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Deep Diffusion Maps

Sergio García-Heredia, Ángela Fernández, Carlos M. Alaíz

TL;DR

This paper tackles the inefficiencies of Diffusion Maps (DM) for large-scale and out-of-sample data by introducing Deep Diffusion Maps (DDM), a neural-network-based embedding that bypasses explicit spectral decomposition. By formulating DM embedding as an unconstrained minimization and defining a regression-style loss, DDM trains a network to approximate diffusion coordinates $\Psi^{(t)}$ for both training and unseen points. Empirical results across synthetic and real datasets show that DDM qualitatively reproduces DM and Nyström embeddings, with quantitative MRE comparable in many cases and notable advantages in inference and integration with deep learning frameworks. The work highlights a new direction for scalable, out-of-sample diffusion-based representations, with potential gains in speed and flexibility on modern hardware.

Abstract

One of the fundamental problems within the field of machine learning is dimensionality reduction. Dimensionality reduction methods make it possible to combat the so-called curse of dimensionality, visualize high-dimensional data and, in general, improve the efficiency of storing and processing large data sets. One of the best-known nonlinear dimensionality reduction methods is Diffusion Maps. However, despite their virtues, both Diffusion Maps and many other manifold learning methods based on the spectral decomposition of kernel matrices have drawbacks such as the inability to apply them to data outside the initial set, their computational complexity, and high memory costs for large data sets. In this work, we propose to alleviate these problems by resorting to deep learning. Specifically, a new formulation of Diffusion Maps embedding is offered as a solution to a certain unconstrained minimization problem and, based on it, a cost function to train a neural network which computes Diffusion Maps embedding -- both inside and outside the training sample -- without the need to perform any spectral decomposition. The capabilities of this approach are compared on different data sets, both real and synthetic, with those of Diffusion Maps and the Nystrom method.

Deep Diffusion Maps

TL;DR

This paper tackles the inefficiencies of Diffusion Maps (DM) for large-scale and out-of-sample data by introducing Deep Diffusion Maps (DDM), a neural-network-based embedding that bypasses explicit spectral decomposition. By formulating DM embedding as an unconstrained minimization and defining a regression-style loss, DDM trains a network to approximate diffusion coordinates for both training and unseen points. Empirical results across synthetic and real datasets show that DDM qualitatively reproduces DM and Nyström embeddings, with quantitative MRE comparable in many cases and notable advantages in inference and integration with deep learning frameworks. The work highlights a new direction for scalable, out-of-sample diffusion-based representations, with potential gains in speed and flexibility on modern hardware.

Abstract

One of the fundamental problems within the field of machine learning is dimensionality reduction. Dimensionality reduction methods make it possible to combat the so-called curse of dimensionality, visualize high-dimensional data and, in general, improve the efficiency of storing and processing large data sets. One of the best-known nonlinear dimensionality reduction methods is Diffusion Maps. However, despite their virtues, both Diffusion Maps and many other manifold learning methods based on the spectral decomposition of kernel matrices have drawbacks such as the inability to apply them to data outside the initial set, their computational complexity, and high memory costs for large data sets. In this work, we propose to alleviate these problems by resorting to deep learning. Specifically, a new formulation of Diffusion Maps embedding is offered as a solution to a certain unconstrained minimization problem and, based on it, a cost function to train a neural network which computes Diffusion Maps embedding -- both inside and outside the training sample -- without the need to perform any spectral decomposition. The capabilities of this approach are compared on different data sets, both real and synthetic, with those of Diffusion Maps and the Nystrom method.
Paper Structure (20 sections, 1 theorem, 50 equations, 12 figures, 2 tables)

This paper contains 20 sections, 1 theorem, 50 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Kernel-based spectral methods
  3. Laplacian Eigenmaps
  4. Diffusion Maps
  5. Nyström method for Diffusion Maps
  6. Related works
  7. Deep Diffusion Maps
  8. Experiments
  9. Methodology
  10. Data sets
  11. Neural network architectures
  12. Hyperparameters
  13. Metrics
  14. Results
  15. Conclusions
  16. ...and 5 more sections

Key Result

Theorem 4.1

The matrix whose columns are the embedding vectors, $\Bqty{\vb*{\gamma}_i}_{i=1}^N$, of dimension $d$, given by Diffusion Maps is $\vb{R}\vb*{\Gamma}^\star$, with $\vb{R}\in O(d)$ a certain orthogonal matrixThat is, a rotation and/or reflection. and where $\vb{A}$ is given by eq:matrix_A, $\vb*{\Pi} = \operatorname{diag}(\pi_1, \ldots, \pi_N)$ and $\vb*{\pi}$ is the stationary distribution eq:sta

Figures (12)

  • Figure 1: Subset $\mathcal{D}_a$ for each data set.
  • Figure 2: Transition probabilities in $t$ steps with respect to the yellow point for the set $\mathcal{D}_a$.
  • Figure 3: Diffusion distances in $t$ steps with respect to the yellow point for the set $\mathcal{D}_a$.
  • Figure 4: Embedding of $\mathcal{D}_b$ for Swiss Roll.
  • Figure 5: Embedding of $\mathcal{D}_b$ for S Curve.
  • ...and 7 more figures

Theorems & Definitions (2)

  • Theorem 4.1
  • proof