Computational and Statistical Boundaries for Submatrix Localization in a Large Noisy Matrix

We study in this paper computational and statistical boundaries for submatrix localization. Given one observation of (one or multiple nonoverlapping) signal submatrix (of magnitude λ and size km×kn) embedded in a large noise matrix (of size m × n), the goal is to optimal identify the support of the signal submatrix computationally and statistically. Two transition thresholds for the signal-to-noise ratio λ/σ are established in terms of m, n, km and kn. The first threshold, SNRc, corresponds to the computational boundary. We introduce a new linear time spectral algorithm that identifies the submatrix with high probability when the signal strength is above the threshold SNRc. Below this threshold, it is shown that no polynomial time algorithm can succeed in identifying the submatrix, under the hidden clique hypothesis. The second threshold, SNRs, captures the statistical boundary, below which no method can succeed in localization with probability going to one in the minimax sense. The exhaustive search method successfully finds the submatrix above this threshold. In marked contrast to submatrix detection and sparse PCA, the results show an interesting phenomenon that SNRc is always significantly larger than SNRs, which implies an essential gap between statistical optimality and computational efficiency for submatrix localization

Structural Obstacles for Women in Academia: Availability and Costs of Campus Child Care

Women face tremendous obstacles to success in academic institutions. While we have witnessed incredible progress in some areas of representation of students, staff, or faculty who are women, outcomes by gender continue to be impacted by structural challenges in higher education. One structural barrier is the availability of child care. The article examines the availability and characteristics of child care centers at institutions with a public service commitment to social equity, as evidenced by offering degree programs accredited by the Network of Schools of Public Policy, Public Affairs, and Public Administration (NASPAA). Findings indicate that, of the 173 schools with NASPAA-accredited programs, 127 schools (73%) provide some type of child care for students, faculty, or staff members. However, the average full-time cost per child exceeds affordability guidelines which indicates a significant structural factor in child-care accessibility. While findings are descriptive, this study provides evidence of institutional barriers for women in academia

Geometric Inference for General High-Dimensional Linear Inverse Problems

This paper presents a unified geometric framework for the statistical analysis of a general ill-posed linear inverse model which includes as special cases noisy compressed sensing, sign vector recovery, trace regression, orthogonal matrix estimation and noisy matrix completion. We propose computationally feasible convex programs for statistical inference including estimation, confidence intervals and hypothesis testing. A theoretical framework is developed to characterize the local estimation rate of convergence and to provide statistical inference guarantees. Our results are built based on the local conic geometry and duality. The difficulty of statistical inference is captured by the geometric characterization of the local tangent cone through the Gaussian width and Sudakov estimate