Computational barriers in minimax submatrix detection

This paper studies the minimax detection of a small submatrix of elevated mean in a large matrix contaminated by additive Gaussian noise. To investigate the tradeoff between statistical performance and computational cost from a complexity-theoretic perspective, we consider a sequence of discretized models which are asymptotically equivalent to the Gaussian model. Under the hypothesis that the planted clique detection problem cannot be solved in randomized polynomial time when the clique size is of smaller order than the square root of the graph size, the following phase transition phenomenon is established: when the size of the large matrix $p\to\infty$, if the submatrix size $k=\Theta(p^{\alpha})$ for any $\alpha\in(0,{2}/{3})$, computational complexity constraints can incur a severe penalty on the statistical performance in the sense that any randomized polynomial-time test is minimax suboptimal by a polynomial factor in $p$; if $k=\Theta(p^{\alpha})$ for any $\alpha\in({2}/{3},1)$, minimax optimal detection can be attained within constant factors in linear time. Using Schatten norm loss as a representative example, we show that the hardness of attaining the minimax estimation rate can crucially depend on the loss function. Implications on the hardness of support recovery are also obtained.Comment: Published at http://dx.doi.org/10.1214/14-AOS1300 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Optimal Estimation and Rank Detection for Sparse Spiked Covariance Matrices

This paper considers sparse spiked covariance matrix models in the high-dimensional setting and studies the minimax estimation of the covariance matrix and the principal subspace as well as the minimax rank detection. The optimal rate of convergence for estimating the spiked covariance matrix under the spectral norm is established, which requires significantly different techniques from those for estimating other structured covariance matrices such as bandable or sparse covariance matrices. We also establish the minimax rate under the spectral norm for estimating the principal subspace, the primary object of interest in principal component analysis. In addition, the optimal rate for the rank detection boundary is obtained. This result also resolves the gap in a recent paper by Berthet and Rigollet [1] where the special case of rank one is considered

Sparse PCA: Optimal rates and adaptive estimation

Principal component analysis (PCA) is one of the most commonly used statistical procedures with a wide range of applications. This paper considers both minimax and adaptive estimation of the principal subspace in the high dimensional setting. Under mild technical conditions, we first establish the optimal rates of convergence for estimating the principal subspace which are sharp with respect to all the parameters, thus providing a complete characterization of the difficulty of the estimation problem in term of the convergence rate. The lower bound is obtained by calculating the local metric entropy and an application of Fano's lemma. The rate optimal estimator is constructed using aggregation, which, however, might not be computationally feasible. We then introduce an adaptive procedure for estimating the principal subspace which is fully data driven and can be computed efficiently. It is shown that the estimator attains the optimal rates of convergence simultaneously over a large collection of the parameter spaces. A key idea in our construction is a reduction scheme which reduces the sparse PCA problem to a high-dimensional multivariate regression problem. This method is potentially also useful for other related problems.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1178 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Class-Imbalanced Learning on Graphs: A Survey

The rapid advancement in data-driven research has increased the demand for effective graph data analysis. However, real-world data often exhibits class imbalance, leading to poor performance of machine learning models. To overcome this challenge, class-imbalanced learning on graphs (CILG) has emerged as a promising solution that combines the strengths of graph representation learning and class-imbalanced learning. In recent years, significant progress has been made in CILG. Anticipating that such a trend will continue, this survey aims to offer a comprehensive understanding of the current state-of-the-art in CILG and provide insights for future research directions. Concerning the former, we introduce the first taxonomy of existing work and its connection to existing imbalanced learning literature. Concerning the latter, we critically analyze recent work in CILG and discuss urgent lines of inquiry within the topic. Moreover, we provide a continuously maintained reading list of papers and code at https://github.com/yihongma/CILG-Papers.Comment: submitted to ACM Computing Survey (CSUR

Efficient non-collinear antiferromagnetic state switching induced by orbital Hall effect in chromium

Recently orbital Hall current has attracted attention as an alternative method to switch the magnetization of ferromagnets. Here we present our findings on electrical switching of antiferromagnetic state in Mn3Sn/Cr, where despite the much smaller spin Hall angle of Cr, the switching current density is comparable to heavy metal based heterostructures. On the other hand, the inverse process, i.e., spin-to-charge conversion in Cr-based heterostructures is much less efficient than the Pt-based equivalents, as manifested in the almost one order of magnitude smaller terahertz emission intensity and spin current induced magnetoresistance in Cr-based structures. These results in combination with the slow decay of terahertz emission against Cr thickness (diffusion length of ~11 nm) suggest that the observed magnetic switching can be attributed to orbital current generation in Cr, followed by efficient conversion to spin current. Our work demonstrates the potential of light metals like Cr as an efficient orbital/spin current source for antiferromagnetic spintronics.Comment: 19 pages, 4 figure