We consider differentially private approximate singular vector computation.
Known worst-case lower bounds show that the error of any differentially private
algorithm must scale polynomially with the dimension of the singular vector. We
are able to replace this dependence on the dimension by a natural parameter
known as the coherence of the matrix that is often observed to be significantly
smaller than the dimension both theoretically and empirically. We also prove a
matching lower bound showing that our guarantee is nearly optimal for every
setting of the coherence parameter. Notably, we achieve our bounds by giving a
robust analysis of the well-known power iteration algorithm, which may be of
independent interest. Our algorithm also leads to improvements in worst-case
settings and to better low-rank approximations in the spectral norm.

研究了不同 ially private 近似奇异向量计算，用矩阵的连通度取代奇异向量的维度，证明了保证是近乎最优的，并给出了强大的功率迭代算法分析，这种算法在保证最差情形的同时，对于低秩近似也有改进。

私有奇异向量计算中超出最坏情况分析

Beyond Worst-Case Analysis in Private Singular Vector Computation

In this paper, a new class of circulant matrices built from deterministic
sequences is proposed for convolution-based compressed sensing (CS). In
contrast to random convolution, the coefficients of the underlying filter are
given by the discrete Fourier transform of a deterministic sequence with good
autocorrelation. Both uniform recovery and non-uniform recovery of sparse
signals are investigated, based on the coherence parameter of the proposed
sensing matrices. Many examples of the sequences are investigated, particularly
the Frank-Zadoff-Chu (FZC) sequence, the \textit{m}-sequence and the Golay
sequence. A salient feature of the proposed sensing matrices is that they can
not only handle sparse signals in the time domain, but also those in the
frequency and/or or discrete-cosine transform (DCT) domain.

本文提出了一类基于确定性序列构建的循环矩阵，用于卷积压缩感知。这些矩阵可以处理时间，频率和离散余弦转换（DCT）域内的稀疏信号， 并研究了统一恢复和非统一恢复的情况。