Low-rank-plus-diagonal matrix representations

Assumed audience:

People with undergrad linear algebra

Specifically \((\mathrm{K}=\mathrm{Z} \mathrm{Z}^{\top}+\sigma^2\mathrm{I})\) where \(\mathrm{K}\in\mathbb{R}^{N\times N}\) and \(\mathrm{Z}\in\mathbb{R}^{N\times D}\) with \(D\ll N\). A workhorse. We might get a cool low rank decompositions like this from matrix factorisation, but they arise everywhere. To pick one example, Gaussian processes.

Lots of fun tricks, mostly because of the Woodbury Identity. See Ken Tay’s intro on that.


Specifically, solving \(\mathrm{X}=\mathrm{K}\mathrm{B}\) for \(\mathrm{X}\) and, in particular solving it efficiently, in the sense that we

  1. exploit the computational efficiency of the low rank structure of \(\mathrm{K}\) so that it costs less than \(\mathcal{O}(D^3M)\) to compute \(\mathrm{K}^{-1}\mathrm{B}\).
  2. avoid every forming the explicit inverse matrix \(\mathrm{K}^{-1}\) which requires storage \(\mathcal{O}(D^2)\).

This is possible using the following useful trick. Applying the Woodbury identity, \[\begin{align*} \mathrm{K}^{-1}=\sigma^{-2}\mathrm{I}-\sigma^{-4} \mathrm{Z}\left(\mathrm{I}+\sigma^{-2}\mathrm{Z}^{\top} \mathrm{Z}\right)^{-1} \mathrm{Z}^{\top} \end{align*}\] we compute the lower Cholesky decomposition \(\mathrm{L} \mathrm{L}^{\top}=\left(\mathrm{I}+\sigma^{-2}\mathrm{Z}^{\top} \mathrm{Z}\right)^{-1}\) at a cost of \(\mathcal{O}(N^3)\), and define \(\mathrm{R}=\sigma^{-2}\mathrm{Z} \mathrm{L}\). We use this to discover \[ \mathrm{K}^{-1}=\sigma^{-2}\mathrm{I}-\mathrm{R} \mathrm{R}^{\top}, \] and we may thus compute the solution by matrix multiplication \[\begin{aligned} \mathrm{K}^{-1}\mathrm{B} &=\left(\mathrm{Z} \mathrm{Z}^{\top}+\sigma^2 \mathrm{I}\right)^{-1}\mathrm{B}\\ &=\left(\sigma^{-2}\mathrm{I}-\mathrm{R} \mathrm{R}^{\top}\right)\mathrm{B}\\ &=\underbrace{\sigma^{-2}\mathrm{B}}_{D \times M} - \underbrace{\mathrm{R}}_{D\times N} \underbrace{\mathrm{R}^{\top}\mathrm{B}}_{N\times M} \end{aligned}\]

The solution of the linear system is available at cost which looks something like \(\mathcal{O}\left(N^2 D + NDM +N^3\right)\) (hmm, should check that). Generalising from \(\sigma^2\mathrm{I}\) to arbitrary diagonal is easy.

TODO: discuss positive-definiteness.

TOD: Centered version (Ameli and Shadden 2023; D. Harville 1976; D. A. Harville 1977; Henderson and Searle 1981).


Nakatsukasa (2019) observes that

The nonzero eigenvalues of \(\mathrm{Y} \mathrm{Z}\) are equal to those of \(\mathrm{Z} \mathrm{Y}\) : an identity that holds as long as the products are square, even when \(\mathrm{Y}, \mathrm{Z}\) are rectangular. This fact naturally suggests an efficient algorithm for computing eigenvalues and eigenvectors of a low-rank matrix \(\mathrm{K}=\mathrm{Y} \mathrm{Z}\) with \(\mathrm{Y}, \mathrm{Z}^{\top} \in \mathbb{C}^{N \times r}, N \gg r\) : form the small \(r \times r\) matrix \(\mathrm{Z} \mathrm{Y}\) and find its eigenvalues and eigenvectors. For nonzero eigenvalues, the eigenvectors are related by \(\mathrm{Y} \mathrm{Z} v=\lambda v \Leftrightarrow \mathrm{Z} \mathrm{Y} w=\lambda w\) with \(w=\mathrm{Z} v\)[…]

Concerting that to the case of \(\mathrm{K}\) we have that the nonzero eigenvalues of \(\mathrm{Z} \mathrm{Z}^{\top}\) are equal to those of \(\mathrm{Z}^{\top} \mathrm{Z}\); to compute eigenvalues and eigenvectors of a low-rank matrix \(X=\mathrm{Z} \mathrm{Z}^{\top}\): form the small \(N \times N\) matrix \(\mathrm{Z}^{\top} \mathrm{Z}\) and find its eigenvalues and eigenvectors. For nonzero eigenvalues, the eigenvectors are related by \(\mathrm{Z} \mathrm{Z}^{\top} v=\lambda v \Leftrightarrow \mathrm{Z}^{\top} \mathrm{Z} w=\lambda w\) with \(w=\mathrm{Z}^{\top} v\).

A classic piece of lore is cheap eigendecomposition of \(\mathrm{K}\) by exploiting the low rank structure and SVD. I have no idea who invented this, but here goes. First we calculate the SVD of \(\mathrm{Z}\) to obtain \(\mathrm{Z}=\mathrm{U}\mathrm{S}\mathrm{V}^{\top}\), where \(\mathrm{U}\in\mathbb{R}^{D\times N}\) and \(\mathrm{V}\in\mathbb{R}^{N\times N}\) are orthogonal and \(\mathrm{S}\in\mathbb{R}^{N\times N}\) is diagonal. Then we may write \[ \begin{aligned} \mathrm{K} &= \mathrm{Z} \mathrm{Z}^{\top} + \sigma^2 \mathrm{I} \\ &= \mathrm{U} \mathrm{S} \mathrm{V}^{\top} \mathrm{V} \mathrm{S} \mathrm{U}^{\top} + \sigma^2 \mathrm{I} \\ &= \mathrm{U} \mathrm{S}^2 \mathrm{U}^{\top} + \sigma^2 \mathrm{I} \end{aligned} \] Thus the top \(N\) eigenvalues of \(\mathrm{K}\) are \(\sigma^2+s_n^2\), and the corresponding eigenvectors are \(\boldsymbol{u}_n\). The remaining eigenvalues are \(\sigma^2\), and the corresponding eigenvectors are an arbitrary subset in the complement of the \(\mathrm{U}\) eigenvectors.


Louis Tiao, Efficient Cholesky decomposition of low-rank updates summarising Seeger (2004).

Multiplying low-rank-plus-diagonal matrices

Specifically, \((\mathrm{Y} \mathrm{Y}^{\top}+\sigma^2\mathrm{I})(\mathrm{Z} \mathrm{Z}^{\top}+\sigma^2\mathrm{I})\). Are low rank products cheap?

\[ \begin{aligned} (\mathrm{Y} \mathrm{Y}^{\top}+\sigma^2\mathrm{I})(\mathrm{Z} \mathrm{Z}^{\top}+\sigma^2\mathrm{I}) &=\mathrm{Y} \mathrm{Y}^{\top} \mathrm{Z} \mathrm{Z}^{\top}+\sigma^2\mathrm{Y} \mathrm{Y}^{\top}+\sigma^2\mathrm{Z} \mathrm{Z}^{\top}+\sigma^4\mathrm{I}\\ &=\mathrm{Y} (\mathrm{Y}^{\top} \mathrm{Z} )\mathrm{Z}^{\top}+\sigma^2\mathrm{Y} \mathrm{Y}^{\top}+\sigma^2\mathrm{Z} \mathrm{Z}^{\top}+\sigma^4\mathrm{I} \end{aligned} \] which is still a sum of low-rank approximations. At this point it might be natural to consider a tensor decomposition.

Multiplying inverse low-rank-plus-diagonal matrices

Suppose the low-rank inverse factors of \(\mathrm{Y}\) and \(\mathrm{X}\) are, respectively, \(\mathrm{R}\) and \(\mathrm{C}\). Then we have

\[ \begin{aligned} &(\mathrm{Y} \mathrm{Y}^{\top}+\sigma^2\mathrm{I})^{-1}(\mathrm{Z} \mathrm{Z}^{\top}+\sigma^2\mathrm{I})^{-1}\\ &=(\sigma^{-2}\mathrm{I}-\mathrm{R} \mathrm{R}^{\top})(\sigma^{-2}\mathrm{I}-\mathrm{C} \mathrm{C}^{\top})\\ &=\sigma^{-4}\mathrm{I}-\sigma^{-4}\mathrm{R} \mathrm{R}^{\top}-\sigma^{-4}\mathrm{C} \mathrm{C}^{\top}+\sigma^{-4}\mathrm{R} (\mathrm{R}^{\top}\mathrm{C}) \mathrm{C}^{\top}\\ \end{aligned} \]

Once again, cheap to evaluate, but not so obviously nice.



Suppose we want to measure the Frobenius distance between \(\mathrm{K}_{\mathrm{U},\sigma^2}\) and \(\mathrm{K}_{\mathrm{R},\gamma^2}\). We recall that we might expect things to be nice if they are exactly low rank because, e.g. \[ \begin{aligned} \|\mathrm{U}\mathrm{U}^{\top}\|_F^2 =\operatorname{tr}\left(\mathrm{U}\mathrm{U}^{\top}\mathrm{U}\mathrm{U}^{\top}\right) =\|\mathrm{U}^{\top}\mathrm{U}\|_F^2 \end{aligned} \] How does it come out as a distance between two low-rank-plus-diagonaly matrices. The answer may be found without forming the full matrices. For compactness, we define \(\delta^2=\sigma^2-\gamma^2\). \[ \begin{aligned} &\|\mathrm{U}\mathrm{U}^{\dagger}+\sigma^{2}\mathrm{I}-\mathrm{R}\mathrm{R}^{\dagger}+\gamma^{2}\mathrm{I}\|_F^2\\ &=\left\|\mathrm{U}\mathrm{U}^{\dagger} -\mathrm{R}\mathrm{R}^{\dagger} + \delta^2\mathrm{I}\right\|_{F}^2\\ &=\left\|\mathrm{U}\mathrm{U}^{\dagger}+i\mathrm{R}i\mathrm{R}^{\dagger} + \delta^2\mathrm{I} \right\|_{F}^2\\ &=\left\|\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}^{\dagger} + \delta^2\mathrm{I} \right\|_{F}^2\\ &=\left\langle\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}^{\dagger} + \delta^2\mathrm{I},\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}^{\dagger} + \delta^2\mathrm{I} \right\rangle_{F}\\ &=\left\langle\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}^{\dagger},\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}^{\dagger}\right\rangle_{F} +\left\langle\delta^2\mathrm{I}, \delta^2\mathrm{I} \right\rangle_{F}\\ &\quad+2\operatorname{Re}\left(\left\langle\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}^{\dagger}, \delta^2\mathrm{I} \right\rangle_{F}\right)\\ &=\operatorname{Tr}\left(\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}^{\dagger}\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}^{\dagger}\right) +\delta^4D\\ &\quad+2\delta^2\operatorname{Re}\operatorname{Tr}\left(\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}^{\dagger}\right)\\ &=\operatorname{Tr}\left(\left(\mathrm{U}\mathrm{U}^{\dagger} -\mathrm{R}\mathrm{R}^{\dagger}\right)\left(\mathrm{U}\mathrm{U}^{\dagger} -\mathrm{R}\mathrm{R}^{\dagger}\right)\right) +\delta^4D\\ &\quad+2\delta^2\operatorname{Tr}\left(\mathrm{U}\mathrm{U}^{\dagger} -\mathrm{R}\mathrm{R}^{\dagger}\right)\\ &=\operatorname{Tr}\left(\mathrm{U}\mathrm{U}^{\dagger}\mathrm{U}\mathrm{U}^{\dagger}\right) -2\operatorname{Tr}\left(\mathrm{U}\mathrm{U}^{\dagger}\mathrm{R}\mathrm{R}^{\dagger}\right) + \operatorname{Tr}\left(\mathrm{R}\mathrm{R}^{\dagger}\mathrm{R}\mathrm{R}^{\dagger}\right) +\delta^4D \\ &\quad+2\delta^2\left(\operatorname{Tr}\left(\mathrm{U}\mathrm{U}^{\dagger}\right) -\operatorname{Tr}\left(\mathrm{R}\mathrm{R}^{\dagger}\right)\right)\\ &=\operatorname{Tr}\left(\mathrm{U}^{\dagger}\mathrm{U}\mathrm{U}^{\dagger}\mathrm{U}\right) -2\operatorname{Tr}\left(\mathrm{U}^{\dagger}\mathrm{R}\mathrm{R}^{\dagger}\mathrm{U}\right) + \operatorname{Tr}\left(\mathrm{R}^{\dagger}\mathrm{R}\mathrm{R}^{\dagger}\mathrm{R}\right) +\delta^4D \\ &\quad+2\delta^2\left(\operatorname{Tr}\left(\mathrm{U}^{\dagger}\mathrm{U}\right) -\operatorname{Tr}\left(\mathrm{R}^{\dagger}\mathrm{R}\right)\right)\\ &=\left\|\mathrm{U}^{\dagger}\mathrm{U}\right\|^2_F -2\left\|\mathrm{U}^{\dagger}\mathrm{R}\right\|^2_F + \left\|\mathrm{R}^{\dagger}\mathrm{R}\right\|^2_F +\delta^4D +2\delta^2\left(\left\|\mathrm{U}\right\|^2_F -\left\|\mathrm{R}\right\|^2_F\right) \end{aligned} \]


Mostly I use pytorch’s linear algebra.


Akimoto, Youhei. 2017. β€œFast Eigen Decomposition for Low-Rank Matrix Approximation.” arXiv.
Ameli, Siavash, and Shawn C. Shadden. 2023. β€œA Singular Woodbury and Pseudo-Determinant Matrix Identities and Application to Gaussian Process Regression.” Applied Mathematics and Computation 452 (September): 128032.
Babacan, S. Derin, Martin Luessi, Rafael Molina, and Aggelos K. Katsaggelos. 2012. β€œSparse Bayesian Methods for Low-Rank Matrix Estimation.” IEEE Transactions on Signal Processing 60 (8): 3964–77.
Bach, C, D. Ceglia, L. Song, and F. Duddeck. 2019. β€œRandomized Low-Rank Approximation Methods for Projection-Based Model Order Reduction of Large Nonlinear Dynamical Problems.” International Journal for Numerical Methods in Engineering 118 (4): 209–41.
Bach, Francis R. 2013. β€œSharp Analysis of Low-Rank Kernel Matrix Approximations.” In COLT, 30:185–209.
Barbier, Jean, Nicolas Macris, and LΓ©o Miolane. 2017. β€œThe Layered Structure of Tensor Estimation and Its Mutual Information.” arXiv:1709.10368 [Cond-Mat, Physics:math-Ph], September.
Bauckhage, Christian. 2015. β€œK-Means Clustering Is Matrix Factorization.” arXiv:1512.07548 [Stat], December.
Berry, Michael W., Murray Browne, Amy N. Langville, V. Paul Pauca, and Robert J. Plemmons. 2007. β€œAlgorithms and Applications for Approximate Nonnegative Matrix Factorization.” Computational Statistics & Data Analysis 52 (1): 155–73.
Brand, Matthew. 2002. β€œIncremental Singular Value Decomposition of Uncertain Data with Missing Values.” In Computer Vision β€” ECCV 2002, edited by Anders Heyden, Gunnar Sparr, Mads Nielsen, and Peter Johansen, 2350:707–20. Berlin, Heidelberg: Springer Berlin Heidelberg.
β€”β€”β€”. 2006. β€œFast Low-Rank Modifications of the Thin Singular Value Decomposition.” Linear Algebra and Its Applications, Special Issue on Large Scale Linear and Nonlinear Eigenvalue Problems, 415 (1): 20–30.
Chen, Yudong, and Yuejie Chi. 2018. β€œHarnessing Structures in Big Data via Guaranteed Low-Rank Matrix Estimation: Recent Theory and Fast Algorithms via Convex and Nonconvex Optimization.” IEEE Signal Processing Magazine 35 (4): 14–31.
Chi, Yuejie, Yue M. Lu, and Yuxin Chen. 2019. β€œNonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview.” IEEE Transactions on Signal Processing 67 (20): 5239–69.
Cichocki, A., N. Lee, I. V. Oseledets, A.-H. Phan, Q. Zhao, and D. Mandic. 2016. β€œLow-Rank Tensor Networks for Dimensionality Reduction and Large-Scale Optimization Problems: Perspectives and Challenges PART 1.” arXiv:1609.00893 [Cs], September.
Drineas, Petros, and Michael W. Mahoney. 2005. β€œOn the NystrΓΆm Method for Approximating a Gram Matrix for Improved Kernel-Based Learning.” Journal of Machine Learning Research 6 (December): 2153–75.
Fasi, Massimiliano, Nicholas J. Higham, and Xiaobo Liu. 2023. β€œComputing the Square Root of a Low-Rank Perturbation of the Scaled Identity Matrix.” SIAM Journal on Matrix Analysis and Applications 44 (1): 156–74.
Flammia, Steven T., David Gross, Yi-Kai Liu, and Jens Eisert. 2012. β€œQuantum Tomography via Compressed Sensing: Error Bounds, Sample Complexity, and Efficient Estimators.” New Journal of Physics 14 (9): 095022.
Ghashami, Mina, Edo Liberty, Jeff M. Phillips, and David P. Woodruff. 2015. β€œFrequent Directions : Simple and Deterministic Matrix Sketching.” arXiv:1501.01711 [Cs], January.
Gordon, Geoffrey J. 2002. β€œGeneralizedΒ² LinearΒ² Models.” In Proceedings of the 15th International Conference on Neural Information Processing Systems, 593–600. NIPS’02. Cambridge, MA, USA: MIT Press.
Gross, D. 2011. β€œRecovering Low-Rank Matrices From Few Coefficients in Any Basis.” IEEE Transactions on Information Theory 57 (3): 1548–66.
Gross, David, Yi-Kai Liu, Steven T. Flammia, Stephen Becker, and Jens Eisert. 2010. β€œQuantum State Tomography via Compressed Sensing.” Physical Review Letters 105 (15).
Halko, Nathan, Per-Gunnar Martinsson, and Joel A. Tropp. 2010. β€œFinding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions.” arXiv.
Harbrecht, Helmut, Michael Peters, and Reinhold Schneider. 2012. β€œOn the Low-Rank Approximation by the Pivoted Cholesky Decomposition.” Applied Numerical Mathematics, Third Chilean Workshop on Numerical Analysis of Partial Differential Equations (WONAPDE 2010), 62 (4): 428–40.
Harville, David. 1976. β€œExtension of the Gauss-Markov Theorem to Include the Estimation of Random Effects.” The Annals of Statistics 4 (2): 384–95.
Harville, David A. 1977. β€œMaximum Likelihood Approaches to Variance Component Estimation and to Related Problems.” Journal of the American Statistical Association 72 (358): 320–38.
Hastie, Trevor, Rahul Mazumder, Jason D. Lee, and Reza Zadeh. 2015. β€œMatrix Completion and Low-Rank SVD via Fast Alternating Least Squares.” In Journal of Machine Learning Research, 16:3367–3402.
Henderson, H. V., and S. R. Searle. 1981. β€œOn Deriving the Inverse of a Sum of Matrices.” SIAM Review 23 (1): 53–60.
Hoaglin, David C., and Roy E. Welsch. 1978. β€œThe Hat Matrix in Regression and ANOVA.” The American Statistician 32 (1): 17–22.
Kannan, Ramakrishnan. 2016. β€œScalable and Distributed Constrained Low Rank Approximations,” April.
Kannan, Ramakrishnan, Grey Ballard, and Haesun Park. 2016. β€œA High-Performance Parallel Algorithm for Nonnegative Matrix Factorization.” In Proceedings of the 21st ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming, 9:1–11. PPoPP ’16. New York, NY, USA: ACM.
Kim, H., and H. Park. 2008. β€œNonnegative Matrix Factorization Based on Alternating Nonnegativity Constrained Least Squares and Active Set Method.” SIAM Journal on Matrix Analysis and Applications 30 (2): 713–30.
Kumar, N. Kishore, and Jan Shneider. 2016. β€œLiterature Survey on Low Rank Approximation of Matrices.” arXiv:1606.06511 [Cs, Math], June.
Liberty, Edo, Franco Woolfe, Per-Gunnar Martinsson, Vladimir Rokhlin, and Mark Tygert. 2007. β€œRandomized Algorithms for the Low-Rank Approximation of Matrices.” Proceedings of the National Academy of Sciences 104 (51): 20167–72.
Lim, Yew Jin, and Yee Whye Teh. 2007. β€œVariational Bayesian Approach to Movie Rating Prediction.” In Proceedings of KDD Cup and Workshop, 7:15–21. Citeseer.
Lin, Zhouchen. 2016. β€œA Review on Low-Rank Models in Signal and Data Analysis.”
Liu, T., and D. Tao. 2015. β€œOn the Performance of Manhattan Nonnegative Matrix Factorization.” IEEE Transactions on Neural Networks and Learning Systems PP (99): 1–1.
Lu, Jun. 2022. β€œA Rigorous Introduction to Linear Models.” arXiv.
Mahoney, Michael W. 2010. Randomized Algorithms for Matrices and Data. Vol. 3.
Martinsson, Per-Gunnar. 2016. β€œRandomized Methods for Matrix Computations and Analysis of High Dimensional Data.” arXiv:1607.01649 [Math], July.
Minka, Thomas P. 2000. Old and new matrix algebra useful for statistics.
Nakajima, Shinichi, and Masashi Sugiyama. 2012. β€œTheoretical Analysis of Bayesian Matrix Factorization.” Journal of Machine Learning Research, 66.
Nakatsukasa, Yuji. 2019. β€œThe Low-Rank Eigenvalue Problem.” arXiv.
Nowak, W., and A. Litvinenko. 2013. β€œKriging and Spatial Design Accelerated by Orders of Magnitude: Combining Low-Rank Covariance Approximations with FFT-Techniques.” Mathematical Geosciences 45 (4): 411–35.
Petersen, Kaare Brandt, and Michael Syskind Pedersen. 2012. β€œThe Matrix Cookbook.”
Rokhlin, Vladimir, Arthur Szlam, and Mark Tygert. 2009. β€œA Randomized Algorithm for Principal Component Analysis.” SIAM J. Matrix Anal. Appl. 31 (3): 1100–1124.
Salakhutdinov, Ruslan, and Andriy Mnih. 2008. β€œBayesian Probabilistic Matrix Factorization Using Markov Chain Monte Carlo.” In Proceedings of the 25th International Conference on Machine Learning, 880–87. ICML ’08. New York, NY, USA: ACM.
Saul, Lawrence K. 2023. β€œA Geometrical Connection Between Sparse and Low-Rank Matrices and Its Application to Manifold Learning.” Transactions on Machine Learning Research, January.
Seeger, Matthias, ed. 2004. Low Rank Updates for the Cholesky Decomposition.
Seshadhri, C., Aneesh Sharma, Andrew Stolman, and Ashish Goel. 2020. β€œThe Impossibility of Low-Rank Representations for Triangle-Rich Complex Networks.” Proceedings of the National Academy of Sciences 117 (11): 5631–37.
Shi, Jiarong, Xiuyun Zheng, and Wei Yang. 2017. β€œSurvey on Probabilistic Models of Low-Rank Matrix Factorizations.” Entropy 19 (8): 424.
Srebro, Nathan, Jason D. M. Rennie, and Tommi S. Jaakkola. 2004. β€œMaximum-Margin Matrix Factorization.” In Advances in Neural Information Processing Systems, 17:1329–36. NIPS’04. Cambridge, MA, USA: MIT Press.
Sundin, Martin. 2016. β€œBayesian Methods for Sparse and Low-Rank Matrix Problems.” PhD Thesis, KTH Royal Institute of Technology.
Tropp, Joel A., Alp Yurtsever, Madeleine Udell, and Volkan Cevher. 2016. β€œRandomized Single-View Algorithms for Low-Rank Matrix Approximation.” arXiv:1609.00048 [Cs, Math, Stat], August.
β€”β€”β€”. 2017. β€œPractical Sketching Algorithms for Low-Rank Matrix Approximation.” SIAM Journal on Matrix Analysis and Applications 38 (4): 1454–85.
Tufts, D. W., and R. Kumaresan. 1982. β€œEstimation of Frequencies of Multiple Sinusoids: Making Linear Prediction Perform Like Maximum Likelihood.” Proceedings of the IEEE 70 (9): 975–89.
TΓΌrkmen, Ali Caner. 2015. β€œA Review of Nonnegative Matrix Factorization Methods for Clustering.” arXiv:1507.03194 [Cs, Stat], July.
Udell, M., and A. Townsend. 2019. β€œWhy Are Big Data Matrices Approximately Low Rank?” SIAM Journal on Mathematics of Data Science 1 (1): 144–60.
Wilkinson, William J., Michael Riis Andersen, Joshua D. Reiss, Dan Stowell, and Arno Solin. 2019. β€œEnd-to-End Probabilistic Inference for Nonstationary Audio Analysis.” arXiv:1901.11436 [Cs, Eess, Stat], January.
Woodruff, David P. 2014. Sketching as a Tool for Numerical Linear Algebra. Foundations and Trends in Theoretical Computer Science 1.0. Now Publishers.
Woolfe, Franco, Edo Liberty, Vladimir Rokhlin, and Mark Tygert. 2008. β€œA Fast Randomized Algorithm for the Approximation of Matrices.” Applied and Computational Harmonic Analysis 25 (3): 335–66.
Xinghao Ding, Lihan He, and L. Carin. 2011. β€œBayesian Robust Principal Component Analysis.” IEEE Transactions on Image Processing 20 (12): 3419–30.
Yang, Linxiao, Jun Fang, Huiping Duan, Hongbin Li, and Bing Zeng. 2018. β€œFast Low-Rank Bayesian Matrix Completion with Hierarchical Gaussian Prior Models.” IEEE Transactions on Signal Processing 66 (11): 2804–17.
Yin, M., J. Gao, and Z. Lin. 2016. β€œLaplacian Regularized Low-Rank Representation and Its Applications.” IEEE Transactions on Pattern Analysis and Machine Intelligence 38 (3): 504–17.
Yu, Chenhan D., William B. March, and George Biros. 2017. β€œAn \(N \log N\) Parallel Fast Direct Solver for Kernel Matrices.” In arXiv:1701.02324 [Cs].
Yu, Hsiang-Fu, Cho-Jui Hsieh, Si Si, and Inderjit S. Dhillon. 2012. β€œScalable Coordinate Descent Approaches to Parallel Matrix Factorization for Recommender Systems.” In IEEE International Conference of Data Mining, 765–74.
β€”β€”β€”. 2014. β€œParallel Matrix Factorization for Recommender Systems.” Knowledge and Information Systems 41 (3): 793–819.
Zhang, Kai, Chuanren Liu, Jie Zhang, Hui Xiong, Eric Xing, and Jieping Ye. 2017. β€œRandomization or Condensation?: Linear-Cost Matrix Sketching Via Cascaded Compression Sampling.” In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 615–23. KDD ’17. New York, NY, USA: ACM.
Zhang, Xiao, Lingxiao Wang, and Quanquan Gu. 2017. β€œStochastic Variance-Reduced Gradient Descent for Low-Rank Matrix Recovery from Linear Measurements.” arXiv:1701.00481 [Stat], January.
Zhou, Tianyi, and Dacheng Tao. 2011. β€œGodec: Randomized Low-Rank & Sparse Matrix Decomposition in Noisy Case.”
β€”β€”β€”. 2012. β€œMulti-Label Subspace Ensemble.” Journal of Machine Learning Research.

No comments yet. Why not leave one?

GitHub-flavored Markdown & a sane subset of HTML is supported.