Journal of Algorithms
Lower bounds on arithmetic circuits via partial derivatives
Computational Complexity
Learning functions represented as multiplicity automata
Journal of the ACM (JACM)
Randomness efficient identity testing of multivariate polynomials
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Testing polynomials which are easy to compute (Extended Abstract)
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
Deterministic polynomial identity testing in non-commutative models
Computational Complexity
Derandomizing polynomial identity tests means proving circuit lower bounds
Computational Complexity
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
Towards Dimension Expanders over Finite Fields
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
Arithmetic Circuits: A Chasm at Depth Four
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Arithmetic Circuits: A survey of recent results and open questions
Foundations and Trends® in Theoretical Computer Science
Blackbox identity testing for bounded top fanin depth-3 circuits: the field doesn't matter
Proceedings of the forty-third annual ACM symposium on Theory of computing
On the Power of Adaptivity in Sparse Recovery
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Proving lower bounds via pseudo-random generators
FSTTCS '05 Proceedings of the 25th international conference on Foundations of Software Technology and Theoretical Computer Science
Tensor codes for the rank metric
IEEE Transactions on Information Theory - Part 2
Shift-register synthesis and BCH decoding
IEEE Transactions on Information Theory
Maximum-rank array codes and their application to crisscross error correction
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Quasi-polynomial hitting-set for set-depth-Δ formulas
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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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 [36]), 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 the first quasi-polynomial sized hitting sets for arbitrary tensors. 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 R) to recover a matrix M from few measurements, under the promise that M is rank ≤ r. In this work, we restrict our attention to recovering matrices that are exactly rank ≤ r using deterministic, non-adaptive, linear measurements, that are free from noise. Over R, we provide a set (of size 4nr) of such measurements, from which M can be recovered in O(rn2+r3n) 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 x n matrices of rank ≤ r, making 2nm non-adaptive measurements. Furthermore, if the sparse-recovery algorithm runs in time τ, then the low-rank recovery algorithm runs in time O(rn2+nτ). 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 M and N is rank(M-N), 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.