Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution

Recent years have seen a flurry of activities in designing provably efficient nonconvex procedures for solving statistical estimation problems. Due to the highly nonconvex nature of the empirical loss, state-of-the-art procedures often require proper regularization (e.g. trimming, regularized cost, projection) in order to guarantee fast convergence. For vanilla procedures such as gradient descent, however, prior theory either recommends highly conservative learning rates to avoid overshooting, or completely lacks performance guarantees. This paper uncovers a striking phenomenon in nonconvex optimization: even in the absence of explicit regularization, gradient descent enforces proper regularization implicitly under various statistical models. In fact, gradient descent follows a trajectory staying within a basin that enjoys nice geometry, consisting of points incoherent with the sampling mechanism. This "implicit regularization" feature allows gradient descent to proceed in a far more aggressive fashion without overshooting, which in turn results in substantial computational savings. Focusing on three fundamental statistical estimation problems, i.e. phase retrieval, low-rank matrix completion, and blind deconvolution, we establish that gradient descent achieves near-optimal statistical and computational guarantees without explicit regularization. In particular, by marrying statistical modeling with generic optimization theory, we develop a general recipe for analyzing the trajectories of iterative algorithms via a leave-one-out perturbation argument. As a byproduct, for noisy matrix completion, we demonstrate that gradient descent achieves near-optimal error control --- measured entrywise and by the spectral norm --- which might be of independent interest.Comment: accepted to Foundations of Computational Mathematics (FOCM

APEX2S: A Two-Layer Machine Learning Model for Discovery of host-pathogen protein-protein Interactions on Cloud-based Multiomics Data

Presented by the avalanche of biological interactions data, computational biology is now facing greater challenges on big data analysis and solicits more studies to mine and integrate cloud-based multiomics data, especially when the data are related to infectious diseases. Meanwhile, machine learning techniques have recently succeeded in different computational biology tasks. In this article, we have calibrated the focus for host-pathogen protein-protein interactions study, aiming to apply the machine learning techniques for learning the interactions data and making predictions. A comprehensive and practical workflow to harness different cloud-based multiomics data is discussed. In particular, a novel two-layer machine learning model, namely APEX2S, is proposed for discovery of the protein-protein interactions data. The results show that our model can better learn and predict from the accumulated host-pathogen protein-protein interactions

Coexisting Innominate Vein Compression Syndrome and May-Thurner Syndrome

AbstractInnominate vein compression syndrome and May-Thurner syndrome (also called iliac vein compression syndrome) are venous compression syndromes caused by normal anatomic structures. Here, we present a case in which these two conditions were found in the same patient using multidetector row computed tomography. This case is significant for two reasons: (1) it is, to the best of our knowledge, the first case study in the literature to report coexisting innominate vein compression syndrome and May-Thurner syndrome; and (2) it shows that multidetector row computed tomography has powerful diagnostic ability for venous diseases

The Color Octet Effect from e+e−→J/ψ+X+γe^+ e^-\to{J/\psi}+X+\gamma at B Factory

We study the initial state radiation process $e^+ e^-\to{J/\psi}+X+\gamma$ for $J/\psi$ production at B factory, and find the cross section is 61% larger than it's Born one for color octet part and is about half as it's Born one for color singlet part. Furthermore, the color singlet and color octet signal are very clearly separated in it's $E_\gamma$ spectra due to kinematics difference. We suggest to measure this $E_\gamma$ spectra at B factory to determine the color octet effect.Comment: 4 pages, 4 figures and 1 tabl