An Incidence Geometry approach to Dictionary Learning

We study the Dictionary Learning (aka Sparse Coding) problem of obtaining a sparse representation of data points, by learning \emph{dictionary vectors} upon which the data points can be written as sparse linear combinations. We view this problem from a geometry perspective as the spanning set of a subspace arrangement, and focus on understanding the case when the underlying hypergraph of the subspace arrangement is specified. For this Fitted Dictionary Learning problem, we completely characterize the combinatorics of the associated subspace arrangements (i.e.\ their underlying hypergraphs). Specifically, a combinatorial rigidity-type theorem is proven for a type of geometric incidence system. The theorem characterizes the hypergraphs of subspace arrangements that generically yield (a) at least one dictionary (b) a locally unique dictionary (i.e.\ at most a finite number of isolated dictionaries) of the specified size. We are unaware of prior application of combinatorial rigidity techniques in the setting of Dictionary Learning, or even in machine learning. We also provide a systematic classification of problems related to Dictionary Learning together with various algorithms, their assumptions and performance

Two-dimensional Glasses

abstract: The structure of glass has been the subject of many studies, however some details remained to be resolved. With the advancement of microscopic imaging techniques and the successful synthesis of two-dimensional materials, images of two-dimensional glasses (bilayers of silica) are now available, confirming that this glass structure closely follows the continuous random network model. These images provide complete in-plane structural information such as ring correlations, and intermediate range order and with computer refinement contain indirect information such as angular distributions, and tilting. This dissertation reports the first work that integrates the actual atomic coordinates obtained from such images with structural refinement to enhance the extracted information from the experimental data. The correlations in the ring structure of silica bilayers are studied and it is shown that short-range and intermediate-range order exist in such networks. Special boundary conditions for finite experimental samples are designed so atoms in the bulk sense they are part of an infinite network. It is shown that bilayers consist of two identical layers separated by a symmetry plane and the tilted tetrahedra, two examples of added value through the structural refinement. Finally, the low-temperature properties of glasses in two dimensions are studied. This dissertation presents a new approach to find possible two-level systems in silica bilayers employing the tools of rigidity theory in isostatic systems.Dissertation/ThesisDoctoral Dissertation Physics 201