Our thesis is that a geometric perspective yields insights into the structure of fundamental problems, and thereby suggests efficient algorithms for them. As evidence, we develop new geometric models and general-purpose tools for removing outliers from numeric data, reducing dimensionality, and counting combinatorial sets. Then we apply these techniques to a set of old problems to obtain polynomial-time algorithms. These include: (1) learning noisy linear-threshold functions (half-spaces), (2) learning the intersection of half-spaces, (3) clustering text corpora, and (4) counting lattice points in a convex body.

We supplement some of our theorems with experimental studies.

86 pages

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