ^2 n)$, demonstrating a smooth tradeoff between sparsity and runtime. Our family of algorithms interpolates smoothly between two existing algorithms: the polynomial-time diagonal thresholding algorithm and the $\exp(ρn)$-time exhaustive search algorithm. Furthermore, by analyzing the low-degree likelihood ratio, we give rigorous evidence suggesting that the tradeoff achieved by our algorithms is optimal.

math.ST↗

arXiv2022

Linear Programming and Community Detection

The problem of community detection with two equal-sized communities is closely related to the minimum graph bisection problem over certain random graph models. In the stochastic block model distribution over networks with community structure, a well-known semidefinite programming (SDP) relaxation of the minimum bisection problem recovers the underlying communities whenever possible. Motivated by their superior scalability, we study the theoretical performance of linear programming (LP) relaxations of the minimum bisection problem for the same random models. We show that unlike the SDP relaxation that undergoes a phase transition in the logarithmic average-degree regime, the LP relaxation exhibits a transition from recovery to non-recovery in the linear average-degree regime. We show that in the logarithmic average-degree regime, the LP relaxation fails in recovering the planted bisection with high probability.

math.OC↗

arXiv2022

The spectrum of the Grigoriev-Laurent pseudomoments

Grigoriev (2001) and Laurent (2003) independently showed that the sum-of-squares hierarchy of semidefinite programs does not exactly represent the hypercube $\{\pm 1\}^n$ until degree at least $n$ of the hierarchy. Laurent also observed that the pseudomoment matrices her proof constructs appear to have surprisingly simple and recursively structured spectra as $n$ increases. While several new proofs of the Grigoriev-Laurent lower bound have since appeared, Laurent's observations have remained unproved. We give yet another, representation-theoretic proof of the lower bound, which also yields exact formulae for the eigenvalues of the Grigoriev-Laurent pseudomoments. Using these, we prove and elaborate on Laurent's observations. Our arguments have two features that may be of independent interest. First, we show that the Grigoriev-Laurent pseudomoments are a special case of a Gram matrix construction of pseudomoments proposed by Bandeira and Kunisky (2020). Second, we find a new realization of the irreducible representations of the symmetric group corresponding to Young diagrams with two rows, as spaces of multivariate polynomials that are multiharmonic with respect to an equilateral simplex.

math.OC↗