The paper improves confidence regions for band-limited functions using tighter norm bounds and majority voting.
problem Constructing reliable confidence regions for band-limited functions from noisy data.
method Improved norm bounds using Hoeffding's inequality and empirical Bernstein bound, majority voting to aggregate intervals.
result Confidence intervals retain their simultaneous coverage guarantee even when aggregated from random subsamples.
The paper creates nonparametric confidence bands for band-limited functions.
problem Estimating confidence bands for band-limited functions with finite samples and unknown noise.
method Uses Paley-Wiener reproducing kernel Hilbert spaces and gradient-perturbation methods.
result Non-asymptotic guarantees for confidence regions without assuming a parametric model.
We study the volume distribution of nodal domains of random band-limited functions on generic manifolds, and find that in the high energy limit a typical instance obeys a deterministic universal law, independent of the manifold. Some of the basic qualitative properties of this law, such as its support, monotonicity and…
Let (M,g) be a smooth compact Riemannian surface with no boundary. Given a smooth vector field V with finitely many zeroes on M, we study the distribution of the number of tangencies to V of the nodal components of random band-limited functions. It is determined that in the high-energy limit, these obey a unive…
In this paper, a nonparametric maximum likelihood (ML) estimator for band-limited (BL) probability density functions (pdfs) is proposed. The BLML estimator is consistent and computationally efficient. To compute the BLML estimator, three approximate algorithms are presented: a binary quadratic programming (BQP) algorit…
Wiatowski and Bölcskei, 2015, proved that deformation stability and vertical translation invariance of deep convolutional neural network-based feature extractors are guaranteed by the network structure per se rather than the specific convolution kernels and non-linearities. While the translation invariance result appli…
The convolutional layers are core building blocks of neural network architectures. In general, a convolutional filter applies to the entire frequency spectrum of the input data. We explore artificially constraining the frequency spectra of these filters and data, called band-limiting, during training. The frequency dom…
This work uses sampling theory to analyze smoothness and error bounds of finite neural networks.
problem Analyzing the function space of finite neural networks and providing error bounds.
method Applying sampling theory to finite neural networks with non-expansive activation functions, considering both deterministic and random sampling.
result Novel error bounds for univariate neural networks under band-limited input assumption, highlighting the advantage of deterministic uniform sampling.
The paper improves nonparametric confidence bands for band-limited functions.
problem Constructing nonparametric simultaneous confidence bands with nonasymptotic and distribition-free guarantees.
method Based on Paley-Wiener reproducing kernel Hilbert spaces, the paper relaxes assumptions, improves noise estimation, and tightens constraints.
result Enhanced confidence bands with improved efficiency and tighter constraints.
We propose a novel class of Gaussian processes (GPs) whose spectra have compact support, meaning that their sample trajectories are almost-surely band limited. As a complement to the growing literature on spectral design of covariance kernels, the core of our proposal is to model power spectral densities through a rect…
Band-limited SAC improves learning efficiency and stability in simulated environments.
problem Improving sample efficiency and stability in SAC algorithms.
method Artificially bandlimiting the target critic's spatial resolution using a convolutional filter.
result Bandlimited SAC outperforms classic twin-critic SAC in various Gym environments and is more stable.
Deep convolutional neural networks (CNNs) used in practice employ potentially hundreds of layers and 10,000s of nodes. Such network sizes entail significant computational complexity due to the large number of convolutions that need to be carried out; in addition, a large number of parameters needs to be learned and…
The class of non-rigid registration methods proposed in the framework of PDE-constrained Large Deformation Diffeomorphic Metric Mapping is a particularly interesting family of physically meaningful diffeomorphic registration methods. PDE-constrained LDDMM methods are formulated as constrained variational problems, wher…
Study proper sampling for X-ray transforms on simple surfaces.
problem Proper discretizing and sampling issues related to geodesic X-ray transforms on simple surfaces.
method Provide minimal sampling rates for faithful reconstruction, quantify sampling quality, and predict artifacts.
result Minimal sampling rates and artifact prediction for geodesic X-ray transforms on simple surfaces.
Compression of Neural Networks (NN) has become a highly studied topic in recent years. The main reason for this is the demand for industrial scale usage of NNs such as deploying them on mobile devices, storing them efficiently, transmitting them via band-limited channels and most importantly doing inference at scale. I…
This paper presents a bias-variance tradeoff of graph Laplacian regularizer, which is widely used in graph signal processing and semi-supervised learning tasks. The scaling law of the optimal regularization parameter is specified in terms of the spectral graph properties and a novel signal-to-noise ratio parameter, whi…
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.
Developed a diffusion model on spherical data, addressing geometric and stochastic challenges.
problem Diffusion models on spherical data face unique geometric and stochastic issues.
method Extended spectral diffusion to spherical harmonics, introducing modified stochastic differential equations.
result Introduced a geometry-dependent inductive bias in spectral diffusion models.
GNNs outperform NNs in interpolating bandlimited functions on Euclidean cubes.
problem Interpolating bandlimited functions on Euclidean cubes using GNNs vs. NNs.
method Investigates optimal GNN configurations and weights for function interpolation.
result GNNs require fewer weights and samples to interpolate bandlimited functions compared to NNs.
In data science, it is often required to estimate dependencies between different data sources. These dependencies are typically calculated using Pearson's correlation, distance correlation, and/or mutual information. However, none of these measures satisfy all the Granger's axioms for an "ideal measure". One such ideal…
Deep convolutional neural networks have led to breakthrough results in numerous practical machine learning tasks such as classification of images in the ImageNet data set, control-policy-learning to play Atari games or the board game Go, and image captioning. Many of these applications first perform feature extraction …
New method constrains CNN filter frequencies to improve robustness.
problem CNN bias towards low frequency components, leading to poor performance in scenario transformations.
method Frequency domain regularization by constraining filter spectra, training valid frequency range end-to-end.
result Demonstrated effectiveness in defending adversarial perturbations, reducing generalization gap, and improving transfer learning.
In this paper, we show synchronization for a group of output passive agents that communicate with each other according to an underlying communication graph to achieve a common goal. We propose a distributed event-triggered control framework that will guarantee synchronization and considerably decrease the required comm…
We develop methods to learn dictionaries invariant under group symmetries, useful in cryo-EM and tracking.
problem Learning dictionaries invariant under group symmetries.
method Representation theory, non-abelian Fourier analysis, matrix orbitopes, alternating minimization.
result Effective dictionary learning for SO(3) symmetries with guarantees.
New method uses CNN for seismic inversion uncertainty quantification.
problem Uncertainty quantification in seismic inversion for noisy data.
method Plug-and-Play Stein Variational Gradient Descent (PnP-SVGD) with CNN denoiser.
result High-resolution, trustworthy posterior samples for subsurface structures.
Paper advances sparse regularisation theory for measures with new kernel insights.
problem Estimating sparse measures from noisy observations using continuous sparse regularisation.
method Develops new continuous sparse regularisation theory on measures with Beurling-LASSO, introduces kernel switch analysis.
result Proves the ``sinc-4'' kernel satisfies a technical LPC assumption for error bounds.
AaSP improves audio self-supervised learning by addressing aliasing issues.
problem Alias issues in audio spectrogram transformers.
method AaSP combines aliasing-aware patch representation, teacher-student masked modeling, cross-attention predictor, and contrastive regularization.
result AaSP learns more stable representations that integrate high-frequency cues.
Dual-space sampling tackles ill-conditioned inverse problems with Bayesian methods.
problem Bayesian inference in constrained inverse problems with ill-conditioned solutions.
method Dual-space posterior sampling using ADMM and SVGD.
result Well-calibrated uncertainty estimates and posterior contraction with increasing data.
New neural network models for complex functional data analysis.
problem Complex relations between functional predictors and responses.
method Function-on-Function regression models using neural networks with continuous hidden layers.
result Demonstrated power and flexibility in handling complex functional models.
Distance function to a finite set is a topological Morse function.
problem Characterizing the topological Morse function of a finite set.
method Analyzing the distance function to a finite set in \(\mathbb{R}^n\).
result Distance function is a topological Morse function, with precise critical points and indices.
Introduces new weighted floating functions and affine surface areas.
problem Developing new mathematical concepts for convex bodies.
method Introducing weighted floating functions and weighted functional affine surface areas.
result New relations to traditional and classical affine surface areas.
Develops methods for selecting and estimating smooth functional coefficients in high-dimensional multivariate functional data.
problem Functional predictor selection and estimation of smooth functional coefficients in high-dimensional multivariate functional data.
method Functional group-sparse regression methods in a generic Hilbert space of infinite dimension.
result Consistency of estimation and selection (oracle property) under infinite-dimensional Hilbert spaces.
Neural networks can approximate functionals on RKHS with error bounds.
problem Approximating functionals on RKHS using neural networks.
method Interpolating orthogonal projections in RKHS using point evaluations.
result Explicit error bounds for various kernels (inverse multiquadric, Gaussian, Sobolev).
FFBO optimizes functions as inputs and outputs, improving on existing BO methods.
problem Optimizing functions as both inputs and outputs in complex systems.
method Function-on-function Gaussian process (FFGP) model with a separable operator-valued kernel, scalar upper confidence bound (UCB) acquisition function, and scalable functional gradient ascent algorithm (FGA).
result FFBO outperforms existing methods in synthetic and real-world data.
Chirped sinosoids and interferometric phase plots are functions that are not periodic, but are the composition of a smooth function and a periodic function. These functions functions factor into a pair of maps: from their domain to a circle, and from a circle to their codomain. One can easily imagine replacing the circ…
The Fridman function is bounded by the injectivity radius for certain hyperbolic manifolds.
problem Bounding the Fridman function for hyperbolic manifolds.
method Analyzing the relationship between the Fridman function and the injectivity radius function.
result The Fridman function is bounded above by the injectivity radius function for certain hyperbolic manifolds.
Optimally estimates a functional using nuisance function tuning and sample splitting.
problem Estimating optimal rates for a doubly robust functional.
method Combines nuisance function tuning and sample splitting strategies.
result Shows optimal rates of convergence for various estimators.
The paper proves isoparametric functions on Finsler space forms under specific conditions.
problem Understanding isoparametric functions in Finsler space forms.
method Proving transnormal functions as isoparametric functions and constructing global and local isoparametric functions using the distance function.
result Generalization of Theorem B to Finsler space forms.
Paper introduces a nonparametric functional graphical model for random functions.
problem Estimating probabilistic conditional independence in functional graphical models.
method Functional sufficient dimension reduction to relax Gaussian or copula Gaussian assumptions.
result Enhances estimation accuracy and retains probabilistic conditional independence.
Robustifies elicitable functionals to handle small distribution misspecifications.
problem Determining uniquely optimal forecasts under distributional misspecification.
method Integrates statistical robustness into elicitable functionals using Kullback-Leibler divergence.
result Robust elicitable functionals admit unique solutions at the boundary of uncertainty regions.
The paper characterizes strong Hamel functions using symmetries and proves their preservation properties.
problem Characterizing strong Hamel functions and their symmetries in Finsler spaces.
method Analyzing geodesic spray, strong dual symmetries, and strong dynamical symmetries.
result Strong Hamel functions can be characterized in terms of strong dual symmetries and strong dynamical symmetries.
Study biharmonic functions on vector bundles with spherical symmetry.
problem Investigate biharmonic functions on vector bundles with spherically symmetric metrics.
method Analyze vertical lifts and radial functions of functions on vector bundle manifolds.
result Construct an infinite two-parameter family of proper biharmonic functions.
Two new methods improve forecasting of functional time series data.
problem Forecasting of functional time-dependent data.
method Functional Singular Spectrum Analysis (FSFA) based forecasting methods.
result Our methods outperform existing algorithms for periodic stochastic processes.
This paper introduces the concept of functional current as a mathematical framework to represent and treat functional shapes, i.e. sub-manifold supported signals. It is motivated by the growing occurrence, in medical imaging and computational anatomy, of what can be described as geometrico-functional data, that is a da…
Study stabilizers of smooth functions on surfaces, focusing on Morse-Bott functions.
problem Understanding the homotopy type of stabilizers of smooth functions on surfaces.
method Analyzing the homotopy properties of stabilizers for a specific class of smooth functions.
result The homotopy type of the connected component of the identity map of the stabilizer is completely described for Morse-Bott functions.
The paper connects convex functions to p-subharmonic functions and proves their equivalence.
problem Understanding the relationship between convex functions and p-subharmonic functions.
method Average principle, variational methods, and PDE techniques.
result Convex functions on R^n are p-subharmonic for every p > 1.
A new deep neural network tackles nonlinear functional regression with improved dimensionality reduction.
problem Nonlinear functional regression in infinite-dimensional functional data analysis.
method Functional deep neural network with adaptive kernel embedding and projection steps.
result Explicit rates of approximating nonlinear smooth functionals are derived, and the network is shown to be effective in both simulated and real datasets.
New model for network analysis using functional data.
problem Existing network models treat nodes as functions, but this paper introduces functional edges.
method Transform adjacency matrix into functional adjacency tensor, apply Tucker decomposition, regularize basis matrices, and solve tensor completion problem.
result The model effectively captures community structure and handles irregular functional edge data.