Pengwen Chen

Pengwen Chen, Albert Fannjiang, Gi-Ren Liu

Alternating projection (AP) of various forms, including the Parallel AP (PAP), Real-constrained AP (RAP) and the Serial AP (SAP), are proposed to solve phase retrieval with at most two coded diffraction patterns. The proofs of geometric convergence are given with sharp bounds on the rates of convergence in terms of a spectral gap condition. To compensate for the local nature of convergence, the null initialization is proposed for initial guess and proved to produce asymptotically accurate initialization for the case of Gaussian random measurement. Numerical experiments show that the null initialization produces more accurate initial guess than the spectral initialization and that AP converges faster to the true object than other iterative schemes for non-convex optimization such as the Wirtinger Flow. In numerical experiments, AP with the null initialization converges globally to the true object.

Chester Holtz, Pengwen Chen, Alexander Cloninger et al.

Motivated by the need to address the degeneracy of canonical Laplace learning algorithms in low label rates, we propose to reformulate graph-based semi-supervised learning as a nonconvex generalization of a \emph{Trust-Region Subproblem} (TRS). This reformulation is motivated by the well-posedness of Laplacian eigenvectors in the limit of infinite unlabeled data. To solve this problem, we first show that a first-order condition implies the solution of a manifold alignment problem and that solutions to the classical \emph{Orthogonal Procrustes} problem can be used to efficiently find good classifiers that are amenable to further refinement. To tackle refinement, we develop the framework of Sequential Subspace Optimization for graph-based SSL. Next, we address the criticality of selecting supervised samples at low-label rates. We characterize informative samples with a novel measure of centrality derived from the principal eigenvectors of a certain submatrix of the graph Laplacian. We demonstrate that our framework achieves lower classification error compared to recent state-of-the-art and classical semi-supervised learning methods at extremely low, medium, and high label rates.

Pengwen Chen, Albert Fannjiang

Uniqueness of solution is proved for any ptychographic scheme with a random masks under a minimum overlap condition and local geometric convergence analysis is given for the alternating projection (AP) and Douglas-Rachford (DR) algorithms. DR is shown to possess a unique fixed point in the object domain and for AP a simple criterion for distinguishing the true solution among possibly many fixed points is given. A minimalist scheme is proposed where the adjacent masks overlap 50\% of area and each pixel of the object is illuminated by exactly four times during the whole measurement process. Such a scheme is conveniently parametrized by the number $q$ of shifted masks in each direction. The lower bound $1-C/q^2$ is proved for the geometric convergence rate of the minimalist scheme, predicting a poor performance with large $q$ which is confirmed by numerical experiments. Extensive numerical experiments are performed to explore what the general features of a well-performing mask are like, what the best-performing values of $q$ for a given mask are, how robust the minimalist scheme is with respect to measurement noise and what the significant factors affecting the noise stability are.