Learning to Embed Distributions via Maximum Kernel Entropy (2408.00549v2)

Abstract: Empirical data can often be considered as samples from a set of probability distributions. Kernel methods have emerged as a natural approach for learning to classify these distributions. Although numerous kernels between distributions have been proposed, applying kernel methods to distribution regression tasks remains challenging, primarily because selecting a suitable kernel is not straightforward. Surprisingly, the question of learning a data-dependent distribution kernel has received little attention. In this paper, we propose a novel objective for the unsupervised learning of data-dependent distribution kernel, based on the principle of entropy maximization in the space of probability measure embeddings. We examine the theoretical properties of the latent embedding space induced by our objective, demonstrating that its geometric structure is well-suited for solving downstream discriminative tasks. Finally, we demonstrate the performance of the learned kernel across different modalities.

• The paper introduces a novel unsupervised entropy-maximizing objective for learning distribution-dependent kernels using quantum entropy.
• It develops a theoretical foundation showing that maximizing the quantum entropy of the covariance operator enhances the discriminative structure of latent spaces.
• Empirical results on flow cytometry, MNIST, and 20 Newsgroups confirm superior performance over traditional kernel methods in classification and regression tasks.

Learning to Embed Distributions via Maximum Kernel Entropy

The paper "Learning to Embed Distributions via Maximum Kernel Entropy" by Oleksii Kachaiev and Stefano Recanatesi presents a novel approach for unsupervised learning of data-dependent distribution kernels, a significant advancement in kernel methods for classification and regression tasks involving probability distributions. The methodology proposed by the authors hinges on the principle of entropy maximization within the space of probability measure embeddings.

Key Contributions

The authors address several crucial areas within distribution regression, an essential task in machine learning where one needs to predict a response variable based on empirical distributions derived from various input features. Their contributions revolve around three main innovations:

1. Entropy-Maximizing Objective: The primary contribution is the introduction of a new unsupervised objective for learning distribution-dependent kernels. This objective leverages the concept of maximizing quantum entropy, specifically the second-order Rènyi entropy, of the empirical covariance operator embeddings derived from the datasets.
2. Theoretical Foundation: The paper explores the theoretical properties of the latent embedding space fostered by the proposed entropy maximization. The authors establish that the geometric structure resulting from this approach is inherently conducive to discriminative tasks. They provide theoretical insights showing that this method enhances the distributional variance, ensuring an optimal configuration of the dataset embeddings.
3. Empirical Validation: The authors empirically demonstrate the effectiveness of their approach through classification and regression tasks across different modalities, including datasets with images, text, and biological data.

Methodological Details

The methodology is grounded in projecting input distributions onto a latent space, guided by a differentiable objective aimed at maximizing the entropy of the corresponding covariance operator. This process involves two primary components:

• Kernel Mean Embedding: The first step is mapping each input distribution to a mean embedding within an RKHS. This mapping preserves crucial characteristics of the original data distributions and allows for efficient computation of kernel metrics.
• Covariance Operator and Quantum Entropy: The second component introduces a new representation for the dataset, termed the dataset embedding, which leverages the covariance operator of the kernel mean embeddings. By maximizing the quantum entropy of this covariance operator, the method ensures that the latent space configuration is optimal for downstream discriminative tasks.

To achieve this, the paper proposes the Maximum Distribution Kernel Entropy (MDKE) objective: $$\mathcal{L}_\text{MDKE}(\theta) \coloneqq \log{ \| \frac{1}{M} K_\mathcal{D} \|_F^2 }$$ where $K_\mathcal{D}$ is the kernel Gram matrix computed from the dataset embeddings, and the Frobenius norm $\|\cdot\|_F$ ensures computational tractability. The encoder parameters $\theta$ are tuned to minimize this objective through a gradient-based optimization, promoting a latent distribution space with geometrical properties conducive to efficient classification and regression.

Empirical Results

The paper validates the proposed approach empirically across various datasets:

• Flow Cytometry: The authors show superior performance in classifying tissue samples and predicting leukemia presence, significantly outperforming traditional methods such as Gaussian Mixture Model Fisher vectors and Sliced Wasserstein kernels.
• MNIST and Fashion-MNIST: By treating images as histograms of pixel intensities, the authors demonstrate noteworthy improvements in classification accuracy with the entropy-optimized latent space representations.
• 20 Newsgroups: In text classification, treating sentences as distributions over a vocabulary space showed substantial accuracy gains, underscoring the method's robustness across different domains.

Implications and Future Directions

This research has significant implications both theoretically and practically. Theoretically, it extends the capability of kernel methods by integrating a principled approach to learning distribution-dependent kernels, enhancing their adaptability and efficiency in tasks involving distributions.

Practically, the developed framework can be employed as a pre-training step to boost the performance in various learning tasks, especially in domains where data points are inherently treated as distributions, such as genomics, neuroscience, and natural language processing.

Future work could explore:

• Scalability: Adapting the method to handle larger-scale datasets efficiently.
• Alternative Kernels: Investigating other kernel functions to enhance the method's robustness and applicability.
• Integration with Supervised Learning: Extending the framework to a semi-supervised or fully supervised learning setup to further leverage label information during kernel learning.

Conclusion

The paper "Learning to Embed Distributions via Maximum Kernel Entropy" introduces a highly innovative and theoretically grounded approach to learning data-dependent distribution kernels. By maximizing quantum entropy in the RKHS embedding space, the method ensures optimal representation for distributional regression, facilitating superior performance across a range of datasets and applications. This work marks a notable advancement in the application of entropy-based optimization techniques within the kernel methods paradigm.