Veniamin I. Morgenshtern

Angel Villar-Corrales, Veniamin I. Morgenshtern

In the last few years, large improvements in image clustering have been driven by the recent advances in deep learning. However, due to the architectural complexity of deep neural networks, there is no mathematical theory that explains the success of deep clustering techniques. In this work we introduce Projected-Scattering Spectral Clustering (PSSC), a state-of-the-art, stable, and fast algorithm for image clustering, which is also mathematically interpretable. PSSC includes a novel method to exploit the geometric structure of the scattering transform of small images. This method is inspired by the observation that, in the scattering transform domain, the subspaces formed by the eigenvectors corresponding to the few largest eigenvalues of the data matrices of individual classes are nearly shared among different classes. Therefore, projecting out those shared subspaces reduces the intra-class variability, substantially increasing the clustering performance. We call this method Projection onto Orthogonal Complement (POC). Our experiments demonstrate that PSSC obtains the best results among all shallow clustering algorithms. Moreover, it achieves comparable clustering performance to that of recent state-of-the-art clustering techniques, while reducing the execution time by more than one order of magnitude. In the spirit of reproducible research, we publish a high quality code repository along with the paper.

Matthias Sonntag, Veniamin I. Morgenshtern

In this paper, we propose a method to segment regions in three-dimensional point clouds. We assume that (i) the shape and the number of regions in the point cloud are not known and (ii) the point cloud may be noisy. The method consists of two steps. In the first step we use a deep neural network to predict the probability that a pair of small patches from the point cloud belongs to the same region. In the second step, we use a convex-optimization based method to improve the predictions of the network by enforcing consistency constraints. We evaluate the accuracy of our method on a custom dataset of convex polyhedra, where the regions correspond to the faces of the polyhedra. The method can be seen as a robust and flexible alternative to the famous region growing segmentation algorithm. All reported results are reproducible and come with easy to use code that could serve as a baseline for future research.

Veniamin I. Morgenshtern, Emmanuel J. Candes

In single-molecule microscopy it is necessary to locate with high precision point sources from noisy observations of the spectrum of the signal at frequencies capped by $f_c$, which is just about the frequency of natural light. This paper rigorously establishes that this super-resolution problem can be solved via linear programming in a stable manner. We prove that the quality of the reconstruction crucially depends on the Rayleigh regularity of the support of the signal; that is, on the maximum number of sources that can occur within a square of side length about $1/f_c$. The theoretical performance guarantee is complemented with a converse result showing that our simple convex program convex is nearly optimal. Finally, numerical experiments illustrate our methods.