New algorithm minimizes convex functions with smooth and non-smooth parts.
problem Minimizing convex functions with smooth and non-smooth components.
method Proximal stochastic quasi-Newton method incorporating Hessian and multistage variance reduction.
result Achieves linear rate of convergence.
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.
Characterizes smooth functions on manifolds with simple Reeb spaces.
problem Understanding the structure of Reeb spaces for smooth functions.
method Analyzes smooth functions on closed manifolds to determine their Reeb spaces' structure.
result Characterizes smooth functions whose Reeb spaces are finite graphs.
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.
Variant of previous work on smooth algebraic functions with compact and non-compact preimages.
problem Constructing smooth algebraic functions with specific preimage properties.
method Explicit construction of smooth real algebraic functions with controlled preimage compactness.
result New results in singularity theory and real algebraic geometry.
We show that for any smooth Hausdorff manifolds M and N, which are not necessarily second countable, paracompact or connected, any isomorphism from the algebra of smooth (real or complex) functions on N to the algebra of smooth functions on M is given by composition with a unique diffeomorphism from M to N. An analogou…
Reidemeister's theorem proved using smooth functions and transversality.
problem Proving Reidemeister's theorem
method Using smooth functions and transversality
result Reidemeister's theorem proved
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.
New method constructs smooth functions for given Reeb graphs.
problem Construct smooth functions with specific Reeb graphs.
method Explicit construction on 3D closed manifold, new ideas for surfaces.
result Smooth function on 3D manifold induces Reeb graph isomorphic to given graph.
New algorithm adapts to unknown smoothness in contextual bandits.
problem Adapting to unknown smoothness in non-parametric multi-armed bandits.
method Develops a self-similarity condition-based policy to adapt to unknown smoothness.
result Matches known smoothness case's regret rate for differentiable and non-differentiable payoff functions.
New algorithm POO optimizes noisy, unknown-smooth functions.
problem Optimizing functions with unknown smoothness and noisy evaluations.
method Adaptive optimization algorithm POO.
result POO performs nearly as well as known algorithms with smoothness knowledge, and works for broader classes of functions.
Smooth Busemann functions found in harmonic Finsler spaces.
problem Analyzing Busemann functions in Finsler manifolds.
method Investigation of Busemann functions in general and asymptotically harmonic Finsler manifolds.
result Smoothness of Busemann functions on asymptotically harmonic Finsler manifolds.
Among all C ∞ C^\infty C ∞ -algebras we characterize those which are algebras of smooth functions on smooth separable Hausdorff manifolds.
Smooth functions on fat closed sets extend to smooth functions on all of space.
problem Extending smooth functions from fat closed sets to the entire space.
method Investigating arc-smooth functions on fat closed sets with Hölder boundary and subanalytic properties.
result Arc-smooth functions on fat closed sets extend to smooth functions on all of space.
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.
Smooth manifolds have functions with exactly two critical values.
problem Characterizing manifolds with specific Reeb functions.
method Proving existence of Reeb functions with prescribed critical values.
result Characterization of manifolds in dimensions 3 and n≥5 using Reeb functions.
The paper explores the structure of Reeb spaces for smooth functions on manifolds.
problem Understanding the structure of Reeb spaces for smooth functions on manifolds.
method Proving the structure of Reeb spaces and showing that any graph can be realized as a Reeb space.
result The Reeb space of a smooth function on a closed manifold with finitely many critical values has a graph structure.
Topologies on algebraic and equational theories are used to define germ determined, near-point determined, and point determined rings of smooth functions, without requiring them to be finitely generated. It is proved, that any commutative algebra morphism (without requiring continuity) between near-point determined rin…
The folk questions in Lorentzian Geometry, which concerns the smoothness of time functions and slicings by Cauchy hypersurfaces, are solved by giving simple proofs of: (a) any globally hyperbolic spacetime ( M , g ) (M,g) ( M , g ) admits a smooth time function τ τ τ whose levels are spacelike Cauchy hyperfurfaces and, thus, also a smooth…
Bagging smooths statistical functionals by reducing prediction error.
problem Reducing prediction error in statistical functionals.
method Draws bootstrap samples, applies learning algorithm, averages predictions.
result Bagged statistical functionals are always smooth with finite von Mises expansion.
Algebras of smooth functions help reconstruct bulk topological types.
problem Reconstructing the smooth topological type of a compact manifold from its boundary.
method Introducing subalgebras of boundary functions and proving their tensor product reconstruction of the bulk algebra.
result The topological algebras A ( v ) \mathcal A(v) A ( v ) and B ( f ) \mathcal B(f) B ( f ) allow for the recovery of the smooth topological type of the bulk X X X . Decomposes smooth manifolds into algebraic submanifolds.
problem Understanding the structure of smooth manifolds induced by continuous selections.
method Generic continuous selection of smooth functions provides stratification of the manifold.
result Stratification leads to local topological structure with nondegenerate critical points.
New smooth functions constructed on 3D manifolds with specific Reeb graphs.
problem Constructing smooth functions with specific Reeb graphs.
method Explicit constructive methods on 3D closed orientable manifolds.
result New smooth functions induced by graphs on 3D manifolds.
Deep ReLU networks can approximate and learn smooth functions efficiently.
problem Efficiently approximating and learning smooth functions using deep ReLU neural networks.
method Extending recent results to anisotropic and mixed smooth function classes, establishing approximation rates.
result Deep ReLU networks achieve minimax optimal rates up to logarithmic factors for various smooth function classes.
Smooth approximations of causal functions derived from Lyapunov.
problem Approximating causal functions with smoothness constraints.
method Using Lyapunov functions in closed cone fields.
result Set of causal functions approximable by smooth Lyapunov functions identified.
New method for faster convergence in non-convex optimization with unbounded smoothness.
problem Finding first-order stationary points of non-convex functions with unbounded smoothness.
method Developed a stopped analysis technique to prove convergence rates for ( L 0 , L 1 ) (L_0,L_1) ( L 0 , L 1 ) -smooth functions. result Achieved O ( p o l y log ( T ) T ) \mathcal{O}(\frac{\mathrm{poly}\log(T)}{\sqrt{T}}) O ( T poly l o g ( T ) ) convergence rates without uniform noise bounds. Smooth activations enable optimal error rates in neural networks for Sobolev function classes.
problem Achieving optimal approximation and estimation error rates for neural networks in Sobolev function classes.
method Study of neural networks with smooth activations, proving optimal rates via approximation and statistical properties.
result Constant-depth networks with smooth activations achieve optimal rates of approximation and estimation, demonstrating smoothness adaptivity.
We discuss generic smooth maps from smooth manifolds to smooth surfaces, which we call "Morse 2-functions", and homotopies between such maps. The two central issues are to keep the fibers connected, in which case the Morse 2-function is "fiber-connected", and to avoid local extrema over 1-dimensional submanifolds of th…
Regularized MFPCA smooths multivariate functional data for clearer patterns.
problem Challenges in controlling roughness of multivariate functional PCs.
method ReMFPCA incorporates a roughness penalty in a penalized framework to smooth PCs.
result Smoothed multivariate functional PCs reveal clearer patterns.
The paper introduces methods to learn smooth functions on hypergraphs with sparsity.
problem Learning smooth functions on hypergraphs with sparsity.
method General framework for smoothness measures, sparse learning on hypergraphs.
result Proposes sparsely smooth formulations that induce sparsity on hypergraphs and show benefits in handling irrelevant or noisy data.
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.
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 C k C^k C k functions. result Diffeological and Sikorski smoothness notions do not coincide for all convex sets.
Paper improves adaptive regret for convex and smooth functions.
problem Online convex optimization in changing environments.
method Develops adaptive algorithms exploiting both convexity and smoothness.
result Regret bounds are comparable to worst-case results but tighter when comparators have small losses.
Deep, wide ConvResNets can approximate functions and their smoothness.
problem Function approximation and smoothness in deep networks.
method Analyzing ConvResNets, proving their ability to approximate functions and their smoothness.
result Large ConvResNets can approximate functions and exhibit sufficient first-order smoothness.
We generalize stochastic smoothing for gradient estimation of non-differentiable functions.
problem Gradient estimation for non-differentiable functions.
method Developed a general framework for relaxation and gradient estimation of non-differentiable black-box functions using stochastic smoothing with reduced assumptions.
result Empirically validated the effectiveness of variance reduction strategies for various non-differentiable tasks.
SFM generates smooth functional data without exposing real data.
problem Challenges in statistical analysis of functional data.
method Copula framework and smooth flow construction.
result SFM produces high-quality synthetic functional data.
In this paper we prove that there exists a smooth classical solution to the HJB equation for a large class of constrained problems with utility functions that are not necessarily differentiable or strictly concave. The value function is smooth if admissible controls satisfy an integrability condition or if it is contin…
Deep neural networks excel at learning non-smooth functions.
problem Understanding why deep neural networks perform better for non-smooth functions.
method Theoretical analysis of statistical properties of deep neural networks for non-smooth functions.
result Deep neural networks achieve almost optimal generalization error for non-smooth functions.
New method constructs smooth functions with specific Reeb graphs and preimages on 3D manifolds.
problem Construct smooth functions with prescribed Reeb graphs and preimages on 3D closed manifolds.
method Develops a new approach to realize graphs as Reeb graphs of smooth functions on 3D closed manifolds.
result Provides a best possible solution for functions on 3D closed manifolds.
Proposes a neural network autoencoder for smoothing and representation learning of functional data.
problem Lack of sufficient nonlinear representations in existing methods for functional data analysis.
method Develops a neural network autoencoder architecture to process functional data directly, learning both smoothing and representation.
result Outperforms traditional methods in prediction, classification, and computational efficiency.
A new method for functional data clustering using varying smoothing parameters.
problem Determining dissimilarity between subjects in functional data.
method Measuring dissimilarity based on varying curve estimates with commutation of smoothing parameters pair-by-pair.
result The method effectively clusters subjects and has practical advantages.
Proposes a spectral method for jointly smooth functions on multiple manifolds.
problem Registering measurements from different sensors and rejecting noise.
method Two steps: kernel subspace span and spectral method.
result Guaranteed orthogonal functions that are as jointly smooth as possible.
Smoothness of value function in affine control problems proven.
problem Regularity of value function in affine optimal control problems.
method Proved continuity and smoothness on open dense subsets without singular minimizers.
result Value function is smooth on an open dense subset of the interior of the attainable set.
Paper offers efficient methods for nonconvex functions.
problem Minimizing smooth quasar-convex functions.
method Near-optimal accelerated gradient descent method.
result Near-optimal number of function and gradient evaluations.
Deep neural networks with various activation functions can approximate Hölder smooth functions.
problem Expressivity of deep neural networks with general activation functions.
method Investigates approximation ability of deep neural networks with a broad class of activation functions, including Hölder smooth functions.
result Derives the required depth, width, and sparsity of deep neural networks to approximate Hölder smooth functions.
This paper improves convergence guarantees for SGD algorithms in non-convex smooth functions.
problem Theoretical convergence properties of SGD algorithms for non-convex smooth functions.
method Analysis of SGD algorithms with arbitrary data ordering for non-convex smooth functions.
result Enhanced convergence guarantees for incremental gradient and single shuffle SGD, improving the optimization term of convergence guarantee.
Study on cosine function in smooth normed spaces, proving symmetry and characterizing planes.
problem Understanding cosine function properties in smooth normed spaces.
method Proved symmetry and derived cosine function in terms of norm's Gateaux derivative.
result Cosine function is symmetric if and only if space is Euclidean.
The minimal number of critical points is studied for smooth functions on closed manifolds.
problem Determining the minimal number of critical points for smooth functions on closed manifolds.
method Investigates cylindrical ball neighborhoods and exotic critical points, proving the conjecture for certain types of critical points.
result The minimal number of critical points is the same for smooth functions without exotic critical points on closed manifolds of dimension at least 6.