The paper studies continuous submodular functions and their optimization.
problem Maximizing continuous submodular functions in poly. time.
method Characterization of continuous submodularity, operations preserving it, and algorithms for constrained maximization.
result Continuous submodularity is equivalent to a weak DR property, leading to continuous DR-submodular functions with the full DR property.
Study restrictions on digitally continuous functions and their effects.
problem Understanding effects of restrictions on digitally continuous functions.
method Analyzing digitally continuous functions and their modifications.
result Analogous result for topological spaces derived from digitally continuous functions.
Smooth approximations for continuous functions on orbit spaces.
problem Approximating continuous functions on orbit spaces.
method Study of subcartesian spaces and proper Lie group actions.
result Continuous functions can be approximated by smooth functions.
Neural networks with integer weights approximate continuous functions efficiently.
problem Approximating continuous functions using neural networks with integer weights.
method Integrates superexpressive activation functions and integer weights.
result Convergence rate of order n2β+d−2βlog2n for neural network regression. Study continuity of complex Sobolev functions, with applications to Kaehler metrics.
problem Continuity of functions in complex Sobolev spaces.
method Analysis of function regularity in Sobolev spaces, with applications to Kaehler metrics.
result Hermitian generalizations of recent results on Kaehler metrics.
Paper proves weak unique continuation for harmonic functions on RCD spaces but finds counterexample for strong uniqueness.
problem Unique continuation of harmonic functions on RCD spaces, especially strong uniqueness.
method Establishes weak unique continuation theorem and provides counterexample for strong uniqueness.
result Found counterexample for strong unique continuation in RCD(K,N) spaces for N≥4 and K∈R.
Zeta functions for non-unitary twists are shown to have analytic continuation.
problem Analytic continuation of zeta functions for non-unitary twists.
method Analytic continuation for compact locally-symmetric spaces with non-unitary twists.
result Zeta functions admit analytic continuation as meromorphic functions.
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.
Study optimal stopping problems with finite-time horizon and proves continuity and strict monotonicity of the boundary.
problem Optimal stopping problems with finite-time horizon and state-dependent discounting.
method Linear diffusion process, time-homogeneous gain function, fine regularity properties, continuity and strict monotonicity proof.
result Proves continuity and strict monotonicity of the optimal stopping boundary under mild assumptions.
Three-hidden-layer neural networks can approximate Hölder continuous functions uniformly with exponential rate.
problem Approximating Hölder continuous functions with neural networks.
method Introduced Floor-Exponential-Step (FLES) networks with three hidden layers.
result Uniform approximation of Hölder continuous functions with an exponential rate.
In this paper, we consider the problem of black box continuous submodular maximization where we only have access to the function values and no information about the derivatives is provided. For a monotone and continuous DR-submodular function, and subject to a bounded convex body constraint, we propose Black-box Contin…
The Lebesgue property (order-continuity) of a monotone convex function on a solid vector space of measurable functions is characterized in terms of (1) the weak inf-compactness of the conjugate function on the order-continuous dual space, (2) the attainment of the supremum in the dual representation by order-continuous…
Study on heat equation and eigenfunctions on RCD spaces, proving unique continuation.
problem Unique continuation for caloric functions and eigenfunctions on RCD spaces.
method Establish weak unique continuation theorem for caloric functions and eigenfunctions on compact RCD(K,2) spaces.
result Existence of non-trivial eigenfunctions and caloric solutions vanishing up to infinite order at one point.
GroupSort neural networks can approximate Lipschitz continuous functions.
problem Understanding and improving the expressive power of neural networks with Lipschitz constraints.
method Introduced and studied GroupSort neural networks with constraints on weights, proving their ability to approximate Lipschitz continuous functions.
result GroupSort networks can represent any Lipschitz continuous piecewise linear functions and are well-suited for approximating general Lipschitz continuous functions.
This paper tackles discontinuous neural networks for better approximation of piecewise continuous functions.
problem Limitation of neural networks in approximating piecewise continuous functions due to discontinuities.
method Proposes a decoupled two-step procedure to train a discontinuous deep neural network model.
result Provides approximation guarantees for the proposed model in piecewise continuous function spaces.
New algorithm optimizes Hölder continuous functions efficiently.
problem Optimizing Hölder continuous multivariate functions.
method Uses a query creation rule for global optimization, avoiding proxy functions.
result Achieves an average regret bound of $O(T^{- racα{n}})$ for Hölder exponent α. Paper constructs continuous families of topological Morse functions.
problem Existence and deformability of topological Morse functions.
method Simple construction of continuous families of topological Morse functions.
result Gives a construction of continuous families of topological Morse functions.
CGNNs use wavelets for continuous function generation in infinite-dimensional spaces.
problem Generating continuous functions in infinite-dimensional spaces for applications like inverse problems.
method Inspired by DCGAN, CGNNs use wavelet multiresolution analysis with convolutional and nonlinear layers.
result CGNNs can be injective under certain conditions on filters and nonlinearity, leading to Lipschitz stability estimates.
Geodesically convex functions are continuous on Riemannian manifolds.
problem Continuity of geodesically convex functions on Riemannian manifolds.
method Proof of continuity using geodesic convexity and addressing a gap in existing proof.
result All geodesically convex functions are continuous in the interior of their domain on Riemannian manifolds.
Proves Hölder continuity of complex Monge-Ampère solutions.
problem Global Hölder continuity of solutions to complex Monge-Ampère equation.
method Analyzes Dirichlet problem on strictly pseudoconvex domains or Hermitian manifolds.
result Proves global Hölder continuity of solutions under given conditions.
The paper explores continuous inverse ambiguous functions on various Lie groups.
problem Existence of continuous inverse ambiguous functions on Lie groups.
method Investigation of continuous inverse ambiguous functions on specific Lie groups.
result Existence of continuous inverse ambiguous functions on various Lie groups.
New RBF networks can approximate any continuous function.
problem Approximating any continuous function on a compact subset.
method Replacing smoothing factors with shifts in RBF networks and proving approximation under certain conditions.
result RBF networks can approximate any continuous function on any compact subset.
Improves risk and variability measures continuity and consistency.
problem Improving the continuity and consistency of risk and variability measures.
method Analyzes convex and order bounded above functionals on Frechet lattices and Orlicz spaces.
result Order-continuous, law-invariant functionals on Orlicz spaces are strongly consistent everywhere.
Continuous-time Q-learning theory developed for reinforcement learning.
problem Continuous-time reinforcement learning challenges.
method Entropy-regularized, exploratory diffusion process formulation; first-order approximation of Q-function; martingale conditions.
result Developed a q-learning theory independent of time discretization.
Unified routing and arbitrage with concave continuation.
problem Combining routing and arbitrage in financial markets.
method Extending AMM trade functions to negative inputs via concave continuation.
result Unified approach unifies routing and arbitrage.
Simple neural networks approximate any continuous function with fixed neurons.
problem Approximating arbitrary continuous functions with limited neurons.
method Developed simple feed-forward neural networks with a specific activation function.
result Proven that networks with 36d(2d+1) neurons and depth 11 can approximate any continuous function.
George Cybenko's landmark 1989 paper showed that there exists a feedforward neural network, with exactly one hidden layer (and a finite number of neurons), that can arbitrarily approximate a given continuous function f on the unit hypercube. The paper did not address how to find the weight/parameters of such a networ…
With the renewed and growing interest in geometric continuity in mind, this article gives a general definition of geometrically continuous polygonal surfaces and geometrically continuous spline functions on them. Polynomial splines defined by G1 gluing data in terms of rational functions are analyzed further. A general…
Let M be a complete Riemannian manifold possessing a strictly convex Lipschitz continuous exhaustion function. We show that the isoperimetric profile of M is a continuous and non-decreasing function. Particular cases are Hadamard manifolds and complete non-compact manifolds with strictly positive sectional curvatur…
Proposes a continuous relaxation for discrete Bayesian optimization.
problem Efficiently optimizing discrete data with limited target observations.
method Continuous relaxation of objective function, incorporating prior knowledge.
result Optimization can be computationally tractable with few observations.
Kolmogorov neural networks can represent various types of functions.
problem Representing different types of functions with neural networks.
method Continuous, discontinuous bounded or unbounded activation functions in a two hidden layer model.
result Kolmogorov neural networks can represent continuous, discontinuous bounded and all unbounded multivariate functions.
The paper interprets policy-gradient algorithms using continuation theory.
problem Optimizing nonconvex functions in reinforcement learning.
method Formulates policy optimization as optimization by continuation, interprets policy-gradient algorithms as implicitly optimizing deterministic policies.
result Exploration in policy-gradient algorithms is seen as computing a continuation of the return of the policy.
Continuous-time analysis shows SGD with noise prefers flat minima.
problem Optimizing neural networks using SGD with noise.
method Continuous-time model for SGD with noise analysis.
result Optimization prefers flat minima in certain noise regimes.
Study proves finiteness for distance functions on curved surfaces with controlled curvature.
problem Understanding distance functions on curved surfaces with Hölder continuous curvature.
method Proves a finiteness principle using Whitney extension theory for geodesics and points on Riemannian surfaces with Hölder continuous curvature.
result Establishes a finiteness principle for isometric embedding of metric spaces into Riemannian surfaces with controlled curvature.
Proves regularity of extremal function on compact Kähler manifolds.
problem Regularity of extremal function on compact Kähler manifolds.
method Local property analysis and equivalence of continuity and Hölder continuity.
result Equivalence of classical notions of local L-regularity and locally Hölder continuous property. Generative models learn distributions of continuous functions.
problem Training generative models on discretized grids limits model size and data type.
method Parameterize data points by continuous functions, learn distributions over these functions.
result Models can learn rich distributions of functions independently of data type and resolution.
Improved density estimation for mixed discrete-continuous data.
problem Inconsistent density estimation for mixtures of continuous and discrete data.
method Modification of existing nonparametric density estimation methods to handle mixed discrete-continuous data.
result Improved consistency and empirical performance for mixed discrete-continuous data.
Improved neural network predicts spectral functions more accurately than traditional methods.
problem Reconstructing real-time spectral functions from imaginary-time Green's functions is ill-posed and challenging.
method Feature Learning Network (FL-net) for enhanced prediction accuracy.
result FL-net achieves at least 20% improvement over traditional methods like MEM.
This paper introduces first order Sobolev spaces on certain rectifiable varifolds. These complete locally convex spaces are contained in the generally nonlinear class of generalised weakly differentiable functions and share key functional analytic properties with their Euclidean counterparts. Assuming the varifold to s…
Adaptive tensor modeling preserves continuity in multidimensional data.
problem Discretization of continuous multidimensional data loses important information.
method Functional Tucker decomposition (FTD) with RKHS modeling.
result FTD enables adaptive and expressive tensor modeling.
TVS-FNNs can approximate any continuous function on expanded input spaces.
problem Processing a broader range of inputs like sequences and matrices.
method Proving a universal approximation theorem for TVS-FNNs.
result TVS-FNNs can approximate any continuous function on expanded input spaces.
Extends holomorphic functions on complex manifolds to larger spaces.
problem Extending holomorphic functions on complex manifolds.
method Proving the existence of a larger space B(S,X) for continuous maps that allows holomorphic continuation. result Bounded holomorphic functions on C(S,X) can be extended to holomorphic functions on B(S,X). Study continuation maps for Morse fundamental group properties.
problem Properties of continuation maps for Morse fundamental group.
method Analysis of continuation maps for Morse fundamental group, functoriality, and isomorphism to relative fundamental group.
result Continuation maps are isomorphic to relative fundamental groups.
Solves complex Monge-Ampère equations with Hölder continuous solutions in Kähler manifolds.
problem Finding Hölder continuous solutions to complex Monge-Ampère equations.
method Analyzes the complex Monge-Ampère equation in Kähler manifolds using Sobolev spaces and Hölder continuity.
result Hölder continuity of solutions is equivalent to the measure's Hölder continuity in a complex Sobolev space.
Study Cowen-Douglas operators from analytic function spaces.
problem Analytic continuation and spectrum of Cowen-Douglas operators.
method Investigate Banach spaces of analytic functions and their operators.
result Analytic continuations of functions relate to the spectrum of Cowen-Douglas operators.
The Brouwer fixed point theorem says that any continuous function from disc to itself has a fixed point. By using simple geometrical technique we have generalized the result in manifold and proved that any continuous function on the boundary of a bounded convex domain of a 2-dimensional Riemannian manifold with a pol…
Binary encoding enables neural networks to extrapolate periodic functions.
problem Extrapolating periodic functions without prior knowledge of their form.
method Normalized Base-2 Encoding (NB2E) for continuous numerical values.
result MLPs using NB2E can successfully extrapolate diverse periodic signals.
We present sufficient conditions for topological stability of continuous functions f:R→R having finitely many local extrema with respect to averagings by discrete measures with finite supports.