Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Wasserstein Nonnegative Tensor Factorization with Manifold Regularization

Jianyu Wang, Linruize Tang

TL;DR

The paper addresses the limitation of equal-feature losses in nonnegative tensor factorization by introducing GWNTF, which minimizes a Wasserstein (optimal transport) distance between the input tensor and its nonnegative CP reconstruction $\hat{\mathcal{X}}$, while enforcing manifold structure through a graph regularizer. The objective combines a smooth Wasserstein tensor distance $W_T(\mathcal{X}, \hat{\mathcal{X}})$ with KL-based alignment terms and a graph-based regularizer, and is optimized via Maximization-Minimization (MM) to derive update rules for the transport tensor $\mathcal{T}$ and factor matrices $\mathbf{A}^{(n)}$. The method yields improved clustering performance on real-world image datasets (e.g., COIL20 and PIE_pose27) compared with six baselines, illustrating the benefits of incorporating both feature correlations via Wasserstein distances and local geometry via graph regularization. The approach advances feature-aware, structure-preserving tensor factorization with practical impact on high-order data analysis.

Abstract

Nonnegative tensor factorization (NTF) has become an important tool for feature extraction and part-based representation with preserved intrinsic structure information from nonnegative high-order data. However, the original NTF methods utilize Euclidean or Kullback-Leibler divergence as the loss function which treats each feature equally leading to the neglect of the side-information of features. To utilize correlation information of features and manifold information of samples, we introduce Wasserstein manifold nonnegative tensor factorization (WMNTF), which minimizes the Wasserstein distance between the distribution of input tensorial data and the distribution of reconstruction. Although some researches about Wasserstein distance have been proposed in nonnegative matrix factorization (NMF), they ignore the spatial structure information of higher-order data. We use Wasserstein distance (a.k.a Earth Mover's distance or Optimal Transport distance) as a metric and add a graph regularizer to a latent factor. Experimental results demonstrate the effectiveness of the proposed method compared with other NMF and NTF methods.

Wasserstein Nonnegative Tensor Factorization with Manifold Regularization

TL;DR

The paper addresses the limitation of equal-feature losses in nonnegative tensor factorization by introducing GWNTF, which minimizes a Wasserstein (optimal transport) distance between the input tensor and its nonnegative CP reconstruction , while enforcing manifold structure through a graph regularizer. The objective combines a smooth Wasserstein tensor distance with KL-based alignment terms and a graph-based regularizer, and is optimized via Maximization-Minimization (MM) to derive update rules for the transport tensor and factor matrices . The method yields improved clustering performance on real-world image datasets (e.g., COIL20 and PIE_pose27) compared with six baselines, illustrating the benefits of incorporating both feature correlations via Wasserstein distances and local geometry via graph regularization. The approach advances feature-aware, structure-preserving tensor factorization with practical impact on high-order data analysis.

Abstract

Nonnegative tensor factorization (NTF) has become an important tool for feature extraction and part-based representation with preserved intrinsic structure information from nonnegative high-order data. However, the original NTF methods utilize Euclidean or Kullback-Leibler divergence as the loss function which treats each feature equally leading to the neglect of the side-information of features. To utilize correlation information of features and manifold information of samples, we introduce Wasserstein manifold nonnegative tensor factorization (WMNTF), which minimizes the Wasserstein distance between the distribution of input tensorial data and the distribution of reconstruction. Although some researches about Wasserstein distance have been proposed in nonnegative matrix factorization (NMF), they ignore the spatial structure information of higher-order data. We use Wasserstein distance (a.k.a Earth Mover's distance or Optimal Transport distance) as a metric and add a graph regularizer to a latent factor. Experimental results demonstrate the effectiveness of the proposed method compared with other NMF and NTF methods.
Paper Structure (12 sections, 2 theorems, 18 equations, 2 tables, 1 algorithm)

This paper contains 12 sections, 2 theorems, 18 equations, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminary and Problem formulation
  3. Notations
  4. Wasserstein Distance
  5. Objective function
  6. Optimization of GWNTF
  7. Update rule for $\mathcal{T}$
  8. Update rule for $\mathbf{A}^{(n)}$
  9. Comparison Framework and Results
  10. Experimental settings.
  11. Main Results.
  12. Conclusion

Key Result

Lemma 1

(Auxiliary function fevotte2011algorithms): Given two functions $G(\mathbf{A}|\hat{\mathbf{A}})$ and $F(\mathbf{A})$, if $\forall \mathbf{A} \in \mathbb{R}$, there are $F(\mathbf{A}) = G(\mathbf{A}|{\mathbf{A}})$ and $F(\mathbf{A}) \le G(\mathbf{A}|\hat{\mathbf{A}})$. The mapping $G(\mathbf{A}|\hat{

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 1
  • Theorem 1
  • Proof 1