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Publications (359)
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 $\exp(n^\epsilon)$ for every $\epsilon >0$, impro...
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....
Given a bipartite graph G, the graphical matrix space \(\cal{S}_{G}\) 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...
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, nam...
We give an efficient algorithm that transforms any bounded degree expander graph into another that achieves almost optimal (namely, near-quadratic, $d \leq 1/\lambda^{2+o(1)}$) trade-off between (any desired) spectral expansion $\lambda$ and degree $d$. Furthermore, the algorithm is local: every vertex can compute its new neighbors as a subset of i...
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 Lov\'asz (FCT, 1979) observed connections between perfect matchings in $G$ and f...
The following multi-determinantal algebraic variety plays a central role in algebra, algebraic geometry and computational complexity theory: SING n , m {{\rm SING}_{n,m}} , consisting of all m -tuples of n × n {n\times n} complex matrices which span only singular matrices. In particular, an efficient deterministic algorithm testing membership in SI...
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...
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 p...
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. Thes...
We present a polynomial-time algorithm that, given as a input the description of a game with incomplete information and any number of players, produces a protocol for playing the game that leaks no partial information, provided the majority of the players is honest.
Our algorithm automatically solves all the multi-party protocol problems addressed...
Every function of n inputs can be efficiently computed by a complete network of n processors in such a way that:
1. If no faults occur, no set of size t < n/2 of players gets any additional information (other than the function value),
2. Even if Byzantine faults are allowed, no set of size t < n/3 can either disrupt the computation or get additiona...
Quite complex cryptographic machinery has been developed based on the assumption that one-way functions exist, yet we know of only a few possible such candidates. It is important at this time to find alternative foundations to the design of secure cryptography. We introduce a new model of generalized interactive proofs as a step in this direction....
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 deve...
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 ${\rm SING}_{n,m}$...
In this paper, we present a deterministic polynomial time algorithm for testing whether a symbolic matrix in non-commuting variables over 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 pr...
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...
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 Lov\'asz [Lov] in his study of flats in matroids, and proved a duality theorem putting this problem in $NP \cap co...
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 [16] in his study of flats in matroids, and proved a duality theorem putting this problem in \(NP \cap coNP...
We introduce a simple logical inference structure we call a $\textsf{spanoid}$ (generalizing the notion of a matroid), which captures well-studied problems in several areas. These include combinatorial geometry, algebra (arrangements of hypersurfaces and ideals), statistical physics (bootstrap percolation) and coding theory. We initiate a thorough...
We propose a new second-order method for geodesically convex optimization on the natural hyperbolic metric over positive definite matrices. We apply it to solve the operator scaling problem in time polynomial in the input size and logarithmic in the error. This is an exponential improvement over previous algorithms which were analyzed in the usual...
We present a polynomial time algorithm to approximately scale tensors of any format to arbitrary prescribed marginals (whenever possible). This unifies and generalizes a sequence of past works on matrix, operator and tensor scaling. Our algorithm provides an efficient weak membership oracle for the associated moment polytopes, an important family o...
We propose a new second-order method for geodesically convex optimization on the natural hyperbolic metric over positive definite matrices. We apply it to solve the operator scaling problem in time polynomial in the input size and logarithmic in the error. This is an exponential improvement over previous algorithms which were analyzed in the usual...
The celebrated Brascamp–Lieb (BL) inequalities [BL76,Lie90], and their reverse form of Barthe [Bar98], are an important mathematical tool, unifying and generalizing numerous inequalities in analysis, convex geometry and information theory, with many used in computer science. While their structural theory is very well understood, far less is known a...
Alternating minimization heuristics seek to solve a (difficult) global optimization task through iteratively solving a sequence of (much easier) local optimization tasks on different parts (or blocks) of the input parameters. While popular and widely applicable, very few examples of this heuristic are rigorously shown to converge to optimality, and...
Arithmetic complexity is considered simpler to understand than Boolean complexity, namely computing Boolean functions via logical gates. And indeed, we seem to have significantly more lower bound techniques and results in arithmetic complexity than in Boolean complexity. Despite many successes and rapid progress, however, challenges like proving su...
In this work we study the quantitative relation between VC-dimension and two other basic parameters related to learning and teaching. Namely, the quality of sample compression schemes and of teaching sets for classes of low VC-dimension. Let C be a binary concept class of size m and VC-dimension d. Prior to this work, the best known upper bounds fo...
The celebrated Brascamp-Lieb (BL) inequalities [BL76, Lie90], and their reverse form of Barthe [Bar98], are an important mathematical tool, unifying and generalizing numerous in- equalities in analysis, convex geometry and information theory, with many used in computer science. While their structural theory is very well understood, far less is know...
We develop several efficient algorithms for the classical \emph{Matrix Scaling} problem, which is used in many diverse areas, from preconditioning linear systems to approximation of the permanent. On an input $n\times n$ matrix $A$, this problem asks to find diagonal (scaling) matrices $X$ and $Y$ (if they exist), so that $X A Y$ $\varepsilon$-appr...
The celebrated Brascamp-Lieb (BL) inequalities (and their extensions) are an important mathematical tool, unifying and generalizing numerous inequalities in analysis, convex geometry and information theory. While their structural theory is very well understood, far less is known about computing their main parameters. We give polynomial time algorit...
We give upper and lower bounds on the power of subsystems of the Ideal Proof System (IPS), the algebraic proof system recently proposed by Grochow and Pitassi, where the circuits comprising the proof come from various restricted algebraic circuit classes. This mimics an established research direction in the boolean setting for subsystems of Extende...
The sensitivity of a Boolean function f is the maximum over all inputs x, of the number of sensitive coordinates of x. The well-known sensitivity conjecture of Nisan (see also Nisan and Szegedy) states that every sensitivity-s Boolean function can be computed by a polynomial over the reals of degree poly(s). The best known upper bounds on degree, h...
A natural measure of smoothness of a Boolean function is its sensitivity (the largest number of Hamming neighbors of a point which differ from it in function value). The structure of smooth or equivalently low-sensitivity functions is still a mystery. A well-known conjecture states that every such Boolean function can be computed by a shallow decis...
We initiate the study of the complexity of arithmetic circuits with division gates over non-commuting variables. Such circuits and formulae compute non-commutative rational functions, which, despite their name, can no longer be expressed as ratios of polynomials. We prove some lower and upper bounds, completeness and simulation results, as follows....
In this paper we present a deterministic polynomial time algorithm for
testing if a symbolic matrix in {\emph 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...
A natural measure of smoothness of a Boolean function is its sensitivity (the
largest number of Hamming neighbors of a point which differ from it in function
value). The structure of smooth or equivalently low-sensitivity functions is
still a mystery. A well-known conjecture states that every such Boolean
function can be computed by a shallow decis...
In this work we study the quantitative relation between VC-dimension and two
other basic parameters related to learning and teaching. We present relatively
efficient constructions of {\em sample compression schemes} and {\em teaching
sets} for classes of low VC-dimension. Let $C$ be a finite boolean concept
class of VC-dimension $d$. Set $k = O(d 2...
This paper studies the parameters for which binary Reed-Muller (RM) codes can be decoded successfully on the BEC and BSC, and in particular when can they achieve capacity for these two classical channels. Necessarily, the paper also studies properties of evaluations of multi-variate GF(2) polynomials on random sets of inputs. For erasures, we prove...
We initiate the study of the complexity of arithmetic circuits with division gates over non-commuting variables. Such circuits and formulas compute non-commutative rational functions, which, despite their name, can no longer be expressed as ratios of polynomials. We prove some lower and upper bounds, completeness and simulation results, as follows....
We prove that 3-query linear locally correctable codes over the Reals of dimension d require block length n > d2+λ for some fixed, positive λ > 0. Geometrically, this means that if n vectors in Rd are such that each vector is spanned by a linear number of disjoint triples of others, then it must be that n > d2+λ. This improves the known quadratic l...
In a transport system, data is reliably transported from a sender to a receiver by organizing the data to be transported into data blocks, wherein each data block comprises a plurality of encoding units, transmitting encoding units of a first data block from the sender to the receiver, and detecting, at the sender, acknowledgments of receipt of enc...
We study interactive proofs with sublinear-time verifiers. These proof systems can be used to ensure approximate correctness for the results of computations delegated to an untrusted server. Following the literature on property testing, we seek proof systems where with high probability the verifier accepts every input in the language, and rejects e...
Let ${{\sigma }_{\mathbb{Z}}}\left( k \right)$ be the smallest $n$ such that there exists an identity $$\left( x_{1}^{2}\,+\,x_{2}^{2}\,+\,\cdot \cdot \cdot \,+\,x_{k}^{2} \right)\,\cdot \,\left( y_{1}^{2}\,+\,y_{2}^{2}\,+\,\cdot \cdot \cdot \,+\,y_{k}^{2} \right)\,=\,f_{1}^{2}\,+\,f_{2}^{2}\,+\,\cdot \cdot \cdot \,+\,f_{n}^{2},$$
with ${{f}_{1}},....
We study the rank of complex sparse matrices in which the supports of
different columns have small intersections. The rank of these matrices, called
design matrices, was the focus of a recent work by Barak et. al. (BDWY11) in
which they were used to answer questions regarding point configurations. In
this work we derive near-optimal rank bounds for...
We study questions in incidence geometry where the precise position of points
is `blurry' (e.g. due to noise, inaccuracy or error). Thus lines are replaced
by narrow tubes, and more generally affine subspaces are replaced by their
small neighborhood. We show that the presence of a sufficiently large number of
approximately collinear triples in a se...
The main result of this paper is an explicit disperser for two independent sources on n bits, each of min-entropy k = 2 log 1 �0 n , for some small absolute constant α0 > 0). Put differently, setting N = 2 n and K = 2 k , we construct an explicit N × N Boolean matrix for which no K × K sub-matrix is monochromatic. Viewed as the adjacency matrix of...
We study several problems in which an unknown distribution over an unknown population of vectors needs to be recovered from partial or noisy samples, each of which nearly completely erases or obliterates the original vector. For example, consider a distribution p over a population V ⊆ {0, 1}n. A noisy sample v' is obtained by choosing v according t...
Foam problems are concerned with how one can partition space into ―bubbles‖ which minimize surface area. We investigate the case where one unit-volume bubble is required to tile d-dimensional space in a periodic fashion according to the standard, cubical lattice. While a cube requires surface area 2d, we construct such a bubble having surface area...
We prove fractional analogs of the classical Sylvester-Gallai theorem. Our theorems translate local information about collinear triples in a set of points into global bounds on the dimension of the set. Specifically, we show that if for every points v in a finite set , there are at least δ|V| other points u∈V for which the line through v,u contains...
What are the limits of mathematical knowledge? The purpose of this chapter is to introduce the main concepts from computational complexity theory that are relevant to algorithmic accessibility of mathematical understanding. In particular, I'll discuss the P versus NP problem, its possible impact on research in mathematics, and how interested Gödel...
We introduce a notion of non-black-box access to computational devices (such as circuits, formulas, decision trees, and so forth) that we call restriction access. Restrictions are partial assignments to input variables. Each restriction simplifies the device, and yields a new device for the restricted function on the unassigned variables. On one ex...
How complex is a given multivariate polynomial? The main point of this survey is that one can learn a great deal about the structure and complexity of polynomials by studying (some of) their partial derivatives. The bulk of the survey shows that partial derivatives provide essential ingredients in proving both upper and lower bounds for computing p...
A (q,k,t)-design matrix is an m x n matrix whose pattern of zeros/non-zeros
satisfies the following design-like condition: each row has at most q
non-zeros, each column has at least k non-zeros and the supports of every two
columns intersect in at most t rows. We prove that the rank of any
(q,k,t)-design matrix over a field of characteristic zero (...
This paper extends Valiant's work on VP and VNP to the settings in which variables are not multiplicatively commutative and/or associative. Our main result is a theory of completeness for these algebraic worlds. We define analogs of Valiant's classes VP and VNP, as well as of the polynomials permanent and determinant, in these worlds. We then prove...
We initiate a direction for proving lower bounds on the size of non-commutative arithmetic circuits. This direction is based on a connection between lower bounds on the size of non-commutative arithmetic circuits and a problem about commutative degree four polynomials, the classical sum-of-squares problem: find the smallest n such that there exists...
This paper attempts to broaden the foundations of public-key cryptography. We construct new public-key encryption schemes based on new hardness-on-average assumptions for natural combinatorial NP-hard optimization problems. We consider the following assumptions: It is infeasible to solve a random set of sparse linear equations mod 2, of which a sma...
A distribution X over binary strings of length n has min-entropy k if every string has probability at most 2-k in X. We say that X is a δ-source if its rate k⁄n is at least δ.We give the following new explicit instructions (namely, poly(n)- time computable functions) of deterministicextractors, dispersers and related objects. All work for any fixed...
The main purpose of this work is to formally define monotone expanders and motivate their study with (known and new) connections to other graphs and to several computational and pseudorandomness problems. In particular we explain how monotone expanders of constant degree lead to: (1) Constant degree dimension expanders in finite fields, resolving a...
The classical Direct-Product Theorem for circuits says that if a Boolean function f: {0,1}n -> {0,1} is somewhat hard to compute on average by small circuits, then the corresponding k-wise direct product function fk(x1,...,xk)=(f(x1),...,f(xk)) (where each xi -> {0,1}n) is significantly harder to compute on average by slightly smaller circuits. We...
Coding theoretic and complexity theoretic considerations naturally lead to the question of generating symmetric, sparse, redundant linear systems. This paper provides new way of constructions with better parameters and new lower bounds.
Low Density Parity Check (LDPC) codes are linear codes defined by short constraints (a property essential for loc...
Classical results from the 1970's state that w.h.p. a random subspace of N-dimensional Euclidean space of proportional (linear in N) dimension is ¿well-spread¿ in the sense that vectors in the subspace have their f<sub>2</sub> mass smoothly spread over a linear number of coordinates. Such well-spread subspaces are intimately connected to low dist...
In this paper we study the one-way multiparty communication model, in which every party speaks exactly once in its turn. For every k, we prove a tight lower bound of Ω(n 1/(k−1)}) on the probabilistic communication complexity of pointer jumping in a k-layered tree, where the pointers of the i-th layer reside on the forehead of the i-th party to spe...
We study solution sets to systems of generalized linear equations of the form ℓi(x1,x2, middot;middot;middot;, xn) ∈ Ai (mod m) where ℓ1,⋯, ℓt are linear forms in n Boolean variables, each Ai is an arbitrary subset of ℤm, and m is a composite integer that is a product of two distinct primes, like 6. Our main technical result is that such solution s...
We propose the study of graphs that are defined by low-complexity distributed and deterministic agents. We suggest that this
viewpoint may help introduce the element of individual choice in models of large scale social networks. This viewpoint may also provide interesting new classes of graphs for which to
design algorithms.
We focus largely on th...
ABSTRACT The \direct product code" of a function f gives its values on all k-tuples (f(x1);:::;f(xk)). This basic construct under- lies \hardness amplication" in cryptography, circuit com- plexity and PCPs. Goldreich and Safra [12] pioneered its local testing and its PCP application. A recent result by Dinur and Goldenberg [5] enabled for the,rst t...
On Saturday, May 30, one day before the start of the regular STOC 2009 program, a workshop was held in celebration of Leslie Valiant's 60th birthday. Talks were given by Jin-Yi Cai, Stephen Cook, Vitaly Feldman, Mark Jerrum, Michael Kearns, Mike Paterson, Michael Rabin, Rocco Servedio, Paul Valiant, Vijay Vazirani, and Avi Wigderson. The workshop w...
Any proof of P≠NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linear-size circuits) that overcome both barriers simultaneously. So the question arises of whether there is a third barrier to progress on the central questions in...
Randomness extractors are efficient algorithms which convert weak random sources into nearly perfect ones. While such purification of randomness was the original motivation for constructing extractors, these constructions turn out to have strong pseudorandom properties which found applications in diverse areas of computer science and combinatorics....
The classical Direct-Product Theorem for circuits says that if a Boolean function f: {0, 1} n → {0, 1} is somewhat hard to compute on average by small circuits, then the corresponding k-wise direct product function f k (x1,..., xk) = (f(x1),..., f(xk)) (where each xi ∈ {0, 1} n) is significantly harder to compute on average by slightly smaller circ...
A merger is a probabilistic procedure which extracts the randomness out of any (arbitrarily correlated) set of random variables, as long as one of them is uniform. Our main result is an efficient, simple, optimal (to constant factors) merger, which, for k random vairables on n bits each, uses a O(log(nk)) seed, and whose error is 1/nk. Our merger c...
Man has grappled with the meaning and utility of randomness for centuries. Research in the Theory of Computation in the last thirty years has enriched this study considerably. This lecture will describe two main aspects of this research on randomness, demonstrating its power and weakness respectively.
Randomness is paramount to computational effici...
Any proof of P!=NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linear-size circuits) that overcome both barriers simultaneously. So the question arises of whether there is a third barrier to progress on the central questions in...
Ahlswede and Winter (IEEE Trans. Inf. Th. 2002) introduced a Chernoff bound for matrix-valued random variables, which is a non-trivial generalization of the usual Chernoff bound for real-valued random variables. We present an efficient derandomization of their bound using the method of pessimistic estimators (see Raghavan (JCSS 1988)). As a consequ...
This paper presents a unified and simple treatment of basic questions concern- ing two computational models: multiparty communication complexity and polynomials over GF(2). The key is the use of (known) norms on Boolean functions, which capture their proximity to each of these models (and are closely related to property testers of this proximity)....
It is well-known that \(\mathbb R^N\) has subspaces of dimension proportional to N on which the ℓ1 and ℓ2 norms are uniformly equivalent, but it is unknown how to construct them explicitly. We show that, for any δ> 0, such a subspace can be generated using only N
δ
random bits. This improves over previous constructions of Artstein-Avidan and Milman...
Any proof of P ≠ NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (e.g., that PP does not have linear-size circuits) that overcome both barriers simultaneously. So the question arises of whether there is a third barrier to progress on the central questions in compl...
In this paper we construct explicit deterministic extractors from polynomial sources, which are distributions sampled by low degree multivariate polynomials over finite fields. This naturally generalizes previous work on extraction from affine sources (which are degree 1 polynomials). A direct consequence is a deterministic extractor for distributi...
This paper presents a unified and simple treatment of basic questions concerning two computational models: multiparty communication complexity and GF(2) polynomials. The key is the use of (known) norms on Boolean functions, which capture their approximability in each of these models. The main contributions are new XOR lemmas. We show that if a Bool...
We study the communication complexity of the disjointness function,
in which each of two players holds a
$k$-subset of a universe of size $n$ and the goal is to determine
whether the sets are disjoint. In the model of a common random
string we prove that
$O(k)$ communication bits
are sufficient, regardless of $n$.
In the model of private random co...
The Nisan–Wigderson pseudo-random generator [19] was constructed to derandomize probabilistic algorithms under the assumption that there exist explicit functions which are hard for small circuits. We give the first explicit construction of a pseudo-random generator with asymptotically optimal seed length even when given a function which is hard for...
We prove that two-party randomized communication complexity satisfies a strong direct product property, so long as the communication lower bound is proved by a "corruption" or "one-sided discrep- ancy" method over a rectangular distribution. We use this to prove new n(1) lower bounds for 3-player number-on-the-forehead protocols in which the first...
The main result of this paper is an explicit disperser for two independent sources on n bits, each of entropy k = no(1). Put differently, setting N = 2n and K = 2k, we construct explicit N × N Boolean matrices for which no K × K sub- matrix is monochromatic. Viewed as adjacency matrices of bipartite graphs, this gives an explicit construction of K-...