Learning functions through Diffusion Maps

We propose a data-driven method for approximating real-valued functions on smooth manifolds, building on the Diffusion Maps framework under the manifold hypothesis. Given pointwise evaluations of a function, the method constructs a smooth extension to the ambient space by exploiting diffusion geometry and its connection to the heat equation and the Laplace-Beltrami operator. To address the computational challenges of high-dimensional data, we introduce a dimensionality reduction strategy based on the low-rank structure of the distance matrix, revealed via singular value decomposition (SVD). In addition, we develop an online updating mechanism that enables efficient incorporation of new data, thereby improving scalability and reducing computational cost. Numerical experiments, including applications to sparse CT reconstruction, demonstrate that the proposed methodology outperforms classical feedforward neural networks and interpolation methods in terms of both accuracy and efficiency.

Hodge Laplacians and Hodge Diffusion Maps

We introduce Hodge Diffusion Maps, a novel manifold learning algorithm designed to analyze and extract topological information from high-dimensional data-sets. This method approximates the exterior derivative acting on differential forms, thereby providing an approximation of the Hodge Laplacian operator. Hodge Diffusion Maps extend existing non-linear dimensionality reduction techniques, including vector diffusion maps, as well as the theories behind diffusion maps and Laplacian Eigenmaps. Our approach captures higher-order topological features of the data-set by projecting it into lower-dimensional Euclidean spaces using the Hodge Laplacian. We develop a theoretical framework to estimate the approximation error of the exterior derivative, based on sample points distributed over a real manifold. Numerical experiments support and validate the proposed methodology.

Diffusion Representation for Asymmetric Kernels

We extend the diffusion-map formalism to data sets that are induced by asymmetric kernels. Analytical convergence results of the resulting expansion are proved, and an algorithm is proposed to perform the dimensional reduction. In this work we study data sets in which its geometry structure is induced by an asymmetric kernel. We use a priori coordinate system to represent this geometry and, thus, be able to improve the computational complexity of reducing the dimensionality of data sets. A coordinate system connected to the tensor product of Fourier basis is used to represent the underlying geometric structure obtained by the diffusion-map, thus reducing the dimensionality of the data set and making use of the speedup provided by the two-dimensional Fast Fourier Transform algorithm (2-D FFT). We compare our results with those obtained by other eigenvalue expansions, and verify the efficiency of the algorithms with synthetic data, as well as with real data from applications including climate change studies.

A diffusion-map-based algorithm for gradient computation on manifolds and applications

We recover the gradient of a given function defined on interior points of a submanifold with boundary of the Euclidean space based on a (normally distributed) random sample of function evaluations at points in the manifold. This approach is based on the estimates of the Laplace-Beltrami operator proposed in the theory of Diffusion-Maps. Analytical convergence results of the resulting expansion are proved, and an efficient algorithm is proposed to deal with non-convex optimization problems defined on Euclidean submanifolds. We test and validate our methodology as a post-processing tool in Cryogenic electron microscopy (Cryo-EM). We also apply the method to the classical sphere packing problem.