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Linear optimal transport subspaces for point set classification

Mohammad Shifat E Rabbi, Naqib Sad Pathan, Shiying Li, Yan Zhuang, Abu Hasnat Mohammad Rubaiyat, Gustavo K Rohde

TL;DR

The paper tackles point-set classification under spatial deformations, focusing on affine-like transformations. It introduces Linear Optimal Transport (LOT) embeddings to map point sets into a linear LOT space, where class variations become (approximately) convex and amenable to a nearest-subspace classifier. By training enriched subspaces that encode invariances to translation, scaling, and shear, the method achieves non-iterative, parameter-light classification with strong data efficiency and robustness to distribution shifts in deformation magnitudes. Empirical results on synthetic and real 3D datasets show competitive accuracy against state-of-the-art methods, with notable advantages in out-of-distribution scenarios, and the approach offers a flexible framework for extending set-based recognition in higher dimensions.

Abstract

Learning from point sets is an essential component in many computer vision and machine learning applications. Native, unordered, and permutation invariant set structure space is challenging to model, particularly for point set classification under spatial deformations. Here we propose a framework for classifying point sets experiencing certain types of spatial deformations, with a particular emphasis on datasets featuring affine deformations. Our approach employs the Linear Optimal Transport (LOT) transform to obtain a linear embedding of set-structured data. Utilizing the mathematical properties of the LOT transform, we demonstrate its capacity to accommodate variations in point sets by constructing a convex data space, effectively simplifying point set classification problems. Our method, which employs a nearest-subspace algorithm in the LOT space, demonstrates label efficiency, non-iterative behavior, and requires no hyper-parameter tuning. It achieves competitive accuracies compared to state-of-the-art methods across various point set classification tasks. Furthermore, our approach exhibits robustness in out-of-distribution scenarios where training and test distributions vary in terms of deformation magnitudes.

Linear optimal transport subspaces for point set classification

TL;DR

The paper tackles point-set classification under spatial deformations, focusing on affine-like transformations. It introduces Linear Optimal Transport (LOT) embeddings to map point sets into a linear LOT space, where class variations become (approximately) convex and amenable to a nearest-subspace classifier. By training enriched subspaces that encode invariances to translation, scaling, and shear, the method achieves non-iterative, parameter-light classification with strong data efficiency and robustness to distribution shifts in deformation magnitudes. Empirical results on synthetic and real 3D datasets show competitive accuracy against state-of-the-art methods, with notable advantages in out-of-distribution scenarios, and the approach offers a flexible framework for extending set-based recognition in higher dimensions.

Abstract

Learning from point sets is an essential component in many computer vision and machine learning applications. Native, unordered, and permutation invariant set structure space is challenging to model, particularly for point set classification under spatial deformations. Here we propose a framework for classifying point sets experiencing certain types of spatial deformations, with a particular emphasis on datasets featuring affine deformations. Our approach employs the Linear Optimal Transport (LOT) transform to obtain a linear embedding of set-structured data. Utilizing the mathematical properties of the LOT transform, we demonstrate its capacity to accommodate variations in point sets by constructing a convex data space, effectively simplifying point set classification problems. Our method, which employs a nearest-subspace algorithm in the LOT space, demonstrates label efficiency, non-iterative behavior, and requires no hyper-parameter tuning. It achieves competitive accuracies compared to state-of-the-art methods across various point set classification tasks. Furthermore, our approach exhibits robustness in out-of-distribution scenarios where training and test distributions vary in terms of deformation magnitudes.
Paper Structure (16 sections, 2 theorems, 19 equations, 4 figures)

This paper contains 16 sections, 2 theorems, 19 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear optimal transport embeddings
  4. Discrete implementation for point sets
  5. Transport based Classification Problem statement
  6. Proposed solution
  7. Training phase
  8. Testing phase
  9. Results
  10. Experimental setup
  11. Accuracy in synthetic case
  12. Accuracy and efficiency in real datasets
  13. Out-of-distribution robustness
  14. Comparison with set-embedding-based methods
  15. Discussion
  16. ...and 1 more sections

Key Result

Proposition 2.1

Let $\mathcal{G}\subseteq \mathcal{T}_L$ be convex. Given $\mu\in \mathcal{P}_2(\mathbb{R}^L)$, define $\mathcal{G}_{\sharp}\mu := \{g_{\sharp}\mu: g\in \mathcal{G}\}$. If $\forall g\in \mathcal{G}$, CompositionProp holds, then $\widehat{\mathcal{G}_{\sharp}\mu}:= \{\widehat{\nu}: \nu \in \mathcal{G

Figures (4)

  • Figure 1: Percentage test accuracy comparison of different methods on synthetic datasets.
  • Figure 2: The accuracy of different methods as a function of the number of training samples per class, evaluated on MNIST, ModelNet, and ShapeNet datasets.
  • Figure 3: Performance assessment under an out-of-distribution experimental setup where training and test distributions vary in terms of deformation magnitudes. The performance of the methods was assessed in terms of percentage test accuracy and plotted against the number of training images per class.
  • Figure 4: Comparative analysis of the percentage test accuracy results attained by the proposed method and the conventional machine learning techniques implemented across different feature embedding spaces.

Theorems & Definitions (2)

  • Proposition 2.1: Lemma A.2 in moosmuller2023linear
  • Proposition 2.2