On Probabilistic Embeddings in Optimal Dimension Reduction (2408.02433v1)

Abstract: Dimension reduction algorithms are a crucial part of many data science pipelines, including data exploration, feature creation and selection, and denoising. Despite their wide utilization, many non-linear dimension reduction algorithms are poorly understood from a theoretical perspective. In this work we consider a generalized version of multidimensional scaling, which is posed as an optimization problem in which a mapping from a high-dimensional feature space to a lower-dimensional embedding space seeks to preserve either inner products or norms of the distribution in feature space, and which encompasses many commonly used dimension reduction algorithms. We analytically investigate the variational properties of this problem, leading to the following insights: 1) Solutions found using standard particle descent methods may lead to non-deterministic embeddings, 2) A relaxed or probabilistic formulation of the problem admits solutions with easily interpretable necessary conditions, 3) The globally optimal solutions to the relaxed problem actually must give a deterministic embedding. This progression of results mirrors the classical development of optimal transportation, and in a case relating to the Gromov-Wasserstein distance actually gives explicit insight into the structure of the optimal embeddings, which are parametrically determined and discontinuous. Finally, we illustrate that a standard computational implementation of this task does not learn deterministic embeddings, which means that it learns sub-optimal mappings, and that the embeddings learned in that context have highly misleading clustering structure, underscoring the delicate nature of solving this problem computationally.

• The paper demonstrates that conventional particle descent methods can lead to non-deterministic, suboptimal embeddings in non-linear dimension reduction.
• It formulates a relaxed probabilistic optimization problem that leverages optimal transportation theory to enhance the interpretability of dimension reduction techniques.
• The study proves that despite initial probabilistic formulations, the globally optimal solutions are deterministic, ensuring reliable and accurate embedding mappings.

On Probabilistic Embeddings in Optimal Dimension Reduction

The paper "On Probabilistic Embeddings in Optimal Dimension Reduction" by Ryan Murray and Adam Pickarski provides a theoretical investigation into the nature of solutions for dimension reduction problems with a specific focus on non-linear algorithms. Many non-linear dimension reduction methods, despite their widespread use, lack a comprehensive theoretical understanding. This work situates a generalized version of multidimensional scaling (MDS) as an optimization problem in which the objective is to preserve either inner products or norms of a distribution from a high-dimensional feature space to a lower-dimensional embedding space.

Overview of the Paper

The crux of this paper revolves around three primary insights:

1. Non-Deterministic Embeddings via Particle Descent Methods: Traditional particle descent methods can lead to solutions that produce non-deterministic embeddings. This means that the learned mappings can be probabilistic rather than deterministic, potentially leading to suboptimal representations.
2. Probabilistic Formulation and Relaxed Problem Solutions: A relaxed, or probabilistic, formulation of the dimension reduction problem admits solutions with necessary conditions that are easier to interpret. This approach mirrors the classical development of optimal transportation problems.
3. Optimal Solutions and Deterministic Embeddings: The globally optimal solutions to the relaxed problem are shown to be deterministic. This indicates that while naive optimization routines might learn non-deterministic embeddings, theoretically optimal solutions must be deterministic.

Key Contributions

Non-Deterministic Behavior in Practical Learning

The paper illustrates through numerical examples that standard computational implementations might often result in non-deterministic embeddings, which means that they might learn sub-optimal mappings. This non-deterministic behavior can lead to misleading clustering structures in the embedded space.

Relaxed Formulation Using Measure Theory

To address the inherent difficulties in solving the high-dimensional optimization problem, the authors propose a relaxed version that involves probabilistic mapping. This formulation translates the original problem into finding a mapping in the space of probability measures, effectively allowing the embedding of a single point in feature space into a distribution over the embedding space.

Deterministic Nature of Optimal Solutions

Despite the probabilistic nature of the relaxed problem, the authors prove that the optimal solutions to this relaxed problem are necessarily deterministic. This deterministic nature of the optimal embeddings is significant as it highlights the theoretical limitations of probabilistic embeddings found via common computation methods like particle descent.

Implications and Future Directions

Theoretical Contributions:

The paper makes substantial contributions by linking the theory of optimal transportation to dimension reduction problems. This linkage enables the use of established mathematical frameworks to analyze and solve non-linear dimension reduction problems, providing a new perspective on understanding these methods.

Practical Relevance:

Practitioners need to be cautious of potential pitfalls when using standard optimization routines for dimension reduction. Probabilistic embeddings can lead to a flawed interpretation of the data structure, especially in applications involving data visualization and clustering.

Computational Improvements:

The authors suggest that improvements in computational algorithms are necessary. These improvements should likely involve methodologies capable of performing more sophisticated perturbations beyond smooth adaptations, to find globally optimal deterministic embeddings.

Broader Applications:

The principles outlined in this paper can be extended to other areas of unsupervised learning where embedding and optimization problems feature prominently. The insights gained here could stimulate further research into overcoming non-convexity and developing efficient algorithms that ensure deterministic solutions.

Conclusion

This paper contributes a significant theoretical framework for understanding non-linear dimension reduction and highlights the conditions under which deterministic embeddings are guaranteed. By addressing both practical computational challenges and theoretical properties, it sets the groundwork for future explorations into more robust and theoretically sound dimension reduction techniques. These findings underscore the importance of ensuring that computational solutions align with the theoretically optimal deterministic embeddings to avoid potential misinterpretations in data analysis and visualization tasks.