New Null Space Results and Recovery Thresholds for Matrix Rank Minimization

Nuclear norm minimization (NNM) has recently gained significant attention for its use in rank minimization problems. Similar to compressed sensing, using null space characterizations, recovery thresholds for NNM have been studied in \cite{arxiv,Recht_Xu_Hassibi}. However simulations show that the thresholds are far from optimal, especially in the low rank region. In this paper we apply the recent analysis of Stojnic for compressed sensing \cite{mihailo} to the null space conditions of NNM. The resulting thresholds are significantly better and in particular our weak threshold appears to match with simulation results. Further our curves suggest for any rank growing linearly with matrix size $n$ we need only three times of oversampling (the model complexity) for weak recovery. Similar to \cite{arxiv} we analyze the conditions for weak, sectional and strong thresholds. Additionally a separate analysis is given for special case of positive semidefinite matrices. We conclude by discussing simulation results and future research directions.

💡 Research Summary

The paper addresses the problem of low‑rank matrix recovery via nuclear‑norm minimization (NNM), a convex surrogate for rank minimization that has become a cornerstone in many signal‑processing and machine‑learning applications. Earlier works—most notably the arXiv preprint and the Recht‑Xu‑Hassibi analysis—derived recovery thresholds by examining null‑space conditions of the measurement operator. Those analyses, however, relied on coarse geometric arguments (e.g., restricted isometry property) and consequently produced overly pessimistic bounds, especially when the target rank is a small fraction of the matrix dimension.

To close this gap, the authors import the high‑dimensional probabilistic geometry framework introduced by Stojnic for compressed sensing. Stojnic’s “escape through a mesh” technique quantifies the probability that a random Gaussian measurement matrix’s null space avoids intersecting a given convex cone by computing the cone’s Gaussian width. By translating the NNM null‑space condition into a cone‑avoidance problem—specifically, the tangent cone of the nuclear‑norm ball at a low‑rank matrix—the authors are able to evaluate the Gaussian width analytically as a function of matrix size (n) and rank (r).

The analysis proceeds in three regimes:

1. Weak recovery – a fixed low‑rank matrix (X_0) is to be recovered uniquely. The authors show that the required number of measurements (m) satisfies
   \