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arXiv · 1711.07271

Positive semi-definite embedding for dimensionality reduction and out-of-sample extensions

Michaël Fanuel·Antoine Aspeel·Jean-Charles Delvenne·Johan A. K. Suykens
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Abstract

In machine learning or statistics, it is often desirable to reduce the dimensionality of a sample of data points in a high dimensional space $\mathbb{R}^d$. This paper introduces a dimensionality reduction method where the embedding coordinates are the eigenvectors of a positive semi-definite kernel obtained as the solution of an infinite dimensional analogue of a semi-definite program. This embedding is adaptive and non-linear. We discuss this problem both with weak and strong smoothness assumptions about the learned kernel. A main feature of our approach is the existence of an out-of-sample extension formula of the embedding coordinates in both cases. This extrapolation formula yields an extension of the kernel matrix to a data-dependent Mercer kernel function. Our empirical results indicate that this embedding method is more robust with respect to the influence of outliers, compared with a spectral embedding method.

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Michaël Fanuel, Antoine Aspeel, Jean-Charles Delvenne, Johan A. K. Suykens. 2021-11-27. Positive semi-definite embedding for dimensionality reduction and out-of-sample extensions. https://arxiv.org/abs/1711.07271

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