We show how to solve a number of problems in numerical linear algebra, such as least squares regression, lp-regression for any p ≥ 1, low rank approximation, and kernel regression, in time T(A)poly(log(nd)), where for a given input matrix A ∈ Rn×d, T(A) is the time needed to compute A · y for an arbitrary vector y ∈ Rd. Since T(A) ≤ O(nnz(A)), where nnz(A) denotes the number of non-zero entries of A, the time is no worse, up to polylogarithmic factors, as all of the recent advances for such problems that run in input-sparsity time. However, for many applications, T(A) can be much smaller than nnz(A), yielding significantly sublinear time algorithms. For example, in the overconstrained (1+ε)-approximate polynomial interpolation problem, A is a Vandermonde matrix and T(A) = O(n log n); in this case our running time is n · poly (log n) + poly (d/ε) and we recover the results of Avron, Sindhwani, and Woodruff (2013) as a special case. For overconstrained autoregression, which is a common problem arising in dynamical systems, T(A) = O(n log n), and we immediately obtain n· poly (log n) + poly(d/ε) time. For kernel autoregression, we significantly improve the running time of prior algorithms for general kernels. For the important case of autoregression with the polynomial kernel and arbitrary target vector b ∈ Rn, we obtain even faster algorithms. Our algorithms show that, perhaps surprisingly, most of these optimization problems do not require much more time than that of a polylogarithmic number of matrix-vector multiplications.

We consider peer review under a conference setting where there are conflicts between the reviewers and the submissions. Under such conflicts, reviewers can manipulate their reviews in a strategic manner to influence the final rankings of their own papers. Present-day peer-review systems are not designed to guard against such strategic behavior, beyond minimal (and insufficient) checks such as not assigning a paper to a conflicted reviewer. In this work, we address this problem through the lens of social choice, and present a theoretical framework for strategyproof and efficient peer review. Given the conflict graph which satisfies a simple property, we first present and analyze a flexible framework for reviewer-assignment and aggregation for the reviews that guarantees not only strategyproofness but also a natural efficiency property (unanimity). Our framework is based on the so-called partitioning method, and can be treated as a generalization of this type of method to conference peer review settings. We then empirically show that the requisite property on the (authorship) conflict graph is indeed satisfied in the ICLR-17 submissions data, and further demonstrate a simple trick to make the partitioning method more practically appealing under conference peer-review settings. Finally, we complement our positive results with negative theoretical results where we prove that under slightly stronger requirements, it is impossible for any algorithm to be both strategyproof and efficient.

In the problem of adaptive compressed sensing, one wants to estimate an approximately k-sparse vector x in R^n from m linear measurements A_1 x, A_2 x,..., A_m x, where A_i can be chosen based on the outcomes A_1 x,..., A_{i-1} x of previous measurements. The goal is to output a vector x^ for which |x-x^|_p <=C * min_{k-sparse x'} |x-x'|_q, with probability at least 2/3, where C > 0 is an approximation factor. Indyk, Price and Woodruff (FOCS'11) gave an algorithm for p=q=2 for C = 1+epsilon with O((k/epsilon) loglog (n/k)) measurements and O(log^*(k) loglog (n)) rounds of adaptivity. We first improve their bounds, obtaining a scheme with O(k * loglog (n/k) + (k/epsilon) * loglog(1/epsilon)) measurements and O(log^*(k) loglog (n)) rounds, as well as a scheme with O((k/epsilon) * loglog (n log (n/k))) measurements and an optimal O(loglog (n)) rounds. We then provide novel adaptive compressed sensing schemes with improved bounds for (p,p) for every 0 < p < 2. We show that the improvement from O(k log(n/k)) measurements to O(k log log (n/k)) measurements in the adaptive setting can persist with a better epsilon-dependence for other values of p and q. For example, when (p,q) = (1,1), we obtain O(k/sqrt{epsilon} * log log n log^3 (1/epsilon)) measurements. We obtain nearly matching lower bounds, showing our algorithms are close to optimal. Along the way, we also obtain the first nearly-optimal bounds for (p,p) schemes for every 0 < p < 2 even in the non-adaptive setting.