Matrix Completion

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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Wang, Peng, Huikang Liu, Anthony Man-Cho So, and Laura Balzano. 2022. “Convergence and Recovery Guarantees of the K-Subspaces Method for Subspace Clustering.” arXiv. https://doi.org/10.48550/arXiv.2206.05553. 1
Du, Zhe, Necmiye Ozay, and Laura Balzano. 2022. “Clustering-Based Mode Reduction for Markov Jump Systems.” In Proceedings of The 4th Annual Learning for Dynamics and Control Conference, 689–701. PMLR. https://proceedings.mlr.press/v168/du22a.html.
Sattar, Yahya, Zhe Du, Davoud Ataee Tarzanagh, Laura Balzano, Necmiye Ozay, and Samet Oymak. 2021. “Identification and Adaptive Control of Markov Jump Systems: Sample Complexity and Regret Bounds.” arXiv. https://doi.org/10.48550/arXiv.2111.07018.
Gilman, Kyle, Sam Burer, and Laura Balzano. 2022. “A Semidefinite Relaxation for Sums of Heterogeneous Quadratics on the Stiefel Manifold.” arXiv. https://doi.org/10.48550/arXiv.2205.13653. 2
Gilman, Kyle, Davoud Ataee Tarzanagh, and Laura Balzano. 2022. “Grassmannian Optimization for Online Tensor Completion and Tracking With the T-SVD.” IEEE Transactions on Signal Processing 70: 2152–67. https://doi.org/10.1109/TSP.2022.3164837.
Zhang, Dejiao, and Laura Balzano. 2022. “Convergence of a Grassmannian Gradient Descent Algorithm for Subspace Estimation From Undersampled Data.” University of Michigan Technical Report, February. https://doi.org/10.7302/4151.
Tarzanagh, Davoud Ataee, Laura Balzano, and Alfred O. Hero. 2021. “Fair Structure Learning in Heterogeneous Graphical Models,” December. https://arxiv.org/abs/2112.05128v1.
Ongie, Greg, Saket Dewangan, Jeffrey A. Fessler, and Laura Balzano. 2017. “Online Dynamic MRI Reconstruction via Robust Subspace Tracking.” In 2017 IEEE Global Conference on Signal and Information Processing (GlobalSIP), 1180–84. https://doi.org/10.1109/GlobalSIP.2017.8309147.
Ledva, Gregory S., Zhe Du, Laura Balzano, and Johanna L. Mathieu. 2018. “Disaggregating Load by Type from Distribution System Measurements in Real Time.” In Energy Markets and Responsive Grids, edited by Sean Meyn, Tariq Samad, Ian Hiskens, and Jakob Stoustrup, 162:413–37. New York, NY: Springer New York. https://doi.org/10.1007/978-1-4939-7822-9_17.
Ongie, Gregory, David Hong, Dejiao Zhang, and Laura Balzano. 2017. “Enhanced Online Subspace Estimation via Adaptive Sensing.” In Asilomar Confernce on Signals, Systems, and Computers. https://pdfs.semanticscholar.org/ba2f/61c45e92ae471552d55a8350f7211b02e6b0.pdf.
Bower, Amanda, and Laura Balzano. 2020. “Preference Modeling with Context-Dependent Salient Features.” In International Conference on Machine Learning, 1067–77. PMLR. https://proceedings.mlr.press/v119/bower20a.html.
Hong, David, Kyle Gilman, Laura Balzano, and Jeffrey A. Fessler. 2021. “HePPCAT: Probabilistic PCA for Data with Heteroscedastic Noise.” IEEE Transactions on Signal Processing, 1–1. https://doi.org/10.1109/TSP.2021.3104979.
Du, Zhe, Yahya Sattar, Davoud Ataee Tarzanagh, Laura Balzano, Samet Oymak, and Necmiye Ozay. 2021. “Certainty Equivalent Quadratic Control for Markov Jump Systems.” ArXiv:2105.12358 [Cs, Eess, Math], May. http://arxiv.org/abs/2105.12358. 3
Lipor, John, David Hong, Yan Shuo Tan, and Laura Balzano. 2021. “Subspace Clustering Using Ensembles of K-Subspaces.” Information and Inference: A Journal of the IMA 10 (1): 73–107. https://doi.org/10.1093/imaiai/iaaa031.
Ongie, Greg, Daniel Pimentel-Alarcón, Laura Balzano, Rebecca Willett, and Robert D. Nowak. 2021. “Tensor Methods for Nonlinear Matrix Completion.” SIAM Journal on Mathematics of Data Science, January, 253–79. https://doi.org/10.1137/20M1323448.
Hong, David, Kyle Gilman, Laura Balzano, and Jeffrey A. Fessler. 2021. “HePPCAT: Probabilistic PCA for Data with Heteroscedastic Noise.” ArXiv:2101.03468 [Eess, Math, Stat], January. http://arxiv.org/abs/2101.03468. 4
Ritchie, Alexander, Laura Balzano, and Clayton Scott. 2020. “Supervised PCA: A Multiobjective Approach.” ArXiv:2011.05309 [Cs, Stat], December. http://arxiv.org/abs/2011.05309.
Lyu, Hanbaek, Deanna Needell, and Laura Balzano. 2020. “Online Matrix Factorization for Markovian Data and Applications to Network Dictionary Learning.” Journal of Machine Learning Research 21 (251): 1–49. http://jmlr.org/papers/v21/20-444.html.
Hong, David, Shunbo Lei, Johanna L. Mathieu, and Laura Balzano. 2019. “Exploration of Tensor Decomposition Applied to Commercial Building Baseline Estimation.” In 2019 IEEE Global Conference on Signal and Information Processing (GlobalSIP), 1–5. https://doi.org/10.1109/GlobalSIP45357.2019.8969417.
Bower, Amanda, and Laura Balzano. 2020. “Preference Modeling with Context-Dependent Salient Features.” Accepted to ICML, February. http://arxiv.org/abs/2002.09615.
Du, Zhe, Laura Balzano, and Necmiye Ozay. 2018. “A Robust Algorithm for Online Switched System Identification.” IFAC-PapersOnLine, 18th IFAC Symposium on System Identification SYSID 2018, 51 (15): 293–98. https://doi.org/10.1016/j.ifacol.2018.09.150.
Ongie, Greg, David Hong, Dejiao Zhang, and Laura Balzano. 2018. “Online Estimation of Coherent Subspaces with Adaptive Sampling.” In 2018 IEEE Statistical Signal Processing Workshop (SSP), 841–45. https://doi.org/10.1109/SSP.2018.8450830.
Zhang, Dejiao, Julian Katz-Samuels, Mário A.T. Figueiredo, and Laura Balzano. 2018. “Simultaneous Sparsity and Parameter Tying for Deep Learning Using Ordered Weighted ℓ1 Regularization.” In 2018 IEEE Statistical Signal Processing Workshop (SSP), 65–69. https://doi.org/10.1109/SSP.2018.8450819.
Bower, Amanda, Lalit Jain, and Laura Balzano. 2018. “The Landscape of Non-Convex Quadratic Feasibility.” In 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 3974–78. https://doi.org/10.1109/ICASSP.2018.8461868.
Thong, Tasha, Yutong Wang, Michael D. Brooks, Christopher T. Lee, Clayton Scott, Laura Balzano, Max S. Wicha, and Justin A. Colacino. 2020. “Hybrid Stem Cell States: Insights Into the Relationship Between Mammary Development and Breast Cancer Using Single-Cell Transcriptomics.” Frontiers in Cell and Developmental Biology 8. https://doi.org/10.3389/fcell.2020.00288.
Gilman, Kyle, and Laura Balzano. 2020. “Online Tensor Completion and Free Submodule Tracking With The T-SVD.” In ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 3282–86. https://doi.org/10.1109/ICASSP40776.2020.9053199.
Gilman, Kyle, and Laura Balzano. 2020. “Grassmannian Optimization for Online Tensor Completion and Tracking in the T-SVD Algebra.” ArXiv:2001.11419 [Cs, Eess], January. http://arxiv.org/abs/2001.11419. 5
Du, Zhe, Necmiye Ozay, and Laura Balzano. 2019. “Mode Clustering for Markov Jump Systems.” In 2019 IEEE 8th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), 126–30. https://doi.org/10.1109/CAMSAP45676.2019.9022650.
Hong, David, Laura Balzano, and Jeffrey A. Fessler. 2019. “Probabilistic PCA for Heteroscedastic Data.” In 2019 IEEE 8th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), 26–30. https://doi.org/10.1109/CAMSAP45676.2019.9022436.
Lipor, John, and Laura Balzano. 2020. “Clustering Quality Metrics for Subspace Clustering.” Pattern Recognition, March, 107328. https://doi.org/10.1016/j.patcog.2020.107328.
Du, Zhe, Necmiye Ozay, and Laura Balzano. 2019. “Mode Clustering for Markov Jump Systems.” ArXiv:1910.02193 [Cs, Eess], October. http://arxiv.org/abs/1910.02193.
Ongie, Greg, Naveen Murthy, Laura Balzano, and Jeffrey A. Fessler. 2018. “A Memory-Efficient Algorithm for Large-Scale Sparsity Regularized Image Reconstruction.” Proceedings of the International Conference on Image Formation in X-Ray Computed Tomography. http://arxiv.org/abs/1904.00423.
Gilman, Kyle, and Laura Balzano. 2019. “Panoramic Video Separation with Online Grassmannian Robust Subspace Estimation.” In Proceedings of the IEEE International Conference on Computer Vision Workshops. http://openaccess.thecvf.com/content_ICCVW_2019/html/RSL-CV/Gilman_Panoramic_Video_Separation_with_Online_Grassmannian_Robust_Subspace_Estimation_ICCVW_2019_paper.html.
Ritchie, A., C. Scott, L. Balzano, D. Kessler, and C. S. Sripada. 2019. “Supervised Principal Component Analysis Via Manifold Optimization.” In 2019 IEEE Data Science Workshop (DSW), 6–10. https://doi.org/10.1109/DSW.2019.8755587.
Eftekhari, Armin, Gregory Ongie, Laura Balzano, and Michael B. Wakin. 2019. “Streaming Principal Component Analysis From Incomplete Data.” Journal of Machine Learning Research 20 (86): 1–62. http://jmlr.org/papers/v20/16-627.html.
Wang, Yutong, Tasha Thong, Venkatesh Saligrama, Justin Colacino, Laura Balzano, and Clayton Scott. 2019. “A Gene Filter for Comparative Analysis of Single-Cell RNA-Sequencing Trajectory Datasets.” BioRxiv, May, 637488. https://doi.org/10.1101/637488.
Eftekhari, Armin, Gregory Ongie, Laura Balzano, and Michael B. Wakin. 2019. “Streaming Principal Component Analysis From Incomplete Data.” ArXiv:1612.00904 [Cs, Math]. http://arxiv.org/abs/1612.00904. 6
Gitlin, A., B. Tao, L. Balzano, and J. Lipor. 2018. “Improving $K$-Subspaces via Coherence Pursuit.” IEEE Journal of Selected Topics in Signal Processing 12 (6): 1575–88. https://doi.org/10.1109/JSTSP.2018.2869363.
Ledva, G. S., L. Balzano, and J. L. Mathieu. 2018. “Exploring Connections Between a Multiple Model Kalman Filter and Dynamic Fixed Share with Applications to Demand Response.” In 2018 IEEE Conference on Control Technology and Applications (CCTA), 217–23. https://doi.org/10.1109/CCTA.2018.8511493.
Hong, D., R. P. Malinas, J. A. Fessler, and L. Balzano. 2018. “Learning Dictionary-Based Unions of Subspaces for Image Denoising.” In 2018 26th European Signal Processing Conference (EUSIPCO), 1597–1601. https://doi.org/10.23919/EUSIPCO.2018.8553117.
Hong, David, Jeffrey A. Fessler, and Laura Balzano. 2018. “Optimally Weighted PCA for High-Dimensional Heteroscedastic Data.” ArXiv:1810.12862 [Math, Stat], October. http://arxiv.org/abs/1810.12862. 7
Zhang, Dejiao, Yifan Sun, Brian Eriksson, and Laura Balzano. 2017. “Deep Unsupervised Clustering Using Mixture of Autoencoders.” University of Michigan Technical Report, December. https://deepblue.lib.umich.edu/handle/2027.42/145190.
“Deep Unsupervised Clustering Using Mixture of Autoencoders.” n.d. Accessed October 31, 2018. https://deepblue.lib.umich.edu/handle/2027.42/145190.
“Connector Preferences [Zotero Documentation].” n.d. Accessed October 19, 2018. https://www.zotero.org/support/connector_preferences#proxies_preferences.
Balzano, L., Y. Chi, and Y. M. Lu. 2018. “Streaming PCA and Subspace Tracking: The Missing Data Case.” Proceedings of the IEEE, 1–18. https://doi.org/10.1109/JPROC.2018.2847041.
Hong, David, Laura Balzano, and Jeffrey A. Fessler. 2018. “Asymptotic Performance of PCA for High-Dimensional Heteroscedastic Data.” Journal of Multivariate Analysis 167 (September): 435–52. https://doi.org/10.1016/j.jmva.2018.06.002.
Ledva, G. S., L. Balzano, and J. L. Mathieu. 2018. “Real-Time Energy Disaggregation of a Distribution Feeder’s Demand Using Online Learning.” IEEE Transactions on Power Systems, 1–1. https://doi.org/10.1109/TPWRS.2018.2800535.
Balzano, Laura, Yuejie Chi, and Yue M. Lu. 2018. “Streaming PCA and Subspace Tracking: The Missing Data Case.” Accepted to Proceedings of IEEE, June. http://arxiv.org/abs/1806.04609.
Zhang, Dejiao, Haozhu Wang, Mario Figueiredo, and Laura Balzano. 2018. “Learning to Share: Simultaneous Parameter Tying and Sparsification in Deep Learning.” International Conference on Learning Representations (ICLR), April. https://openreview.net/forum?id=rypT3fb0b.