Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, November 08, 2018, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Micha Sharir (Tel Aviv University)

Radial isotropic position: Theory, algorithms, and applications

A set X of n vectors on the unit sphere in d dimensions is said to be in isotropic position if the sum of the squares of their projections in any direction is constant, independently of the direction. We say that X can be brought to radial isotropic position if there exists a linear tranformation A, such that if we normalize each of the vectors of AX we get a set in isotropic position.

We review the elegant theory that shows that almost any set X can be brought to radial isotropic position, show the strong connection between this notion and singular value decomposition, a basic tool in linear algebra, present iterative algorithms for approximating the desired linear transformation, and discuss combinatorial and algorithmic applications, most notably the recent work of Kane, Lovett and Moran on point location in high-dimensional hyperplane arrangements.

Joint work with Shiri Artstein-Avidan and Haim Kaplan.

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