Inverse spectral theory reveals shapes from sound.
problem Can the shape of a drum be determined by its sound?
method Inverse isospectral techniques applied to specific shapes.
result The regular n-gon can be uniquely determined by its eigenvalues.
SC-Net learns interpretable filters for inverse problems, achieving optimal convergence and super-resolution.
problem Solving ill-posed inverse problems with effective regularization and interpretability.
method SC-Net operates in the spectral domain, learning a pointwise adaptive filter function based on signal-to-noise ratio.
result SC-Net achieves optimal convergence rate and zero-shot super-resolution, matching theoretical bounds.
In this work a spectral theory for 2-dimensional, simply periodic, complex-valued solutions u of the sinh-Gordon equation is developed. Spectral data for such solutions are defined (following Hitchin and Bobenko) and the space of spectral data is described by an asymptotic characterization. Using methods of asymptotic …
We prove inverse spectral results for differential operators on manifolds and orbifolds invariant under a torus action. These inverse spectral results involve the asymptotic equivariant spectrum, which is the spectrum itself together with "very large" weights of the torus action on eigenspaces. More precisely, we show …
This paper provides a full controlled version of algebraic K-theory. This includes a rich array of assembly maps; the controlled assembly isomorphism theorem identifying the controlled group with homology; and the stability theorem describing the behavior of the inverse limit as the control parameter goes to 0. There…
The paper defines surface area for graphs and derives spectral estimates.
problem Understanding connectivity measures and spectral properties of graphs.
method Introducing surface area concepts related to inverse degree and deriving spectral bounds.
result An upper bound on the second eigenvalue for planar graphs.
The paper extends spectral results to non-abelian groups acting on compact Riemannian manifolds.
problem Determining potential functions from spectral data for non-abelian group actions.
method Generalized Legendrian relations and spectral invariants.
result Potential functions are determined by the equivariant spectrum for certain Schrödinger operators.
Analyzes how diffusion models learn, revealing a spectral bias in structure mastery.
problem Understanding the learning dynamics and bias in diffusion models.
method Developed an analytical framework using a Gaussian-equivalence principle to solve gradient-flow dynamics and integrate probability-flow ODEs.
result Exposes a universal inverse-variance spectral law: high-variance structure is mastered faster than low-variance detail.
While sparse inverse covariance matrices are very popular for modeling network connectivity, the value of the dense solution is often overlooked. In fact the L2-regularized solution has deep connections to a number of important applications to spectral graph theory, dimensionality reduction, and uncertainty quantificat…
Spectral portfolio theory links neural networks to wealth dynamics via SGD weight matrices.
problem Understanding wealth dynamics from neural network training.
method Direct identification of weight matrices as portfolio allocation matrices, linking SGD forces to portfolio dynamics.
result Spectral properties of SGD weight matrices transition between additive and multiplicative regimes, influencing wealth dynamics.
This note summarizes results that were obtained by the author in his habilitation thesis (arXiv:1607.08792) concerning the development of a spectral theory for simply periodic, 2-dimensional, complex-valued solutions of the sinh-Gordon equation. Spectral data for such solutions are defined for periodic Cauchy data on a…
Study shows stability of Schrödinger operator spectral data on a manifold.
problem Determining a manifold and potential function from spectral data.
method Approximation of spectral data on a subset to determine manifold and potential.
result Quantitative stability estimate for Schrödinger operator inverse problem.
Paper shows stability of metric reconstruction for orbifolds from spectral data.
problem Determining the metric structure of collapsing orbifolds from spectral data.
method Improved quantitative unique continuation for wave operator on Riemannian manifolds.
result Quantitative stability of inverse problem for Riemannian orbifolds.
Study on Matérn covariance approximations on grids, finding issues with high-frequency aliasing.
problem Issues with high-frequency aliasing in SPDE approximations of Matérn covariance functions.
method Analysis of aliased spectral densities and numerical simulations.
result SPDE approximations assign too much power at high frequencies and do not improve accuracy as grid spacing decreases.
Study shows stability of travel time data reconstruction from closed subsets.
problem Reconstruction of length spaces from travel time data on a closed subset.
method Lipschitz stability proof for certain types of length spaces.
result Reconstruction of length spaces is Lipschitz stable from travel time data on a closed subset.
Researchers approximate spectral targets on manifolds with constant negative curvature.
problem Prescribing an arbitrary finite portion of the Laplace-Beltrami spectrum on manifolds of constant negative curvature.
method Constructing macroscopically heterogeneous hyperbolic covering manifolds in d≥3 and using discrete spectral limit theorems in d=2. result Any finite strictly increasing target list can be approximated to arbitrary precision by a closed manifold of constant negative curvature.
Study magnetic potentials on Anosov manifolds using spectral data.
problem Recover magnetic potentials from spectral data on Anosov manifolds.
method Utilize principal wave trace invariants and magnetic Schrödinger operator.
result Spectral data uniquely determines magnetic and electric potentials on Anosov manifolds.
We construct an inverse system of unstable Vassiliev spectral sequences on the spaces of plumbers' knots, which model the homotopy type of the space of long knots, and show that the limit of these sequences contains the finite type invariants in their usual complexity. Utilizing the cell structure on the discriminant o…
Paper analyzes AIRL in high-dimensional spaces using random matrix theory.
problem AIRL's performance challenges in high-dimensional environments.
method Examined the rank of the matrix derived from transition matrix, applied random matrix theory.
result High-dimensional scenarios reveal transfer limitations not inherent to AIRL framework.
Paper presents a unique method to recover signals from their bispectrum.
problem Retrieving signals accurately from their bispectrum.
method Two-step trust region algorithm that minimizes a non-convex objective function.
result Signals with finite spectral or temporal support can be recovered from at least 3B measurements of their bispectrum.
The paper reformulates regression in infinite dimensions as an inverse problem, showing it's equivalent to compact inverse problems.
problem Learning a linear operator between Hilbert spaces from empirical observations.
method Reformulates regression as an inverse problem, proving equivalence to compact inverse problems under specific conditions.
result The inverse problem is equivalent to compact inverse problems in terms of spectral properties and regularisation theory.
We define a pseudo-inverse for line graphs using linear integer programming.
problem Not all graphs have a corresponding root graph, making the line graph operation non-invertible.
method Propose a linear integer program to edit the smallest number of edges in the line graph to recover a root graph.
result The pseudo-inverse operation is well-behaved and works in practice as shown by empirical experiments.
The paper studies magnetic field effects on surface eigenvalues and spectral properties.
problem Understanding magnetic effects on surface eigenvalues and spectral properties.
method Provided precise spectral asymptotics expansion for the magnetic Dirichlet-to-Neumann map on surfaces.
result The spectrum of the magnetic Dirichlet-to-Neumann map uniquely determines the number and length of boundary components, parallel transport, and magnetic flux.
New methods tackle statistical inverse problems with random data.
problem Statistical inverse problems with random experimental design.
method Spectral regularization, regularization by projection, convex penalties.
result Minimax rates in expectation and probability for convergence.
The main results of this paper are an asymptotic expansion in powers of ℏ for the spectral measure μℏ of a semi-classical Toeplitz operator, Qℏ, and an equivariant version of this result when Qℏ admits an n-torus as a symmetry group. In addition we discuss some inverse spectral consequences…
New method for mixed memberships using symmetrized Laplacian inverse matrix.
problem Mixed memberships in community detection.
method Spectral clustering on symmetrized Laplacian inverse matrix.
result Mixed-SLIM methods outperform state-of-the-art methods.
Many problems in machine learning and statistics can be formulated as (generalized) eigenproblems. In terms of the associated optimization problem, computing linear eigenvectors amounts to finding critical points of a quadratic function subject to quadratic constraints. In this paper we show that a certain class of con…
Proves new inequality linking spectral numbers of Lagrangians and their reductions.
problem Understanding spectral properties of Lagrangian submanifolds.
method Develops inverse reduction inequalities for spectral numbers.
result Proof of inequality between spectral numbers of Lagrangian and its reductions.
In this paper we consider two inverse problems on a closed connected Riemannian manifold (M,g). The first one is a direct analog of the Gel'fand inverse boundary spectral problem. To formulate it, assume that M is divided by a hypersurface Σ into two components and we know the eigenvalues λj of the Laplace ope…
This paper is devoted to an inverse Steklov problem for a particular class of n-dimensional manifolds having the topology of a hollow sphere and equipped with a warped product metric. We prove that the knowledge of the Steklov spectrum determines uniquely the associated warping function up to a natural invariance.
We revisit the dividend payment problem in the dual model of Avanzi et al. ([2], [1], and [3]). Using the fluctuation theory of spectrally positive Lévy processes, we give a short exposition in which we show the optimality of barrier strategies for all such Lévy processes. Moreover, we characterize the optimal barrier …
Study finds finitely many non-congruent polygonal domains with same Steklov spectrum.
problem Inverse Steklov problem on convex polygons.
method Analysis of Steklov eigenvalues and isoperimetric bounds.
result For almost all convex polygonal domains, there exist at most finitely many non-congruent domains with the same Steklov spectrum.
Geometric framework explains and controls implicit bias in machine learning.
problem Understanding and controlling the selection of solutions in overparameterized models.
method Developed a theoretical and constructive framework based on geometric corrections induced by gradient noise and continuous symmetries of the loss.
result Computed the induced bias across various architectures and enabled inverse design to shape the bias.
We study the inverse spectral problem for weighted projective spaces using wave-trace methods. We show that in many cases one can "hear" the weights of a weighted projective space.
Paper analyzes spectral algorithms under covariate shift, providing convergence rates.
problem Addressing distributional mismatch in regression models.
method Incorporates importance weights into spectral algorithms in RKHS.
result Establishes minimax-optimal convergence rates for misspecified cases.
Survey of spectral theory and dynamics for infinite volume hyperbolic manifolds.
problem Understanding infinite volume asymptotically hyperbolic manifolds.
method Survey of geometry, spectral theory, dynamics, and quantum/classical mechanics.
result Recent results, ideas, and conjectures discussed.
Develops trace class operators and inverse Laplacian theory for infinite dimensions.
problem Understanding trace class operators and inverse Laplacian on infinite dimensional spaces.
method Presentation of trace class operators and construction of inverse Laplacian on closed manifolds.
result Original trace computations involving the inverse Laplacian on the torus.
Graph curvature measured by inverse resistance distance.
problem Defining and analyzing curvature in graphs.
method Defining curvature via inverse resistance distance and proving properties.
result Graphs with positive curvature have controlled diameter and spectral properties.
In this paper, we obtain analytical expression for the distribution of the occupation time in the red (below level 0) up to an (independent) exponential horizon for spectrally negative Lévy risk processes and refracted spectrally negative Lévy risk processes. This result improves the existing literature in which only…
We consider the geometric inverse problem of determining a closed Riemannian manifold from measurements of the heat kernel in an open subset of the manifold. In this paper we analyze the stability of this problem in the class of n-dimensional Riemannian manifolds with bounded diameter and sectional curvature. It is w…
Spectral sparsification improves Laplacian-constrained graph learning.
problem Improving accuracy of Laplacian-constrained graph learning.
method Spectral graph sparsification as a post-estimation operation.
result Improved accuracy of Laplacian-constrained graph learning.
We consider the problem of finding sufficient conditions for a locally Lipschitz mapping between Finsler manifolds to be a global homeomorphism. For this purpose, we develop the notion of Clarke generalized differential in this context and, using this, we obtain a version of the Hadamard integral condition for invertib…
A new convolutional spectral kernel network learns hierarchical and local features.
problem Lack of deep learning in non-stationary spectral kernels.
method Introduces convolutional filters and deep architectures into non-stationary spectral kernels, derives generalization error bounds, and introduces regularizers.
result Validated the effectiveness of the convolutional spectral kernel network on real-world datasets.
Dual regularized graph Laplacian improves spectral clustering for community detection.
problem Detecting clusters in networks with improved spectral clustering methods.
method Proposes dual regularized graph Laplacian for three spectral clustering approaches.
result Theoretical analysis shows DRSC and DRSLIM yield stable consistent community detection.
Study reveals how Fisher information changes with network depth, finding it grows linearly.
problem Understanding the trainability of deep neural networks (DNNs).
method Investigates the spectral distribution of the conditional Fisher information matrix (FIM) for fully-connected networks achieving dynamical isometry.
result The conditional FIM's spectrum concentrates around the maximum and grows linearly with depth.
Unified view of spectral networks linking geometry and gauge theory.
problem Understanding BPS states in gauge theories.
method Unified geometric and physical approaches, focusing on spectral networks.
result Spectral networks provide a framework for determining BPS spectra.
Study elliptic operators on glued manifolds, reducing to finite-dimensional systems.
problem Mapping properties of elliptic operators in gluing problems.
method Reduction to finite-dimensional linear systems in the limit Tightarrow∞. result Construction of Fredholm inverses with controlled norms.
In this paper, we develop a general theory of truncated inverse binomial sampling. In this theory, the fixed-size sampling and inverse binomial sampling are accommodated as special cases. In particular, the classical Chernoff-Hoeffding bound is an immediate consequence of the theory. Moreover, we propose a rigorous and…