Low-rank Matrix Completion in a General Non-orthogonal Basis
Provides a theoretical foundation for matrix completion with arbitrary measurement bases, addressing a gap where RIP may fail or be computationally hard to verify.
The paper extends low-rank matrix completion theory to general non-orthogonal bases, proving recovery from O(n r ν log² n) random coefficients under a new correlation condition, without requiring the restricted isometry property.
This paper considers theoretical analysis of recovering a low rank matrix given a few expansion coefficients with respect to any basis. The current approach generalizes the existing analysis for the low-rank matrix completion problem with sampling under entry sensing or with respect to a symmetric orthonormal basis. The analysis is based on dual certificates using a dual basis approach and does not assume the restricted isometry property (RIP). We introduce a condition on the basis called the correlation condition. This condition can be computed in time $O(n^3)$ and holds for many cases of deterministic basis where RIP might not hold or is NP hard to verify. If the correlation condition holds and the underlying low rank matrix obeys the coherence condition with parameter $ν$, under additional mild assumptions, our main result shows that the true matrix can be recovered with very high probability from $O(nrν\log^2n)$ uniformly random expansion coefficients.