Interior-Point Method for Nuclear Norm Approximation with Application to System Identification

By Zhang Liu, Lieven Vandenberghe. 2009

528 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Mechanics, Signal Processing

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Citations in the corpus

Spectral Regularization Algorithms for Learning Large Incomplete Matrices (2010, methodology): “However, since these algorithms are based on second order methods [ LV08 ], they can become prohibitively expensive if the dimensions of the matrix get large [CCS08].”
A Singular Value Thresholding Algorithm for Matrix Completion (2008, methodology): “We refer the reader to [ 42 ] for some recent progress on interior-point methods concerning some special nuclear norm-minimization problems.”
Guaranteed Matrix Completion via Non-Convex Factorization (2014, methodology): “In particular, we show that under similar conditions to those in previous works, many standard optimization algorithms converge to the global optima of a factorization-based formulation and recover the true low-rank matrix.”
Trace Norm Regularization: Reformulations, Algorithms, and Multi-Task Learning (2010, methodology): “Multi-task learning, gene expression pattern analysis, trace norm regularization, convex optimiza- tion, duality, semidefinite programming, proximal gradient method.”
Nuclear Norm Based Matrix Regression with Applications to Face Recognition with Occlusion and Illumination Changes (2014, background): “Fazel et al. [18, 19] found that the convex envelope of rank (X) is the nuclear norm of X, i.e.”
Rate-Optimal Perturbation Bounds for Singular Subspaces with Applications to High-Dimensional Statistics (2016, background): “…include collaborative filtering (the Netflix problem) (Goldberg et al., 1992), multi-task learning (Argyriou et al., 2008), system identification (Liu and Vandenberghe, 2009), and sensor localization (Singer and ∗The research of Tony Cai was supported in part by NSF Grant DMS-1208982 and…”
Matrix Completion With Noise (2009, background): “In the new economy, companies are interested predicting musical preferences (Apple Inc.), literary preferences (Amazon, Barnes and Noble) and many other such things.”
Generalized Low Rank Models (2014, background): “Nuclear norm regularization is often used to encourage solutions of rank less than k, and has applications ranging from graph embedding to linear system identification [46, 92, 102, 130, 107].”
Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization (2007, methodology): “This can be controlled to some extent by exploiting the problem structure when assembling and solving the Newton system, as in the recent work of Liu and Vandenberghe [48].”
An Overview of Low-Rank Matrix Recovery From Incomplete Observations (2016, background): “To name just a few examples, we commonly encounter low-rank matrices in contexts as varied as: • ensembles of signals: the output of a sensor array or network, a collection of video frames, or a sequence of segments of a longer signal can often be highly correlated and represented using a low-rank m”
Robust principal component analysis? (2009, background): “This was suggested for rank minimization in [16, 45] and for low-rank and sparse decomposition [12] (see also [36]).”
Approximation accuracy, gradient methods, and error bound for structured convex optimization (2010, background): “has a more complex structure than (2), and only recently have solution methods, including interior-point method and dual gradient method, been developed [25,76,83] and exactness results been obtained [27,117].”
A unified framework for high-dimensional analysis of $M$-estimators with decomposable regularizers (2009, methodology): “Techniques based on nuclear norm regularization have been studied for different statistical models, including compressed sensing [37, 62], matrix completion [15, 31, 52, 61], multitask regression [4, 10, 51, 63, 77] and system identification [23, 38, 51].”
An alternating direction algorithm for matrix completion with nonnegative factors (2011, background): “Netflix to infer individual preference from an incomplete set of user preferences [13], global positioning, which discovers the positions of nodes in a network from incomplete pair-wise distances [3], system identification and order reduction, which recovers or reduces the dimension of the state vec”
Parallel stochastic gradient algorithms for large-scale matrix completion (2013, methodology): “Keywords Matrix completion · Incremental gradient methods · Parallel computing · Multicore Mathematics Subject Classification 90C06 · 90C25 · 90C15”
Matrix Completion on Graphs (2014, background): “Several important real-world problems can be cast as a matrix completion problem, including remote sensing [24], system identification [16] and recommendation systems [25].”