arXiv subjects

Alvaro Almeida Gomez

Publications and source records attributed to Alvaro Almeida Gomez.

6 recordsLinked to original sources

arXiv2026

Data-Driven Diffusion Processes on Differential Forms via the Projected Ambient Connection Laplacian

We develop a data-driven approximation of the projected ambient connection Laplacian acting on differential forms over smooth Riemannian manifolds sampled by point clouds. The proposed construction extends the classical framework of diffusion maps and Vector Diffusion Maps from scalar functions and tangent vector fields to differential forms of arbitrary degree. Our approach is based on a novel representation of differential forms as alternating differential arrays obtained through an extension of the classical musical isomorphism. This representation enables the construction of a matrix-valued diffusion operator that approximates the projected ambient connection Laplacian directly from point cloud data without requiring a mesh or simplicial complex. The proposed discretization admits the asymptotically optimal kernel bandwidth scaling inherited from diffusion maps, leading to sharper convergence guarantees than previous data-driven approximations of the Hodge Laplacian. Building upon this operator, we derive a fully data-driven explicit Euler scheme for the heat equation on differential forms and validate the proposed methodology through numerical experiments on the unit sphere. The experiments confirm the predicted decay of the analytical solution and demonstrate the effectiveness of the proposed discretization. The proposed framework provides a natural generalization of Vector Diffusion Maps to differential forms of arbitrary degree and establishes a practical foundation for the numerical approximation of geometric partial differential equations directly from point cloud data.

math.NA↗

arXiv2026

A Data-Driven Interpolation Method on Smooth Manifolds via Diffusion Processes and Voronoi Tessellations

We propose a data-driven interpolation framework for reconstructing real-valued functions on smooth manifolds from scattered pointwise observations. The method combines a Gaussian Nadaraya--Watson kernel interpolant with a Voronoi-adaptive bandwidth determined entirely by the geometry of the sampled data, yielding an explicit closed-form construction that requires neither training, iterative optimization, preprocessing, nor parameter tuning. The proposed interpolant satisfies several theoretical properties. It reproduces the observed data exactly, enforces a vanishing intrinsic gradient at every sample point, and, in the dense-sampling limit, attenuates high-frequency oscillatory components through the geometric regularization induced by the adaptive bandwidth. Furthermore, the construction admits an interpretation in terms of minimizing a discrete total variation--type functional, establishing a natural connection with compressed sensing and sparsity-promoting regularization. Unlike classical kernel interpolation methods employing a fixed global bandwidth, the proposed adaptive strategy automatically adjusts to the local sampling geometry through the Voronoi tessellation while preserving an explicit analytical formulation. Because the interpolant is available in closed form, the overall computational cost is entirely determined by the inference stage: evaluating the interpolant at a query point requires only the computation of Gaussian kernel weights and their weighted combination, resulting in linear complexity with respect to the number of sample points. In contrast to many data-driven interpolation approaches, no additional offline computational stage is required before inference.

cs.LG↗

arXiv2025

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.

cs.LG↗

arXiv2024

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.

cs.LG↗

arXiv2024

Recovering the Fragmentation Rate in the Growth-Fragmentation Equation

We consider the inverse problem of determining the fragmentation rate from noisy measurements in the growth-fragmentation equation. We use Fourier transform theory on locally compact groups to treat this problem for general fragmentation probabilities. We develop a regularization method based on spectral filtering, which allows us to deal with the inverse problem in weighted ${L}^2$ spaces. %Our approach regularizes the signal generated by differential operators in the frequency domain. As a result, we obtain a regularization method with error of order $O(\varepsilon^{\frac{2m}{2m+1}})$, where $\varepsilon$ is the noise level and $m>0$ is the {\em a priori} regularity order of the fragmentation rate.

math.NA↗

arXiv2023

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

We recover the Riemannian gradient of a given function defined on interior points of a Riemannian submanifold in the Euclidean space based on a sample of function evaluations at points in the submanifold. This approach is based on the estimates of the Laplace-Beltrami operator proposed in the diffusion-maps theory. The Riemannian gradient estimates do not involve differential terms. Analytical convergence results of the Riemannian gradient expansion are proved. We apply the Riemannian gradient estimate in a gradient-based algorithm providing a derivative-free optimization method. We test and validate several applications, including tomographic reconstruction from an unknown random angle distribution, and the sphere packing problem in dimensions 2 and 3.

cs.LG↗