In recent years, sparse principal component analysis has emerged as an
extremely popular dimension reduction technique for high-dimensional data. The
theoretical challenge, in the simplest case, is to estimate the leading
eigenvector of a population covariance matrix under the assumption that this
eigenvector is sparse. An impressive range of estimators have been proposed;
some of these are fast to compute, while others are known to achieve the
minimax optimal rate over certain Gaussian or sub-Gaussian classes. In this
paper, we show that, under a widely-believed assumption from computational
complexity theory, there is a fundamental trade-off between statistical and
computational performance in this problem. More precisely, working with new,
larger classes satisfying a restricted covariance concentration condition, we
show that there is an effective sample size regime in which no randomised
polynomial time algorithm can achieve the minimax optimal rate. We also study
the theoretical performance of a (polynomial time) variant of the well-known
semidefinite relaxation estimator, revealing a subtle interplay between
statistical and computational efficiency.

通过研究计算复杂性理论，发现在满足一定限制的协方差集中条件下存在有效的样本大小范围，在此范围内无法有随机多项式时间算法达到最佳极小风险率；对著名的半定松弛估计方法的理论性能进行研究，揭示了统计效率和计算效率之间微妙的相互作用，此方法为多维数据稀疏主成分分析提供了一种解决方案。

稀疏主成分估计中的统计和计算折衷

Statistical and computational trade-offs in estimation of sparse  principal components

Principal Component Analysis (PCA) is an important tool of dimension
reduction especially when the dimension (or the number of variables) is very
high. Asymptotic studies where the sample size is fixed, and the dimension
grows [i.e., High Dimension, Low Sample Size (HDLSS)] are becoming increasingly
relevant. We investigate the asymptotic behavior of the Principal Component
(PC) directions. HDLSS asymptotics are used to study consistency, strong
inconsistency and subspace consistency. We show that if the first few
eigenvalues of a population covariance matrix are large enough compared to the
others, then the corresponding estimated PC directions are consistent or
converge to the appropriate subspace (subspace consistency) and most other PC
directions are strongly inconsistent. Broad sets of sufficient conditions for
each of these cases are specified and the main theorem gives a catalogue of
possible combinations. In preparation for these results, we show that the
geometric representation of HDLSS data holds under general conditions, which
includes a $\rho$-mixing condition and a broad range of sphericity measures of
the covariance matrix.

研究 PCA 在高维，低样本大小的情况下的渐近行为，发现在一些充分的条件下，估计的 PC 方向是一致的，其他的方向强不一致，而这些条件在主定理中指定。