# Manifold learning with bi-stochastic kernels

arXiv: https://arxiv.org/abs/1711.06711 · HTML: https://mlsanity.com/paper/1711.06711/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Nov 2017.

## Built on by (most cited first, 20 of 20)

- [Sinkformers: Transformers with Doubly Stochastic Attention](https://mlsanity.com/paper/2110.11773/) (2021)
- [Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation](https://mlsanity.com/paper/2101.09875/) (2021)
- [SNEkhorn: Dimension Reduction with Symmetric Entropic Affinities](https://mlsanity.com/paper/2305.13797/) (2023)
- [Bi-stochastically normalized graph Laplacian: convergence to manifold Laplacian and robustness to outlier noise](https://mlsanity.com/paper/2206.11386/) (2022)
- [Robust Inference of Manifold Density and Geometry by Doubly Stochastic Scaling](https://mlsanity.com/paper/2209.08004/) (2022)
- [Doubly Stochastic Graph-based Non-autoregressive Reaction Prediction](https://mlsanity.com/paper/2306.06119/) (2023)
- [Convergence analysis of t-SNE as a gradient flow for point cloud on a manifold](https://mlsanity.com/paper/2401.17675/) (2024)
- [Beyond kNN: Adaptive, Sparse Neighborhood Graphs via Optimal Transport](https://mlsanity.com/paper/2208.00604/) (2022)
- [Stable generative modeling using Schr"odinger bridges](https://mlsanity.com/paper/2401.04372/) (2024)
- [Entropic Optimal Transport Eigenmaps for Nonlinear Alignment and Joint Embedding of High-Dimensional Datasets](https://mlsanity.com/paper/2407.01718/) (2024)
- [Manifold Learning with Sparse Regularised Optimal Transport](https://mlsanity.com/paper/2307.09816/) (2023)
- [Conditional expectation using compactification operators](https://mlsanity.com/paper/2306.10592/) (2023)
- [Sketching the Heat Kernel: Using Gaussian Processes to Embed Data](https://mlsanity.com/paper/2403.07929/) (2024)
- [Langevin Diffusion Approximation to Same Marginal Schr"odinger Bridge](https://mlsanity.com/paper/2505.07647/) (2025)
- [Image denoising as a conditional expectation](https://mlsanity.com/paper/2505.21546/) (2025)
- [Sinkhorn Normalization of Diffusion Kernels](https://mlsanity.com/paper/2507.06161/) (2025)
- [Euclidean Distance Deflation Under High-Dimensional Heteroskedastic Noise](https://mlsanity.com/paper/2507.18520/) (2025)
- [Digital Twins: McKean-Pontryagin Control for Partially Observed Physical Twins](https://mlsanity.com/paper/2510.00937/) (2025)
- [Robust Tangent Space Estimation via Laplacian Eigenvector Gradient Orthogonalization](https://mlsanity.com/paper/2510.02308/) (2025)
- [Learning manifold diffusion semigroups from graph transition matrices](https://mlsanity.com/paper/2605.25383/) (2026)

## Abstract (arXiv)

In this paper we answer the following question: what is the infinitesimal generator of the diffusion process defined by a kernel that is normalized such that it is bi-stochastic with respect to a specified measure? More precisely, under the assumption that data is sampled from a Riemannian manifold we determine how the resulting infinitesimal generator depends on the potentially nonuniform distribution of the sample points, and the specified measure for the bi-stochastic normalization. In a special case, we demonstrate a connection to the heat kernel. We consider both the case where only a single data set is given, and the case where a data set and a reference set are given. The spectral theory of the constructed operators is studied, and Nystr"om extension formulas for the gradients of the eigenfunctions are computed. Applications to discrete point sets and manifold learning are discussed.