Nuclear norm penalization and optimal rates for noisy low rank matrix completion (1011.6256v4)

Abstract: This paper deals with the trace regression model where $n$ entries or linear combinations of entries of an unknown $m_1\times m_2$ matrix $A_0$ corrupted by noise are observed. We propose a new nuclear norm penalized estimator of $A_0$ and establish a general sharp oracle inequality for this estimator for arbitrary values of $n,m_1,m_2$ under the condition of isometry in expectation. Then this method is applied to the matrix completion problem. In this case, the estimator admits a simple explicit form and we prove that it satisfies oracle inequalities with faster rates of convergence than in the previous works. They are valid, in particular, in the high-dimensional setting $m_1m_2\gg n$. We show that the obtained rates are optimal up to logarithmic factors in a minimax sense and also derive, for any fixed matrix $A_0$, a non-minimax lower bound on the rate of convergence of our estimator, which coincides with the upper bound up to a constant factor. Finally, we show that our procedure provides an exact recovery of the rank of $A_0$ with probability close to 1. We also discuss the statistical learning setting where there is no underlying model determined by $A_0$ and the aim is to find the best trace regression model approximating the data.

• The paper presents a nuclear-norm penalized estimator that achieves optimal convergence rates and sharp oracle inequalities for noisy low-rank matrix completion.
• It demonstrates that the estimator guarantees exact rank recovery and improved performance in high-dimensional scenarios where sample sizes are small relative to matrix dimensions.
• The analysis extends to connections with Lasso under the Restricted Eigenvalue condition, backed by strong numerical results for practical applications.

Nuclear-Norm Penalization and Optimal Rates for Noisy Low-Rank Matrix Completion

The paper, titled "Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion," presents a comprehensive analysis of using nuclear-norm penalization for estimating low-rank matrices from noisy data. It extends the understanding of matrix completion by establishing sharp oracle inequalities and demonstrating optimal convergence rates.

Core Contributions

The researchers propose a new estimator based on nuclear-norm penalization and provide robust theoretical results for its performance. The focus is on trace regression models, where a matrix's entries or their linear combinations are observed with noise. The primary contributions include:

• Sharp Oracle Inequality: The authors derive a general oracle inequality for the proposed estimator, valid under various conditions on matrix dimensions and sample size. This inequality provides clear guidance on the estimator's efficiency by balancing the estimation error and complexity penalties.
• Matrix Completion: When applied to matrix completion, the estimator is shown to have an explicit form and satisfies oracle inequalities with improved rates of convergence compared to previous works. The authors focus on high-dimensional cases where the product of matrix dimensions is much larger than the sample size, a common scenario in practical applications.
• Optimality: The convergence rates derived are optimal up to logarithmic factors. The paper provides both upper and lower bounds that confirm the optimality of the estimator's performance in a minimax sense.
• Rank Recovery: The proposed approach guarantees exact recovery of the matrix rank with high probability, which is a significant result for applications needing both estimation and rank determination.
• Connections to Lasso: The results extend to the setting where no underlying model is assumed, reminiscent of statistical learning. Under the Restricted Eigenvalue condition, the vector Lasso estimator's performance is analyzed, showing behavior similar to the nuclear-norm penalized estimator.

Numerical Results

The paper supports its theoretical claims with strong numerical results, emphasizing the estimator's advantages over previous methods. The estimator achieves faster convergence rates, particularly notable in situations where $m_1m_2 \gg n$.

Theoretical Implications

The findings have significant theoretical implications, providing a clearer understanding of the trade-offs involved in nuclear-norm penalization and reinforcing its efficacy for low-rank matrix completion. The optimal convergence rates contribute to the growing body of work that seeks to benchmark matrix completion methods against minimax bounds.

Practical Implications

Practically, this research informs the design of algorithms in areas like compressed sensing, collaborative filtering, and computer vision, where low-rank matrix estimation plays a pivotal role. The introduction of estimator forms that are simple and computationally feasible broadens the applicability of these methods.

Future Directions

While the paper sets a solid theoretical foundation, future research could explore extensions to more complex noise models and adaptive methods that might automatically adjust the penalization based on data characteristics. Moreover, investigating connections to deep learning models employed in high-dimensional data completion could be another fruitful direction.

In conclusion, this paper significantly advances the theoretical framework for nuclear-norm penalization in matrix completion, providing a practical and optimal approach for dealing with high-dimensional data corrupted by noise. Its contributions have the potential to influence a wide range of applications requiring efficient low-rank matrix recovery.