Frequency of bounded-condition-number matrices in applications

Determine how frequently practical matrix-completion datasets satisfy the bounded-condition-number assumption required by spectral recovery algorithms.

Background

The paper explains that several spectral matrix-completion methods have guarantees whose effectiveness depends strongly on a bounded condition number, meaning that the largest and smallest nonzero singular values are of comparable magnitude. The authors note that rapidly decaying singular values, and hence large condition numbers, occur in examples such as the Yale face database.

The unresolved issue is empirical and structural: the paper does not determine how prevalent the bounded-condition-number regime is in real datasets, even though that prevalence affects the practical relevance of the corresponding algorithms.