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• W3103519306 abstract "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 H, spanned by rank 1 tensors, such that any non-zero tensor in the dual space ker(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 the reals) to recover a matrix M from few measurements, under the promise that M is rank <=r. We also give a 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 nxn matrices of rank <=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+ntau). We obtain this reduction using linear-algebraic techniques, and not using convex optimization, which is more commonly seen in compressed sensing algorithms. By using a dual Reed-Solomon code, we are able to (deterministically) construct low-rank recovery schemes taking 4nr measurements over the reals, such that the measurements can be all rank-1 matrices, or all sparse matrices." @default.
• W3103519306 created "2020-11-23" @default.
• W3103519306 creator A5004182247 @default.
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• W3103519306 date "2012-05-19" @default.
• W3103519306 modified "2023-10-16" @default.
• W3103519306 title "On identity testing of tensors, low-rank recovery and compressed sensing" @default.
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• W3103519306 doi "https://doi.org/10.1145/2213977.2213995" @default.
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