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arXiv · 0911.3633

A Geometric Approach to Sample Compression

Benjamin I. P. Rubinstein·J. Hyam Rubinstein
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Abstract

The Sample Compression Conjecture of Littlestone & Warmuth has remained unsolved for over two decades. This paper presents a systematic geometric investigation of the compression of finite maximum concept classes. Simple arrangements of hyperplanes in Hyperbolic space, and Piecewise-Linear hyperplane arrangements, are shown to represent maximum classes, generalizing the corresponding Euclidean result. A main result is that PL arrangements can be swept by a moving hyperplane to unlabeled d-compress any finite maximum class, forming a peeling scheme as conjectured by Kuzmin & Warmuth. A corollary is that some d-maximal classes cannot be embedded into any maximum class of VC dimension d+k, for any constant k. The construction of the PL sweeping involves Pachner moves on the one-inclusion graph, corresponding to moves of a hyperplane across the intersection of d other hyperplanes. This extends the well known Pachner moves for triangulations to cubical complexes.

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Benjamin I. P. Rubinstein, J. Hyam Rubinstein. 2009-11-18. A Geometric Approach to Sample Compression. https://arxiv.org/abs/0911.3633

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