Derives new orthogonal coordinates for evolving surfaces and curves.
problem Accounting for geometric effects in boundary layer asymptotics.
method Elementary derivation of orthogonal signed-distance coordinates.
result Provides vector calculus identities for these coordinates.
Lowered regularity assumption for a phase-dependent Helfrich energy equation.
problem Analyzing the phase separation line of the Helfrich energy.
method Used a carefully chosen test function with a signed distance function.
result Regularity assumption lowered from C2 to C1,1 for the phase separation line. For a domain Ω⊂Rn, we introduce the concept of a uniformly Cm defining function. We characterize uniformly Cm defining functions in terms of the signed distance function for the boundary and provide a large class of examples of unbounded domains with uniformly Cm defining functions. Some of ou…
Novel neural network approximates exact distance for robust classification.
problem Adversarial attacks on neural networks in safety-critical systems.
method Signed Distance Classifiers (SDCs) and Unitary-Gradient Neural Network.
result Approximates exact distance from classification boundary for certifiable predictions.
In this paper, we present an InSphereNet method for the problem of 3D object classification. Unlike previous methods that use points, voxels, or multi-view images as inputs of deep neural network (DNN), the proposed method constructs a class of more representative features named infilling spheres from signed distance f…
Study heat content on RCD(K,N) spaces with specific boundary conditions.
problem Analyzing heat content in RCD(K,N) spaces with irregular boundaries.
method Proved first-order asymptotics using measured interior geodesic condition.
result Established first-order heat content asymptotics on RCD(K,N) spaces.
This paper improves the scalability of sparse neural network compression.
problem Sparse neural network compression for diverse data modalities.
method State-of-the-art sparsification techniques and meta-learning.
result Meta-learning sparse compression networks achieve new state-of-the-art results.
Let Ω be an open half-space or slab in Rn+1 endowed with a perturbation of the Gaussian measure of the form f(p):=exp(ω(p)−c∣p∣2), where c>0 and ω is a smooth concave function depending only on the signed distance from the linear hyperplane parallel to ∂Ω. In this work we follow a varia…
Equivalence found between smooth and synthetic timelike curvature bounds.
problem Understanding timelike sectional curvature bounds in spacetime geometry.
method Established equivalence between sectional curvature bounds on timelike planes and synthetic timelike bounds.
result Equivalence proved for sectional curvature bounds on timelike planes and synthetic timelike bounds on strongly causal spacetimes.
A celebrated conjecture due to De Giorgi states that any bounded solution of the equation Δu+(1−u2)u=0inRN with $\pp_{y_N}u >0$ must be such that its level sets $\{u=\la\}$ are all hyperplanes, {\em \bf at least} for dimension N≤8. A counterexample for N≥9 has long been believed to exist. …
We investigate the equation (−ΔHn)γw=f(w)inHn, where (−ΔHn)γ corresponds to the fractional Laplacian on hyperbolic space for γ∈(0,1) and f is a smooth nonlinearity that typically comes from a double well potential. We prove the existence of heteroclinic connecti…
One-Class Boundary Peeling detects outliers efficiently and robustly.
problem Unsupervised outlier detection in diverse data distributions.
method One-Class Boundary Peeling uses flexible boundaries generated by one-class SVMs and iteratively peels them.
result One-Class Boundary Peeling outperforms state-of-the-art methods in synthetic data simulations.
Differentiable pipeline replaces non-differentiable CAE components for shape optimization.
problem Gradient-based optimization is limited by non-differentiable components in CAE workflows.
method Surrogate models replace non-differentiable pipeline components, enabling gradient-based optimization.
result Gradient-based shape optimization possible without differentiable solvers.
Study of families of lines on spheres and their focal sets.
problem Characterizing families of lines on spheres and their geometric properties.
method Analyzing submanifolds of TSn and their focal sets, using symplectic structures and sectional curvatures. result Derivation of formulas relating sectional curvatures of focal sets to differences in radii of curvature of generating hypersurfaces.
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.
The study finds a special type of smooth function on connected sums of manifolds.
problem Finding smooth functions that are Morse on preimages of non-extrema values.
method Investigates internally Morse (I-Morse) and neat with respect to Reeb graph (N-Reeb) functions.
result Constructs an IN-Morse-Reeb function on a connected sum of given manifolds.
Function trees simplify complex ML models for better understanding.
problem Understanding and interpreting machine learning model predictions.
method Representing a multivariate function as a tree of simpler functions.
result Function trees reveal the global internal structure of functions.
We study functions whose truncations are convex or quasiconvex.
problem Understanding functions with specific truncation properties.
method Analyzing C2-smooth functions with positive definite Hessians. result Injectivity of restricted gradient in positive definite region.
NeuTSFlow models continuous functions behind time series forecasting.
problem Forecasting treats time series as discrete sequences, ignoring their continuous nature.
method NeuTSFlow uses Neural Operators to learn the transition between historical and future function families.
result NeuTSFlow outperforms traditional methods in forecasting accuracy and robustness.
New spectral functionals for Dirac operators with inner fluctuations computed.
problem Spectral functionals and Dirac operators with inner fluctuations.
method Extension of spectral functionals for Dirac operators with inner fluctuations.
result Computed spectral Einstein functional for Dirac operator with inner fluctuations on even-dimensional spin manifolds.
Regularizers change the geometric properties of loss functions in neural networks.
problem Understanding how different regularizers affect the geometric properties of loss functions in neural networks.
method Examined several regularizers, including weight decay, to determine if the regularized loss function becomes Morse.
result For certain regularizers, the regularized loss function becomes Morse, indicating a change in geometric properties.
A new model approximates complex functions in parameter space.
problem Complex and nonlinear functional regression problems.
method Mapping-to-Parameter function model with B-spline free knot placement.
result Robust knot placement algorithms improve model performance.
The paper generalizes inequalities on almost Kähler manifolds.
problem Generalizing inequalities on almost Kähler manifolds.
method Considered Donaldson gauge functional and twisted Aubin functionals.
result Generalized inequality between Aubin functionals.
As a generalization of geodesic function, in the present paper, we introduce the notion of geodesic φ-convex function and deduce some basic properties of φ-convex function and geodesic φ-convex function. We also introduce the concept of geodesic φ-convex set and φ-epigraph and in…
The diameter function is a topological Morse function.
problem The relationship between systole and diameter functions on Teichmüller space.
method Mapping class group-equivariant topological Morse function approach.
result The diameter function on Teichmüller space is a topological Morse function.
Proposes an L1-regularized functional SVM for binary classification with functional covariates.
problem Binary classification with multivariate functional covariates.
method L1-regularized functional support vector machine (SVM) with an accompanying algorithm.
result The proposed classifier performs well in prediction and feature selection.
Paper defines Farey Recursive Functions and explores their properties.
problem Understanding recursive functions on rationals.
method Defined and studied Farey Recursive Functions using Farey graph.
result Farey Recursive Functions naturally connect to 2-bridge knots and links.
In this paper we propose and study a family of sparsity-inducing penalty functions. Since the penalty functions are related to the kinetic energy in special relativity, we call them \emph{kinetic energy plus} (KEP) functions. We construct the KEP function by using the concave conjugate of a χ2-distance function and …
The paper calculates asymptotic expansions for specific types of oscillatory integrals.
problem Analyzing oscillatory integrals with complex phase functions.
method Using asymptotic expansions of simpler phase functions to derive results for more complex cases.
result Explicit computation of coefficients in asymptotic expansions for certain integrals.