Seminars Sorted by Series

This talk will be a partial survey of the first questions in the complexity theory of geometric topology problems. What is the complexity, or what are known complexity bounds, for distinguishing n-manifolds for various n? For distinguishing knots...

If $x$, $y$, and $z$ are random vectors, what is the expectation of the product of their inner products, $\langle x,y \rangle$, $\langle y, z \rangle$, $\langle z, x\rangle$? If $U$ and $V$ are random unitary matrices, what is the expected trace of...

A matrix A naturally defines a quadratic form x^tAy. If A is of rank <=r, then the rank<=r decomposition of A corresponds naturally to a size ~2nr circuit for computing the quadratic form. It is clear how to perform "white box" polynomial identity testing for such circuits, and the motivating question for this work is to explore black-box identity testing. The probabilistic method shows that there is a size ~2nr hitting set for such polynomials. In this work we match this bound (over large fields), which is optimal up to constant factors. Further, we explore how A can be reconstructed from the evaluations of the quadratic form. Similar probabilistic constructions show that there exist ~4nr evaluations which determine any such matrix A. Again, we match this bound (over large fields) with an explicit construction, and furthermore give a polynomial-time algorithm to reconstruct A from such evaluations. More generally, we show an efficient reduction from (exact) low-rank matrix reconstruction to (exact) sparse recovery. This reduction is novel in the compressed-sensing realm as it is field independent and unrelated to convex optimization. Finally, using matrices as a base case we also derive a quasi-polynomial hitting set for higher-order tensors. Joint work with Michael Forbes.

We prove a strong limitation on the ability of entangled provers to collude in a multiplayer game. Our main result is the first nontrivial lower bound on the class MIP* of languages having multi-prover interactive proofs with entangled provers...

We study algorithms for combinatorial market design problems, where a collection of objects are priced and sold to potential buyers subject to equilibrium constraints. We introduce the notion of a combinatorial Walrasian equilibrium (CWE) as a...

Holant Problems are a broad framework to describe counting problems. The framework generalizes counting Constraint Satisfaction Problems and partition functions of Graph Homomorphisms. We prove a complexity dichotomy theorem for Holant problems over...

Polynomial Identity Testing (PIT) is the problem of identifying whether a given algebraic circuit computes the identically zero polynomial. It is well-known that this problem can be solved with small error probability by testing whether the circuit...

In this talk we will discuss information complexity -- a measure of the amount of information Alice and Bob need to exchange to solve a problem over distributed inputs. We will present an information-theoretically optimal protocol for computing the...

For all practical purposes, the Micali-Vazirani algorithm, discovered in 1980, is still the most efficient known maximum matching algorithm (for very dense graphs, slight asymptotic improvement can be obtained using fast matrix multiplication)...

For a set of polynomials F, we define their bilinear complexity as the smallest k so that F lies in an ideal generated by k bilinear polynomials. The main open problem is to estimate the bilinear complexity of the single polynomial $\sum_{i,j}x_i^2...

In small dimension a random geometric graph behaves very differently from a standard Erdös-Rényi random graph. On the other hand, when the dimension tends to infinity (with the number of vertices being fixed) both models coincide. In this talk we...

We study the problem of constructing extractors for independent weak random sources. The probabilistic method shows that such an extractor exists for two sources on n bits with min-entropy k >= 2 log n. On the other hand, explicit constructions are...

I will describe some recent results and problems regarding influence of sets of variables on Boolean functions: In 1989 Benny Chor conjectured that a balanced Boolean function with n variables has a subset S of size 0.4n with influence (1-c^n) where...

With the notion of interaction with oracles as a unifying theme of much of my dissertation work, I discuss novel models and results for property testing and computational learning, with the use of Fourier analytic and probabilistic methods. One...

Polar codes have recently emerged as a new class of low-complexity codes achieving Shannon capacity. This talk introduces polar codes with emphasis on the probabilistic phenomenon underlying the code construction. New results and connections to...

Since the foundational results of Thomason and Chung-Graham-Wilson on quasirandom graphs over 20 years ago, there has been a lot of effort by many researchers to extend the theory to hypergraphs. I will present some of this history, and then...

We discuss three areas of algorithmic game theory that have grappled with intractability. The first is the complexity of computing game-theoretic equilibria, like Nash equilibria. There is an urgent need for new ideas on this topic, to enable...

The PCP theorem (Arora et. al., J. ACM 45(1,3)) says that every NP-proof can be encoded to another proof, namely, a probabilistically checkable proof (PCP), which can be tested by a verifier that queries only a small part of the PCP. A natural...

Locally decodable codes (LDCs) are error-correcting codes that allow for highly-efficient recovery of "pieces" of information even after arbitrary corruption of a codeword. Locally testable codes (LTCs) are those that allow for highly-efficient...

A trusted source of independent and uniform random bits is a basic resource in many computational tasks, such as cryptography, game theoretic protocols, algorithms and physical simulations. Implementing such a source presents an immediate challenge...

We propose an “analytical” framework for studying parallel repetitions of one-round two-prover games. We define a new relaxation of the value of a game, val+, and prove that it is both multiplicative and a good approximation for the true value of...

We study some problems relating to polynomials evaluated either at random Gaussian or random Bernoulli inputs. We present some new work on a structure theorem for degree-d polynomials with Gaussian inputs. In particular, if p is a given degree-d...

We present practically efficient methods for proving correctness of announced results of a computation while keeping input and intermediate values information theoretically secret. These methods are applied to solve the long standing problem of...

In many distributed systems, the cost of computation is dominated by the cost of communication between the machines participating in the computation. Communication complexity is therefore a very useful tool in understanding distributed computation...

In this talk I will describe nondeterministic reductions which yield new direct product theorems (DPTs) for Boolean circuits. In our theorems one assumes that a function F is "mildly hard" against *nondeterministic* circuits of some size s(n) , and...

Finding cliques in random graphs and the closely related "planted" clique variant, where a clique of size k is planted in a random G(n,1/2) graph, have been the focus of substantial study in algorithm design. Despite much effort, the best known...

The Depth First Search (DFS) algorithm is one of the most standard graph exploration algorithms, used normally to find the connected components of an input graph. Though perhaps less popular than its sister algorithm, Breadth First Search (BFS), the...

Deep learning, a modern version of neural nets, is increasingly seen as a promising way to implement AI tasks such as speech recognition and image recognition. Most current algorithms are heuristic and have no provable guarantees. This talk will...

For a permutation \(p\), let \(S_n(p)\) be the number of permutations on \(n\) letters avoiding \(p\). Stanley and Wilf conjectured that, for each permutation \(p\), \(S_n(p)^{1/n}\) tends to a finite limit \(L(p)\). Marcus and Tardos proved the...

The goal of (general-purpose) program obfuscation is to make an arbitrary computer program "unintelligible" while preserving its functionality. The problem of program obfuscation is well studied both in theory and in practice. Though despite its...

Let \(K\) and \(T\) be convex bodies in the \(n\)-dimensional Euclidean space. The covering number of \(K\) by \(T\) is the minimal number of translates of \(T\) required to cover \(K\) entirely. One open question regarding this classical notion is...

Given data stream \(D = \{p_1,p_2,...,p_m\}\) of size \(m\) of numbers from \(\{1,..., n\}\), the frequency of \(i\) is defined as \(f_i = |\{j: p_j = i\}|\). The \(k\)-th frequency moment of \(D\) is defined as \(F_k = \sum_{i=1}^n f_i^k\). We...

We use critical block sensitivity, a new complexity measure introduced by Huynh and Nordstrom (STOC 2012) to study the communication complexity of search problems. Our main result is a simple proof that if \(S\) is a search problem with high...

Machine learning is often employed as one step in a larger application, serving to perform information extraction or data mining for example. The rules obtained by such learning are then used as inputs to a further analysis. As a consequence of the...

Algebraic geometry and representation theory have played an important role in obtaining lower bounds in algebraic complexity theory. After giving an overview of the general set-up, I will present very recent results that indicate a possible role for...

We will outline the proof that gives a positive solution of to Weaver's conjecture \(KS_2\). That is, we will show that any isotropic collection of vectors whose outer products sum to twice the identity can be partitioned into two parts such that...

Most of the cryptosystems we currently use are highly secure, and cannot be broken with reasonable complexity by mathematical cryptanalysis. However, over the last fifteen years researchers have developed many types of physical attacks on their...

We construct a bi-Lipschitz bijection from the Boolean cube to the Hamming ball of equal volume. More precisely, we show that for all even \(n\), there exists an explicit bijection \(f\) from the \(n\)-dimensional Boolean cube to the Hamming ball of...

In this talk, I'll describe connections between the unique games conjecture (or more precisely, the closely relatedly problem of small-set expansion) and the quantum separability problem. Remarkably, not only are the problems related, but the...

An error-correcting code is called locally decodable if there exists a decoding algorithm that can recover any symbol of the message with high probability by reading only a small number of symbols of the corrupted codeword. There is a fundamental...

One of the key results in Robertson and Seymour's seminal work on graph minors is the Grid-Minor Theorem (also known as the Excluded Grid Theorem). The theorem states that any graph of treewidth at least \(k\) contains a grid minor of size \(f(k)\)...

The Geometric Complexity Theory (GCT) Program is an approach towards P versus NP and other lower bounds using algebraic geometry and representation theory. In this talk, we discuss how essentially all known algebraic circuit lower bounds and...