Ellipsoid Fitting up to a Constant
In [Saunderson, 2011; Saunderson et al., 2013], Saunderson, Parrilo, and Willsky asked the following elegant geometric question: what is the largest m = m(d) such that there is an ellipsoid in ℝ^d that passes through v_1, v_2, …, v_m with high probability when the v_is are chosen independently from the standard Gaussian distribution N(0,I_d)? The existence of such an ellipsoid is equivalent to the existence of a positive semidefinite matrix X such that v_i^⊤ X v_i = 1 for every 1 ⩽ i ⩽ m - a natural example of a random semidefinite program. SPW conjectured that m = (1-o(1)) d²/4 with high probability. Very recently, Potechin, Turner, Venkat and Wein [Potechin et al., 2022] and Kane and Diakonikolas [Kane and Diakonikolas, 2022] proved that m ≳ d²/log^O(1) d via a certain natural, explicit construction.
In this work, we give a substantially tighter analysis of their construction to prove that m ≳ d²/C for an absolute constant C > 0. This resolves one direction of the SPW conjecture up to a constant. Our analysis proceeds via the method of Graphical Matrix Decomposition that has recently been used to analyze correlated random matrices arising in various areas [Barak et al., 2019; Bafna et al., 2022]. Our key new technical tool is a refined method to prove singular value upper bounds on certain correlated random matrices that are tight up to absolute dimension-independent constants. In contrast, all previous methods that analyze such matrices lose logarithmic factors in the dimension.
Semidefinite programming
random matrices
average-case complexity
Theory of computation~Semidefinite programming
78:1-78:20
Track A: Algorithms, Complexity and Games
Jun-Ting
Hsieh
Jun-Ting Hsieh
Carnegie Mellon University, Pittsburgh, PA, USA
https://jthsieh.github.io/
Supported by NSF CAREER Award #2047933.
Pravesh K.
Kothari
Pravesh K. Kothari
Carnegie Mellon University, Pittsburgh, PA, USA
http://praveshkkothari.org/
Supported by NSF CAREER Award #2047933, Alfred P. Sloan Fellowship and a Google Research Scholar Award.
Aaron
Potechin
Aaron Potechin
University of Chicago, IL, USA
http://www.potechin.org/aaronpotechin/
Supported in part by NSF grant CCF-2008920.
Jeff
Xu
Jeff Xu
Carnegie Mellon University, Pittsburgh, PA, USA
https://www.andrew.cmu.edu/user/sichaoxu/
Supported in part by NSF CAREER Award #2047933, and a Cylab Presidential Fellowship.
10.4230/LIPIcs.ICALP.2023.78
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Jun-Ting Hsieh, Pravesh K. Kothari, Aaron Potechin, and Jeff Xu
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