arXiv subjects

Avi Wigderson

Publications and source records attributed to Avi Wigderson.

At least 19 recordsLinked to original sources

arXiv2025

The communication complexity of distributed estimation

We study an extension of the standard two-party communication model in which Alice and Bob hold probability distributions $p$ and $q$ over domains $X$ and $Y$, respectively. Their goal is to estimate \[ \mathbb{E}_{x \sim p,\, y \sim q}[f(x, y)] \] to within additive error $\varepsilon$ for a bounded function $f$, known to both parties. We refer to this as the distributed estimation problem. Special cases of this problem arise in a variety of areas including sketching, databases and learning. Our goal is to understand how the required communication scales with the communication complexity of $f$ and the error parameter $\varepsilon$. The random sampling approach -- estimating the mean by averaging $f$ over $O(1/\varepsilon^2)$ random samples -- requires $O(R(f)/\varepsilon^2)$ total communication, where $R(f)$ is the randomized communication complexity of $f$. We design a new debiasing protocol which improves the dependence on $1/\varepsilon$ to be linear instead of quadratic. Additionally we show better upper bounds for several special classes of functions, including the Equality and Greater-than functions. We introduce lower bound techniques based on spectral methods and discrepancy, and show the optimality of many of our protocols: the debiasing protocol is tight for general functions, and that our protocols for the equality and greater-than functions are also optimal. Furthermore, we show that among full-rank Boolean functions, Equality is essentially the easiest.

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arXiv2025

Constant-Depth Arithmetic Circuits for Linear Algebra Problems

We design polynomial size, constant depth (namely, $\mathsf{AC}^0$) arithmetic formulae for the greatest common divisor (GCD) of two polynomials, as well as the related problems of the discriminant, resultant, Bézout coefficients, squarefree decomposition, and the inversion of structured matrices like Sylvester and Bézout matrices. Our GCD algorithm extends to any number of polynomials. Previously, the best known arithmetic formulae for these problems required super-polynomial size, regardless of depth. These results are based on new algorithmic techniques to compute various symmetric functions in the roots of polynomials, as well as manipulate the multiplicities of these roots, without having access to them. These techniques allow $\mathsf{AC}^0$ computation of a large class of linear and polynomial algebra problems, which include the above as special cases. We extend these techniques to problems whose inputs are multivariate polynomials, which are represented by $\mathsf{AC}^0$ arithmetic circuits. Here too we solve problems such as computing the GCD and squarefree decomposition in $\mathsf{AC}^0$.

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arXiv2024

Sparser Abelian High Dimensional Expanders

We present two new explicit constructions of Cayley high dimensional expanders (HDXs) over the abelian group $\mathbb{F}_2^n$. Our expansion proofs use only linear algebra and combinatorial arguments. The first construction gives local spectral HDXs of any constant dimension and subpolynomial degree $\exp(n^ε)$ for every $ε>0$, improving on a construction by Golowich [Gol23] which achieves $ε=1/2$. [Gol23] derives these HDXs by sparsifying the complete Grassmann poset of subspaces. The novelty in our construction is the ability to sparsify any expanding Grassmannian posets, leading to iterated sparsification and much smaller degrees. The sparse Grassmannian (which is of independent interest in the theory of HDXs) serves as the generating set of the Cayley graph. Our second construction gives a 2-dimensional HDXs of any polynomial degree $\exp(εn$) for any constant $ε> 0$, which is simultaneously a spectral expander and a coboundary expander. To the best of our knowledge, this is the first such non-trivial construction. We name it the Johnson complex, as it is derived from the classical Johnson scheme, whose vertices serve as the generating set of this Cayley graph. This construction may be viewed as a derandomization of the recent random geometric complexes of [LMSY23]. Establishing coboundary expansion through Gromov's "cone method" and the associated isoperimetric inequalities is the most intricate aspect of this construction. While these two constructions are quite different, we show that they both share a common structure, resembling the intersection patterns of vectors in the Hadamard code. We propose a general framework of such "Hadamard-like" constructions in the hope that it will yield new HDXs.

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arXiv2024

Almost Ramanujan Expanders from Arbitrary Expanders via Operator Amplification

We give an efficient algorithm that transforms any bounded degree expander graph into another that achieves almost optimal (namely, near-quadratic, $d \leq 1/λ^{2+o(1)}$) trade-off between (any desired) spectral expansion $λ$ and degree $d$. Furthermore, the algorithm is local: every vertex can compute its new neighbors as a subset of its original neighborhood of radius $O(\log(1/λ))$. The optimal quadratic trade-off is known as the Ramanujan bound, so our construction gives almost Ramanujan expanders from arbitrary expanders. The locality of the transformation preserves structural properties of the original graph, and thus has many consequences. Applied to Cayley graphs, our transformation shows that any expanding finite group has almost Ramanujan expanding generators. Similarly, one can obtain almost optimal explicit constructions of quantum expanders, dimension expanders, monotone expanders, etc., from existing (suboptimal) constructions of such objects. Another consequence is a "derandomized" random walk on the original (suboptimal) expander with almost optimal convergence rate. Our transformation also applies when the degree is not bounded or the expansion is not constant. We obtain our results by a generalization of Ta-Shma's technique in his breakthrough paper [STOC 2017], used to obtain explicit almost optimal binary codes. Specifically, our spectral amplification extends Ta-Shma's analysis of bias amplification from scalars to matrices of arbitrary dimension in a very natural way. Curiously, while Ta-Shma's explicit bias amplification derandomizes a well-known probabilistic argument (underlying the Gilbert--Varshamov bound), there seems to be no known probabilistic (or other existential) way of achieving our explicit ("high-dimensional") spectral amplification.

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arXiv2024

Complexity of Robust Orbit Problems for Torus Actions and the abc-conjecture

When a group acts on a set, it naturally partitions it into orbits, giving rise to orbit problems. These are natural algorithmic problems, as symmetries are central in numerous questions and structures in physics, mathematics, computer science, optimization, and more. Accordingly, it is of high interest to understand their computational complexity. Recently, Bürgisser et al. gave the first polynomial-time algorithms for orbit problems of torus actions, that is, actions of commutative continuous groups on Euclidean space. In this work, motivated by theoretical and practical applications, we study the computational complexity of robust generalizations of these orbit problems, which amount to approximating the distance of orbits in $\mathbb{C}^n$ up to a factor $γ>1$. In particular, this allows deciding whether two inputs are approximately in the same orbit or far from being so. On the one hand, we prove the NP-hardness of this problem for $γ= n^{Ω(1/\log\log n)}$ by reducing the closest vector problem for lattices to it. On the other hand, we describe algorithms for solving this problem for an approximation factor $γ= \exp(\mathrm{poly}(n))$. Our algorithms combine tools from invariant theory and algorithmic lattice theory, and they also provide group elements witnessing the proximity of the given orbits (in contrast to the algebraic algorithms of prior work). We prove that they run in polynomial time if and only if a version of the famous number-theoretic $abc$-conjecture holds -- establishing a new and surprising connection between computational complexity and number theory.

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arXiv2023

On linear-algebraic notions of expansion

A fundamental fact about bounded-degree graph expanders is that three notions of expansion -- vertex expansion, edge expansion, and spectral expansion -- are all equivalent. In this paper, we study to what extent such a statement is true for linear-algebraic notions of expansion. There are two well-studied notions of linear-algebraic expansion, namely dimension expansion (defined in analogy to graph vertex expansion) and quantum expansion (defined in analogy to graph spectral expansion). Lubotzky and Zelmanov proved that the latter implies the former. We prove that the converse is false: there are dimension expanders which are not quantum expanders. Moreover, this asymmetry is explained by the fact that there are two distinct linear-algebraic analogues of graph edge expansion. The first of these is quantum edge expansion, which was introduced by Hastings, and which he proved to be equivalent to quantum expansion. We introduce a new notion, termed dimension edge expansion, which we prove is equivalent to dimension expansion and which is implied by quantum edge expansion. Thus, the separation above is implied by a finer one: dimension edge expansion is strictly weaker than quantum edge expansion. This new notion also leads to a new, more modular proof of the Lubotzky--Zelmanov result that quantum expanders are dimension expanders.

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arXiv2022

Robustly Self-Ordered Graphs: Constructions and Applications to Property Testing

A graph $G$ is called self-ordered (a.k.a asymmetric) if the identity permutation is its only automorphism. Equivalently, there is a unique isomorphism from $G$ to any graph that is isomorphic to $G$. We say that $G=(V,E)$ is robustly self-ordered if the size of the symmetric difference between $E$ and the edge-set of the graph obtained by permuting $V$ using any permutation $π:V\to V$ is proportional to the number of non-fixed-points of $π$. In this work, we initiate the study of the structure, construction and utility of robustly self-ordered graphs. We show that robustly self-ordered bounded-degree graphs exist (in abundance), and that they can be constructed efficiently, in a strong sense. Specifically, given the index of a vertex in such a graph, it is possible to find all its neighbors in polynomial-time (i.e., in time that is poly-logarithmic in the size of the graph). We also consider graphs of unbounded degree, seeking correspondingly unbounded robustness parameters. We again demonstrate that such graphs (of linear degree) exist (in abundance), and that they can be constructed efficiently, in a strong sense. This turns out to require very different tools. Specifically, we show that the construction of such graphs reduces to the construction of non-malleable two-source extractors (with very weak parameters but with some additional natural features). We demonstrate that robustly self-ordered bounded-degree graphs are useful towards obtaining lower bounds on the query complexity of testing graph properties both in the bounded-degree and the dense graph models. One of the results that we obtain, via such a reduction, is a subexponential separation between the query complexities of testing and tolerant testing of graph properties in the bounded-degree graph model.

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arXiv2022

An Optimal "It Ain't Over Till It's Over" Theorem

We study the probability of Boolean functions with small max influence to become constant under random restrictions. Let $f$ be a Boolean function such that the variance of $f$ is $Ω(1)$ and all its individual influences are bounded by $τ$. We show that when restricting all but a $ρ=\tildeΩ((\log(1/τ))^{-1})$ fraction of the coordinates, the restricted function remains nonconstant with overwhelming probability. This bound is essentially optimal, as witnessed by the tribes function $\mathrm{TRIBES}=\mathrm{AND}_{n/C\log n}\circ\mathrm{OR}_{C\log n}$. We extend it to an anti-concentration result, showing that the restricted function has nontrivial variance with probability $1-o(1)$. This gives a sharp version of the "it ain't over till it's over" theorem due to Mossel, O'Donnell, and Oleszkiewicz. Our proof is discrete, and avoids the use of the invariance principle. We also show two consequences of our above result: (i) As a corollary, we prove that for a uniformly random input $x$, the block sensitivity of $f$ at $x$ is $\tildeΩ(\log(1/τ))$ with probability $1-o(1)$. This should be compared with the implication of Kahn, Kalai, and Linial's result, which implies that the average block sensitivity of $f$ is $Ω(\log(1/τ))$. (ii) Combining our proof with a well-known result due to O'Donnell, Saks, Schramm, and Servedio, one can also conclude that: Restricting all but a $ρ=\tildeΩ(1/\sqrt{\log (1/τ) })$ fraction of the coordinates of a monotone function $f$, then the restricted function has decision tree complexity $Ω(τ^{-Θ(ρ)})$ with probability $Ω(1)$.

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arXiv2022

Connections between graphs and matrix spaces

Given a bipartite graph $G$, the graphical matrix space $\mathcal{S}_G$ consists of matrices whose non-zero entries can only be at those positions corresponding to edges in $G$. Tutte (J. London Math. Soc., 1947), Edmonds (J. Res. Nat. Bur. Standards Sect. B, 1967) and Lovász (FCT, 1979) observed connections between perfect matchings in $G$ and full-rank matrices in $\mathcal{S}_G$. Dieudonné ({Arch. Math., 1948) proved a tight upper bound on the dimensions of those matrix spaces containing only singular matrices. The starting point of this paper is a simultaneous generalization of these two classical results: we show that the largest dimension over subspaces of $\mathcal{S}_G$ containing only singular matrices is equal to the maximum size over subgraphs of $G$ without perfect matchings, based on Meshulam's proof of Dieudonné's result (Quart. J. Math., 1985). Starting from this result, we go on to establish more connections between properties of graphs and matrix spaces. For example, we establish connections between acyclicity and nilpotency, between strong connectivity and irreducibility, and between isomorphism and conjugacy/congruence. For each connection, we study three types of correspondences, namely the basic correspondence, the inherited correspondence (for subgraphs and subspaces), and the induced correspondence (for induced subgraphs and restrictions). Some correspondences lead to intriguing generalizations of classical results, such as for Dieudonné's result mentioned above, and for a celebrated theorem of Gerstenhaber regarding the largest dimension of nil matrix spaces (Amer. J. Math., 1958). Finally, we show some implications of our results to quantum information and present open problems in computational complexity motivated by these results.

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arXiv2021

Towards a theory of non-commutative optimization: geodesic first and second order methods for moment maps and polytopes

This paper initiates a systematic development of a theory of non-commutative optimization. It aims to unify and generalize a growing body of work from the past few years which developed and analyzed algorithms for natural geodesically convex optimization problems on Riemannian manifolds that arise from the symmetries of non-commutative groups. These algorithms minimize the moment map (a non-commutative notion of the usual gradient) and test membership in null cones and moment polytopes (a vast class of polytopes, typically of exponential vertex and facet complexity, which arise from this a priori non-convex, non-linear setting). This setting captures a diverse set of problems in different areas of computer science, mathematics, and physics. Several of them were solved efficiently for the first time using non-commutative methods; the corresponding algorithms also lead to solutions of purely structural problems and to many new connections between disparate fields. In the spirit of standard convex optimization, we develop two general methods in the geodesic setting, a first order and a second order method, which respectively receive first and second order information on the "derivatives" of the function to be optimized. These in particular subsume all past results. The main technical work goes into identifying the key parameters of the underlying group actions which control convergence to the optimum in each of these methods. These non-commutative analogues of "smoothness" are far more complex and require significant algebraic and analytic machinery. Despite this complexity, the way in which these parameters control convergence in both methods is quite simple and elegant. We show how to bound these parameters and hence obtain efficient algorithms for null cone membership in several concrete situations. Our work points to intriguing open problems and suggests further research directions.

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arXiv2021

On the Power and Limitations of Branch and Cut

The Stabbing Planes proof system was introduced to model the reasoning carried out in practical mixed integer programming solvers. As a proof system, it is powerful enough to simulate Cutting Planes and to refute the Tseitin formulas -- certain unsatisfiable systems of linear equations mod 2 -- which are canonical hard examples for many algebraic proof systems. In a recent (and surprising) result, Dadush and Tiwari showed that these short refutations of the Tseitin formulas could be translated into quasi-polynomial size and depth Cutting Planes proofs, refuting a long-standing conjecture. This translation raises several interesting questions. First, whether all Stabbing Planes proofs can be efficiently simulated by Cutting Planes. This would allow for the substantial analysis done on the Cutting Planes system to be lifted to practical mixed integer programming solvers. Second, whether the quasi-polynomial depth of these proofs is inherent to Cutting Planes. In this paper we make progress towards answering both of these questions. First, we show that any Stabbing Planes proof with bounded coefficients SP* can be translated into Cutting Planes. As a consequence of the known lower bounds for Cutting Planes, this establishes the first exponential lower bounds on SP*. Using this translation, we extend the result of Dadush and Tiwari to show that Cutting Planes has short refutations of any unsatisfiable system of linear equations over a finite field. Like the Cutting Planes proofs of Dadush and Tiwari, our refutations also incur a quasi-polynomial blow-up in depth, and we conjecture that this is inherent. As a step towards this conjecture, we develop a new geometric technique for proving lower bounds on the depth of Cutting Planes proofs. This allows us to establish the first lower bounds on the depth of Semantic Cutting Planes proofs of the Tseitin formulas.

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arXiv2021

Polynomial time algorithms in invariant theory for torus actions

An action of a group on a vector space partitions the latter into a set of orbits. We consider three natural and useful algorithmic "isomorphism" or "classification" problems, namely, orbit equality, orbit closure intersection, and orbit closure containment. These capture and relate to a variety of problems within mathematics, physics and computer science, optimization and statistics. These orbit problems extend the more basic null cone problem, whose algorithmic complexity has seen significant progress in recent years. In this paper, we initiate a study of these problems by focusing on the actions of commutative groups (namely, tori). We explain how this setting is motivated from questions in algebraic complexity, and is still rich enough to capture interesting combinatorial algorithmic problems. While the structural theory of commutative actions is well understood, no general efficient algorithms were known for the aforementioned problems. Our main results are polynomial time algorithms for all three problems. We also show how to efficiently find separating invariants for orbits, and how to compute systems of generating rational invariants for these actions (in contrast, for polynomial invariants the latter is known to be hard). Our techniques are based on a combination of fundamental results in invariant theory, linear programming, and algorithmic lattice theory.

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arXiv2020

The uncertainty principle: variations on a theme

We show how a number of well-known uncertainty principles for the Fourier transform, such as the Heisenberg uncertainty principle, the Donoho--Stark uncertainty principle, and Meshulam's non-abelian uncertainty principle, have little to do with the structure of the Fourier transform itself. Rather, all of these results follow from very weak properties of the Fourier transform (shared by numerous linear operators), namely that it is bounded as an operator $L^1 \to L^\infty$, and that it is unitary. Using a single, simple proof template, and only these (or weaker) properties, we obtain some new proofs and many generalizations of these basic uncertainty principles, to new operators and to new settings, in a completely unified way. Together with our general overview, this paper can also serve as a survey of the many facets of the phenomena known as uncertainty principles.

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arXiv2020

Search problems in algebraic complexity, GCT, and hardness of generator for invariant rings

We consider the problem of computing succinct encodings of lists of generators for invariant rings for group actions. Mulmuley conjectured that there are always polynomial sized such encodings for invariant rings of $\SL_n(\C)$-representations. We provide simple examples that disprove this conjecture (under standard complexity assumptions). We develop a general framework, denoted \emph{algebraic circuit search problems}, that captures many important problems in algebraic complexity and computational invariant theory. This framework encompasses various proof systems in proof complexity and some of the central problems in invariant theory as exposed by the Geometric Complexity Theory (GCT) program, including the aforementioned problem of computing succinct encodings for generators for invariant rings.

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arXiv2019

Singular tuples of matrices is not a null cone (and, the symmetries of algebraic varieties)

The following multi-determinantal algebraic variety plays a central role in algebra, algebraic geometry and computational complexity theory: ${\rm SING}_{n,m}$, consisting of all $m$-tuples of $n\times n$ complex matrices which span only singular matrices. In particular, an efficient deterministic algorithm testing membership in ${\rm SING}_{n,m}$ will imply super-polynomial circuit lower bounds, a holy grail of the theory of computation. A sequence of recent works suggests such efficient algorithms for memberships in a general class of algebraic varieties, namely the null cones of linear group actions. Can this be used for the problem above? Our main result is negative: ${\rm SING}_{n,m}$ is not the null cone of any (reductive) group action! This stands in stark contrast to a non-commutative analog of this variety, and points to an inherent structural difficulty of ${\rm SING}_{n,m}$. To prove this result we identify precisely the group of symmetries of ${\rm SING}_{n,m}$. We find this characterization, and the tools we introduce to prove it, of independent interest. Our work significantly generalizes a result of Frobenius for the special case $m=1$, and suggests a general method for determining the symmetries of algebraic varieties.

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arXiv2019

More barriers for rank methods, via a "numeric to symbolic" transfer

We prove new barrier results in arithmetic complexity theory, showing severe limitations of natural lifting (aka escalation) techniques. For example, we prove that even optimal rank lower bounds on $k$-tensors cannot yield non-trivial lower bounds on the rank of $d$-tensors, for any constant $d>k$. This significantly extends recent barrier results on the limits of (matrix) rank methods by Efremenko, Garg, Oliveira and Wigderson, which handles the (very important) case $k=2$. Our generalization requires the development of new technical tools and results in algebraic geometry, which are interesting in their own right and possibly applicable elsewhere. The basic issue they probe is the relation between numeric and symbolic rank of tensors, essential in the proofs of previous and current barriers. Our main technical result implies that for every symbolic $k$-tensor (namely one whose entries are polynomials in some set of variables), if the tensor rank is small for every evaluation of the variables, then it is small symbolically. This statement is obvious for $k=2$. To prove an analogous statement for $k>2$ we develop a "numeric to symbolic" transfer of algebraic relations to algebraic functions, somewhat in the spirit of the implicit function theorem. It applies in the general setting of inclusion of images of polynomial maps, in the form appearing in Raz's elusive functions approach to proving VP $\neq$ VNP. We give a toy application showing how our transfer theorem may be useful in pursuing this approach to prove arithmetic complexity lower bounds.

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arXiv2019

Subspace arrangements, graph rigidity and derandomization through submodular optimization

This paper presents a deterministic, strongly polynomial time algorithm for computing the matrix rank for a class of symbolic matrices (whose entries are polynomials over a field). This class was introduced, in a different language, by Lovász [Lov] in his study of flats in matroids, and proved a duality theorem putting this problem in $NP \cap coNP$. As such, our result is another demonstration where ``good characterization'' in the sense of Edmonds leads to an efficient algorithm. In a different paper Lovász [Lov79] proved that all such symbolic rank problems have efficient probabilistic algorithms, namely are in $BPP$. As such, our algorithm may be interpreted as a derandomization result, in the long sequence special cases of the PIT (Polynomial Identity Testing) problem. Finally, Lovász and Yemini [LoYe] showed how the same problem generalizes the graph rigidity problem in two dimensions. As such, our algorithm may be seen as a generalization of the well-known deterministic algorithm for the latter problem. There are two somewhat unusual technical features in this paper. The first is the translation of Lovász' flats problem into a symbolic rank one. The second is the use of submodular optimization for derandomization. We hope that the tools developed for both will be useful for related problems, in particular for better understanding of graph rigidity in higher dimensions.

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arXiv2019

Operator scaling: theory and applications

In this paper we present a deterministic polynomial time algorithm for testing if a symbolic matrix in non-commuting variables over $\mathbb{Q}$ is invertible or not. The analogous question for commuting variables is the celebrated polynomial identity testing (PIT) for symbolic determinants. In contrast to the commutative case, which has an efficient probabilistic algorithm, the best previous algorithm for the non-commutative setting required exponential time (whether or not randomization is allowed). The algorithm efficiently solves the "word problem" for the free skew field, and the identity testing problem for arithmetic formulae with division over non-commuting variables, two problems which had only exponential-time algorithms prior to this work. The main contribution of this paper is a complexity analysis of an existing algorithm due to Gurvits, who proved it was polynomial time for certain classes of inputs. We prove it always runs in polynomial time. The main component of our analysis is a simple (given the necessary known tools) lower bound on central notion of capacity of operators (introduced by Gurvits). We extend the algorithm to actually approximate capacity to any accuracy in polynomial time, and use this analysis to give quantitative bounds on the continuity of capacity (the latter is used in a subsequent paper on Brascamp-Lieb inequalities). Symbolic matrices in non-commuting variables, and the related structural and algorithmic questions, have a remarkable number of diverse origins and motivations. They arise independently in (commutative) invariant theory and representation theory, linear algebra, optimization, linear system theory, quantum information theory, approximation of the permanent and naturally in non-commutative algebra. We provide a detailed account of some of these sources and their interconnections.

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