UC San Diego

In many systems, both physical and mathematical, internal boundaries play an outsized role in dictating the large-scale behavior of the system. This theme is explored here in two parts: by identifying natural boundaries to diffusion on undirected graphs for the improvement of spectral graph methods for clustering and classification, and by imposing boundaries in two dimensional PEPS tensor networks to reduce their computational complexity for image generation. In the graph diffusion work, an algorithm is developed that identifies and removes vertices serving as bridges between well-connected clusters. With the use of this algorithm, the performance of unsupervised spectral clustering on graphs derived from synthetic point cloud data shows excellent cluster separation down to approximately two-thirds the point cloud gap size that standard spectral clustering alone tolerates. The same vertex-removal scheme applied to a diffusion-informed active learning classification algorithm shows an approximately order-of-magnitude best-case reduction in the classification error rate on benchmark hyperspectral imagery data in the low-label-query-limit versus the same active learning algorithm applied without vertex removal. The PEPS work proposes a scheme whereby such networks are cut into overlapping patch networks, whose individual contraction complexity is comparatively low. Feasibility testing is performed showing that in principle, this scheme could be used for image in-painting, and has intuitively appealing model features that are directly reflective of the data.