Lower Bounds on Adaptive Sensing for Matrix Recovery

We study lower bounds on adaptive sensing algorithms for recovering low rank matrices using linear measurements. Given an n × n matrix A, a general linear measurement S(A), for an n × n matrix S, is just the inner product of S and A, each treated as n2-dimensional vectors. By performing as few linear measurements as possible on a rank-r matrix A, we hope to construct a matrix hatA that satisfies |A - hatA|F2 ≤ c|A|F2, for a small constant c. It is commonly assumed that when measuring A with S, the response is corrupted with an independent Gaussian random variable of mean 0 and variance σ2. Cand'es and Plan study non-adaptive algorithms for low rank matrix recovery using random linear measurements. At a certain noise level, it is known that their non-adaptive algorithms need to perform Omega(n2) measurements, which amounts to reading the entire matrix. An important question is whether adaptivity helps in decreasing the overall number of measurements. We show that any adaptive algorithm that uses k linear measurements in each round and outputs an approximation to the underlying matrix with probability ≥ 9/10 must run for t = Omega(log(n2/k)/loglog n) rounds showing that any adaptive algorithm which uses n2-β linear measurements in each round must run for Omega(log n/loglog n) rounds to compute a reconstruction with probability ≥ 9/10. Hence any adaptive algorithm that has o(log n/loglog n) rounds must use an overall Omega(n2) linear measurements. Our techniques also readily extend to obtain lower bounds on adaptive algorithms for tensor recovery and obtain measurement-vs-rounds trade-off for many sensing problems in numerical linear algebra, such as spectral norm low rank approximation, Frobenius norm low rank approximation, singular vector approximation, and more.