Revisiting Memory Efficient Kernel Approximation: An Indefinite Learning Perspective

• URL: http://arxiv.org/abs/2112.09893v1
• Date: Sat, 18 Dec 2021 10:01:34 GMT
• Title: Revisiting Memory Efficient Kernel Approximation: An Indefinite Learning Perspective
• Authors: Simon Heilig, Maximilian M\"unch, Frank-Michael Schleif
• Abstract summary: Matrix approximations are a key element in large-scale machine learning approaches. We extend MEKA to be applicable not only for shift-invariant kernels but also for non-stationary kernels. We present a Lanczos-based estimation of a spectrum shift to develop a stable positive semi-definite MEKA approximation.
• Score: 0.8594140167290097
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Matrix approximations are a key element in large-scale algebraic machine learning approaches. The recently proposed method MEKA (Si et al., 2014) effectively employs two common assumptions in Hilbert spaces: the low-rank property of an inner product matrix obtained from a shift-invariant kernel function and a data compactness hypothesis by means of an inherent block-cluster structure. In this work, we extend MEKA to be applicable not only for shift-invariant kernels but also for non-stationary kernels like polynomial kernels and an extreme learning kernel. We also address in detail how to handle non-positive semi-definite kernel functions within MEKA, either caused by the approximation itself or by the intentional use of general kernel functions. We present a Lanczos-based estimation of a spectrum shift to develop a stable positive semi-definite MEKA approximation, also usable in classical convex optimization frameworks. Furthermore, we support our findings with theoretical considerations and a variety of experiments on synthetic and real-world data.

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