张量一致性检验、低秩恢复与压缩感知
On Identity Testing of Tensors, Low-rank Recovery and Compressed Sensing
摘要：
我们研究的问题，获得有效的，确定性的，黑盒多项式身份测试算法的深度3集多线性电路（在任意字段）。这类电路有一个有效的，确定性的，白盒多项式身份测试算法（由于Raz和Shpilka），但没有已知的黑盒算法。我们将这个问题重新定义为寻找一个低维子空间H，由秩为1的张量张成，使得对偶空间ker（H）中的任何非零张量都具有高秩。我们得到明确的结构，基本上最佳大小的击中2度张量（矩阵），并获得准多项式大小的击中任意张量（但这第二次击中集是不太明确）。我们还显示了连接到执行低秩恢复矩阵的任务，这是在压缩感知领域的研究。低秩恢复要求（比如说，在实数上）在保证M的秩<=r的情况下，从几个测量值中恢复矩阵M。我们还给出了低秩恢复和稀疏（向量）恢复任务之间的正式联系：任何稀疏恢复算法，精确恢复长度为n和稀疏度为2 r的向量，使用m个非自适应测量，产生一个低秩恢复方案，用于精确恢复秩<=r的nxn矩阵，进行2nm非自适应测量。此外，如果稀疏恢复算法在时间τ内运行，则低秩恢复算法在时间O（rn^2+nτ）内运行。我们使用线性代数技术获得这种减少，而不是使用凸优化，这是更常见的压缩感知算法。通过使用对偶Reed-Solomon码，我们能够（确定性地）构造在实数上进行4 nr次测量的低秩恢复方案，使得测量可以是所有秩1矩阵或所有稀疏矩阵。
英文摘要：
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 τ, then the low-rank recovery algorithm runs in time O(rn^2+nτ). 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.
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