Under mild sparsity and overlap assumptions, almost every proportional-size column subset of a sparse matrix has smallest singular value o(1), so constant-sparsity SparseStack maps are not (Ω(k), Ω(1))-OSI.
Interlacing Families III: Sharper Restricted Invertibility Estimates
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abstract
We use the method of interlacing families of polynomials to derive a simple proof of Bourgain and Tzafriri's Restricted Invertibility Principle, and then to sharpen the result in two ways. We show that the stable rank can be replaced by the Schatten 4-norm stable rank and that tighter bounds hold when the number of columns in the matrix under consideration does not greatly exceed its number of rows. Our bounds are derived from an analysis of smallest zeros of Jacobi and associated Laguerre polynomials.
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Well-invertible column subsets of sparse matrices are rare
Under mild sparsity and overlap assumptions, almost every proportional-size column subset of a sparse matrix has smallest singular value o(1), so constant-sparsity SparseStack maps are not (Ω(k), Ω(1))-OSI.