<p>We study the problem of learning a partially observed matrix under the low rank assumption in the presence of fully observed side information that depends linearly on the true underlying matrix. This problem consists of an important generalization of the Matrix Completion problem, a central problem in Statistics, Operations Research and Machine Learning, that arises in applications such as recommendation systems, signal processing, system identification and image denoising. We formalize this problem as an optimization problem with an objective that balances the strength of the fit of the reconstruction to the observed entries with the ability of the reconstruction to be predictive of the side information. We derive a mixed-projection reformulation of the resulting optimization problem and present a strong semidefinite cone relaxation. We design an efficient, scalable alternating direction method of multipliers algorithm that produces high quality feasible solutions to the problem of interest. Our numerical results demonstrate that in the small rank regime (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(k \le 10\)</EquationSource> </InlineEquation>), our algorithm outputs solutions that achieve on average <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(2.3\%\)</EquationSource> </InlineEquation> lower objective value and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(41\%\)</EquationSource> </InlineEquation> lower <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\ell _2\)</EquationSource> </InlineEquation> reconstruction error than the solutions returned by the best performing benchmark method on synthetic data. The runtime of our algorithm is competitive with and often superior to that of the benchmark methods. Our algorithm is able to solve problems with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(n = 10000\)</EquationSource> </InlineEquation> rows and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(m = 10000\)</EquationSource> </InlineEquation> columns in less than a minute. On large scale real world data, our algorithm produces solutions that achieve <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(67\%\)</EquationSource> </InlineEquation> lower out of sample error than benchmark methods in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(97\%\)</EquationSource> </InlineEquation> less execution time.</p>

Predictive Low Rank Matrix Learning Under Partial Observations: Mixed-Projection ADMM

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发表日期：

2026年5月21日

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Dimitris Bertsimas,
Nicholas A. G. Johnson

摘要

We study the problem of learning a partially observed matrix under the low rank assumption in the presence of fully observed side information that depends linearly on the true underlying matrix. This problem consists of an important generalization of the Matrix Completion problem, a central problem in Statistics, Operations Research and Machine Learning, that arises in applications such as recommendation systems, signal processing, system identification and image denoising. We formalize this problem as an optimization problem with an objective that balances the strength of the fit of the reconstruction to the observed entries with the ability of the reconstruction to be predictive of the side information. We derive a mixed-projection reformulation of the resulting optimization problem and present a strong semidefinite cone relaxation. We design an efficient, scalable alternating direction method of multipliers algorithm that produces high quality feasible solutions to the problem of interest. Our numerical results demonstrate that in the small rank regime ( \(k \le 10\) ), our algorithm outputs solutions that achieve on average \(2.3\%\) lower objective value and \(41\%\) lower \(\ell _2\) reconstruction error than the solutions returned by the best performing benchmark method on synthetic data. The runtime of our algorithm is competitive with and often superior to that of the benchmark methods. Our algorithm is able to solve problems with \(n = 10000\) rows and \(m = 10000\) columns in less than a minute. On large scale real world data, our algorithm produces solutions that achieve \(67\%\) lower out of sample error than benchmark methods in \(97\%\) less execution time.