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Aaron Potechin

Publications and source records attributed to Aaron Potechin.

At least 37 records · Page 2Linked to original sources
arXiv2023

Sum-of-Squares Lower Bounds for Densest $k$-Subgraph

Given a graph and an integer $k$, Densest $k$-Subgraph is the algorithmic task of finding the subgraph on $k$ vertices with the maximum number of edges. This is a fundamental problem that has been subject to intense study for decades, with applications spanning a wide variety of fields. The state-of-the-art algorithm is an $O(n^{1/4 + ε})$-factor approximation (for any $ε> 0$) due to Bhaskara et al. [STOC '10]. Moreover, the so-called log-density framework predicts that this is optimal, i.e. it is impossible for an efficient algorithm to achieve an $O(n^{1/4 - ε})$-factor approximation. In the average case, Densest $k$-Subgraph is a prototypical noisy inference task which is conjectured to exhibit a statistical-computational gap. In this work, we provide the strongest evidence yet of hardness for Densest $k$-Subgraph by showing matching lower bounds against the powerful Sum-of-Squares (SoS) algorithm, a meta-algorithm based on convex programming that achieves state-of-art algorithmic guarantees for many optimization and inference problems. For $k \leq n^{\frac{1}{2}}$, we obtain a degree $n^δ$ SoS lower bound for the hard regime as predicted by the log-density framework. To show this, we utilize the modern framework for proving SoS lower bounds on average-case problems pioneered by Barak et al. [FOCS '16]. A key issue is that small denser-than-average subgraphs in the input will greatly affect the value of the candidate pseudoexpectation operator around the subgraph. To handle this challenge, we devise a novel matrix factorization scheme based on the positive minimum vertex separator. We then prove an intersection tradeoff lemma to show that the error terms when using this separator are indeed small.

cs.CC↗
arXiv2023

The Sixth Moment of Random Determinants

In this paper, we determine the sixth moment of the determinant of an asymmetric $n \times n$ random matrix where the entries are drawn independently from an arbitrary distribution $Ω$ with mean $0$. Furthermore, we derive the asymptotic behavior of the sixth moment of the determinant as the size of the matrix tends to infinity.

math.CO↗
arXiv2023

Machinery for Proving Sum-of-Squares Lower Bounds on Certification Problems

In this paper, we construct general machinery for proving Sum-of-Squares lower bounds on certification problems by generalizing the techniques used by Barak et al. [FOCS 2016] to prove Sum-of-Squares lower bounds for planted clique. Using this machinery, we prove degree $n^ε$ Sum-of-Squares lower bounds for tensor PCA, the Wishart model of sparse PCA, and a variant of planted clique which we call planted slightly denser subgraph.

cs.CC↗
arXiv2022

On Mixing Distributions Via Random Orthogonal Matrices and the Spectrum of the Singular Values of Multi-Z Shaped Graph Matrices

In this paper, we introduce and analyze a new operation $\circ_{R}$ which mixes two distributions $Ω$ and $Ω'$ via a random orthogonal matrix. In particular, we take $Ω\circ_R Ω'$ to be the limit as $n \to \infty$ of the distribution of singular values of $DRD'$ where $D$ and $D'$ are $n \times n$ diagonal matrices whose diagonal entries have distributions $Ω$ and $Ω'$ respectively and $R$ is a random $n \times n$ orthogonal matrix. We show that $\circ_R$ has several nice properties. We first observe that $\circ_R$ is commutative and associative and compute the moments of $Ω\circ_R Ω'$ in terms of the moments of $Ω$ and $Ω'$. We then show that $\circ_R$ interacts very nicely with the spectrum of the singular values of Z-shaped and multi-Z-shaped graph matrices. This allows us to answer the question posed by our previous paper of how to describe the spectrum of the singular values of Z-shaped and multi-Z-shaped graph matrices when the input distribution is not $\{-1,1\}$. In our analysis, we show that the moments of our distributions are closely connected to non-crossing partitions and prove a number of new results on non-crossing partitions which may be of independent interest.

math.CO↗
arXiv2021

Almost-Orthogonal Bases for Inner Product Polynomials

In this paper, we consider low-degree polynomials of inner products between a collection of random vectors. We give an almost orthogonal basis for this vector space of polynomials when the random vectors are Gaussian, spherical, or Boolean. In all three cases, our basis admits an interesting combinatorial description based on the topology of the underlying graph of inner products. We also analyze the expected value of the product of two polynomials in our basis. In all three cases, we show that this expected value can be expressed in terms of collections of matchings on the underlying graph of inner products. In the Gaussian and Boolean cases, we show that this expected value is always non-negative. In the spherical case, we show that this expected value can be negative but we conjecture that if the underlying graph of inner products is planar then this expected value will always be non-negative.

math.CO↗
arXiv2021

Sum-of-Squares Lower Bounds for Sparse Independent Set

The Sum-of-Squares (SoS) hierarchy of semidefinite programs is a powerful algorithmic paradigm which captures state-of-the-art algorithmic guarantees for a wide array of problems. In the average case setting, SoS lower bounds provide strong evidence of algorithmic hardness or information-computation gaps. Prior to this work, SoS lower bounds have been obtained for problems in the "dense" input regime, where the input is a collection of independent Rademacher or Gaussian random variables, while the sparse regime has remained out of reach. We make the first progress in this direction by obtaining strong SoS lower bounds for the problem of Independent Set on sparse random graphs. We prove that with high probability over an Erdos-Renyi random graph $G\sim G_{n,\frac{d}{n}}$ with average degree $d>\log^2 n$, degree-$D_{SoS}$ SoS fails to refute the existence of an independent set of size $k = Ω\left(\frac{n}{\sqrt{d}(\log n)(D_{SoS})^{c_0}} \right)$ in $G$ (where $c_0$ is an absolute constant), whereas the true size of the largest independent set in $G$ is $O\left(\frac{n\log d}{d}\right)$. Our proof involves several significant extensions of the techniques used for proving SoS lower bounds in the dense setting. Previous lower bounds are based on the pseudo-calibration heuristic of Barak et al [FOCS 2016] which produces a candidate SoS solution using a planted distribution indistinguishable from the input distribution via low-degree tests. In the sparse case the natural planted distribution does admit low-degree distinguishers, and we show how to adapt the pseudo-calibration heuristic to overcome this. Another notorious technical challenge for the sparse regime is the quest for matrix norm bounds. In this paper, we obtain new norm bounds for graph matrices in the sparse setting.

cs.CC↗
arXiv2021

Exact nuclear norm, completion and decomposition for random overcomplete tensors via degree-4 SOS

In this paper we show that simple semidefinite programs inspired by degree $4$ SOS can exactly solve the tensor nuclear norm, tensor decomposition, and tensor completion problems on tensors with random asymmetric components. More precisely, for tensor nuclear norm and tensor decomposition, we show that w.h.p. these semidefinite programs can exactly find the nuclear norm and components of an $(n\times n\times n)$-tensor $\mathcal{T}$ with $m\leq n^{3/2}/polylog(n)$ random asymmetric components. Unlike most of the previous algorithms, our algorithm provides a certificate for the decomposition, does not require knowledge about the number of components in the decomposition and does not make any assumptions on the sizes of the coefficients in the decomposition. As a byproduct, we show that w.h.p. the nuclear norm decomposition exactly coincides with the minimum rank decomposition for tensors with $m\leq n^{3/2}/polylog(n)$ random asymmetric components. For tensor completion, we show that w.h.p. the semidefinite program, introduced by Potechin & Steurer (2017) for tensors with orthogonal components, can exactly recover an $(n\times n\times n)$-tensor $\mathcal{T}$ with $m$ random asymmetric components from only $n^{3/2}m polylog(n)$ randomly observed entries. For non-orthogonal tensors, this improves the dependence on $m$ of the number of entries needed for exact recovery over all previously known algorithms and provides the first theoretical guarantees for exact tensor completion in the overcomplete regime.

cs.DS↗
arXiv2021

SoS certification for symmetric quadratic functions and its connection to constrained Boolean hypercube optimization

We study the rank of the Sum of Squares (SoS) hierarchy over the Boolean hypercube for Symmetric Quadratic Functions (SQFs) in $n$ variables with roots placed in points $k-1$ and $k$. Functions of this type have played a central role in deepening the understanding of the performance of the SoS method for various unconstrained Boolean hypercube optimization problems, including the Max Cut problem. Recently, Lee, Prakash, de Wolf, and Yuen proved a lower bound on the SoS rank for SQFs of $Ω(\sqrt{k(n-k)})$ and conjectured the lower bound of $Ω(n)$ by similarity to a polynomial representation of the $n$-bit OR function. Using Chebyshev polynomials, we refute the Lee -- Prakash -- de~Wolf -- Yuen conjecture and prove that the SoS rank for SQFs is at most $O(\sqrt{nk}\log(n))$. We connect this result to two constrained Boolean hypercube optimization problems. First, we provide a degree $O( \sqrt{n})$ SoS certificate that matches the known SoS rank lower bound for an instance of Min Knapsack, a problem that was intensively studied in the literature. Second, we study an instance of the Set Cover problem for which Bienstock and Zuckerberg conjectured an SoS rank lower bound of $n/4$. We refute the Bienstock -- Zuckerberg conjecture and provide a degree $O(\sqrt{n}\log(n))$ SoS certificate for this problem.

cs.CC↗
arXiv2021

Graph Matrices: Norm Bounds and Applications

In this paper, we derive nearly tight probabilistic norm bounds for a class of random matrices we call graph matrices. While the classical case of symmetric matrices with independent random entries (Wigner's matrices) is a special case, in general, the entries of our matrices will be dependent in a way that can be specified in terms of a fixed-size graph we refer to as the shape. For Wigner's matrices, this shape is $K_2$, the clique on 2 vertices. To prove our norm bounds, we use the trace power method. In a recent series of papers by Potechin and coauthors, graph matrices played a crucial role in proving average-case lower bounds for the Sum-of-Squares (SoS) hierarchy of proof systems, one of the most powerful, but difficult to analyze, techniques in combinatorial optimization. In particular, graph matrices played a crucial role in proving that low-degree SoS cannot refute the existence of a large clique in a random graph and proving that low-degree SoS cannot prove a tight lower bound on the ground state energy of the Sherrington-Kirkpatrick Hamiltonian. In this paper, we give several additional applications of graph matrices. We show that for several technical lemmas in the literature, while the original analyses were quite involved, we can give direct proofs using graph matrices and our norm bounds.

math.CO↗
arXiv2020

Sum-of-Squares Lower Bounds for Sherrington-Kirkpatrick via Planted Affine Planes

The Sum-of-Squares (SoS) hierarchy is a semi-definite programming meta-algorithm that captures state-of-the-art polynomial time guarantees for many optimization problems such as Max-$k$-CSPs and Tensor PCA. On the flip side, a SoS lower bound provides evidence of hardness, which is particularly relevant to average-case problems for which NP-hardness may not be available. In this paper, we consider the following average case problem, which we call the \emph{Planted Affine Planes} (PAP) problem: Given $m$ random vectors $d_1,\ldots,d_m$ in $\mathbb{R}^n$, can we prove that there is no vector $v \in \mathbb{R}^n$ such that for all $u \in [m]$, $\langle v, d_u\rangle^2 = 1$? In other words, can we prove that $m$ random vectors are not all contained in two parallel hyperplanes at equal distance from the origin? We prove that for $m \leq n^{3/2-ε}$, with high probability, degree-$n^{Ω(ε)}$ SoS fails to refute the existence of such a vector $v$. When the vectors $d_1,\ldots,d_m$ are chosen from the multivariate normal distribution, the PAP problem is equivalent to the problem of proving that a random $n$-dimensional subspace of $\mathbb{R}^m$ does not contain a boolean vector. As shown by Mohanty--Raghavendra--Xu [STOC 2020], a lower bound for this problem implies a lower bound for the problem of certifying energy upper bounds on the Sherrington-Kirkpatrick Hamiltonian, and so our lower bound implies a degree-$n^{Ω(ε)}$ SoS lower bound for the certification version of the Sherrington-Kirkpatrick problem.

cs.CC↗
arXiv2020

Sum of squares bounds for the ordering principle

In this paper, we analyze the sum of squares hierarchy (SOS) on the ordering principle on $n$ elements. We prove that degree $O(\sqrt{n}log(n))$ SOS can prove the ordering principle. We then show that this upper bound is essentially tight by proving that for any $ε> 0$, SOS requires degree $Ω(n^{\frac{1}{2} - ε})$ to prove the ordering principle on $n$ elements.

cs.CC↗
arXiv2020

On the Approximability of Presidential Type Predicates

Given a predicate $P: \{-1, 1\}^k \to \{-1, 1\}$, let $CSP(P)$ be the set of constraint satisfaction problems whose constraints are of the form $P$. We say that $P$ is approximable if given a nearly satisfiable instance of $CSP(P)$, there exists a probabilistic polynomial time algorithm that does better than a random assignment. Otherwise, we say that $P$ is approximation resistant. In this paper, we analyze presidential type predicates, which are balanced linear threshold functions where all of the variables except the first variable (the president) have the same weight. We show that almost all presidential-type predicates $P$ are approximable. More precisely, we prove the following result: for any $δ_0 > 0$, there exists a $k_0$ such that if $k \geq k_0$, $δ\in (δ_0,1 - 2/k]$, and $δk + k - 1$ is an odd integer then the presidential type predicate $P(x) = sign(δk{x_1} + \sum_{i=2}^{k}{x_i})$ is approximable. To prove this, we construct a rounding scheme that makes use of biases and pairwise biases. We also give evidence that using pairwise biases is necessary for such rounding schemes.

cs.CC↗
arXiv2020

A Conjecture on Induced Subgraphs of Cayley Graphs

In this paper, we propose the following conjecture which generalizes a theorem proved by Huang [Hua19] in his recent breakthrough proof of the sensitivity conjecture. We conjecture that for any Cayley graph $X = Γ(G,S)$ on a group $G$ and any generating set $S$, if $U \subseteq G$ has size $|U| > |G|/2$, then the induced subgraph of $X$ on $U$ has maximum degree at least $\sqrt{|S|/2}$. Using a recent idea of Alon and Zheng [AZ20], who proved this conjecture for the special case when $G = Z_2^n$, we prove that this conjecture is true whenever $G$ is abelian. We also observe that for this conjecture to hold for a graph $X$, some symmetry is required: it is insufficient for $X$ to just be regular and bipartite.

math.CO↗
arXiv2018

Sum of squares lower bounds from symmetry and a good story

In this paper, we develop machinery which makes it much easier to prove sum of squares lower bounds when the problem is symmetric under permutations of $[1,n]$ and the unsatisfiability of our problem comes from integrality arguments, i.e. arguments that an expression must be an integer. Roughly speaking, to prove SOS lower bounds with our machinery it is sufficient to verify that the answer to the following three questions is yes: 1. Are there natural pseudo-expectation values for the problem? 2. Are these pseudo-expectation values rational functions of the problem parameters? 3. Are there sufficiently many values of the parameters for which these pseudo-expectation values correspond to the actual expected values over a distribution of solutions which is the uniform distribution over permutations of a single solution? We demonstrate our machinery on three problems, the knapsack problem analyzed by Grigoriev, the MOD 2 principle (which says that the complete graph $K_n$ has no perfect matching when $n$ is odd), and the following Turan type problem: Minimize the number of triangles in a graph $G$ with a given edge density. For knapsack, we recover Grigoriev's lower bound exactly. For the MOD 2 principle, we tighten Grigoriev's linear degree sum of squares lower bound, making it exact. Finally, for the triangle problem, we prove a sum of squares lower bound for finding the minimum triangle density. This lower bound is completely new and gives a simple example where constant degree sum of squares methods have a constant factor error in estimating graph densities.

cs.CC↗
arXiv2018

On the Approximation Resistance of Balanced Linear Threshold Functions

In this paper, we show that there exists a balanced linear threshold function (LTF) which is unique games hard to approximate, refuting a conjecture of Austrin, Benabbas, and Magen. We also show that the almost monarchy predicate on k variables is approximable for sufficiently large k.

cs.CC↗
arXiv2018

Lengths of Words Accepted by Nondeterministic Finite Automata

We consider two natural problems about nondeterministic finite automata. First, given such an automaton M of n states, and a length l, does M accept a word of length l? We show that the classic problem of triangle-free graph recognition reduces to this problem, and give an O(n^ω (log n)^{1+ε} log l)-time algorithm to solve it, where ω is the optimal exponent for matrix multiplication. Second, provided L(M) is finite, we consider the problem of listing the lengths of all words accepted by M. Although this problem seems like it might be significantly harder, we show that this problem can be solved in O(n^ω(log n)^{2+ε}) time. Finally, we give a connection between NFA acceptance and the strong exponential-time hypothesis.

cs.FL↗
arXiv2017

The power of sum-of-squares for detecting hidden structures

We study planted problems---finding hidden structures in random noisy inputs---through the lens of the sum-of-squares semidefinite programming hierarchy (SoS). This family of powerful semidefinite programs has recently yielded many new algorithms for planted problems, often achieving the best known polynomial-time guarantees in terms of accuracy of recovered solutions and robustness to noise. One theme in recent work is the design of spectral algorithms which match the guarantees of SoS algorithms for planted problems. Classical spectral algorithms are often unable to accomplish this: the twist in these new spectral algorithms is the use of spectral structure of matrices whose entries are low-degree polynomials of the input variables. We prove that for a wide class of planted problems, including refuting random constraint satisfaction problems, tensor and sparse PCA, densest-k-subgraph, community detection in stochastic block models, planted clique, and others, eigenvalues of degree-d matrix polynomials are as powerful as SoS semidefinite programs of roughly degree d. For such problems it is therefore always possible to match the guarantees of SoS without solving a large semidefinite program. Using related ideas on SoS algorithms and low-degree matrix polynomials (and inspired by recent work on SoS and the planted clique problem by Barak et al.), we prove new nearly-tight SoS lower bounds for the tensor and sparse principal component analysis problems. Our lower bounds for sparse principal component analysis are the first to suggest that going beyond existing algorithms for this problem may require sub-exponential time.

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