Often a dataset can be viewed as a matrix, and in many situations that matrix is incomplete. Consider for example the Netflix matrix, where every entry is a particular user’s rating of a particular movie. Netflix does not have the ratings for every user on every movie, so this matrix is incomplete. The problem of matrix completion asks, very generally, what kinds of assumptions might we make on that underlying matrix to successfully reconstruct the entire matrix? This paper (and its predecessor by Candes and Recht) provided breakthrough results showing that a low-rank and incoherent matrix can be perfectly reconstructed using a convex optimization problem. Our work showed that a high-rank matrix can also be recovered, if it’s columns lie in a union of subspaces. I am studying the assumptions behind such algorithms, the application of matrix completion to real engineering problems, and new generalizations of the matrix completion problem to other models.
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- P. Wang, H. Liu, L Wang, Q. Qu and L. Balzano. “Symmetric Matrix Completion with ReLU Sampling.” Proceedings of ICML, 2024.
- K. Gilman, D.A. Tarzanagh, and L. Balzano, “Grassmannian Optimization for Online Tensor Completion and Tracking with the t-SVD.” IEEE Transactions on Signal Processing, vol. 70 (2022): 2152-2167.
- G. Ongie, D. Pimentel-Alarcon, R. Nowak, R. Willett, and L. Balzano, “Tensor Methods for Nonlinear Matrix Completion.” SIAM Journal of the Mathematics of Data Science, 3, no. 1 (2021): 253-279.
- R. Ganti, L. Balzano, and R. Willett, “Matrix Completion under Monotonic Single Index Models,” Proceedings of the conference for Neural Information Processing Systems (NeurIPS, Poster presentation 22% acceptance), December 2015.
- V. Tan, L. Balzano, and S. Draper, “Rank Minimization over Finite Fields: Fundamental Limits and Coding-Theoretic Interpretations,’’ IEEE Transactions on Information Theory 58, no. 4 (2012): 2018-2039.
- L. Balzano, B. Eriksson, and R. Nowak, “High-Rank Matrix Completion,’’ Proceedings of the conference on Artificial Intelligence and Statistics (AISTATS, poster presentation 35% acceptance), April 2012.
- L. Balzano, R. Nowak, and B. Recht, “Online and Adaptive Tracking of Subspaces from Highly Incomplete Information,” Proceedings of the Allerton Conference on Communications, Control, and Computing, September 2010.