Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem

By Vin de Silva, Lek-Heng Lim. 2006

963 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Mechanics, Computational Theory and Mathematics, Optimization & Training, Pure Mathematics

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Note that, in contrast to matrices, the set CP( R ) of tensors of rank at most R is in general not closed, which renders the problem of ﬁnding a best low-rank approximation ill-posed [56].”
Identifying Finite Mixtures of Nonparametric Product Distributions and Causal Inference of Confounders (2013, background): “For d > 2, ﬁnding a low-rank approximation of a tensor is an ill-posed problem (De Silva & Lim, 2008).”
Speeding-up Convolutional Neural Networks Using Fine-tuned CP-Decomposition (2014, background): “We suspect that this effect is due to the instability of the low-rank CP-decomposition (De Silva & Lim, 2008).”
Compression of Deep Convolutional Neural Networks for Fast and Low Power Mobile Applications (2015, background): “However it cannot be applied to the entire layers and the instability issue of low-rank CP decomposition is reported (De Silva & Lim, 2008; Lebedev et al., 2015).”
Krylov-Type Methods for Tensor Computations (2010, neutral): “These subspaces are given by Range {A (: , j, k ) The multilinear rank [8, 13] of the tensor is said to be equal to ( p, q, r ) if the dimension of these subspaces are p, q, and r , respectively.”
Factorization strategies for third-order tensors (2011, background): “It may be that the best rank-k approximation does not exist [22, 35, 40].”
Cross approximation in tensor electron density computations (2010, background): “However, canonical decomposition/approximation of a tensor with minimal value of R is a rather ill-posed and computationally unstable problem [20], and among several practical algorithms none is known to be absolutely robust.”
Tensor inversion and its application to the tensor equations with Einstein product (2018, background): “Introduction Tensors are multidimensional generalizations of vectors and matrices, which have important applications in multiway data analysis [1], psychometrics [2], digital image restorations [3], quantum entanglement [4], blind source separation and blind deconvolution in signal processing [5–7] ”
The Convex Geometry of Linear Inverse Problems (2010, background): “ with low-rank tensors including the non-existence in general of a singular value decomposition analogous to that for matrices [45], and the diﬀerence between the rank of a tensor and its border rank [23]. Nonorthogonal factor analysis. Suppose that a data matrix admits a factorization X= AB. The ma”
QTT approximation of elliptic solution operators in higher dimensions (2011, methodology): “Such procedure cannot be used in the canonical format, since approximation by the canonical format is known to be an ill-posed problem [9].”
Direct Solution of the Chemical Master Equation Using Quantized Tensor Trains (2014, background): “One reason is that the problem of tensor approximation (in the Frobenius norm) with a tensor of fixed rank is, in general, ill-posed [30].”
Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination (2014, background): “The ill-posedness of the best low-rank approximation of a tensor was investigated in [25].”
F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig on Minimal Subspaces in Tensor Representations on Minimal Subspaces in Tensor Representations (neutral): “Asking for a solution of inf {v − u : u ∈ R r }, one finds examples of v ∈ V even for finite dimensional V, but d ≥ 3, such that there is no minimiser u best ∈ R r (cf. [6]).”
Tensor-Train Decomposition (2011, background): “fixed canonical rank in the Frobenius norm can be ill-posed [10]; thus the numerical algorithms for computing an approximate representation in such cases might fail.”
Approximation rates for the hierarchical tensor format in periodic Sobolev spaces (2014, methodology): “This phenomenon already occurs in finite dimension, see the article of De Silva and Lim [42], and is suspected of being responsible that no robust general method to calculate almost best approximations from C≤r is known.”