# Theory and Algorithms for the Localized Setting of Learning Kernels

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Yunwen Lei,
Alexander Binder,
Ürün Dogan,
Marius Kloft
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Proceedings of the 1st International Workshop on Feature Extraction: Modern Questions and Challenges at NIPS 2015, PMLR 44:173-195, 2015.

#### Abstract

We analyze the localized setting of learning kernels also known as localized multiple kernel learning. This problem has been addressed in the past using rather heuristic approaches based on approximately optimizing non-convex problem formulations, of which up to now no theoretical learning bounds are known. In this paper, we show generalization error bounds for learning localized kernel classes where the localities are coupled using graph-based regularization. We propose a novel learning localized kernels algorithm based on this hypothesis class that is formulated as a convex optimization problem using a pre-obtained cluster structure of the data. We derive dual representations using Fenchel conjugation theory, based on which we give a simple yet efficient wrapper-based optimization algorithm. We apply the method to problems involving multiple heterogeneous data sources, taken from domains of computational biology and computer vision. The results show that the proposed convex approach to learning localized kernels can achieve higher prediction accuracies than its global and non-convex local counterparts.

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