Identifiable Shared Component Analysis of Unpaired Multimodal Mixtures (2409.19422v2)

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.

• The paper introduces an unaligned shared component analysis model that uses distribution divergence minimization to identify shared components without requiring pairwise sample alignment.
• It establishes relaxed sufficient conditions for identifiability by integrating structural constraints and handling scenarios with limited aligned samples.
• Empirical validations in domain adaptation and sequence analysis confirm the method’s superior accuracy compared to state-of-the-art techniques.

Essay on "Identifiable Shared Component Analysis of Unpaired Multimodal Mixtures"

The paper "Identifiable Shared Component Analysis of Unpaired Multimodal Mixtures" addresses the challenge of identifying shared components in multimodal data when cross-modality samples are unaligned. This issue arises frequently in multimodal learning where samples could be from diverse feature spaces such as text, audio, or images.

The problem setup is predicated on the linear mixture model, wherein data from each modality are assumed to be a linear combination of shareable and private components. In classical scenarios, methods such as Canonical Correlation Analysis (CCA) have been proven effective in identifying these shared components under the assumption that cross-modality data samples are aligned. However, this assumption of alignment is often impractical.

The main contributions of this work can be outlined as follows:

1. Unaligned Shared Component Analysis (SCA) Model:
   • The authors propose a model where shared components can be identified even when the multimodal data are unaligned. This is a significant relaxation from the previous aligned data models.
   • By utilizing a distribution divergence minimization-based loss, the paper derives a suite of sufficient conditions under which the shared components can be reliably identified. This distribution matching framework does not require sample-level alignment, making it more applicable to real-world scenarios where pairings are not readily available.
2. Relaxed Conditions for Identifiability:
   • The paper highlights that the identifiability results for shared components can be achieved under milder conditions compared to existing studies which heavily rely on Independent Component Analysis (ICA).
   • Structural constraints motivated by practical applications are examined to further relax the identifiability conditions. For instance, the paper considers scenarios with a small number of cross-domain aligned samples, providing enhanced flexibility.
3. Empirical Validation:
   • Extensive experiments using synthetic and real-world datasets validate the proposed theoretical claims. The application domains include cross-lingual word retrieval, genetic information alignment, and image data domain adaptation, showcasing the practical utility of the proposed methods.
4. Key Numerical Results:
   • For instance, in domain adaptation tasks, the proposed method consistently improved classification accuracy on the target domain. Notably, the method outperformed state-of-the-art techniques with significant performance margins, demonstrating its efficacy in aligning unpaired multimodal data.
   • In the problem of single-cell sequence analysis, the proposed model achieved significantly higher k-NN accuracy, indicating better alignment of RNA and ATAC sequences compared to existing methods.
5. Theoretical Implications and Future Directions:
   • The theoretical contributions of this paper lie in establishing identifiability conditions under practical scenarios, which is a non-trivial task given the ill-posed nature of linear mixture models.
   • The framework is potentially extendable to nonlinear mixture models, which could be an interesting future direction. Another important future direction mentioned includes finite sample analysis to understand the behavior of the proposed methods with limited data.

In summary, this paper makes substantial progress in the field of multimodal representation learning by addressing the challenge of identifying shared components from unaligned data. The proposed methods significantly advance the theoretical understanding of identifiability in this context and offer practical solutions validated through comprehensive empirical tests. This work lays the groundwork for further explorations into more complex models and real-world applications where alignment of data across multiple modalities remains a fundamental challenge.