Optimal reduced sets for sparse kernel spectral clustering

Abstract

Kernel spectral clustering (KSC) solves a weighted kernel principal component analysis problem in a primal-dual optimization framework. It results in a clustering model using the dual solution of the problem. It has a powerful out-of-sample extension property leading to good clustering generalization w.r.t. the unseen data points. The out-of-sample extension property allows to build a sparse model on a small training set and introduces the first level of sparsity. The clustering dual model is expressed in terms of non-sparse kernel expansions where every point in the training set contributes. The goal is to find reduced set of training points which can best approximate the original solution. In this paper a second level of sparsity is introduced in order to reduce the time complexity of the computationally expensive out-of-sample extension. In this paper we investigate various penalty based reduced set techniques including the Group Lasso, L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> , L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> + L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> penalization and compare the amount of sparsity gained w.r.t. a previous L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> penalization technique. We observe that the optimal results in terms of sparsity corresponds to the Group Lasso penalization technique in majority of the cases. We showcase the effectiveness of the proposed approaches on several real world datasets and an image segmentation dataset.