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Diffusion Representations
Diffusion Maps framework is a kernel based method for manifold learning ...
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Manifold Alignment with Feature Correspondence
We propose a novel framework for combining datasets via alignment of the...
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Extendable and invertible manifold learning with geometry regularized autoencoders
A fundamental task in data exploration is to extract simplified low dime...
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Coarse Graining of Data via Inhomogeneous Diffusion Condensation
Big data often has emergent structure that exists at multiple levels of ...
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Geometry-Based Data Generation
Many generative models attempt to replicate the density of their input d...
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Scalability and robustness of spectral embedding: landmark diffusion is all you need
While spectral embedding is a widely applied dimension reduction techniq...
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Multi-Frequency Vector Diffusion Maps
We introduce multi-frequency vector diffusion maps (MFVDM), a new framew...
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Compressed Diffusion
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 (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.
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