Spherical Discrepancy Minimization and Algorithmic Lower Bounds for Covering the Sphere
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Inspired by the boolean discrepancy problem, we study the following optimization problem which we term Spherical Discrepancy: given m unit vectors v_1, ..., v_m, find another unit vector x that minimizes max_i 〈 x, v_i〉. We show that Spherical Discrepancy is APX-hard and develop a multiplicative weights-based algorithm that achieves nearly optimal worst-case error bounds. We use our algorithm to give the first non-trivial lower bounds for the problem of covering a hypersphere by hyperspherical caps of uniform volume at least 2^-o(√(n)), and to give a lower bound for covering a Gaussian random variable by equal-sized halfspaces. Up to a log factor, our lower bounds match known upper bounds in this regime. Finally, we show how to modify our algorithm to solve a natural version of the Komlós problem for the spherical setting.
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