We study dynamical sampling for graph signals using the spectral theory of the normalized graph Laplacian. Graph Paley-Wiener spaces GPW_w are defined as spectral subspaces associated with a bandwidth parameter w, and reconstruction is studied from measurements generated by iterates of bounded operators leaving these spaces invariant. For the lattice graph Z, we give an explicit spectral decomposition of the normalized Laplacian, identify the corresponding Graph Paley--Wiener spaces with classical bandlimited spaces on the torus, and recover the classical sampling theorem in this setting. Using frames of operator orbits, we construct dynamical frames for GPW_w(Z) and derive reconstruction formulas. We also discuss the extension to Z^n, where the bandwidth region is described by a sublevel set of the Laplacian symbol. Finally, for finite graphs, we formulate dynamical recovery in GPW_w(G)\) in terms of finite frames generated by iterates of the graph operator. Using the path graph P_N as a numerical model, we study how the number of iterates and the spatial distribution of the sampled vertices affect the optimal frame bounds and the conditioning of the resulting dynamical frame.
In this thesis, we study the spectral properties of dynamically defined aperiodic metric and discrete graphs. Our goal is to determine to what extent spectral properties of discrete one-dimensional ergodic Schr\"odinger operators persist when the aperiodicity is manifested through the geometry rather than through a pot...
Spectral parametrizations in operator learning fall into two classes. One learns an arbitrary multiplier on the frequency lattice, as the Fourier neural operator does on the torus; the other learns a scalar function of the Laplace–Beltrami eigenvalue, as recent manifold constructions do. For spectral graph networks the...
Ajay Minj· International journal of re...· 0 citations
This paper investigates spectral minimal partitions for weighted graphs, thus extending the extensive class of results that are currently available on domains and, to a lesser extent, manifolds and metric graphs. We provide a rigorous framework for analyzing graph Laplacians under Dirichlet, Neumann, and boundaryless e...
M. Hofmann, James B Kennedy, Delio Mugnolo et al.· 0 citations
Operator-theoretic generalization bounds for deep multi-output function classes are developed by representing network layers as Koopman composition operators on vector-valued reproducing kernel Hilbert spaces and derive Rademacher complexity bounds for invertible and width-expanding injective architectures.
Mahdi Mohammadigohari, Thomas Borsani, G. Di Fatta· 1 citation
We consider the problem of learning nonlinear operators $G_0: \mathcal X\to \mathcal Y$ from finitely many noisy observations. While finite-regularity classes such as Lipschitz- or $C^k$-operators suffer from the curse of sample complexity (i.e. algebraic sample complexity is impossible), algebraic convergence rates ca...
Invariant manifold theory for partial differential equations (PDEs) is technically challenging as we often lack spectral gaps. Motivated by recent progress using renormalization in various areas of mathematics, we introduce a spectral renormalization method for semilinear parabolic equations that creates an artificial...
C. Kuehn, Jan-Eric Sulzbach· 0 citations
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