Proximity Graphs for Similarity Searches: Experimental Survey and the New Connected-Partition Approach HGraph

Similarity searching is a widely used approach to retrieve complex data (images, videos, time series, etc.). Similarity searches aim at retrieving similar data according to the intrinsic characteristics of the data. Recently, graph-based methods have emerged as a very efficient alternative for similarity retrieval, with reports indicating they have outperformed methods of other categories in several situations. This work presents two main contributions to graph-based methods for similarity searches. The first contribution is a survey on the main graph types currently employed for similarity searches and an experimental evaluation of the most representative graphs in a common platform for exact and approximate search algorithms. The second contribution is a new graph-based method called HGraph, which is a connected-partition approach to build a proximity graph and answer similarity searches. Both of our contributions and results were published and received awards in international conferences.

Graph approximations to the Laplacian spectra

I prove that the spectrum of the Laplace–Beltrami operator with the Neumann boundary condition on a compact Riemannian manifold with boundary admits a fast approximation by the spectra of suitable graph Laplacians on proximity graphs on the manifold, and similar graph approximation works for metric-measure spaces glued out of compact Riemannian manifolds of the same dimension.