Diffusion Maps : Using the Semigroup Property for Parameter Tuning

• URL: http://arxiv.org/abs/2203.02867v1
• Date: Sun, 6 Mar 2022 03:02:24 GMT
• Title: Diffusion Maps : Using the Semigroup Property for Parameter Tuning
• Authors: Shan Shan and Ingrid Daubechies
• Abstract summary: Diffusion maps (DM) are used to reduce data lying on or close to a low-dimensional manifold embedded in a much larger dimensional space. We address the problem of setting a diffusion time t when constructing the diffusion kernel matrix by using the semigroup property of the diffusion operator. Experiments show that this principled approach is effective and robust.
• Score: 1.8782750537161608
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: Diffusion maps (DM) constitute a classic dimension reduction technique, for data lying on or close to a (relatively) low-dimensional manifold embedded in a much larger dimensional space. The DM procedure consists in constructing a spectral parametrization for the manifold from simulated random walks or diffusion paths on the data set. However, DM is hard to tune in practice. In particular, the task to set a diffusion time t when constructing the diffusion kernel matrix is critical. We address this problem by using the semigroup property of the diffusion operator. We propose a semigroup criterion for picking t. Experiments show that this principled approach is effective and robust.

Related papers

• How well behaved is finite dimensional Diffusion Maps? [0.0]
  We derive a series of properties that remain valid after finite-dimensional and almost isometric Diffusion Maps (DM) We establish rigorous bounds on the embedding errors introduced by the DM algorithm is $Oleft(fraclog nn)frac18d+16right$. These results offer a solid theoretical foundation for understanding the performance and reliability of DM in practical applications.
  arXiv  Detail & Related papers  (2024-12-05T09:12:25Z)
• Scaling Riemannian Diffusion Models [68.52820280448991]
  We show that our method enables us to scale to high dimensional tasks on nontrivial manifold. We model QCD densities on $SU(n)$ lattices and contrastively learned embeddings on high dimensional hyperspheres.
  arXiv  Detail & Related papers  (2023-10-30T21:27:53Z)
• Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes [57.396578974401734]
  We introduce a principled framework for building a generative diffusion process on general manifold. Instead of following the denoising approach of previous diffusion models, we construct a diffusion process using a mixture of bridge processes. We develop a geometric understanding of the mixture process, deriving the drift as a weighted mean of tangent directions to the data points.
  arXiv  Detail & Related papers  (2023-10-11T06:04:40Z)
• Reconstructing Graph Diffusion History from a Single Snapshot [87.20550495678907]
  We propose a novel barycenter formulation for reconstructing Diffusion history from A single SnapsHot (DASH) We prove that estimation error of diffusion parameters is unavoidable due to NP-hardness of diffusion parameter estimation. We also develop an effective solver named DIffusion hiTting Times with Optimal proposal (DITTO)
  arXiv  Detail & Related papers  (2023-06-01T09:39:32Z)
• A Heat Diffusion Perspective on Geodesic Preserving Dimensionality Reduction [66.21060114843202]
  We propose a more general heat kernel based manifold embedding method that we call heat geodesic embeddings. Results show that our method outperforms existing state of the art in preserving ground truth manifold distances. We also showcase our method on single cell RNA-sequencing datasets with both continuum and cluster structure.
  arXiv  Detail & Related papers  (2023-05-30T13:58:50Z)
• Decomposed Diffusion Sampler for Accelerating Large-Scale Inverse Problems [64.29491112653905]
  We propose a novel and efficient diffusion sampling strategy that synergistically combines the diffusion sampling and Krylov subspace methods. Specifically, we prove that if tangent space at a denoised sample by Tweedie's formula forms a Krylov subspace, then the CG with the denoised data ensures the data consistency update to remain in the tangent space. Our proposed method achieves more than 80 times faster inference time than the previous state-of-the-art method.
  arXiv  Detail & Related papers  (2023-03-10T07:42:49Z)
• Your diffusion model secretly knows the dimension of the data manifold [0.0]
  A diffusion model approximates the gradient of the log density of a noise-corrupted version of the target distribution for varying levels of corruption. We prove that, if the data concentrates around a manifold embedded in the high-dimensional ambient space, then as the level of corruption decreases, the score function points towards the manifold.
  arXiv  Detail & Related papers  (2022-12-23T23:15:14Z)
• The Manifold Scattering Transform for High-Dimensional Point Cloud Data [16.500568323161563]
  We present practical schemes for implementing the manifold scattering transform to datasets arising in naturalistic systems. We show that our methods are effective for signal classification and manifold classification tasks.
  arXiv  Detail & Related papers  (2022-06-21T02:15:00Z)
• Time-inhomogeneous diffusion geometry and topology [69.55228523791897]
  Diffusion condensation is a time-inhomogeneous process where each step first computes and then applies a diffusion operator to the data. We theoretically analyze the convergence and evolution of this process from geometric, spectral, and topological perspectives. Our work gives theoretical insights into the convergence of diffusion condensation, and shows that it provides a link between topological and geometric data analysis.
  arXiv  Detail & Related papers  (2022-03-28T16:06:17Z)
• Interpreting diffusion score matching using normalizing flow [22.667661526643265]
  We show that diffusion score matching (DSM) (or diffusion Stein discrepancy (DSD)) is equivalent to the original score matching (or Stein discrepancy) evaluated in a normalizing flow. Specifically, we prove that DSM (or DSD) is equivalent to the original score matching (or Stein discrepancy) evaluated in the transformed space defined by the normalizing flow.
  arXiv  Detail & Related papers  (2021-07-21T13:27:32Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.