Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Landmark Alternating Diffusion

Sing-Yuan Yeh, Hau-Tieng Wu, Ronen Talmon, Mao-Pei Tsui

TL;DR

This work tackles the computational bottleneck of diffusion-based sensor fusion via Alternating Diffusion (AD) by introducing Landmark Alternating Diffusion (LAD), a landmark-accelerated variant inspired by ROSELAND. LAD preserves AD’s core mechanism while enabling efficient diffusion through a small landmark set, with a tunable α-normalization that controls dependence on landmark sampling. The authors establish a rigorous manifold-based asymptotic analysis showing that LAD converges to a deformed Laplacian operator, and they validate the approach through simulations and an EEG sleep-stage annotation application, demonstrating major speedups with negligible loss in accuracy. The results highlight a scalable diffusion-map framework for multi-sensor data fusion in high-dimensional settings, with practical implications for real-time or large-scale biomedical and signal-processing tasks.

Abstract

Alternating Diffusion (AD) is a commonly applied diffusion-based sensor fusion algorithm. While it has been successfully applied to various problems, its computational burden remains a limitation. Inspired by the landmark diffusion idea considered in the Robust and Scalable Embedding via Landmark Diffusion (ROSELAND), we propose a variation of AD, called Landmark AD (LAD), which captures the essence of AD while offering superior computational efficiency. We provide a series of theoretical analyses of LAD under the manifold setup and apply it to the automatic sleep stage annotation problem with two electroencephalogram channels to demonstrate its application.

Landmark Alternating Diffusion

TL;DR

This work tackles the computational bottleneck of diffusion-based sensor fusion via Alternating Diffusion (AD) by introducing Landmark Alternating Diffusion (LAD), a landmark-accelerated variant inspired by ROSELAND. LAD preserves AD’s core mechanism while enabling efficient diffusion through a small landmark set, with a tunable α-normalization that controls dependence on landmark sampling. The authors establish a rigorous manifold-based asymptotic analysis showing that LAD converges to a deformed Laplacian operator, and they validate the approach through simulations and an EEG sleep-stage annotation application, demonstrating major speedups with negligible loss in accuracy. The results highlight a scalable diffusion-map framework for multi-sensor data fusion in high-dimensional settings, with practical implications for real-time or large-scale biomedical and signal-processing tasks.

Abstract

Alternating Diffusion (AD) is a commonly applied diffusion-based sensor fusion algorithm. While it has been successfully applied to various problems, its computational burden remains a limitation. Inspired by the landmark diffusion idea considered in the Robust and Scalable Embedding via Landmark Diffusion (ROSELAND), we propose a variation of AD, called Landmark AD (LAD), which captures the essence of AD while offering superior computational efficiency. We provide a series of theoretical analyses of LAD under the manifold setup and apply it to the automatic sleep stage annotation problem with two electroencephalogram channels to demonstrate its application.
Paper Structure (20 sections, 15 theorems, 120 equations, 8 figures, 1 table, 1 algorithm)

This paper contains 20 sections, 15 theorems, 120 equations, 8 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background on Alternating Diffusion (AD)
  3. Proposed Landmark Alternating Diffusion (LAD) Algorithm
  4. Landmark alternating diffusion
  5. Justification of LAD from a linear algebra perspective
  6. Asymptotic Analysis of LAD under the Manifold Setup
  7. Manifold Setting
  8. Simulation Experiment Results
  9. The impact of landmark size on $\alpha$-LAD
  10. The dependence on the initial diffusion
  11. The impact of landmark distribution and $\alpha$-normalization
  12. Application to Automatic Sleep Stage Annotation
  13. Feature Extraction
  14. Classification
  15. Results
  16. ...and 5 more sections

Key Result

Theorem 4.3

Take $f\in C^3(\mathcal{M})$. For $x\in\mathcal{M}$, we have where $w(x)=\frac{1}{3}s(x)-\frac{d}{12|S^{d-1}|}\int_{S^{d-1}}[\Pi^{(2)}_x(\theta,\theta)]^2d\theta$, $s(x)$ is the scalar curvature at $x$, and $\Pi^{(2)}_x$ is the second fundamental form of the embedding at $x$, and $|S^{d-1}|$ is the volume of the canonical $(d-1)$-dim sphere.

Figures (8)

  • Figure 1: From left to right: a comparison of computational time between AD and $1/2$-LAD with different landmark sizes, the difference ratio of the 1st, 3rd, 5th non-trivial eigenvalues between AD and $0.5$-LAD, the inner product between the 1st, 3rd, 5th non-trivial eigenvectors of AD and $0.5$-LAD, where we plot top 6 non-trivial eigenvectors, and the embedding similarity between AD and $0.5$-LAD.
  • Figure 2: Upper left: AD starting from $\mathbb{S}^1$. Upper middle: $0.5$-LAD starting from $\mathbb{S}^1$. Upper Right: DM of dataset on $C$. Lower left: AD starting from Trefoil knot $C$. Lower middle: $0.5$-LAD starting from Trefoil knot $C$. Lower right: DM of dataset on $\mathbb{S}^1$.
  • Figure 3: Top row, from left to right: $\alpha$-LAD with various $\alpha$, the difference ratio of the 2nd, 4th, 6th non-trivial eigenvalues between AD and $\alpha$-LAD, the inner product between the 2nd, 4th, 6th non-trivial eigenvectors of AD and $\alpha$-LAD, where we plot the top 6 eigenvectors, and the embedding similarity between AD and $\alpha$-LAD, where the landmark distribution is non-uniform. Bottom row shows the same thing but when the landmark distribution is uniform.
  • Figure 4: Illustration of various embeddings by $\alpha$-LAD under different landmark distributions. Top row, from left to right: the sampling density function $p^{(2)}$ and four landmark distributions, labeled "Case 1" to "Case 4." Middle row, from left to right: the LAD with $\alpha=0, 1/4, 1/2, 3/4$, and $1$, respectively. The blue curve is the result of AD, which is the same from left to right. The curves in other colors correspond to different landmark distributions. Lower row, from left to right: the pairwise embedding similarity between different embeddings with $\alpha=0, 1/4, 1/2, 3/4$, and $1$, respectively.
  • Figure 5: Left: The $\alpha$-LAD with $p_\mathcal{Z}^{(2)}=p^{(2)}$ and $\alpha=0,1/4,1/2,3/4,1$. Middle left: The difference ratio of the 2nd, 4th, 6th non-trivial eigenvalues of AD and $\alpha$-LAD with $\alpha=0,0.25,0.375,0.5,0.625,0.75,1$. Middle right: The inner product of the 2nd, 4th, 6th non-trivial eigenvectors of AD and $\alpha$-LAD with $\alpha=0,0.25,0.375,0.5,0.625,0.75,1$. Right: The embedding similarity between AD and $\alpha$-LAD with $q=6$.
  • ...and 3 more figures

Theorems & Definitions (22)

  • Definition 4.2
  • Theorem 4.3: Theorem 10 in shen2022
  • Definition 4.4
  • Definition 4.5
  • Theorem 4.6
  • Corollary 4.7
  • Corollary 4.8
  • Corollary 4.9
  • Theorem 4.10
  • Lemma 7.1: talmon2019 Lemma A.2
  • ...and 12 more