Related papers: A Tighter Relation Between Hereditary Discrepancy …
We consider the classic Set Cover problem in the data stream model. For $n$ elements and $m$ sets ($m\geq n$) we give a $O(1/\delta)$-pass algorithm with a strongly sub-linear $\tilde{O}(mn^{\delta})$ space and logarithmic approximation…
The problem of finding \emph{distance} between \emph{pattern} of length $m$ and \emph{text} of length $n$ is a typical way of generalizing pattern matching to incorporate dissimilarity score. For both Hamming and $L_1$ distances only a…
We prove the lower bound R(M_m) \geq 3/2 m^2 - 2 on the border rank of m x m matrix multiplication by exhibiting explicit representation theoretic (occurence) obstructions in the sense of the geometric complexity theory (GCT) program. While…
In the online matching on the line problem, the task is to match a set of requests $R$ online to a given set of servers $S$. The distance metric between any two points in $R\,\cup\, S$ is a line metric and the objective for the online…
We present a new explicit formula for the determinant that contains superexponentially fewer terms than the usual Leibniz formula. As an immediate corollary of our formula, we show that the tensor rank of the $n \times n$ determinant tensor…
Let $\mathbf{A}\in \mathbb{R}^{n\times n}$ be a matrix with diagonal $\text{diag}(\mathbf{A})$ and let $\bar{\mathbf{A}}$ be $\mathbf{A}$ with its diagonal set to all zeros. We show that Hutchinson's estimator run for $m$ iterations returns…
In a recent breakthrough result, Balliu et al. [FOCS'19] proved a deterministic $\Omega(\min(\Delta,\log n /\log \log n))$-round and a randomized $\Omega(\min(\Delta,\log \log n/\log \log \log n))$-round lower bound for the complexity of…
A fooling-set matrix has nonzero diagonal, but at least one in every pair of diagonally opposite entries is 0. Dietzfelbinger et al. '96 proved that the rank of such a matrix is at least $\sqrt n$. It is known that the bound is tight (up to…
We establish sharp non-asymptotic probabilistic bounds for the star discrepancy of double-infinite random matrices -- a canonical model for sequences of random point sets in high dimensions. By integrating the recently proved…
Consider an infinite sequence of independent, uniformly chosen points from $[0,1]^d$. After looking at each point in the sequence, an overseer is allowed to either keep it or reject it, and this choice may depend on the locations of all…
We consider the problem of minimizing a continuous function f over a compact set K. We analyze a hierarchy of upper bounds proposed by Lasserre in [SIAM J. Optim. 21(3) (2011), pp. 864--885], obtained by searching for an optimal probability…
We provide a variety of lower bounds for the well-known shortcut set problem: how much can one decrease the diameter of a directed graph on $n$ vertices and $m$ edges by adding $O(n)$ or $O(m)$ of shortcuts from the transitive closure of…
We show that the set of maximal lower bounds of two symmetric matrices with respect to the L\"owner order can be identified to the quotient set $O(p,q)/(O(p)\times O(q))$. Here, $(p,q)$ denotes the inertia of the difference of the two…
We investigate the minimum distance of structured binary Low-Density Parity-Check (LDPC) codes whose parity-check matrices are of the form $[\mathbf{C} \vert \mathbf{M}]$ where $\mathbf{C}$ is circulant and of column weight $2$, and…
We overcome two major bottlenecks in the study of low rank approximation by assuming the low rank factors themselves are sparse. Specifically, (1) for low rank approximation with spectral norm error, we show how to improve the best known…
Chaos, in quantum systems, can be diagnosed by certain out-of-time-order correlators (OTOCs) that obey the chaos bound of Maldacena, Shenker, and Stanford (MSS). We begin by deriving a dispersion relation for this class of OTOCs, implying…
This paper develops a new technique for proving amortized, randomized cell-probe lower bounds on dynamic data structure problems. We introduce a new randomized nondeterministic four-party communication model that enables "accelerated",…
Recent research has introduced a key notion of $H$-consistency bounds for surrogate losses. These bounds offer finite-sample guarantees, quantifying the relationship between the zero-one estimation error (or other target loss) and the…
The classic exact pattern matching problem, given two strings -- a pattern $P$ of length $m$ and a text $T$ of length $n$ -- asks whether $P$ occurs as a substring of $T$. A property tester for the problem needs to distinguish (with high…
We study the equivalence testing problem where the goal is to determine if the given two unknown distributions on $[n]$ are equal or $\epsilon$-far in the total variation distance in the conditional sampling model (CFGM, SICOMP16; CRS,…