Fourier Transform
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Rotation-Parameterized Graph Fractional Fourier Transform: Definition, Properties, and Optimal Filtering
Announce Type: replace-cross Abstract: Graph spectral representations are fundamental in graph signal processing, providing a rigorous frameworkforanalyzing graph-structured data. The graph fractional Fourier transform (GFRFT) extends the graph Fourier transform (GFT) through a fractional-order parameter, enabling flexible spectral analysis with mathematical consistency. The angular graph Fourier transform (AGFT) further introduces angular control by rotating GFT eigenvectors; however,...
Low-Rank Acceleration of the Operator Fourier Transform
arXiv:2606.09689v1 Announce Type: new Abstract: We develop a numerical algorithm for the efficient solution or approximation of solutions to the Helmholtz equation on a structured grid in two dimensions. We make use of the Operator Fourier Transform (OFT) and a low-rank cross approximation scheme (Cross-DEIM) to decompose the problem into an integral over a pseudo-time of solutions to the Schr\"odinger equation. The OFT is a framework for solving operator equations like fractional Laplacian...
Revisiting Neural Processes via Fourier Transform and Volterra Series
arXiv:2606.01172v1 Announce Type: new Abstract: Modeling unknown latent functions from finite, irregularly sampled measurements is a recurring challenge across science and engineering. Neural processes (NPs), a family of probabilistic functional models, are promising solutions -- especially when endowed with domain-specific symmetries like translation equivariance, which improve sample efficiency and generalization. Yet existing translation-equivariant NPs face two limitations: (i) they...
When Attention Beats Fourier: Multi-Scale Transformers for PDE Solving on Irregular Domains
arXiv:2605.08318v2 Announce Type: replace Abstract: We study the problem of \emph{architecture selection} for deep learning models trained to solve partial differential equations (PDEs), asking when transformer-based architectures with learned attention outperform Fourier-domain neural operators. We introduce the \textbf{Multi-Scale Attention Transformer} (\msat{}), a deep learning architecture that encodes spatiotemporal solution histories as token sequences and trains end-to-end via a...
When Attention Beats Fourier: Multi-Scale Transformers for PDE Solving on Irregular Domains
arXiv:2605.08318v2 Announce Type: replace-cross Abstract: We study the problem of \emph{architecture selection} for deep learning models trained to solve partial differential equations (PDEs), asking when transformer-based architectures with learned attention outperform Fourier-domain neural operators. We introduce the \textbf{Multi-Scale Attention Transformer} (\msat{}), a deep learning architecture that encodes spatiotemporal solution histories as token sequences and trains end-to-end via...
On the Role of the Double Fourier Sphere Method in Fast Algorithms on SO(3)
arXiv:2602.06677v3 Announce Type: replace Abstract: We analyze the Double Fourier Sphere (DFS) method on the rotation group $\mathcal{SO}(3)$ in the frequency domain and demonstrate its central role in fast algorithms. Fast Fourier algorithms on $\mathcal{SO}(3)$ are commonly formulated as a Wigner transform - mapping harmonic to Fourier coefficients - followed by a Fourier transform. We revisit this formulation and interpret the Wigner transform as an explicit realization of the DFS method,...
Fourier Neural Operators with rank-1 lattice points and hyperbolic cross
Announce Type: new Abstract: The \emph{Fourier neural operator} (FNO) is a neural network architecture that learns mappings between function spaces. Its efficient implementation is based on the multi-dimensional Fourier transform. By deriving general regularity bounds for the FNO with respect to both the spatial and parametric variables, we prove that the generalization error of the FNO can be improved by replacing spatial tensor product grids with purpose-built rank-1 lattice points, and by...
MOSAIC: A Workload-Driven Simulation and Design-Space Exploration Framework for Heterogeneous NPUs
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Global Plane Waves From Local Gaussians: Periodic Charge Densities in a Blink
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