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.
The paper links set cuspidality to function regularity and flatness.
problem Linking set cuspidality to function regularity and flatness.
method Analyzes arc-smooth functions and their properties on various sets.
result Establishes a precise link between set cuspidality and function regularity.
Deep Sets approximates functions on sets with high-dimensional latent space.
problem Modeling functions of sets (permutation-invariant functions).
method Deep Sets, a method known to be a universal approximator for continuous set functions.
result Deep Sets' universal approximation property is only guaranteed with a sufficiently high-dimensional latent space.
A new convex loss function optimizes set predictions with balanced size and coverage.
problem Optimizing set predictions with balanced size and coverage.
method Proposes a convex loss function using Choquet integrals for nondecreasing subset-valued functions.
result Optimal trade-offs between conditional probabilistic coverage and set size.
Study on transnormal functions and their level sets on Finsler manifolds.
problem Understanding transnormal functions and their geometric properties on Finsler manifolds.
method Proving smoothness of focal varieties and regular level sets of transnormal functions.
result Focal varieties of a C2 transnormal function are smooth submanifolds and regular level sets are tubes over these varieties.
The paper defines wave-front singularities using explicit analytic functions.
problem Characterizing the images of wave-front singularities.
method Explicit resultant computations to construct main-analytic functions.
result Explicit formulas for main-analytic functions of wave-front singularities of types A, D, and E.
Recently, it has been shown that many functions on sets can be represented by sum decompositions. These decompositons easily lend themselves to neural approximations, extending the applicability of neural nets to set-valued inputs---Deep Set learning. This work investigates a core component of Deep Set architecture: ag…
To each unit complex number with positive imaginary part there is defined a Tristram-Levine knot signature function. The set of all such signature functions is linearly independent as a set of functions defined on the set of all knots. The set of averaged signature functions forms a linearly independent set of homomoro…
New algorithms learn sparse set functions in non-orthogonal Fourier bases.
problem Learning sparse set functions in non-orthogonal Fourier bases.
method Novel algorithms using non-orthogonal Fourier transforms.
result At most nk−klog2k+k queries for k non-zero Fourier coefficients. Study the geometry of bifurcation sets for specific types of functions.
problem Understanding the structure of bifurcation sets for specific types of functions.
method Using blow-ups and parametrization, investigate the Gaussian curvature, principal curvatures, and curve behavior.
result Bifurcation sets of D4±-functions can be parametrized as surfaces in R3. Study rationality of meromorphic functions between real algebraic sets in the plane.
problem Understanding rationality of meromorphic functions mapping real algebraic sets to complex algebraic sets.
method Using Schwarz reflection functions and theory of meromorphic functions.
result For certain real algebraic sets, meromorphic functions are rational.
The abstract discusses convergence properties of Lipschitz functions and sets defined by equations.
problem Convergence of Lipschitz functions and sets defined by equations.
method Painlevé-Kuratowski convergence applied to Lipschitz functions and sets defined by equations.
result Generalizations and reverses of classical theorems on convergence of functions and sets.
Study shows observability from a measurable set for Gevrey functions.
problem Determining observability from a subset for Gevrey functions.
method Used measurable sets and inequalities for Gevrey regular functions.
result Established observability estimates from measurable sets for Gevrey functions.
Study sharp decay of capacity for subharmonic functions on compact Hermitian manifolds.
problem Sharp decay of capacity of sublevel sets of (ω,m)-subharmonic functions. method Generalizes previous results on Kähler manifolds and obtains full characterizations of polar sets.
result Full characterizations of polar sets and extremal functions.
Study approximates unknown function levels with queries.
problem Approximating unknown function levels through sequential queries.
method Introduce Bisect and Approximate algorithms to reduce to local function approximation.
result Rate-optimal sample complexity guarantees for H{ö}lder functions.
Study finds critical points in perimeter functional for fixed volume sets.
problem Finding critical points in perimeter functional for sets of fixed volume.
method Utilizes Mazurwoski--Zhou techniques and new Cacciopoli set connectedness results.
result Constructs smooth almost embedded hypersurfaces with non-zero constant mean curvature.
Families of hypersurfaces that are level-set families of harmonic functions free of critical points are characterized by a local differential-geometric condition. Harmonic functions with a specified level-set family are constructed from geometric data. As a by-product, it is shown that the evolution of the gradient of …
In this paper, we consider the problem of learning functions over sets, i.e., functions that are invariant to permutations of input set items. Recent approaches of pooling individual element embeddings can necessitate extremely large embedding sizes for challenging functions. We address this challenge by allowing stand…
Characterizes learnability of multioutput functions in various settings.
problem Learning multioutput function classes in batch and online settings.
method Characterizes learnability based on single-output restrictions.
result Complete characterization of learnability in multioutput classification and regression.
Study shows singular set of distance functions is delta-convex.
problem Understanding singular set of distance functions in Finsler manifolds.
method Proved singular set is delta-convex hypersurfaces or Jordan arcs up to exceptional sets.
result Optimal results in Finsler manifolds, even in Euclidean space.
Paper presents a robust transfer learning method for active level set estimation.
problem Efficiently identifying regions of a black-box function with limited function evaluations.
method Incorporates prior knowledge from a related function while locally adapting it.
result The method achieves better convergence of level sets compared to standard transfer learning.
Study examines harmonic functions in sub-Riemannian and RCD settings.
problem Characterizing harmonic functions in sub-Riemannian and RCD settings.
method Analyzes weak and strong asymptotically mean value harmonic functions.
result Weakly amv-harmonic functions are equivalent to harmonicity in Carnot groups.
The study classifies rational isoparametric functions on Damek-Ricci spaces.
problem Classifying isoparametric functions on Damek-Ricci spaces.
method Using polynomial functions divided by t for classification. result New isoparametric functions discovered and their properties studied.
We present a novel class of convolutional neural networks (CNNs) for set functions, i.e., data indexed with the powerset of a finite set. The convolutions are derived as linear, shift-equivariant functions for various notions of shifts on set functions. The framework is fundamentally different from graph convolutions b…
Neural networks are dense among Lipschitz functions with fixed Lipschitz constant.
problem Characterizing neural network approximations to Lipschitz functions.
method Analyzing L-Lipschitz neural networks and their density in L-Lipschitz functions. result One layer neural networks are dense in the set of all L-Lipschitz functions. Proves a function's locally least gradient property if its level sets are minimal laminations.
problem Understanding the relationship between 1-harmonic functions and minimal laminations.
method Analyzes minimal laminations and their convergence properties, then applies to 1-harmonic functions.
result Proves a function is 1-harmonic if its level sets are minimal laminations.
In binary classification problems, mainly two approaches have been proposed; one is loss function approach and the other is uncertainty set approach. The loss function approach is applied to major learning algorithms such as support vector machine (SVM) and boosting methods. The loss function represents the penalty of …
The logcosh loss function helps neural networks learn set-valued functions better.
problem Learning set-valued functions with neural networks.
method Using artificial neural networks with logcosh loss.
result Neural networks with logcosh loss can classify samples based on set-valued functions.
Introduces epistemic deep learning for better uncertainty estimation in neural networks.
problem Uncertainty quantification in deep neural networks.
method Random-set convolutional neural networks with belief function-based loss functions.
result Epistemic approach produces better performance in uncertainty estimation.
Given only information in the form of similarity triplets "Object A is more similar to object B than to object C" about a data set, we propose two ways of defining a kernel function on the data set. While previous approaches construct a low-dimensional Euclidean embedding of the data set that reflects the given similar…
Study on harmonic functions on nonnegative curvature 3D manifolds.
problem Analyzing harmonic functions on specific 3D manifolds.
method Inspired by Miao, developed a monotonic quantity for level sets of harmonic functions on (R3∖{0},g) with nonnegative scalar curvature. result Established a rigidity result for the derived monotonic quantity.
In this note we discuss a few properties of transnormal Finsler functions, i.e., the natural generalization of distance functions and isoparametric Finsler functions. In particular, we prove that critical level sets of an analytic transnormal function are submanifolds, and the partition of M into level sets is a Fins…
Study robust regression learning under adversarial attacks.
problem Understanding which function classes are learnable in the presence of adversarial attacks.
method Introduced a novel agnostic sample compression scheme and used fat-shattering dimension to construct adversarially robust sample compression schemes.
result Finite fat-shattering dimension classes are learnable in both realizable and agnostic settings.
In this paper we consider the problems of supervised classification and regression in the case where attributes and labels are functions: a data is represented by a set of functions, and the label is also a function. We focus on the use of reproducing kernel Hilbert space theory to learn from such functional data. Basi…
Study functional confounders in causal inference, enabling estimable effects.
problem Causal inference challenges with functional confounders violating positivity.
method Functional interventions, functional positivity, gradient fields, Level-set Orthogonal Descent Estimation (LODE).
result Valid causal effect estimation under certain conditions.
This paper reviews the functional aspects of statistical learning theory. The main point under consideration is the nature of the hypothesis set when no prior information is available but data. Within this framework we first discuss about the hypothesis set: it is a vectorial space, it is a set of pointwise defined fun…
We establish a few formulas that compute the volume of the zero-set (or nodal set) of a function on a compact Riemannian manifold as integrals of functionals of the function and its derivatives.
The paper uses MDM theory to analyze multifiltering functions on simplicial complexes.
problem Understanding multifiltering functions through discrete Morse theory.
method Applying multiparameter discrete Morse theory to vector-valued multifiltering functions.
result Any multifiltering function can be approximated by a compatible MDM function.
FuDGE estimates differences between functional graphs in high-dimensional settings.
problem Estimating differences between two undirected functional graphical models with shared structures.
method FuDGE: A method that directly estimates the functional differential graph without first estimating individual graphs.
result FuDGE consistently estimates the functional differential graph in high-dimensional settings.
We consider a class of sparsity-inducing regularization terms based on submodular functions. While previous work has focused on non-decreasing functions, we explore symmetric submodular functions and their \lova extensions. We show that the Lovasz extension may be seen as the convex envelope of a function that depends …
The study examines exceptional sets for radial limits of superharmonic functions on curved manifolds.
problem Understanding exceptional sets for radial limits of superharmonic functions on curved manifolds.
method Analysis of radial geodesic rays, Poisson integrals, Green potentials, and Riesz decomposition.
result Sharp bounds on Hausdorff dimensions of exceptional sets for superharmonic functions.
The paper explores different smooth map notions on convex sets and their relationships.
problem Exploring and comparing different smooth map notions on convex sets.
method Constructing a function that doesn't extend to a smooth function on any open neighborhood but does for Ck functions. result Diffeological and Sikorski smoothness notions do not coincide for all convex sets.
Sublinear functionals of random variables are known as sublinear expectations; they are convex homogeneous functionals on infinite-dimensional linear spaces. We extend this concept for set-valued functionals defined on measurable set-valued functions (which form a nonlinear space), equivalently, on random closed sets. …
New condition for reconstructing Morse functions on 3D manifolds.
problem Reconstructing Morse functions with specific level sets.
method Studied a necessary and sufficient condition for reconstruction.
result New condition strengthens previous sufficient conditions.
Survey on neural networks for set-based data.
problem Efficient processing of set-based inputs in machine learning.
method Exploration of Deep Sets and Transformers for set functions approximation.
result Deep Sets can be generalized by differences in aggregation function.
The paper extends properties of smooth functions to closed sets and maps.
problem Properties of smooth functions on closed sets and maps.
method Extending properties of smooth functions to closed sets and maps, proving isomorphisms with natural topologies.
result Bornological isomorphisms of function spaces are established.
Federated learning (FL) is a distributed learning approach where a set of end-user devices participate in the learning process by acting on their isolated local data sets. Here, we process local data sets of users where worst-case optimization theory is used to reformulate the FL problem where the impact of local data …
Set-functions appear in many areas of computer science and applied mathematics, such as machine learning, computer vision, operations research or electrical networks. Among these set-functions, submodular functions play an important role, similar to convex functions on vector spaces. In this tutorial, the theory of sub…