Sparse Dictionary Learning for Measure-Valued Data via Linearized Optimal Transport: Efficient Algorithms and Sample Complexity
We study dictionary learning for probability measure-valued data by embedding measures into Hilbert spaces using features derived from optimal transport geometry.
This paper introduces a novel framework for dictionary learning applied to probability measure-valued data, leveraging optimal transport geometry and sparse representation techniques.
What happened
The study presents a computational and theoretical framework for representing probability measures through sparse dictionary learning. It combines optimal transport geometry with sparse representation, demonstrating effectiveness across multiple data modalities while establishing mathematical guarantees for generalization and sample complexity.
Why it matters
The framework enables more efficient and interpretable representation of complex data types (e.g., distributions, point clouds) in AI systems, with theoretical guarantees that support scalable deployment in real-world applications.
- Optimal transport geometry enhances sparse representation learning for probability measures
- Theoretical bounds ensure reliability for practical implementations
- The framework supports diverse data modalities through flexible dictionary learning
- Sparse representations enable efficient computation without sacrificing interpretability
- New mathematical tools improve understanding of measure-valued data complexity
The bigger picture
The approach advances representation learning for measure-valued data, offering a scalable method with mathematical rigor that could improve applications in computer vision, natural language processing, and statistical modeling.
This work establishes a robust framework for dictionary learning of measure-valued data, combining geometric insights from optimal transport with sparse representation techniques, while providing rigorous theoretical analysis to support its application in AI systems.
Technical details
- Hilbert space embeddings of probability measures
- Sparse autoencoder architectures for dictionary learning
- Optimal transport and entropic transport map dictionaries
- Metric complexity and covering number bounds for generalization
- $L^2$-valued measure embeddings
FAQ
What is the main contribution of this research?
The paper introduces a novel framework for dictionary learning of probability measure-valued data using optimal transport geometry, combined with sparse representation techniques and theoretical generalization guarantees.
What types of data does this method apply to?
The approach is applicable to diverse data modalities including hyperspectral images, geometric point clouds, and document embeddings in language model latent spaces.
What theoretical advancements does this work provide?
It establishes generalization bounds for $L^2$-valued measure embeddings and derives new covering number bounds for optimal transport and entropic transport map dictionaries.
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Originally published by arXiv - stat.ML on Oct 12, 2026.
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