Structure Preserving Encoding of Non-euclidean Similarity Data

Maximilian Münch, Christoph Raab, Michael Biehl, Frank-Michael Schleif

Abstract:

Domain-specific proximity measures, like divergence measures in signal processing or alignment scores in
bioinformatics, often lead to non-metric, indefinite similarities or dissimilarities. However, many classical
learning algorithms like kernel machines assume metric properties and struggle with such metric violations.
For example, the classical support vector machine is no longer able to converge to an optimum. One possible
direction to solve the indefiniteness problem is to transform the non-metric (dis-)similarity data into positive
(semi-)definite matrices. For this purpose, many approaches have been proposed that adapt the eigenspectrum
of the given data such that positive definiteness is ensured. Unfortunately, most of these approaches modify
the eigenspectrum in such a strong manner that valuable information is removed or noise is added to the data.
In particular, the shift operation has attracted a lot of interest in the past few years despite its frequently re-
occurring disadvantages. In this work, we propose a modified advanced shift correction method that enables
the preservation of the eigenspectrum structure of the data by means of a low-rank approximated nullspace
correction. We compare our advanced shift to classical eigenvalue corrections like eigenvalue clipping, flip-
ping, squaring, and shifting on several benchmark data. The impact of a low-rank approximation on the data’s
eigenspectrum is analyzed