A lifting approach to ParaTuck-2 tensor decompositions

TL;DR

This paper introduces a lifting-based, algebraic framework to recover ParaTuck-2 decompositions (PT2D) and its symmetric counterpart DEDICOM under best-known uniqueness conditions. By lifting the frontal slices into a higher-dimensional space and analyzing a structured matrix Φ, the authors obtain left kernels from which the factor matrices A and B can be recovered (up to permutation and scaling), with F,G,H extracted subsequently via core-factorization techniques. The approach yields constructive, algebraic algorithms for both the nonsymmetric PT2D and the symmetric DEDICOM/PARAFAC-2 cases, and it extends to approximate decompositions through SVD/EVD-based specializations. The results relax several prior assumptions (e.g., nonzero F and full-rank G/H) and provide a unified, principled route to identifiability and computation, albeit with computational complexity that grows with R and S. Numerical experiments demonstrate exact recovery in synthetic settings and competitive performance in noisy scenarios, indicating practical viability for core tensor decompositions in multiway data analysis.

Abstract

The ParaTuck-2 decomposition (PT2D) of third-order tensor is a two-layer generalization of the well-known canonical polyadic decomposition (CPD).While being more flexible than the CPD, the PT2D also possesses similar uniqueness properties.In this paper, we show than under the best known uniqueness conditions, the exact PT2D can be computed by an algebraic algorithm (i.e., can the PT2D problems can be reduced to computing nullspaces and eigenvalues of certain matrices).We do so by lifting the slices of the tensor to higher-dimensional space, which also allows for refining the existing uniqueness conditions.The algorithms are developed for general PT2D and its symmetric version (DEDICOM), which leads to an algebraic algorithm for another generalization of the CPD, the PARAFAC2 decomposition.Our methods are also applicable in the approximation scenario, as shown by the numerical experiments.

Figures (8)

• Figure 1: Product of three matrices.
• Figure 1: Equations for core tensor via in terms of the elements of the $2\times 2\times 2$ subtensor $\boldsymbol{\mathcal{C}}_{(i,r),(j,s),(k,t)}$.
• Figure 1: Performance of the plain ALS for the $2\times 2$ example. Left: convergence plots for $100$ initializations. Right: $SSS(\boldsymbol{A},\widehat{\boldsymbol{A}})$ versus $SSS(\boldsymbol{B},\widehat{\boldsymbol{B}})$ after 5000 ALS iterations.
• Figure 2: Performance of \ref{['alg:paratuck_2x2_complete']} for random examples with $(R,S) = (4,4)$, boxplots for 1000 realizations. Left: relative error in logarithmic scale $\log_{10}(MSE(\boldsymbol{\mathcal{T}},\widehat{\boldsymbol{\mathcal{T}}})/ \|\boldsymbol{\mathcal{T}}\|^2)$, middle: $SSS(\boldsymbol{A},\widehat{\boldsymbol{A}})$. Right: $SSS(\boldsymbol{B},\widehat{\boldsymbol{B}})$.
• Figure 3: Performance of \ref{['alg:paratuck_2x2_complete']} with respect to additive noise, for 100 realizations (plots in logarithmic scale for easier visualization). Left: relative error $\log_{10}(MSE(\boldsymbol{\mathcal{T}},\widehat{\boldsymbol{\mathcal{T}}})/ \|\boldsymbol{\mathcal{T}}\|^2)$. Middle: $\log_{10}(SSS(\boldsymbol{A},\widehat{\boldsymbol{A}}))$. Right: $\log_{10}(SSS(\boldsymbol{B},\widehat{\boldsymbol{B}}))$. For each of the dB levels, the pair of columns is given: without and with HOOI improvement (10 iterations).

Theorems & Definitions (79)

• Definition 3.1
• Example 3.2
• Remark 3.3
• Remark 3.4
• Example 3.5
• Lemma 3.6
• Proof 1
• Definition 3.7
• Theorem 3.8: HarsL96:uniqueness
• Remark 3.9: On genericity of assumptions