🤖 AI Summary
This work addresses the problem of establishing lower bounds on the smallest singular value of random matrices whose entries are low-degree polynomials of a small number of base random variables—bypassing fundamental limitations of classical smoothed analysis under strong anti-concentration assumptions. The method introduces a novel anti-concentration framework grounded in the well-conditionedness of polynomial mappings’ Jacobians, integrating hierarchical ε-nets, higher-order matrix lifting, and spectral analysis of linear operators. This yields the first singular-value criterion applicable to matrices with strong algebraic dependencies. The contribution resolves long-standing open problems—including power-sum decomposition and robust subspace entanglement certification—by providing the first smoothed-analysis guarantees for such settings. Crucially, it extends the scope of smoothed analysis from fully independent random matrices to non-robust, algebraically dependent ensembles, thereby furnishing new theoretical tools and foundations for algorithmic smoothed analysis.
📝 Abstract
We develop new techniques for proving lower bounds on the least singular value of random matrices with limited randomness. The matrices we consider have entries that are given by polynomials of a few underlying base random variables. This setting captures a core technical challenge for obtaining smoothed analysis guarantees in many algorithmic settings. Least singular value bounds often involve showing strong anti-concentration inequalities that are intricate and much less understood compared to concentration (or large deviation) bounds. First, we introduce a general technique for proving anti-concentration that uses well-conditionedness properties of the Jacobian of a polynomial map, and show how to combine this with a hierarchical є-net argument to prove least singular value bounds. Our second tool is a new statement about least singular values to reason about higher-order lifts of smoothed matrices and the action of linear operators on them. Apart from getting simpler proofs of existing smoothed analysis results, we use these tools to now handle more general families of random matrices. This allows us to produce smoothed analysis guarantees in several previously open settings. These new settings include smoothed analysis guarantees for power sum decompositions and certifying robust entanglement of subspaces, where prior work could only establish least singular value bounds for fully random instances or only show non-robust genericity guarantees.