TR11-147 Authors: Michael Forbes, Amir Shpilka

Publication: 2nd November 2011 22:00

Downloads: 2664

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We study the problem of obtaining efficient, deterministic, black-box polynomial identity testing algorithms for depth-3 set-multilinear circuits (over arbitrary fields). This class of circuits has an efficient, deterministic, white-box polynomial identity testing algorithm (due to Raz and Shpilka), but has no known such black-box algorithm. We recast this problem as a question of finding a low-dimensional subspace $\mathcal{H}$, spanned by rank 1 tensors, such that any non-zero tensor in the dual space $\ker(\mathcal{H})$ has high rank. We obtain explicit constructions of essentially optimal-size hitting sets for tensors of degree 2 (matrices), and obtain quasi-polynomial sized hitting sets for arbitrary tensors (but this second hitting set is less explicit).

We also show connections to the task of performing low-rank recovery of matrices, which is studied in the field of compressed sensing. Low-rank recovery asks (say, over $\mathbb{R}$) to recover a matrix $M$ from few measurements, under the promise that $M$ is rank $\le r$. In this work, we restrict our attention to recovering matrices that are exactly rank $\le r$ using deterministic, non-adaptive, linear measurements, that are free from noise. Over $\mathbb{R}$, we provide a set (of size $4nr$) of such measurements, from which $M$ can be recovered in $O(rn^2+r^3n)$ field operations, and the number of measurements is essentially optimal. Further, the measurements can be taken to be all rank-1 matrices, or all sparse matrices. To the best of our knowledge no explicit constructions with those properties were known prior to this work.

We also give a more formal connection between low-rank recovery and the task of sparse (vector) recovery: any sparse-recovery algorithm that exactly recovers vectors of length $n$ and sparsity $2r$, using $m$ non-adaptive measurements, yields a low-rank recovery scheme for exactly recovering $n\times n$ matrices of rank $\le r$, making $2nm$ non-adaptive measurements. Furthermore, if the sparse-recovery algorithm runs in time $\tau$, then the low-rank recovery algorithm runs in time $O(rn^2+n\tau)$. We obtain this reduction using linear-algebraic techniques, and not using convex optimization, which is more commonly seen in compressed sensing algorithms.

Finally, we also make a connection to rank-metric codes, as studied in coding theory. These are codes with codewords consisting of matrices (or tensors) where the distance of matrices $A$ and $B$ is $\text{rank}(A-B)$, as opposed to the usual hamming metric. We obtain essentially optimal-rate codes over matrices, and provide an efficient decoding algorithm. We obtain codes over tensors as well, with poorer rate, but still with efficient decoding.