A Simplified Approach to Recovery Conditions for Low Rank Matrices

Recovering sparse vectors and low-rank matrices from noisy linear measurements has been the focus of much recent research. Various reconstruction algorithms have been studied, including $\ell_1$ and nuclear norm minimization as well as $\ell_p$ minimization with $p<1$. These algorithms are known to succeed if certain conditions on the measurement map are satisfied. Proofs of robust recovery for matrices have so far been much more involved than in the vector case. In this paper, we show how several robust classes of recovery conditions can be extended from vectors to matrices in a simple and transparent way, leading to the best known restricted isometry and nullspace conditions for matrix recovery. Our results rely on the ability to “vectorize” matrices through the use of a key singular value inequality.

💡 Research Summary

The paper addresses the fundamental problem of recovering low‑rank matrices from noisy linear measurements, a task that parallels the well‑studied recovery of sparse vectors. While numerous algorithms—ℓ₁ minimization, nuclear‑norm minimization, and non‑convex ℓₚ (p < 1) minimization—have been shown to succeed under certain conditions on the measurement operator, the existing proofs for matrices are considerably more intricate than their vector counterparts. The authors propose a unifying and dramatically simpler framework that lifts robust recovery conditions from the vector setting to matrices with minimal technical overhead.

The central idea is to “vectorize” a matrix by means of its singular values. For any matrix X, let σ(X) denote the vector of its singular values sorted in non‑increasing order. The authors prove a key singular‑value inequality: for any linear measurement map 𝔄, the Euclidean norm of the measurements depends only on σ(X), i.e., ‖𝔄(X)‖₂ = ‖𝔄(diag(σ(X)))‖₂. Consequently, any Restricted Isometry Property (RIP) that holds for vectors automatically holds for matrices when expressed in terms of σ(X). In particular, if 𝔄 satisfies the r‑restricted RIP with constant δ, then for every matrix X of rank ≤ r, \