Binary, or one-bit, representations of data arise naturally in many applications, and are appealing in both hardware implementations and algorithm design. In this talk, we provide a brief background to sparsity and 1-bit measurements, and then present new results on the problem of data classification from binary data that proposes a stochastic framework with low computation and resource costs. We illustrate the utility of the proposed approach through stylized and realistic numerical experiments, provide a theoretical analysis for a simple case, and discuss future directions. 

Many problems of contemporary interest in signal processing and machine learning involve highly non-convex optimization problems. While nonconvex problems are known to be intractable in general, simple local search heuristics such as (stochastic) gradient descent are often surprisingly effective at finding global optima on real or randomly generated data. In this talk I will discuss some results explaining the success of these heuristics by connecting convergence of nonconvex optimization algorithms to supremum of certain stochastic processes. I will focus on two problems.

The first problem, concerns the recovery of a structured signal from under-sampled random quadratic measurements. I will show that projected gradient descent on a natural nonconvex formulation finds globally optimal solutions with a near minimal number of samples, breaking through local sample complexity barriers that have emerged in recent literature. I will also discuss how these new mathematical developments pave the way for a new generation of data-driven phaseless imaging systems that can utilize prior information to significantly reduce acquisition time and enhance image reconstruction, enabling nano-scale imaging at unprecedented speeds and resolutions. The second problem is about learning the optimal weights of the shallowest of neural networks consisting of a single Rectified Linear Unit (ReLU). I will discuss this problem in the high-dimensional regime where the number of observations are fewer than the ReLU weights. I will show that projected gradient descent on a natural least-squares objective, when initialization at 0, converges at a linear rate to globally optimal weights with a number of samples that is optimal up to numerical constants.

We show that a four-dimensional complete gradient shrinking Ricci
soliton with positive isotropic curvature is either a quotient of S^4 or
a quotient of S^3 x R. We also give a classification result on
four-dimensional gradient shrinking Ricci solitons with non-negative
isotropic curvature. This is joint work with Lei Ni and Kui Wang.

We describe Riemannian (non-Kahler) Ricci flow solutions that develop finite-time Type-I singularities whose parabolic dilations converge to a shrinking Kahler–Ricci soliton singularity model. More specifically, the singularity model for these solutions is the “blowdown soliton” discovered in 2003 by Feldman, Ilmanen, and the speaker. Our results support the conjecture that the blowdown soliton is stable under Ricci flow. This work also provides the first set of rigorous examples of non-Kahler solutions of Ricci flow that become asymptotically Kahler, in suitable space-time neighborhoods of developing singularities, at rates that break scaling invariance. These results support the conjectured stability of the subspace of Kahler metrics under Ricci flow.

 In this talk, I will propose a multi-layered interface system which can be formally derived by the singular limit of the weakly coupled system of the Allen-Cahn equation.  By using the level set approach, this system can be written as a quasi-monotone degenerate parabolic system. We give results of the well-posedness of viscosity solutions, and study the singularity of each layers. 

This is a joint work with H. Ninomiya, K. Todoroki.