The multiplayer promise set disjointness is one of the most widely used problems from communication complexity in applications. In this problem there are k players with subsets S¹, …, S^k, each drawn from {1, 2, …, n}, and we are promised that either the sets are (1) pairwise disjoint, or (2) there is a unique element j occurring in all the sets, which are otherwise pairwise disjoint. The total communication of solving this problem with constant probability in the blackboard model is Ω(n/k). We observe for most applications, it instead suffices to look at what we call the "mostly" set disjointness problem, which changes case (2) to say there is a unique element j occurring in at least half of the sets, and the sets are otherwise disjoint. This change gives us a much simpler proof of an Ω(n/k) randomized total communication lower bound, avoiding Hellinger distance and Poincare inequalities. Our proof also gives strong lower bounds for high probability protocols, which are much larger than what is possible for the set disjointness problem. Using this we show several new results for data streams: 1) for 𝓁₂-Heavy Hitters, any O(1)-pass streaming algorithm in the insertion-only model for detecting if an ε-𝓁₂-heavy hitter exists requires min(1/(ε²)log((ε²n)/δ), 1/(ε)n^{1/2}) bits of memory, which is optimal up to a log n factor. For deterministic algorithms and constant ε, this gives an Ω(n^{1/2}) lower bound, improving the prior Ω(log n) lower bound. We also obtain lower bounds for Zipfian distributions. 2) for 𝓁_p-Estimation, p > 2, we show an O(1)-pass Ω(n^{1-2/p} log(1/δ)) bit lower bound for outputting an O(1)- approximation with probability 1-δ, in the insertion-only model. This is optimal, and the best previous lower bound was Ω(n^{1-2/p} + log(1/δ)). 3) for low rank approximation of a sparse matrix in ℝ^{d× n}, if we see the rows of a matrix one at a time in the row-order model, each row having O(1) non-zero entries, any deterministic algorithm requires Ω(√d) memory to output an O(1)-approximate rank-1 approximation. Finally, we consider strict and general turnstile streaming models, and show separations between sketching lower bounds and non-sketching upper bounds for the heavy hitters problem.

The goal of compressed sensing is to learn a structured signal $x$ from a limited number of noisy linear measurements $y \approx Ax$. In traditional compressed sensing, “structure” is represented by sparsity in some known basis. Inspired by the success of deep learning in modeling images, recent work starting with Bora-Jalal-Price-Dimakis’17 has instead considered structure to come from a generative model $G: \mathbb{R}^k \to \mathbb{R}^n$. We present two results establishing the difficulty and strength of this latter task, showing that existing bounds are tight: First, we provide a lower bound matching the Bora et.al upper bound for compressed sensing with $L$-Lipschitz generative models $G$ which holds even for the more relaxed goal of \emph{non-uniform} recovery. Second, we show that generative models generalize sparsity as a representation of structure by constructing a ReLU-based neural network with $2$ hidden layers and $O(n)$ activations per layer whose range is precisely the set of all $k$-sparse vectors.

The goal of adaptive sparse recovery is to estimate an approximately sparse vector x from a series of linear measurements A1x, A2x, …, ARx, where each matrix Ai may depend on the previous observations. With an unlimited number of rounds R, it is known that O(k log log n) measurements suffice for O(1)-approximate k-sparse recovery in ℝn, and that Ω(k + log log n) measurements are necessary. We initiate the study of what happens with a constant number of rounds of adaptivity. Previous techniques could not give nontrivial bounds using less than 5 rounds of adaptivity, and were inefficient for any constant R. We give nearly matching upper and lower bounds for any constant number of rounds R. Our lower bound shows that measurements are necessary for any k < ; significantly, this is the first lower bound that combines k and n in an adaptive setting. Our upper bound shows that measurements suffice. The O(log* k) gap between the two bounds comes from a similar gap for nonadaptive sparse recovery in the high-SNR regime, and would be reduced to constant factors with improvements to nonadaptive high-SNR sparse recovery.

Pseudorandom functions (PRFs) play a fundamental role in symmetric-key cryptography. However, they are inherently complex and cannot be implemented in the class AC0 (MOD2). Weak pseudorandom functions (weak PRFs) do not suffer from this complexity limitation, yet they suffice for many cryptographic applications.