Improved Bounds for Universal One-Bit Compressive Sensing

Unlike compressive sensing where the measurement outputs are assumed to be real-valued and have infinite precision, in "one-bit compressive sensing", measurements are quantized to one bit, their signs. In this work, we show how to recover the support of sparse high-dimensional vectors in the one-bit compressive sensing framework with an asymptotically near-optimal number of measurements. We also improve the bounds on the number of measurements for approximately recovering vectors from one-bit compressive sensing measurements. Our results are universal, namely the same measurement scheme works simultaneously for all sparse vectors. Our proof of optimality for support recovery is obtained by showing an equivalence between the task of support recovery using 1-bit compressive sensing and a well-studied combinatorial object known as Union Free Families.Comment: 14 page

Testing Poisson Binomial Distributions

A Poisson Binomial distribution over $n$ variables is the distribution of the sum of $n$ independent Bernoullis. We provide a sample near-optimal algorithm for testing whether a distribution $P$ supported on $\{0,...,n\}$ to which we have sample access is a Poisson Binomial distribution, or far from all Poisson Binomial distributions. The sample complexity of our algorithm is $O(n^{1/4})$ to which we provide a matching lower bound. We note that our sample complexity improves quadratically upon that of the naive "learn followed by tolerant-test" approach, while instance optimal identity testing [VV14] is not applicable since we are looking to simultaneously test against a whole family of distributions.Comment: To appear in ACM-SIAM Symposium on Discrete Algorithms (SODA) 201

Testing Poisson Binomial Distributions

A Poisson Binomial distribution over n variables is the distribution of the sum of n independent Bernoullis. We provide a sample near-optimal algorithm for testing whether a distribution P supported on {0, …, n} to which we have sample access is a Poisson Binomial distribution, or far from all Poisson Binomial distributions. The sample complexity of our algorithm is O(n[superscript 1/4]) to which we provide a matching lower bound. We note that our sample complexity improves quadratically upon that of the naive “learn followed by tolerant-test” approach, while instance optimal identity testing [VV14] is not applicable since we are looking to simultaneously test against a whole family of distributions.Shell-MITEI Seed Fund ProgramAlfred P. Sloan Foundation (Fellowship)Microsoft Research (Faculty Fellowship)National Science Foundation (U.S.) (CAREER Award CCF-0953960)National Science Foundation (U.S.) (Award CCF-1101491

The role of shared randomness in quantum state certification with unentangled measurements

Given $n$ copies of an unknown quantum state $\rho\in\mathbb{C}^{d\times d}$, quantum state certification is the task of determining whether $\rho=\rho_0$ or $\|\rho-\rho_0\|_1>\varepsilon$, where $\rho_0$ is a known reference state. We study quantum state certification using unentangled quantum measurements, namely measurements which operate only on one copy of $\rho$ at a time. When there is a common source of shared randomness available and the unentangled measurements are chosen based on this randomness, prior work has shown that $\Theta(d^{3/2}/\varepsilon^2)$ copies are necessary and sufficient. This holds even when the measurements are allowed to be chosen adaptively. We consider deterministic measurement schemes (as opposed to randomized) and demonstrate that ${\Theta}(d^2/\varepsilon^2)$ copies are necessary and sufficient for state certification. This shows a separation between algorithms with and without shared randomness. We develop a unified lower bound framework for both fixed and randomized measurements, under the same theoretical framework that relates the hardness of testing to the well-established L\"uders rule. More precisely, we obtain lower bounds for randomized and fixed schemes as a function of the eigenvalues of the L\"uders channel which characterizes one possible post-measurement state transformation.Comment: 29 pages, 2 tables. Comments welcom