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Property Inheritance for Subtensors in Tensor Train Decompositions

HanQin Cai, Longxiu Huang

TL;DR

The paper addresses how essential tensor properties pass to subtensors in tensor-train (TT) decompositions when subtensors are obtained by fiber-wise sampling. It develops rank-preservation results across TT unfoldings and derives explicit bounds on incoherence and the condition number for subtensors, expressed via sampling-dependent parameters $\alpha_{i,t}$, $\alpha_i$, and $\beta_i$. Theoretical results are complemented by numerical experiments showing these parameters remain small under uniform sampling without replacement, indicating good property inheritance. These findings provide analytic tools for efficient TT-based tensor dimensionality reduction and extend Nyström-style ideas to TT representations, informing scalable tensor analysis methods.

Abstract

Tensor dimensionality reduction is one of the fundamental tools for modern data science. To address the high computational overhead, fiber-wise sampled subtensors that preserve the original tensor rank are often used in designing efficient and scalable tensor dimensionality reduction. However, the theory of property inheritance for subtensors is still underdevelopment, that is, how the essential properties of the original tensor will be passed to its subtensors. This paper theoretically studies the property inheritance of the two key tensor properties, namely incoherence and condition number, under the tensor train setting. We also show how tensor train rank is preserved through fiber-wise sampling. The key parameters introduced in theorems are numerically evaluated under various settings. The results show that the properties of interest can be well preserved to the subtensors formed via fiber-wise sampling. Overall, this paper provides several handy analytic tools for developing efficient tensor analysis methods.

Property Inheritance for Subtensors in Tensor Train Decompositions

TL;DR

The paper addresses how essential tensor properties pass to subtensors in tensor-train (TT) decompositions when subtensors are obtained by fiber-wise sampling. It develops rank-preservation results across TT unfoldings and derives explicit bounds on incoherence and the condition number for subtensors, expressed via sampling-dependent parameters , , and . Theoretical results are complemented by numerical experiments showing these parameters remain small under uniform sampling without replacement, indicating good property inheritance. These findings provide analytic tools for efficient TT-based tensor dimensionality reduction and extend Nyström-style ideas to TT representations, informing scalable tensor analysis methods.

Abstract

Tensor dimensionality reduction is one of the fundamental tools for modern data science. To address the high computational overhead, fiber-wise sampled subtensors that preserve the original tensor rank are often used in designing efficient and scalable tensor dimensionality reduction. However, the theory of property inheritance for subtensors is still underdevelopment, that is, how the essential properties of the original tensor will be passed to its subtensors. This paper theoretically studies the property inheritance of the two key tensor properties, namely incoherence and condition number, under the tensor train setting. We also show how tensor train rank is preserved through fiber-wise sampling. The key parameters introduced in theorems are numerically evaluated under various settings. The results show that the properties of interest can be well preserved to the subtensors formed via fiber-wise sampling. Overall, this paper provides several handy analytic tools for developing efficient tensor analysis methods.

Paper Structure

This paper contains 5 sections, 5 theorems, 31 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Main Results
  4. Numerical Evaluation
  5. Conclusion and Future Work

Key Result

Theorem 1

Suppose $\bm{M}\in\mathbb{R}^{n_1\times n_2}$ is rank-$r$ and $\{\mu_{1,\bm{M}},\mu_{2,\bm{M}}\}$-incoherent. Choose index set $I\subseteq[n_1]$ such that the row submatrix $\bm{R}=\bm{M}(I,:)$ is also rank-$r$. Then it holds where $\alpha:= {\hbox{$\sqrt{\frac{|I|}{n_1}\,}$}}\left\|{\bm{W}}_{\bm{M}}(I,:)^\dagger\right\|_2$. Similarly, choose index set $J\subseteq[n_2]$ such that the column subma

Figures (3)

  • Figure 1: yin2021towards. Visual representation of tensor train (TT) decomposition for a 4-order tensor. Note that $r_0=r_4=1$.
  • Figure 2: Boxplot for $\alpha_{i,t}$ as introduced in \ref{['thm:mu_of_R']}. Each box represents the distribution of parameter values over 20 trials, showing the median (center line), interquartile range (box), and potential outliers (red $+$). The whiskers (top and bottom horizontal lines) extend to the most extreme data points within 1.5 times the interquartile range. The dashed blue line indicates the mean of the parameter values. Left: Gaussian generation; Middle: Hadamard generation; Right: Uniform generation.
  • Figure 3: Boxplot for $\alpha_i$ and $\beta_i$ as introduced in \ref{['thm:mu_of_C']}. The setup of the boxplot is the same as \ref{['fig:thmR']}. Left: Gaussian generation; Middle: Hadamard generation; Right: Uniform generation.

Theorems & Definitions (15)

  • Definition 1: Matrix incoherence and condition number
  • Theorem 1: Section 3 of cai2021robust
  • Definition 2: Mode-$k$ product
  • Definition 3: TT rank and $i$-th tensor unfolding
  • Definition 4: TT decomposition
  • Definition 5: TT incoherence
  • Theorem 2
  • proof
  • Corollary 3
  • proof
  • ...and 5 more