Dimension reduction and coefficient estimation in multivariate linear regression

By Ming Yuan, Ali Ekici, Zhaosong Lu, Renato D. C. Monteiro. 2007

350 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/423a7a629266906f066e6d1c9e16c42ca9e30c06

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In Statistics and Probability

Citations in the corpus

When is there a representer theorem? Vector versus matrix regularizers (2008, background): “…area of interest in machine learning, especially during the past few years [Abernethy et al., 2006, Argyriou et al., 2006, 2007a,b, Candès and Recht, 2008, Cavallanti et al., 2008, Izenman, 1975, Maurer, 2006a,b, Srebro et al., 2005, Wolf et al., 2007, Xiang and Bennett, 2005, Yuan et al., 2007].”
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.”
Scalable Matrix-valued Kernel Learning for High-dimensional Nonlinear Multivariate Regression and Granger Causality (2012, methodology): “Results are shown in Table 1 where we compare our methods in terms of mean test RMSE against standard linear regression (OLS) and linear Lasso independently applied to each output coordinate, and the sparse multivariate regression with covariance estimation approaches of [31, 39], labeled MRCE and F”
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].”
On Spectral Learning (2010, methodology): “This occurs, for example, in multi-task learning and problems closely related to it (Abernethy et al., 2009; Argyriou et al., 2007a,b; Cand̀es and Recht, 2008; Cavallanti et al., 2008; Izenman, 1975; Maurer, 2006a,a; Srebro et al., 2005; Yuan et al., 2007, etc.).”
Approximation accuracy, gradient methods, and error bound for structured convex optimization (2010, background): “This problem is also NP-hard, and a convex relaxation has been proposed whereby rank(x) is replaced by the nuclear/trace norm ‖x‖nuc (the 1-norm of the singular values of x) [25,27,52,76,83,117,157].”
Sparse Multivariate Regression With Covariance Estimation (2010, methodology): “The results for the FES method where copied from Table 3 in Yuan et al. (2007). OLS sep.”
An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems (2009, methodology): “For example, the problem (7) arises naturally in simultaneous dimension reduction and coefficient estimation in multivariate linear regression [45].”
Regularized Multivariate Regression for Identifying Master Predictors with Application to Integrative Genomics Study of Breast Cancer (2008, methodology): “Moreover, we say a false positive trans-predictor (FPP) occurs if a cis-predictor is incorrectly identiﬁed as a trans-predictor ; we say a false negative trans-predictor (FNP) occurs if it is the other way around.”
Noisy matrix decomposition via convex relaxation: Optimal rates in high dimensions (2011, background): “, that is, the vectors β∗ j ∈ Rd2—exhibit some degree of shared structure across tasks [2, 32, 21, 27].”
Handbook of Big Data (2016, background): “, 2010), multivariate regression (Yuan et al., 2007), and graphical modeling in the presence of latent variables (Chandrasekaran et al.”
Estimation of (near) low-rank matrices with noise and high-dimensional scaling (2009, background): “[53] provide nonasymptotic bounds on the operator norm error of the estimate in the multi-task setting, provided that the design matrices are orthogonal.”
Fast global convergence rates of gradient methods for high-dimensional statistical recovery (2010, background): “In many contexts, it is natural to assume that Θ is exactly or approximately low rank; applications include collaborative filtering and matrix completion [7, 15], compressed sensing [16], and multitask learning [19, 10, 17].”
Reduced rank stochastic regression with a sparse singular value decomposition (2012, methodology): “The rank of C can also be estimated by various methods; see, for example Anderson (2002, 2003), Camba-Mendez et al. (2003) and Yuan et al. (2007).”