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Identifiable Shared Component Analysis of Unpaired Multimodal Mixtures

Subash Timilsina, Sagar Shrestha, Xiao Fu

TL;DR

This paper tackles the identifiability of shared components in unaligned multimodal linear mixtures by introducing a distribution-divergence based loss that aligns transformed data across modalities without sample-level pairing. It proves identifiability under milder conditions than prior ICA-based approaches and strengthens results with structural constraints such as homogeneous mixing and weak supervision. The key contributions include an identifiable learning loss, theoretical conditions guaranteeing recovery of the shared component up to a common rotation, and practical extensions to private components; these are validated on synthetic data and real-world tasks like domain adaptation, single-cell data alignment, and cross-lingual retrieval. The approach offers a scalable, post-processing-friendly framework for extracting modality-invariant representations and improving downstream tasks in settings where cross-modal pairs are scarce or unavailable.

Abstract

A core task in multi-modal learning is to integrate information from multiple feature spaces (e.g., text and audio), offering modality-invariant essential representations of data. Recent research showed that, classical tools such as {\it canonical correlation analysis} (CCA) provably identify the shared components up to minor ambiguities, when samples in each modality are generated from a linear mixture of shared and private components. Such identifiability results were obtained under the condition that the cross-modality samples are aligned/paired according to their shared information. This work takes a step further, investigating shared component identifiability from multi-modal linear mixtures where cross-modality samples are unaligned. A distribution divergence minimization-based loss is proposed, under which a suite of sufficient conditions ensuring identifiability of the shared components are derived. Our conditions are based on cross-modality distribution discrepancy characterization and density-preserving transform removal, which are much milder than existing studies relying on independent component analysis. More relaxed conditions are also provided via adding reasonable structural constraints, motivated by available side information in various applications. The identifiability claims are thoroughly validated using synthetic and real-world data.

Identifiable Shared Component Analysis of Unpaired Multimodal Mixtures

TL;DR

This paper tackles the identifiability of shared components in unaligned multimodal linear mixtures by introducing a distribution-divergence based loss that aligns transformed data across modalities without sample-level pairing. It proves identifiability under milder conditions than prior ICA-based approaches and strengthens results with structural constraints such as homogeneous mixing and weak supervision. The key contributions include an identifiable learning loss, theoretical conditions guaranteeing recovery of the shared component up to a common rotation, and practical extensions to private components; these are validated on synthetic data and real-world tasks like domain adaptation, single-cell data alignment, and cross-lingual retrieval. The approach offers a scalable, post-processing-friendly framework for extracting modality-invariant representations and improving downstream tasks in settings where cross-modal pairs are scarce or unavailable.

Abstract

A core task in multi-modal learning is to integrate information from multiple feature spaces (e.g., text and audio), offering modality-invariant essential representations of data. Recent research showed that, classical tools such as {\it canonical correlation analysis} (CCA) provably identify the shared components up to minor ambiguities, when samples in each modality are generated from a linear mixture of shared and private components. Such identifiability results were obtained under the condition that the cross-modality samples are aligned/paired according to their shared information. This work takes a step further, investigating shared component identifiability from multi-modal linear mixtures where cross-modality samples are unaligned. A distribution divergence minimization-based loss is proposed, under which a suite of sufficient conditions ensuring identifiability of the shared components are derived. Our conditions are based on cross-modality distribution discrepancy characterization and density-preserving transform removal, which are much milder than existing studies relying on independent component analysis. More relaxed conditions are also provided via adding reasonable structural constraints, motivated by available side information in various applications. The identifiability claims are thoroughly validated using synthetic and real-world data.
Paper Structure (34 sections, 7 theorems, 61 equations, 7 figures, 8 tables)

This paper contains 34 sections, 7 theorems, 61 equations, 7 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Proposed Approach
  4. Enhanced Identifiability via Structural Constraints
  5. Related Works
  6. Numerical Validation
  7. Conclusion
  8. Notation
  9. Proof of Theorem \ref{['thm:unstructured']}
  10. Linearly transformed content identification
  11. Considering Assumption (a)
  12. Considering Assumption (b)
  13. Proof of Theorem \ref{['thm:identical_A']}
  14. Proof of Theorem \ref{['thm:weaksupervision']}
  15. Detailed Identifiability Conditions of Existing Results
  16. ...and 19 more sections

Key Result

Theorem 1

Under Assumption assump:style_var and the generative model in eq:sigmod, denote any solution of eq:shared_formulation as $\widehat{\boldsymbol{Q}}^{(q)}$$q=1,2$. Then, if the mixing matrices $\boldsymbol{A}^{(q)}$ are full column ranks and $\mathbb{E}[\bm c\bm c^{\!\top\!}])$ is full rank, we have $ Then, we have $\widehat{\boldsymbol{Q}}^{(q)}\boldsymbol{x}^{(q)} =\bm \Theta \bm c,$ i.e., $\bm \T

Figures (7)

  • Figure 1: Scatter plots of matched distribution $\bm \Theta^{(1)}\bm c$ (left) and $\bm \Theta^{(2)}\bm c$ (right) when $\bm c$ follows the Gaussian distribution. Colors in the scatter plot represent alignment; same color represent the data are aligned.
  • Figure 3: Validation of Theorem \ref{['thm:unstructured']}. Top row: results under assumption (a). Bottom row: results under assumption (b).
  • Figure 4: $k$-NN accuracy at top-$k$ neighbors.
  • Figure 5: Validation of Theorem \ref{['thm:weaksupervision']}$d_{\rm C}=3$ and $d_{\rm P}^{(1)}=1$.
  • Figure 6: Clip features $d^{(q)} =768$.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Theorem 4: Identifiability of Aligned SCA via CCA ibrahim2021cell
  • Theorem 5: Identifiability of Unaligned SCA via ICA sturma2024unpaired
  • Theorem 6
  • proof