Compressed Diffusion

Abstract

Diffusion maps are a commonly used kernel-based method for manifold learning, which can reveal intrinsic structures in data and embed them in low dimensions. However, as with most kernel methods, its implementation requires a heavy computational load, reaching up to cubic complexity in the number of data points. This limits its usability in modern data analysis. Here, we present a new approach to computing the diffusion geometry, and related embeddings, from a compressed diffusion process between data regions rather than data points. Our construction is based on an adaptation of the previously proposed measure-based Gaussian correlation (MGC) kernel that robustly captures the local geometry around data points. We use this MGC kernel to efficiently compress diffusion relations from pointwise to data region resolution. Finally, a spectral embedding of the data regions provides coordinates that are used to interpolate and approximate the pointwise diffusion map embedding of data. We analyze theoretical connections between our construction and the original diffusion geometry of diffusion maps, and demonstrate the utility of our method in analyzing big datasets, where it outperforms competing approaches. † These authors contributed equally. † † These authors contributed equally. § Corresponding

Visualizing structure and transitions in high-dimensional biological data

“…However, since storing the entries of the powered diffusion operator in memory is also an issue, we employ the use of landmarks earlier in the process. It has also been shown that "compressing" the process of diffusion through landmarks in the fashion described here performs better than simply applying Nystrom extension (which includes landmark MDS [66]) to diffusion maps [68].…”

Visualizing Structure and Transitions for Biological Data Exploration

“…However, since storing the entries of the powered diffusion operator in memory is also an issue, we employ the use of landmarks earlier in the process. It has also been shown that "compressing" the process of diffusion through landmarks in the fashion described here performs better than simply applying Nystrom extension (which includes landmark MDS [82]) to diffusion maps [84].…”

Multiscale PHATE Exploration of SARS-CoV-2 Data Reveals Multimodal Signatures of Disease

“…On this initial coarse-graining we compute the diffusion potential coordinates by employing landmarking as developed in [19]. Landmarking refers to the idea that instead of computing diffusion probabilities between every pair of points, we can compute diffusion probabilities from points to a well-chosen set of central “landmarks” that maintain the geometry of the data.…”

“…We have shown in [13] that this leads to high quality approximations of the diffusion operator which lead to near-identical visualizations with PHATE. In addition, we examined in [19] that this leads to low error approximations of diffusion operators in general. We use this fast approach to compute a low error diffusion potential system for our coarse graining process.…”