Diffusion Maps is not Dimensionality Reduction

Diffusion maps (DMAP) are often used as a dimensionality-reduction tool, but more precisely they provide a spectral representation of the intrinsic geometry rather than a complete charting method. To illustrate this distinction, we study a Swiss roll with known isometric coordinates and compare DMAP, Isomap, and UMAP across latent dimensions. For each representation, we fit an oracle affine readout to the ground-truth chart and measure reconstruction error. Isomap most efficiently recovers the low-dimensional chart, UMAP provides an intermediate tradeoff, and DMAP becomes accurate only after combining multiple diffusion modes. Thus the correct chart lies in the span of diffusion coordinates, but standard DMAP do not by themselves identify the appropriate combination.

LISA: Laplacian In-context Spectral Analysis

We propose Laplacian In-context Spectral Analysis (LISA), a method for inference-time adaptation of Laplacian-based time-series models using only an observed prefix. LISA combines delay-coordinate embeddings and Laplacian spectral learning to produce diffusion-coordinate state representations, together with a frozen nonlinear decoder for one-step prediction. We introduce lightweight latent-space residual adapters based on either Gaussian-process regression or an attention-like Markov operator over context windows. Across forecasting and autoregressive rollout experiments, LISA improves over the frozen baseline and is often most beneficial under changing dynamics. This work links in-context adaptation to nonparametric spectral methods for dynamical systems.

Fast ($\sim N$) Diffusion Map Algorithm

In this work we explore parsimonious manifold learning techniques, specifically for Diffusion-maps. We demonstrate an algorithm and it's implementation with computational complexity (in both time and memory) of $\sim N$, with $N$ representing the number-of-samples. These techniques are essential for large-scale unsupervised learning tasks without any prior assumptions, due to sampling theorem limitations.