Hardy-Muckenhoupt Bounds for Laplacian Eigenvalues

We present two graph quantities Psi(G,S) and Psi_2(G) which give constant factor estimates to the Dirichlet and Neumann eigenvalues, lambda(G,S) and lambda_2(G), respectively. Our techniques make use of a discrete Hardy-type inequality due to Muckenhoupt

Sharpness and well-conditioning of nonsmooth convex formulations in statistical signal recovery

We study a sample complexity vs. conditioning tradeoff in modern signal recovery problems where convex optimization problems are built from sampled observations. We begin by introducing a set of condition numbers related to sharpness in $\ell_p$ or Schatten-p norms ($p\in[1,2]$) based on nonsmooth reformulations of a class of convex optimization problems, including sparse recovery, low-rank matrix sensing, covariance estimation, and (abstract) phase retrieval. In each of the recovery tasks, we show that the condition numbers become dimension independent constants once the sample size exceeds some constant multiple of the recovery threshold. Structurally, this result ensures that the inaccuracy in the recovered signal due to both observation noise and optimization error is well-controlled. Algorithmically, such a result ensures that a new first-order method for solving the class of sharp convex functions in a given $\ell_p$ or Schatten-p norm, when applied to the nonsmooth formulations, achieves nearly-dimension-independent linear convergence

Electric field control and optical signature of entanglement in quantum dot molecules

The degree of entanglement of an electron with a hole in a vertically coupled self-assembled dot molecule is shown to be tunable by an external electric field. Using atomistic pseudopotential calculations followed by a configuration interaction many-body treatment of correlations, we calculate the electronic states, degree of entanglement and optical absorption. We offer a novel way to spectroscopically detect the magnitude of electric field needed to maximize the entanglement.Comment: 4 pages, 6 figure

Accelerated first-order methods for a class of semidefinite programs

This paper introduces a new storage-optimal first-order method (FOM), CertSDP, for solving a special class of semidefinite programs (SDPs) to high accuracy. The class of SDPs that we consider, the exact QMP-like SDPs, is characterized by low-rank solutions, a priori knowledge of the restriction of the SDP solution to a small subspace, and standard regularity assumptions such as strict complementarity. Crucially, we show how to use a certificate of strict complementarity to construct a low-dimensional strongly convex minimax problem whose optimizer coincides with a factorization of the SDP optimizer. From an algorithmic standpoint, we show how to construct the necessary certificate and how to solve the minimax problem efficiently. We accompany our theoretical results with preliminary numerical experiments suggesting that CertSDP significantly outperforms current state-of-the-art methods on large sparse exact QMP-like SDPs