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Trace Ratio Based Manifold Learning with Tensor Data

Mohammed Bouallala, Franck Dufrenois, khalide jbilou, Ahmed Ratnani

TL;DR

The paper generalizes trace-ratio based manifold learning to multidimensional data by leveraging the t-product, introducing Laplacian tensors and a Tensor Newton-QR solver. It formulates tensor versions of LE, LLE, and LDE, deriving their optimization as generalized eigenproblems in the transform domain and solving them with the proposed algorithm. The approach demonstrates that tensor-based methods outperform traditional n-mode counterparts in accuracy, with nonlinear tensor methods offering particularly strong performance on complex, multi-class datasets. This work enables efficient, scalable tensorized dimensionality reduction with practical impact on multi-modal and high-dimensional data analysis.

Abstract

In this paper, we propose an extension of trace ratio based Manifold learning methods to deal with multidimensional data sets. Based on recent progress on the tensor-tensor product, we present a generalization of the trace ratio criterion by using the properties of the t-product. This will conduct us to introduce some new concepts such as Laplacian tensor and we will study formally the trace ratio problem by discuting the conditions for the exitence of solutions and optimality. Next, we will present a tensor Newton QR decomposition algorithm for solving the trace ratio problem. Manifold learning methods such as Laplacian eigenmaps, linear discriminant analysis and locally linear embedding will be formulated in a tensor representation and optimized by the proposed algorithm. Lastly, we will evaluate the performance of the different studied dimension reduction methods on several synthetic and real world data sets.

Trace Ratio Based Manifold Learning with Tensor Data

TL;DR

The paper generalizes trace-ratio based manifold learning to multidimensional data by leveraging the t-product, introducing Laplacian tensors and a Tensor Newton-QR solver. It formulates tensor versions of LE, LLE, and LDE, deriving their optimization as generalized eigenproblems in the transform domain and solving them with the proposed algorithm. The approach demonstrates that tensor-based methods outperform traditional n-mode counterparts in accuracy, with nonlinear tensor methods offering particularly strong performance on complex, multi-class datasets. This work enables efficient, scalable tensorized dimensionality reduction with practical impact on multi-modal and high-dimensional data analysis.

Abstract

In this paper, we propose an extension of trace ratio based Manifold learning methods to deal with multidimensional data sets. Based on recent progress on the tensor-tensor product, we present a generalization of the trace ratio criterion by using the properties of the t-product. This will conduct us to introduce some new concepts such as Laplacian tensor and we will study formally the trace ratio problem by discuting the conditions for the exitence of solutions and optimality. Next, we will present a tensor Newton QR decomposition algorithm for solving the trace ratio problem. Manifold learning methods such as Laplacian eigenmaps, linear discriminant analysis and locally linear embedding will be formulated in a tensor representation and optimized by the proposed algorithm. Lastly, we will evaluate the performance of the different studied dimension reduction methods on several synthetic and real world data sets.
Paper Structure (19 sections, 5 theorems, 91 equations, 5 figures, 1 table, 5 algorithms)

This paper contains 19 sections, 5 theorems, 91 equations, 5 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Multilinear algebra concepts
  3. The t-product
  4. Trace-Ratio Tensor problem
  5. Existence and uniqueness of a solution
  6. Necessary conditions for optimality
  7. Newton-QR algorithm
  8. Trace-Ratio tensor methods
  9. Graphs and Multidimensional data
  10. Definition of the weights
  11. Multilinear Local Discriminant Embedding
  12. Multilinear Laplacian Eigenmaps
  13. Locally Multilinear Embedding
  14. Multilinear Reconstruction by the Neighbors
  15. Multilinear Embedding
  16. ...and 4 more sections

Key Result

Lemma 2.6

Given a real vector $v \in \mathbb{R}^{n}$, the corresponding DFT vector $\widehat{v}=F_{n} v$ satisfies In this context, conj is used to represent the complex conjugation operator, while $\left[\frac{n_{3}+1}{2}\right]$ indicates the integer part of $\frac{n_{3}+1}{2}$.

Figures (5)

  • Figure 5.1: FEI data set.
  • Figure 5.2: AR data set.
  • Figure 5.3: Novel COVID-19 Chestxray Repository.
  • Figure 5.4: Brain Tumor MRI Dataset.
  • Figure 5.5: Summary of performance accuracy % of methods across various datasets.

Theorems & Definitions (27)

  • Definition 2.1: Kronecker product
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4: face-wise product
  • Definition 2.5
  • Lemma 2.6: rojo2004some
  • Definition 2.7
  • Definition 2.8: Identity tensor
  • Definition 2.9: Tensor transpose
  • Definition 2.10: Positive Definite (Semi-Definite) Tensor
  • ...and 17 more