# Spatiotemporal Analysis Using Riemannian Composition of Diffusion Operators

arXiv: https://arxiv.org/abs/2201.08530 · HTML: https://mlsanity.com/paper/2201.08530/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 84 references that are in the map.
- First posted on arXiv 21 Jan 2022.

## Builds on (most cited first, 16 of 16)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Discrete Signal Processing on Graphs: Frequency Analysis](https://mlsanity.com/paper/1307.0468/) (2013)
- [From Manifold to Manifold: Geometry-Aware Dimensionality Reduction for SPD Matrices](https://mlsanity.com/paper/1407.1120/) (2014)
- [Nonparametric Canonical Correlation Analysis](https://mlsanity.com/paper/1511.04839/) (2015)
- [Parallel Transport on the Cone Manifold of SPD Matrices for Domain Adaptation](https://mlsanity.com/paper/1807.10479/) (2018)
- [Convergence of Graph Laplacian with kNN Self-tuned Kernels](https://mlsanity.com/paper/2011.01479/) (2020)
- [Latent common manifold learning with alternating diffusion: analysis and applications](https://mlsanity.com/paper/1602.00078/) (2016)
- [Impact of signal-to-noise ratio and bandwidth on graph Laplacian spectrum from high-dimensional noisy point cloud](https://mlsanity.com/paper/2011.10725/) (2020)
- [Recovering Hidden Components in Multimodal Data with Composite Diffusion Operators](https://mlsanity.com/paper/1808.07312/) (2018)
- [MultiView Diffusion Maps](https://mlsanity.com/paper/1508.05550/) (2015)
- [A common variable minimax theorem for graphs](https://mlsanity.com/paper/2107.14747/) (2021)
- [Manifold Learning with Contracting Observers for Data-driven Time-series Analysis](https://mlsanity.com/paper/1604.04492/) (2016)
- [Efficient fetal-maternal ECG signal separation from two channel maternal abdominal ECG via diffusion-based channel selection](https://mlsanity.com/paper/1702.02025/) (2017)
- [Spectral Flow on the Manifold of SPD Matrices for Multimodal Data Processing](https://mlsanity.com/paper/2009.08062/) (2020)

## Built on by (most cited first, 6 of 6)

- [Log-Euclidean Signatures for Intrinsic Distances Between Unaligned Datasets](https://mlsanity.com/paper/2202.01671/) (2022)
- [Tree-Wasserstein Distance for High Dimensional Data with a Latent Feature Hierarchy](https://mlsanity.com/paper/2410.21107/) (2024)
- [ManiFeSt: Manifold-based Feature Selection for Small Data Sets](https://mlsanity.com/paper/2207.08574/) (2022)
- [Landmark Alternating Diffusion](https://mlsanity.com/paper/2404.19649/) (2024)
- [On Learning what to Learn: heterogeneous observations of dynamics and establishing (possibly causal) relations among them](https://mlsanity.com/paper/2406.06812/) (2024)
- [Complex Interpolation of Matrices with an application to Multi-Manifold Learning](https://mlsanity.com/paper/2604.14118/) (2026)

## Abstract (arXiv)

Multivariate time-series have become abundant in recent years, as many data-acquisition systems record information through multiple sensors simultaneously. In this paper, we assume the variables pertain to some geometry and present an operator-based approach for spatiotemporal analysis. Our approach combines three components that are often considered separately: (i) manifold learning for building operators representing the geometry of the variables, (ii) Riemannian geometry of symmetric positive-definite matrices for multiscale composition of operators corresponding to different time samples, and (iii) spectral analysis of the composite operators for extracting different dynamic modes. We propose a method that is analogous to the classical wavelet analysis, which we term Riemannian multi-resolution analysis (RMRA). We provide some theoretical results on the spectral analysis of the composite operators, and we demonstrate the proposed method on simulations and on real data.