arXiv subjects
Publications and source records attributed to Sidhanth Mohanty.
We prove that the simulated annealing algorithm efficiently samples from the Sherrington-Kirkpatrick model at inverse temperature $β< 1$ to an arbitrary inverse polynomial accuracy in total variation.
We study the problem of constructing explicit codes whose rate and distance match the Gilbert-Varshamov bound in the low-rate, high-distance regime. In 2017, Ta-Shma gave an explicit family of codes where every pair of codewords has relative distance $\frac{1-\varepsilon}{2}$, with rate $Ω(\varepsilon^{2+o(1)})$, matching the Gilbert-Varshamov bound up to a factor of $\varepsilon^{o(1)}$. Ta-Shma's construction was based on starting with a good code and amplifying its bias with walks arising from the $s$-wide-replacement product. In this work, we give a simpler almost-optimal construction, based on what we call free expander walks: ordinary expander walks where each step is taken on a distinct expander from a carefully chosen sequence. This sequence of expanders is derived from the construction of near-$X$-Ramanujan graphs due to O'Donnell and Wu. We additionally discuss some additional applications of near-$X$-Ramanujan graphs to "on average" lossless expansion and rotating expanders.
Over the past decade, the low-degree heuristic has been used to estimate the algorithmic thresholds for a wide range of average-case planted vs null distinguishing problems. Such results rely on the hypothesis that if the low-degree moments of the planted and null distributions are sufficiently close, then no efficient (noise-tolerant) algorithm can distinguish between them. This hypothesis is appealing due to the simplicity of calculating the low-degree likelihood ratio (LDLR) -- a quantity that measures the similarity between low-degree moments. However, despite sustained interest in the area, it remains unclear whether low-degree indistinguishability actually rules out any interesting class of algorithms. In this work, we initiate the study and develop technical tools for translating LDLR upper bounds to rigorous lower bounds against concrete algorithms. As a consequence, we prove: for any permutation-invariant distribution $\mathsf{P}$, 1. If $\mathsf{P}$ is over $\{0,1\}^n$ and is low-degree indistinguishable from $U = \mathrm{Unif}(\{0,1\}^n)$, then a noisy version of $\mathsf{P}$ is statistically indistinguishable from $U$. 2. If $\mathsf{P}$ is over $\mathbb{R}^n$ and is low-degree indistinguishable from the standard Gaussian ${N}(0, 1)^n$, then no statistic based on symmetric polynomials of degree at most $O(\log n/\log \log n)$ can distinguish between a noisy version of $\mathsf{P}$ from ${N}(0, 1)^n$. 3. If $\mathsf{P}$ is over $\mathbb{R}^{n\times n}$ and is low-degree indistinguishable from ${N}(0,1)^{n\times n}$, then no constant-sized subgraph statistic can distinguish between a noisy version of $\mathsf{P}$ and ${N}(0, 1)^{n\times n}$.
We introduce a new technique to prove bounds for the spectral radius of a random matrix, based on using Jensen's formula to establish the zerofreeness of the associated characteristic polynomial in a region of the complex plane. Our techniques are entirely non-asymptotic, and we instantiate it in three settings: (i) The spectral radius of non-asymptotic Girko matrices -- these are asymmetric matrices $\mathbf{M} \in \mathbb{C}^{n \times n}$ whose entries are independent and satisfy $\mathbb{E} \mathbf{M}_{ij} = 0$ and $\mathbb{E} |\mathbf{M}_{ij}^2| \le \frac{1}{n}$. (ii) The spectral radius of non-asymptotic Wigner matrices -- these are symmetric matrices $\mathbf{M} \in \mathbb{C}^{n \times n}$ whose entries above the diagonal are independent and satisfy $\mathbb{E} \mathbf{M}_{ij} = 0$, $\mathbb{E} |\mathbf{M}_{ij}^2| \le \frac{1}{n}$, and $\mathbb{E} |\mathbf{M}_{ij}^4| \le \frac{1}{n}$. (iii) The second eigenvalue of the adjacency matrix of a random $d$-regular graph on $n$ vertices, as drawn from the configuration model. In all three settings, we obtain constant-probability eigenvalue bounds that are tight up to a constant. Applied to specific random matrix ensembles, we recover classic bounds for Wigner matrices, as well as results of Bordenave--Chafaï--García-Zelada, Bordenave--Lelarge--Massoulié, and Friedman, up to constants.
A classic result in graph theory, due to Batson, Spielman, and Srivastava (STOC 2009) shows that every graph admits a $(1 \pm \varepsilon)$ cut (or spectral) sparsifier which preserves only $O(n / \varepsilon^2)$ reweighted edges. However, when applying this result to \emph{Cayley graphs}, the resulting sparsifier is no longer necessarily a Cayley graph -- it can be an arbitrary subset of edges. Thus, a recent line of inquiry, and one which has only seen minor progress, asks: for any group $G$, do all Cayley graphs over the group $G$ admit sparsifiers which preserve only $\mathrm{polylog}(|G|)/\varepsilon^2$ many re-weighted generators? As our primary contribution, we answer this question in the affirmative, presenting a proof of the existence of such Cayley graph spectral sparsifiers, along with an efficient algorithm for finding them. Our algorithm even extends to \emph{directed} Cayley graphs, if we instead ask only for cut sparsification instead of spectral sparsification. We additionally study the sparsification of linear equations over non-abelian groups. In contrast to the abelian case, we show that for non-abelian valued equations, super-polynomially many linear equations must be preserved in order to approximately preserve the number of satisfied equations for any input. Together with our Cayley graph sparsification result, this provides a formal separation between Cayley graph sparsification and sparsifying linear equations.
Many natural Markov chains fail to mix to their stationary distribution in polynomially many steps. Often, this slow mixing is inevitable since it is computationally intractable to sample from their stationary measure. Nevertheless, Markov chains can be shown to always converge quickly to measures that are locally stationary, i.e., measures that don't change over a small number of steps. These locally stationary measures are analogous to local minima in continuous optimization, while stationary measures correspond to global minima. While locally stationary measures can be statistically far from stationary measures, do they enjoy provable theoretical guarantees that have algorithmic implications? We study this question in this work and demonstrate three algorithmic applications of locally stationary measures: 1. We show that Glauber dynamics on the hardcore model can be used to find independent sets of size $Ω\left(\frac{\log d}{d} \cdot n\right)$ in triangle-free graphs of degree at most $d$. 2. Let $W$ be a symmetric real matrix with bounded spectral diameter and $v$ be a unit vector. Given the matrix $M = λvv^\top + W$ with a planted rank-one spike along vector $v$, for sufficiently large constant $λ$, Glauber dynamics on the Ising model defined by $M$ samples vectors $x \in \{\pm 1\}^n$ that have constant correlation with the vector $v$. 3. Let $M = A_{\mathbf{G}} - \frac{d}{n}\mathbf{1}\mathbf{1}^\top$ be a centered version of the adjacency matrix where the graph $\mathbf{G}$ is drawn from a sparse 2-community stochastic block model. We show that for sufficiently large constant signal-to-noise ratio, Glauber dynamics on the Ising model defined by $M$ samples vectors $x \in \{\pm 1\}^n$ that have constant correlation with the hidden community vector $\mathbfσ$.
We give the first construction of explicit constant-degree lossless vertex expanders. Specifically, for any $\varepsilon > 0$ and sufficiently large $d$, we give an explicit construction of an infinite family of $d$-regular graphs where every small set $S$ of vertices has $(1-\varepsilon)d|S|$ neighbors (which implies $(1-2\varepsilon)d|S|$ unique-neighbors). Our results also extend naturally to construct biregular bipartite graphs of any constant imbalance, where small sets on each side have strong expansion guarantees. The graphs we construct admit a free group action, and hence realize new families of quantum LDPC codes of Lin and M. Hsieh with a linear time decoding algorithm. Our construction is based on taking an appropriate product of a constant-sized lossless expander with a base graph constructed from Ramanujan Cayley cubical complexes.
Given a $k$-uniform hypergraph $H$ on $n$ vertices, an even cover in $H$ is a collection of hyperedges that touch each vertex an even number of times. Even covers are a generalization of cycles in graphs and are equivalent to linearly dependent subsets of a system of linear equations modulo $2$. As a result, they arise naturally in the context of well-studied questions in coding theory and refuting unsatisfiable $k$-SAT formulas. Analogous to the irregular Moore bound of Alon, Hoory, and Linial (2002), in 2008, Feige conjectured an extremal trade-off between the number of hyperedges and the length of the smallest even cover in a $k$-uniform hypergraph. This conjecture was recently settled up to a multiplicative logarithmic factor in the number of hyperedges (Guruswami, Kothari, and Manohar 2022 and Hsieh, Kothari, and Mohanty 2023). These works introduce the new technique that relates hypergraph even covers to cycles in the associated Kikuchi graphs. Their analysis of these Kikuchi graphs, especially for odd $k$, is rather involved and relies on matrix concentration inequalities. In this work, we give a simple and purely combinatorial argument that recovers the best-known bound for Feige's conjecture for even $k$. We also introduce a novel variant of a Kikuchi graph which together with this argument improves the logarithmic factor in the best-known bounds for odd $k$. As an application of our ideas, we also give a purely combinatorial proof of the improved lower bounds (Alrabiah, Guruswami, Kothari and Manohar, 2023) on 3-query binary linear locally decodable codes.
There has been a recent surge of powerful tools to show rapid mixing of Markov chains, via functional inequalities such as Poincaré inequalities. In many situations, Markov chains fail to mix rapidly from a worst-case initialization, yet are expected to approximately sample from a random initialization. For example, this occurs if the target distribution has metastable states, small clusters accounting for a vanishing fraction of the mass that are essentially disconnected from the bulk of the measure. Under such conditions, a Poincaré inequality cannot hold, necessitating new tools to prove sampling guarantees. We develop a framework to analyze simulated annealing, based on establishing so-called weak Poincaré inequalities. These inequalities imply mixing from a suitably warm start, and simulated annealing provides a way to chain such warm starts together into a sampling algorithm. We further identify a local-to-global principle to prove weak Poincaré inequalities, mirroring the spectral independence and localization schemes frameworks for analyzing mixing times of Markov chains. As our main application, we prove that simulated annealing samples from the Gibbs measure of a spherical spin glass for inverse temperatures up to a natural threshold, matching recent algorithms based on algorithmic stochastic localization. This provides the first Markov chain sampling guarantee that holds beyond the uniqueness threshold for spherical spin glasses, where mixing from a worst-case initialization is provably slow due to the presence of metastable states. As an ingredient in our proof, we prove bounds on the operator norm of the covariance matrix of spherical spin glasses in the full replica-symmetric regime. Additionally, we resolve a question related to sampling using data-based initializations.
We construct the first explicit two-sided vertex expanders that bypass the spectral barrier. Previously, the strongest known explicit vertex expanders were given by $d$-regular Ramanujan graphs, whose spectral properties imply that every small subset of vertices $S$ has at least $0.5d|S|$ distinct neighbors. However, it is possible to construct Ramanujan graphs containing a small set $S$ with no more than $0.5d|S|$ neighbors. In fact, no explicit construction was known to break the $0.5 d$-barrier. In this work, we give an explicit construction of an infinite family of $d$-regular graphs (for large enough $d$) where every small set expands by a factor of $\approx 0.6d$. More generally, for large enough $d_1,d_2$, we give an infinite family of $(d_1,d_2)$-biregular graphs where small sets on the left expand by a factor of $\approx 0.6d_1$, and small sets on the right expand by a factor of $\approx 0.6d_2$. In fact, our construction satisfies an even stronger property: small sets on the left and right have unique-neighbor expansion $0.6d_1$ and $0.6d_2$ respectively. Our construction follows the tripartite line product framework of Hsieh, McKenzie, Mohanty & Paredes, and instantiates it using the face-vertex incidence of the $4$-dimensional Ramanujan clique complex as its base component. As a key part of our analysis, we derive new bounds on the triangle density of small sets in the Ramanujan clique complex.
Motivated by the community detection problem in Bayesian inference, as well as the recent explosion of interest in spin glasses from statistical physics, we study the classical Glauber dynamics for sampling from Ising models with sparse random interactions. It is now well-known that when the interaction matrix has spectral diameter less than $1$, Glauber dynamics mixes in $O(n\log n)$ steps. Unfortunately, such criteria fail dramatically for interactions supported on arguably the most well-studied sparse random graph: the Erdős--Rényi random graph $G(n,d/n)$, due to the presence of almost linearly many outlier eigenvalues of unbounded magnitude. We prove that for the \emph{Viana--Bray spin glass}, where the interactions are supported on $G(n,d/n)$ and randomly assigned $\pmβ$, Glauber dynamics mixes in $n^{1+o(1)}$ time with high probability as long as $β\le O(1/\sqrt{d})$, independent of $n$. We further extend our results to random graphs drawn according to the $2$-community stochastic block model, as well as when the interactions are given by a "centered" version of the adjacency matrix. The latter setting is particularly relevant for the inference problem in community detection. Indeed, we use this to show that Glauber dynamics succeeds at recovering communities in the stochastic block model in a companion paper [LMR+24]. The primary technical ingredient in our proof is showing that with high probability, a sparse random graph can be decomposed into two parts -- a \emph{bulk} which behaves like a graph with bounded maximum degree and a well-behaved spectrum, and a \emph{near-forest} with favorable pseudorandom properties. We then use this decomposition to design a localization procedure that interpolates to simpler Ising models supported only on the near-forest, and then execute a pathwise analysis to establish a modified log-Sobolev inequality.
We study the problem of $\textit{robust community recovery}$: efficiently recovering communities in sparse stochastic block models in the presence of adversarial corruptions. In the absence of adversarial corruptions, there are efficient algorithms when the $\textit{signal-to-noise ratio}$ exceeds the $\textit{Kesten--Stigum (KS) threshold}$, widely believed to be the computational threshold for this problem. The question we study is: does the computational threshold for robust community recovery also lie at the KS threshold? We answer this question affirmatively, providing an algorithm for robust community recovery for arbitrary stochastic block models on any constant number of communities, generalizing the work of Ding, d'Orsi, Nasser & Steurer on an efficient algorithm above the KS threshold in the case of $2$-community block models. There are three main ingredients to our work: (i) The Bethe Hessian of the graph is defined as $H_G(t) \triangleq (D_G-I)t^2 - A_Gt + I$ where $D_G$ is the diagonal matrix of degrees and $A_G$ is the adjacency matrix. Empirical work suggested that the Bethe Hessian for the stochastic block model has outlier eigenvectors corresponding to the communities right above the Kesten-Stigum threshold. We formally confirm the existence of outlier eigenvalues for the Bethe Hessian, by explicitly constructing outlier eigenvectors from the community vectors. (ii) We develop an algorithm for a variant of robust PCA on sparse matrices. Specifically, an algorithm to partially recover top eigenspaces from adversarially corrupted sparse matrices under mild delocalization constraints. (iii) A rounding algorithm to turn vector assignments of vertices into a community assignment, inspired by the algorithm of Charikar \& Wirth \cite{CW04} for $2$XOR.
We study the problem of constructing explicit sparse graphs that exhibit strong vertex expansion. Our main result is the first two-sided construction of imbalanced unique-neighbor expanders, meaning bipartite graphs where small sets contained in both the left and right bipartitions exhibit unique-neighbor expansion, along with algebraic properties relevant to constructing quantum codes. Our constructions are obtained from instantiations of the tripartite line product of a large tripartite spectral expander and a sufficiently good constant-sized unique-neighbor expander, a new graph product we defined that generalizes the line product in the work of Alon and Capalbo and the routed product in the work of Asherov and Dinur. To analyze the vertex expansion of graphs arising from the tripartite line product, we develop a sharp characterization of subgraphs that can arise in bipartite spectral expanders, generalizing results of Kahale, which may be of independent interest. By picking appropriate graphs to apply our product to, we give a strongly explicit construction of an infinite family of $(d_1,d_2)$-biregular graphs $(G_n)_{n\ge 1}$ (for large enough $d_1$ and $d_2$) where all sets $S$ with fewer than a small constant fraction of vertices have $Ω(d_1\cdot |S|)$ unique-neighbors (assuming $d_1 \leq d_2$). Additionally, we can also guarantee that subsets of vertices of size up to $\exp(Ω(\sqrt{\log |V(G_n)|}))$ expand losslessly.
For every constant $d \geq 3$ and $ε> 0$, we give a deterministic $\mathrm{poly}(n)$-time algorithm that outputs a $d$-regular graph on $Θ(n)$ vertices that is $ε$-near-Ramanujan; i.e., its eigenvalues are bounded in magnitude by $2\sqrt{d-1} + ε$ (excluding the single trivial eigenvalue of~$d$).
Consider a random geometric 2-dimensional simplicial complex $X$ sampled as follows: first, sample $n$ vectors $\boldsymbol{u_1},\ldots,\boldsymbol{u_n}$ uniformly at random on $\mathbb{S}^{d-1}$; then, for each triple $i,j,k \in [n]$, add $\{i,j,k\}$ and all of its subsets to $X$ if and only if $\langle{\boldsymbol{u_i},\boldsymbol{u_j}}\rangle \ge τ, \langle{\boldsymbol{u_i},\boldsymbol{u_k}}\rangle \ge τ$, and $\langle \boldsymbol{u_j}, \boldsymbol{u_k}\rangle \ge τ$. We prove that for every $\varepsilon > 0$, there exists a choice of $d = Θ(\log n)$ and $τ= τ(\varepsilon,d)$ so that with high probability, $X$ is a high-dimensional expander of average degree $n^\varepsilon$ in which each $1$-link has spectral gap bounded away from $\frac{1}{2}$. To our knowledge, this is the first demonstration of a natural distribution over $2$-dimensional expanders of arbitrarily small polynomial average degree and spectral link expansion better than $\frac{1}{2}$. All previously known constructions are algebraic. This distribution also furnishes an example of simplicial complexes for which the trickle-down theorem is nearly tight. En route, we prove general bounds on the spectral expansion of random induced subgraphs of arbitrary vertex transitive graphs, which may be of independent interest. For example, one consequence is an almost-sharp bound on the second eigenvalue of random $n$-vertex geometric graphs on $\mathbb{S}^{d-1}$, which was previously unknown for most $n,d$ pairs.
The hypergraph Moore bound is an elegant statement that characterizes the extremal trade-off between the girth - the number of hyperedges in the smallest cycle or even cover (a subhypergraph with all degrees even) and size - the number of hyperedges in a hypergraph. For graphs (i.e., $2$-uniform hypergraphs), a bound tight up to the leading constant was proven in a classical work of Alon, Hoory and Linial [AHL02]. For hypergraphs of uniformity $k>2$, an appropriate generalization was conjectured by Feige [Fei08]. The conjecture was settled up to an additional $\log^{4k+1} n$ factor in the size in a recent work of Guruswami, Kothari and Manohar [GKM21]. Their argument relies on a connection between the existence of short even covers and the spectrum of a certain randomly signed Kikuchi matrix. Their analysis, especially for the case of odd $k$, is significantly complicated. In this work, we present a substantially simpler and shorter proof of the hypergraph Moore bound. Our key idea is the use of a new reweighted Kikuchi matrix and an edge deletion step that allows us to drop several involved steps in [GKM21]'s analysis such as combinatorial bucketing of rows of the Kikuchi matrix and the use of the Schudy-Sviridenko polynomial concentration. Our simpler proof also obtains tighter parameters: in particular, the argument gives a new proof of the classical Moore bound of [AHL02] with no loss (the proof in [GKM21] loses a $\log^3 n$ factor), and loses only a single logarithmic factor for all $k>2$-uniform hypergraphs. As in [GKM21], our ideas naturally extend to yield a simpler proof of the full trade-off for strongly refuting smoothed instances of constraint satisfaction problems with similarly improved parameters.
In the random geometric graph model $\mathsf{Geo}_d(n,p)$, we identify each of our $n$ vertices with an independently and uniformly sampled vector from the $d$-dimensional unit sphere, and we connect pairs of vertices whose vectors are ``sufficiently close'', such that the marginal probability of an edge is $p$. We investigate the problem of testing for this latent geometry, or in other words, distinguishing an Erdős-Rényi graph $\mathsf{G}(n, p)$ from a random geometric graph $\mathsf{Geo}_d(n, p)$. It is not too difficult to show that if $d\to \infty$ while $n$ is held fixed, the two distributions become indistinguishable; we wish to understand how fast $d$ must grow as a function of $n$ for indistinguishability to occur. When $p = \fracα{n}$ for constant $α$, we prove that if $d \ge \mathrm{polylog} n$, the total variation distance between the two distributions is close to $0$; this improves upon the best previous bound of Brennan, Bresler, and Nagaraj (2020), which required $d \gg n^{3/2}$, and further our result is nearly tight, resolving a conjecture of Bubeck, Ding, Eldan, \& Rácz (2016) up to logarithmic factors. We also obtain improved upper bounds on the statistical indistinguishability thresholds in $d$ for the full range of $p$ satisfying $\frac{1}{n}\le p\le \frac{1}{2}$, improving upon the previous bounds by polynomial factors. Our analysis uses the Belief Propagation algorithm to characterize the distributions of (subsets of) the random vectors {\em conditioned on producing a particular graph}. In this sense, our analysis is connected to the ``cavity method'' from statistical physics. To analyze this process, we rely on novel sharp estimates for the area of the intersection of a random sphere cap with an arbitrary subset of the sphere, which we prove using optimal transport maps and entropy-transport inequalities on the unit sphere.
Let $G$ be a random $d$-regular graph. We prove that for every constant $α> 0$, with high probability every eigenvector of the adjacency matrix of $G$ with eigenvalue less than $-2\sqrt{d-2}-α$ has $Ω(n/$polylog$(n))$ nodal domains.