Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, May 18, 2021, 12:15 pm

Duration: 30 minutes

Location: Zoom: conference room

Speaker: Stefan Tiegel

SoS Degree Reduction with Applications to Clustering and Robust Moment Estimation

In the past few years, the sum-of-squares hierarchy of semidefinite programs has been successfully applied to a wide range of estimation problems, often yielding the first efficient algorithms. The main measure of complexity used to analyze these algorithms is the so-called "degree" or level of the hierarchy we use. The higher degree we allow, the more powerful our algorithms become, however at the cost of a larger running time. In this work, we develop a general framework to significantly reduce this degree. To illustrate the power of our framework, we use it to speed up previous algorithms based on sum-of-squares for two important estimation problems, clustering and robust moment estimation. The resulting algorithms offer the same statistical guarantees as the previous best algorithms but have significantly faster running times. Roughly speaking, given a sample of n points in dimension d, our algorithms can exploit order-l moments in time d^O(l) * n^O(1), whereas a naive implementation requires time (d*n)^O(l). Since for the aforementioned applications, the typical sample size is d^Theta(l), our framework improves running times from d^O(l^2) to d^O(l). Based on joint work with David Steurer.

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