Characterization of the subdifferential of some matrix norms

By G. Alistair Watson. 1992

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Citations in the corpus

Robust Large Margin Deep Neural Networks (2016, methodology): “In this case the computation of the subgradient of the spectral norm requires the calculation of a SVD decomposition [44], which makes the proposed regularizer inefficient.”
The Power of Convex Relaxation: Near-Optimal Matrix Completion (2009, background): “It is known [16], [25] that There is a more compact way to write (III.2).”
Novel Methods for Multilinear Data Completion and De-noising Based on Tensor-SVD (2014, background): “where Equation (8) is least-squares projection onto the constraint and the solution to Equation (9) is given by the singular value thresholding[19, 3].”
Linearized augmented Lagrangian and alternating direction methods for nuclear norm minimization (2012, unknown): “, [3, 52]) ∂∥X∥∗ = {UV ⊤ +W : U⊤W = 0, WV = 0, ∥W∥2 ≤ 1}, where ∥W∥2 represents the operator norm of W , i.”
Robust PCA via Outlier Pursuit (2010, background): “In this and the next section, we prove Theorems 1 and 2.”
Trace Norm Regularised Deep Multi-Task Learning (2016, methodology): “A more numerical stable method instead of computing the inverse matrix square root is X(XTX)− 1 2 = UV T where U and V are obtained from SVD: X = UΣV T (Watson, 1992).”
Global Convergence of ADMM in Nonconvex Nonsmooth Optimization (2015, background): “Then the general subgradient of |||A|||qq [45,64] is ∂|||A|||qq = U1DV T 1 + {U2ΓV T 2 ∣”
Latent variable graphical model selection via convex optimization (2010, background): “Elements of the subdifferentials with respect to the 1 norm and the nuclear norm at a matrix M have the key property that they decompose with respect to the tangent spaces (M) and T (M) [34].”
Decomposition-Based Transfer Distance Metric Learning for Image Classification (2014, background): “The sub-gradient of F at h (see [43]) is deﬁned as (24)”
Nuclear norm penalization and optimal rates for noisy low rank matrix completion (2010, background): “Finally, Section 6 is devoted to the control of the stochastic term appearing in the proof of the upper bound.”
Analysis of elliptical copula correlation factor model with Kendall's tau (2013, unknown): “We deﬁne, for any constant c ≥ 1, (For the form of ∂ Θ ∗ , the subdiﬀerential of the nuclear norm, see [24].)”
Tensor Robust Principal Component Analysis with a New Tensor Nuclear Norm (2018, background): “It is known that any subgradient of the nuclear norm at A is of the form UV ∗ + W , where U ∗ W = 0 , W V = 0 and (cid:107) W (cid:107) ≤ 1 [32].”
Exact Tensor Completion Using t-SVD (2015, background): “Moreover, Y is a subgradient of the nuclear norm at M0 if and only if Y is of the form [35]”
Rank-Sparsity Incoherence for Matrix Decomposition (2009, background): “We also have that Q ∈ ∂kB k if and only if [33]”