Proportional Volume Sampling and Approximation Algorithms for A-Optimal Design [article]

Aleksandar Nikolov and Mohit Singh and Uthaipon Tao Tantipongpipat
2018 arXiv   pre-print
We study the optimal design problems where the goal is to choose a set of linear measurements to obtain the most accurate estimate of an unknown vector in d dimensions. We study the A-optimal design variant where the objective is to minimize the average variance of the error in the maximum likelihood estimate of the vector being measured. The problem also finds applications in sensor placement in wireless networks, sparse least squares regression, feature selection for k-means clustering, and
more » ... trix approximation. In this paper, we introduce proportional volume sampling to obtain improved approximation algorithms for A-optimal design. Our main result is to obtain improved approximation algorithms for the A-optimal design problem by introducing the proportional volume sampling algorithm. Our results nearly optimal bounds in the asymptotic regime when the number of measurements done, k, is significantly more than the dimension d. We also give first approximation algorithms when k is small including when k=d. The proportional volume-sampling algorithm also gives approximation algorithms for other optimal design objectives such as D-optimal design and generalized ratio objective matching or improving previous best known results. Interestingly, we show that a similar guarantee cannot be obtained for the E-optimal design problem. We also show that the A-optimal design problem is NP-hard to approximate within a fixed constant when k=d.
arXiv:1802.08318v5 fatcat:dirb5tbukzavho356x7s23nftm