Sampling algebraic varieties for sum of squares programs
Diego Cifuentes, Pablo A. Parrilo

TL;DR
This paper introduces a sampling-based approach for sum of squares relaxations over algebraic varieties, leveraging geometric sampling to simplify SDPs especially for complex varieties with easy sampling methods.
Contribution
It proposes a novel sampling methodology that bypasses algebraic descriptions, reducing SDP size and connecting SOS optimization with numerical algebraic geometry.
Findings
Effective for varieties with easy sampling but complex defining equations
Reduces SDP size by exploiting coordinate ring structure
Applicable to varieties like $SO(n)$, Grassmannians, and rank $k$ tensors
Abstract
We study sum of squares (SOS) relaxations to optimize polynomial functions over a set , where is a complex algebraic variety. We propose a new methodology that, rather than relying on some algebraic description, represents with a generic set of complex samples. This approach depends only on the geometry of , avoiding representation issues such as multiplicity and choice of generators. It also takes advantage of the coordinate ring structure to reduce the size of the corresponding semidefinite program (SDP). In addition, the input can be given as a straight-line program. Our methods are particularly appealing for varieties that are easy to sample from but for which the defining equations are complicated, such as , Grassmannians or rank tensors. For arbitrary varieties we can obtain the required samples by using the tools of numerical algebraic geometry.…
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