Recovering low-rank matrices from few coefficients in any basis (0910.1879v5)

Abstract: We present novel techniques for analyzing the problem of low-rank matrix recovery. The methods are both considerably simpler and more general than previous approaches. It is shown that an unknown (n x n) matrix of rank r can be efficiently reconstructed from only O(n r nu log^2 n) randomly sampled expansion coefficients with respect to any given matrix basis. The number nu quantifies the "degree of incoherence" between the unknown matrix and the basis. Existing work concentrated mostly on the problem of "matrix completion" where one aims to recover a low-rank matrix from randomly selected matrix elements. Our result covers this situation as a special case. The proof consists of a series of relatively elementary steps, which stands in contrast to the highly involved methods previously employed to obtain comparable results. In cases where bounds had been known before, our estimates are slightly tighter. We discuss operator bases which are incoherent to all low-rank matrices simultaneously. For these bases, we show that O(n r nu log n) randomly sampled expansion coefficients suffice to recover any low-rank matrix with high probability. The latter bound is tight up to multiplicative constants.

• The paper introduces a convex optimization technique that reconstructs low-rank matrices from a limited set of observed coefficients.
• It employs large deviation bounds and a recursive 'golfing scheme' to provide strong probabilistic guarantees for recovery.
• The method generalizes matrix completion to any orthonormal basis, enhancing sampling efficiency in applications like quantum tomography and signal processing.

Recovering Low-Rank Matrices From Few Coefficients In Any Basis

David Gross's paper, "Recovering Low-Rank Matrices From Few Coefficients In Any Basis," presents an innovative and general framework for matrix recovery that simplifies previous methods while widening their applicability. Specifically, the paper tackles the problem of reconstructing an unknown low-rank matrix $\rho$ from a limited number of observed expansion coefficients. This is achieved using a convex optimization technique and leverages the concept of "incoherence" between the matrix and the basis in which the coefficients are represented.

Summary and Methodology

Gross introduces a significant methodological refinement: a matrix with rank $r$ can be reconstructed from $O(nr \nu \ln^2 n)$ randomly sampled expansion coefficients in any given basis, where $\nu$ denotes the degree of incoherence between the basis and the matrix. This result generalizes prior work that focused primarily on matrix completion—recovering a low-rank matrix from its randomly selected elements—by extending the approach to arbitrary orthonormal bases.

The incoherence parameter $\nu$ plays a crucial role. It measures how well the basis vectors align with the row and column spaces of the matrix. The key insight is that matrices incoherent with respect to a given basis can be recovered more efficiently. Gross's results are tighter and derived from less complex methods compared to previous techniques.

The primary steps and results of the paper can be summarized as follows:

1. Sampling Operator Definition: Define a sampling operator $R$ that projects the matrix onto randomly sampled basis elements.
2. Convex Optimization: Formulate the matrix recovery problem as a convex optimization problem, minimizing the trace norm (sum of singular values) subject to the constraint that the known coefficients remain fixed.
3. Incoherence and Large Deviation Bound: Introduce the notion of coherence and use a large deviation bound to provide probabilistic guarantees on the solution's accuracy.
4. Hierarchy of Incoherence: Discuss special bases (Fourier-type bases) where the recovery bounds are proven to be near-optimal, i.e., $O(nr \nu \ln n)$ coefficients suffice.
5. Novel Techniques: Employ a "golfing scheme"—a recursive strategy to construct an approximate solution based on past observations—and utilize operator Chernoff bounds to control deviations.

Theoretical and Practical Implications

The implications of Gross's work are twofold:

1. Theoretical Contribution: The innovative application of large deviation theorems to matrix recovery, along with the generalized incoherence parameter, provides a robust theoretical underpinning. The proof techniques, which simplify and improve upon earlier methods, could inspire further research in high-dimensional statistics and random matrix theory.
2. Practical Applications: Practically, this research has profound implications for various fields, such as signal processing, quantum state tomography, and machine learning, where efficient recovery of low-rank structures is critical. The reduction to $O(nr \nu \ln n)$ coefficients for Fourier-type bases is particularly noteworthy as it offers computational and sampling efficiency.

Future Directions

The results suggest several avenues for future research:

• Noise Resilience: Extending the framework to noisy observations is a crucial next step, with initial indications that the methods are robust under certain noise models.
• Extension to Non-Hermitian Matrices: Gross briefly indicates that the methods extend to non-Hermitian matrices by embedding the problem in a higher-dimensional Hermitian space. This warrants further investigation to fully leverage non-Hermitian settings.
• Applications to Larger Classes of Bases: Exploring other classes of bases beyond orthonormal and Fourier-type bases could provide new insights. The generality of the approach makes it adaptable to specific applications with tailored bases.

Conclusion

David Gross's paper represents a significant stride in the theory of low-rank matrix recovery, offering simpler, more general methods than previously available. The introduction of large deviation bounds for matrix coefficients and the efficient golfing scheme are notable innovations. These theoretical advancements not only improve our understanding of matrix recovery but also hold substantial practical promise, particularly in fields that rely on efficient processing of large, high-dimensional data.