Parametric UMAP embeddings for representation and semi-supervised learning (2009.12981v4)

Abstract: UMAP is a non-parametric graph-based dimensionality reduction algorithm using applied Riemannian geometry and algebraic topology to find low-dimensional embeddings of structured data. The UMAP algorithm consists of two steps: (1) Compute a graphical representation of a dataset (fuzzy simplicial complex), and (2) Through stochastic gradient descent, optimize a low-dimensional embedding of the graph. Here, we extend the second step of UMAP to a parametric optimization over neural network weights, learning a parametric relationship between data and embedding. We first demonstrate that Parametric UMAP performs comparably to its non-parametric counterpart while conferring the benefit of a learned parametric mapping (e.g. fast online embeddings for new data). We then explore UMAP as a regularization, constraining the latent distribution of autoencoders, parametrically varying global structure preservation, and improving classifier accuracy for semi-supervised learning by capturing structure in unlabeled data. Google Colab walkthrough: https://colab.research.google.com/drive/1WkXVZ5pnMrm17m0YgmtoNjM_XHdnE5Vp?usp=sharing

• The paper introduces a parametric mapping approach using neural networks to extend UMAP for rapid online inference of new data.
• It integrates autoencoders to capture both local and global data structures, thereby improving reconstruction quality of embeddings.
• The method enhances semi-supervised learning by regularizing classifiers with efficiently computed low-dimensional representations.

Overview of Parametric UMAP for Representation and Semi-Supervised Learning

The paper discusses an extension to the UMAP (Uniform Manifold Approximation and Projection) algorithm by introducing a parametric form known as Parametric UMAP. Unlike the traditional non-parametric UMAP, which relies on a graph-based dimensionality reduction approach, Parametric UMAP integrates deep learning methodologies to achieve a learned parametric mapping between data and their low-dimensional embeddings.

UMAP operates based on two primary stages: graph construction and graph embedding. The authors extend the graph embedding stage to optimize neural network weights, allowing for the generation of embeddings with the benefits of parametric mappings. This extension enhances UMAP's utility by providing faster online embeddings for novel data and facilitating semi-supervised learning tasks.

Algorithmic Insights

UMAP begins with the construction of a fuzzy simplicial complex representing the data's local structure. This probabilistic graphical model captures relationships between data points. In UMAP, distances between points are computed using an adapted Riemannian geometry framework, focusing on the local connectivity defined by nearest neighbors. Parametric UMAP inherits this mechanism but replaces the direct optimization of embeddings with a neural network-based approach.

Key Contributions

1. Parametric Mapping: Parametric UMAP maintains comparable performance to the non-parametric version while enabling rapid online inference for new data points. This extends the usability of UMAP in real-time applications like brain-machine interfacing.
2. Regularization with Autoencoders: By integrating UMAP with autoencoders, Parametric UMAP enhances reconstruction quality and captures additional global data structure. This combination aids in structuring embeddings that reflect global and local relationships more faithfully.
3. Semi-Supervised Learning: Parametric UMAP is positioned as a regularization tool to improve classifier accuracy in semi-supervised learning scenarios by capturing intrinsic structures within unlabeled data.

Comparative Analysis

The paper offers a comparative paper involving both parametric and non-parametric algorithms. Parametric UMAP demonstrates promising results, particularly when utilized for tasks requiring fast embeddings and real-time structure learning. It competes closely with methods like t-SNE in embedding trustworthiness and clustering quality while surpassing in computational efficiency for embedding new data.

Theoretical and Practical Implications

The theoretical advancement of combining UMAP's manifold learning capabilities with neural networks introduces a versatile tool for capturing data structures in scalable and dynamic environments. Practically, it extends the application of UMAP into domains requiring rapid processing and adaptability, such as real-time data analysis and control systems.

Future Directions

Potential avenues for future work include exploring alternative global structure preservation techniques beyond pairwise distances. Moreover, the application of more sophisticated metrics such as the Fisher information metric could refine embeddings directly in relation to task-specific objectives. Enhancements in the neural network architecture tailored for different data types could further optimize performance across diverse datasets.

In summary, Parametric UMAP stands as a robust extension of the UMAP algorithm, merging topological data analysis with deep learning, thereby opening new possibilities in multidimensional data representation and learning frameworks.