Spherical Structured Feature Maps for Kernel Approximation

Yueming Lyu
Proceedings of the 34th International Conference on Machine Learning, PMLR 70:2256-2264, 2017.

Abstract

We propose Spherical Structured Feature (SSF) maps to approximate shift and rotation invariant kernels as well as bth-order arc-cosine kernels (Cho \& Saul, 2009). We construct SSF maps based on the point set on d−1 dimensional sphere Sd−1. We prove that the inner product of SSF maps are unbiased estimates for above kernels if asymptotically uniformly distributed point set on Sd−1 is given. According to (Brauchart \& Grabner, 2015), optimizing the discrete Riesz s-energy can generate asymptotically uniformly distributed point set on Sd−1. Thus, we propose an efficient coordinate decent method to find a local optimum of the discrete Riesz s-energy for SSF maps construction. Theoretically, SSF maps construction achieves linear space complexity and loglinear time complexity. Empirically, SSF maps achieve superior performance compared with other methods.

Cite this Paper

BibTeX
@InProceedings{pmlr-v70-lyu17a, title = {Spherical Structured Feature Maps for Kernel Approximation}, author = {Yueming Lyu}, booktitle = {Proceedings of the 34th International Conference on Machine Learning}, pages = {2256--2264}, year = {2017}, editor = {Precup, Doina and Teh, Yee Whye}, volume = {70}, series = {Proceedings of Machine Learning Research}, month = {06--11 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v70/lyu17a/lyu17a.pdf}, url = {https://proceedings.mlr.press/v70/lyu17a.html}, abstract = {We propose Spherical Structured Feature (SSF) maps to approximate shift and rotation invariant kernels as well as $b^{th}$-order arc-cosine kernels (Cho \& Saul, 2009). We construct SSF maps based on the point set on $d-1$ dimensional sphere $\mathbb{S}^{d-1}$. We prove that the inner product of SSF maps are unbiased estimates for above kernels if asymptotically uniformly distributed point set on $\mathbb{S}^{d-1}$ is given. According to (Brauchart \& Grabner, 2015), optimizing the discrete Riesz s-energy can generate asymptotically uniformly distributed point set on $\mathbb{S}^{d-1}$. Thus, we propose an efficient coordinate decent method to find a local optimum of the discrete Riesz s-energy for SSF maps construction. Theoretically, SSF maps construction achieves linear space complexity and loglinear time complexity. Empirically, SSF maps achieve superior performance compared with other methods.} }
Endnote
%0 Conference Paper %T Spherical Structured Feature Maps for Kernel Approximation %A Yueming Lyu %B Proceedings of the 34th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2017 %E Doina Precup %E Yee Whye Teh %F pmlr-v70-lyu17a %I PMLR %P 2256--2264 %U https://proceedings.mlr.press/v70/lyu17a.html %V 70 %X We propose Spherical Structured Feature (SSF) maps to approximate shift and rotation invariant kernels as well as $b^{th}$-order arc-cosine kernels (Cho \& Saul, 2009). We construct SSF maps based on the point set on $d-1$ dimensional sphere $\mathbb{S}^{d-1}$. We prove that the inner product of SSF maps are unbiased estimates for above kernels if asymptotically uniformly distributed point set on $\mathbb{S}^{d-1}$ is given. According to (Brauchart \& Grabner, 2015), optimizing the discrete Riesz s-energy can generate asymptotically uniformly distributed point set on $\mathbb{S}^{d-1}$. Thus, we propose an efficient coordinate decent method to find a local optimum of the discrete Riesz s-energy for SSF maps construction. Theoretically, SSF maps construction achieves linear space complexity and loglinear time complexity. Empirically, SSF maps achieve superior performance compared with other methods.
APA
Lyu, Y.. (2017). Spherical Structured Feature Maps for Kernel Approximation. Proceedings of the 34th International Conference on Machine Learning, in Proceedings of Machine Learning Research 70:2256-2264 Available from https://proceedings.mlr.press/v70/lyu17a.html.

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