L1-Norm Tucker Tensor Decomposition

Abstract

Tucker decomposition is a common method for the analysis of multi-way/tensor data. Standard Tucker has been shown to be sensitive against heavy corruptions, due to its L2-norm-based formulation which places squared emphasis to peripheral entries.In this work, we explore L1-Tucker, an L1-norm based reformulation of standard Tucker decomposition. After formulating the problem, we present two algorithms for its solution, namely L1-norm Higher-Order Singular Value Decomposition (L1-HOSVD) and L1-norm Higher-Order Orthogonal Iterations (L1-HOOI). The presented algorithms are accompanied by complexity and convergence analysis. Our numerical studies on tensor reconstruction and classification corroborate that L1-Tucker, implemented by means of the proposed methods, attains similar performance to standard Tucker when the processed data are corruption-free, while it exhibits sturdy resistance against heavily corrupted entries.

Dynamic L1-Norm Tucker Tensor Decomposition

“…where [22,32,33] approximates the solution to L1-Tucker by N parallel L1-PCA problems. That is, for every n ∈ [N ], it finds Q n by solving (approximately or exactly) the L1-PCA max.…”

“…On the other hand, L1-HOOI is an iterative process that provably attains a higher L1-Tucker metric when initialized at the solution of L1-HOSVD [22,34]. Initialized at {Q n,0 } n∈[N ] (typically by means of L1-HOSVD), at every iteration i ≥ 1, L1-HOOI updates Q n,i by solving max.…”

“…For instance, Higher-Order Robust PCA (HoRPCA) [20] models and decomposes the processed tensor as the sum of a low multi-linear rank tensor (nominal data) and a sparse tensor (outliers). Another straightforward robust reformulation is L1-Tucker [21,22], which derives by simple substitution of the L2-norm in the Tucker formulation by the more robust L1-norm (not to be confused with sparsity-inducing L1-norm regularization schemes). Algorithms for the (approximate) solution of L1-Tucker have been proposed in [21][22][23][24][25][26][27].…”

Hamiltonian and Q-Inspired Neural Network-Based Machine Learning

“…It is well known that in data mining, signal processing, and machine learning, two transforms known as principal component analysis (PCA) [21], [22], [23] and independent component analysis (ICA) [24] are commonly used. PCA can be categorized as an orthogonal, and ICA can be categorized as a biorthogonal transform.…”

Dynamic L1-norm Tucker Tensor Decomposition

“…On the other hand, L1-HOOI is an iterative process that provably attains a higher L1-Tucker metric when initialized at the solution of L1-HOSVD [22,34]. Initialized at {Q n,0 } n∈[N ] (typically by means of L1-HOSVD), at every…”