Curvature Regularized Similarity Based Ricci Flow Embedding

Abstract

This work proposes a novel curvature regularization method to regularize the individual sectional curvatures in Similarity-Based Ricci Flow Embedding (SBRFE) to reduce the non-Euclidean artefacts and compute the Euclidean embedding of similarity-based data sets. In pattern recognition, pairwise similarity or dissimilarity data have been used as alternative to the more conventional feature vector representation. While similarity representations are rarely Euclidean, most methods involving statistical analysis or learning of such data require them to be Euclidean. In SBRFE, each similarity is considered an individual entity, and the respective sectional curvature is calculated and updated separately. This compromises the smoothness of the manifold and causes numerical instability. To overcome this problem, we used regularized Newton’s method (RNM) to regularize the sectional curvatures of each patch obtained from the initial curvature computation. It ensures numerical stability in the embedding and smoothens the manifold. It also preserves both the local and global geometry of the original data sets. Results show that proposed curvature regularized similarity-based Ricci Flow Embedding (CRRFE) is able to estimate the Euclidean embedding of similarity data sets with much lower computation cost and time complexity than the existing regularization method. Comparison results show that our proposed methodology outperforms other existing embedding methods in most data sets with a lower classification error rate.