AES learns feasible domains in unbounded spaces with bounded query budget.
problem Learning feasible domains in unbounded input spaces with limited query budget.
method Active Expansion Sampling (AES) progressively expands knowledge of the input space, switching between learning decision boundaries and searching for new feasible domains.
result AES has a misclassification loss guarantee within the explored region, independent of iterations or labeled samples.
Narrow neural networks have unbounded decision regions.
problem Understanding decision regions of narrow neural networks.
method Analyzing decision regions of neural networks with width ≤ input dimension.
result All connected components of decision regions are unbounded.
The paper proves homotopy equivalences for spaces of unbounded Fredholm operators.
problem Spaces of unbounded Fredholm operators and their properties.
method Analyzing the spaces and proving homotopy equivalences.
result Natural maps between four spaces of unbounded Fredholm operators are homotopy equivalences.
Unbounded convex domains have zero mean curvature on disconnected boundaries.
problem Understanding mean curvature in unbounded convex domains.
method Analyzing mean curvature on disconnected boundary components.
result Mean curvature is zero on disconnected boundary components of unbounded mean convex domains.
Constructs unbounded Kasparov product for sphere embeddings into Euclidean space.
problem Embedding spheres into Euclidean space and their associated Kasparov cycles.
method Constructs unbounded Kasparov cycles, equips with connections, computes unbounded Kasparov product with Dirac operator, identifies index cycles.
result Spectral triple for algebra C ( S n ) C(\mathbb S^n) C ( S n ) differs from round sphere Dirac operator by index cycle. Study on surfaces with unbounded mean curvature and singularities.
problem Understanding surfaces with unbounded mean curvature and singularities.
method Analyzing surfaces given by Kenmotsu-type formula.
result Characterization of singularities in surfaces with unbounded mean curvature.
Paper tackles robust deep learning from weakly dependent data with unbounded loss and input.
problem Tackles robust deep learning from weakly dependent data with unbounded loss and input.
method Establishes non-asymptotic bounds for expected excess risk under strong mixing and ψ ψ ψ -weak dependence assumptions. result Derives a relationship between bounds and r r r , and shows convergence rate close to i.i.d. results for r = ∞ r=\infty r = ∞ . Study ancient solutions on graphs with unbounded Laplacians, generalizing previous results.
problem Understanding ancient solutions on graphs with unbounded Laplacians.
method Generalizing Colding and Minicozzi's theorem and Hua's result to graphs with unbounded Laplacians.
result The dimension of the space of ancient solutions of polynomial growth is bounded by the dimension of harmonic functions with the same growth.
Symplectic method solves infinite-dimensional Schrödinger equations.
problem Solving Schrödinger equations on infinite-dimensional Hilbert spaces with unbounded Hamiltonians.
method Analytic vectors, manifolds modelled on normed spaces, symplectic differential geometry, Marsden--Weinstein reduction.
result Mapped t t t -dependent Schrödinger equations onto projective spaces. We construct a complete, embedded minimal surface in euclidean 3-space which has unbounded Gaussian curvature. It has infinite genus, infinitely many catenoidal type ends and one limit end.
New algorithm tackles multiclass transductive online learning with unbounded labels.
problem Characterizing optimal mistake bound for unbounded label spaces.
method Introducing new combinatorial dimensions (Level-constrained Littlestone and Branching dimensions) to characterize online learnability.
result Established trichotomy of possible minimax rates for unbounded label spaces: Θ ( T ) Θ(T) Θ ( T ) , Θ ( log T ) Θ(\log T) Θ ( log T ) , or Θ ( 1 ) Θ(1) Θ ( 1 ) . Novel SVM approach for extreme quantile regression with heavy tailed inputs.
problem Learning from extreme values in quantile regression.
method Support Vector Machine framework for handling high-dimensional and nonlinear settings.
result Established finite-sample learning guarantees under mild regularity assumptions.
New neural network rates for unbounded domains with weighted Sobolev spaces.
problem Improving neural network approximation rates for unbounded domains.
method Embedding results for weighted Fourier-Lebesgue spaces in weighted Sobolev spaces, followed by asymptotic approximation rates.
result Asymptotic approximation rates for shallow neural networks without curse of dimensionality for unbounded domains and Muckenhoupt weights.
Short note shows unbounded dimensions in Fano K-moduli spaces.
problem Unboundedness of Fano K-moduli space dimensions.
method Analyzes Fano varieties and their K-moduli spaces.
result Dimension of K-moduli space can be unbounded.
Spectral algorithms improve under covariate shift with novel weighted techniques.
problem Improving spectral algorithms' performance under covariate shift.
method Analysis of spectral algorithms in non-parametric regression over RKHS, proposing a weighted spectral algorithm with clipped weights.
result Normalized weighted spectral algorithm achieves optimal capacity-independent convergence rates, and clipped weights can approach optimal capacity-dependent rates.
New degree theory for perturbed self-adjoint operators.
problem Equivariant gradient perturbations of unbounded self-adjoint operators.
method Equivariant gradient degree for perturbations of self-adjoint operators with discrete spectrum.
result Definition and application of new degree theory.
New approach finds solutions to games with unbounded controls.
problem Existence of equilibrium in mean-field games with unbounded controls.
method Weak formulation and new existence/stability results for quadratic-growth generalized McKean-Vlasov BSDEs.
result Existence of equilibrium result for non-Markovian mean-field games with unbounded control space.
New method for manifold fitting under unbounded noise.
problem Noise blurs tangent space estimation, leading to inaccurate manifold recovery.
method Directly estimate tangent spaces at projected points on the manifold, not at sample points.
result Theoretical convergence in high probability for upper bound of manifold distance.
Ricci flow from spaces with cylindrical ends and unbounded curvature.
problem Existence and stability of Ricci flow starting from complex singular metrics.
method Proving existence and local stability of Ricci flow for specific metrics.
result Local stability allows gluing to other manifolds for broader applications.
Geometric model of unbounded sl3 laminations with tropical coordinates.
problem Modeling unbounded laminations in cluster varieties.
method Introducing tropical cluster coordinates and geometric gluing procedures.
result Established a geometric gluing procedure for unbounded sl3 laminations.
New RL policy for unbounded state space with stability guarantee.
problem Traditional RL methods fail for unbounded state space.
method Proposes stability as performance metric, uses Sparse-Sampling-based Monte Carlo Oracle.
result Proposed policy ensures state dynamics remain bounded with high probability.
New approach to concentration inequalities for unbounded state space dynamical systems.
problem Concentration inequalities for unbounded state space dynamical systems.
method Functional analytic framework, transport-entropy inequality.
result Exponential concentration inequalities for sampling from stationary distribution.
Surveying Ricci flow and its applications to Ricci limit spaces.
problem Understanding Ricci flows with unbounded curvature.
method Analyzing Ricci flows that can have infinite curvature at infinity and singular times.
result Developed insights into Ricci limit spaces.
New analysis shows a gap between Gaussian RKHS and neural networks on unbounded domains.
problem Understanding the function space bias of neural networks compared to Gaussian RKHS.
method Infinite-center asymptotic analysis of neural network Banach space and Gaussian RKHS on unbounded domains.
result Certain functions in Gaussian RKHS have infinite norm in neural network Banach space on unbounded domains.
Solves open problem on universally consistent online learning with unbounded losses.
problem Open problem on universally consistent online learning with unbounded losses.
method Constructs random measurable partitions of the instance space.
result Simple memorization rule is optimistically universal for any unbounded loss.
Using results from our companion article [arXiv:1112.4824v2] on a Schauder approach to existence of solutions to a degenerate-parabolic partial differential equation, we solve three intertwined problems, motivated by probability theory and mathematical finance, concerning degenerate diffusion processes. We show that th…
The paper proves density of smooth functions in Sobolev spaces on certain manifolds.
problem Density of smooth functions in Sobolev spaces on manifolds with unbounded geometry.
method Distance-like function with bounded gradient and mild growth of Hessian, proving density results.
result Smooth compactly supported functions are dense in W 2 , p W^{2,p} W 2 , p on the considered manifolds. New findings show unbounded gaps between ordinary and equivariant Dehn surgery numbers.
problem Understanding differences between ordinary and equivariant Dehn surgery numbers.
method Proved existence of pairs ( Y k , τ k ) (Y_k,τ_k) ( Y k , τ k ) with specific gap between $\DS(Y_k)$ and $\EDS(Y_k,τ_k)$ . result Difference $\EDS(Y,τ)-\DS(Y)$ is unbounded even for involutions.
Two new algorithms improve performance in adversarial bandits with unbounded losses.
problem Adversarial Multi-Armed Bandits with unbounded losses.
method Developed UMAB-NN and UMAB-G for non-negative and general unbounded losses respectively.
result UMAB-NN achieves the first adaptive and scale-free regret bound for non-negative unbounded losses.
New framework for conformal equivariant cycles in KK-theory.
problem Tackles conformal equivariance in unbounded KK-theory.
method Extends unbounded Kasparov theory with novel perturbation theory.
result Defines new unbounded representatives of Kasparov classes.
GP-PSRL achieves sublinear regret for continuous control with unbounded state space.
problem Analyzing regret bounds for GP-PSRL in continuous control with unbounded state space.
method Recursive application of Borell-Tsirelson-Ibragimov-Sudakov inequality and chaining method.
result Sublinear regret bound of O ~ ( H γ T T ) \widetilde{\mathcal{O}}(H\sqrt{γ_TT}) O ( H γ T T ) for GP-PSRL. We produce solutions to the Kähler-Ricci flow emerging from complete initial metrics g 0 g_0 g 0 which are C 0 C^0 C 0 Hermitian limits of Kähler metrics. Of particular interest is when g 0 g_0 g 0 is Kähler with unbounded curvature. We provide such solutions for a wide class of U ( n ) U(n) U ( n ) -invariant Kähler metrics g 0 g_0 g 0 on n n n dimensional c…
LLA shows strong performance in Bayesian optimization but has unbounded search space issues.
problem Applying LLA in unbounded search spaces for Bayesian optimization.
method Linearized-Laplace approximation applied to Bayesian optimization problems.
result LLA demonstrates strong performance but also presents unbounded search space challenges.
Algorithm learns diffusion processes with high-dimensional state spaces.
problem Stochastic control of unbounded diffusion processes with high-dimensional state spaces.
method Adaptive partitioning and learning algorithm that refines discretization based on estimation bias and statistical confidence.
result Established regret bounds that depend on problem parameters, extending to unbounded diffusion processes.
We consider the notion of dimension in four categories: the category of (unbounded) separable metric spaces and (metrically proper) Lipschitz maps, and the category of (unbounded) separable metric spaces and (metrically proper) uniform maps. A unified treatment is given to the large scale dimension and the small scale …
Study utility maximization in financial markets with bounded and unbounded payoffs.
problem Utility maximization in financial markets with constraints and unbounded payoffs.
method Combines quadratic backward stochastic differential equations and convex duality.
result Established utility indifference valuation, regime switching, and consumption-investment problems in unbounded markets.
New groups found in hyperbolic space with infinite fields of definition.
problem Finding maximal reflection groups in hyperbolic space.
method Developed new quasi-arithmetic reflection groups for hyperbolic 2-space.
result Infinitely many maximal quasi-arithmetic reflection groups with unbounded field degrees.
Develops thermodynamic formalism for quasimorphisms on negatively curved spaces.
problem Analyzing quasimorphisms on negatively curved spaces.
method Thermodynamic formalism framework, Banach isomorphism, weak Livšic cohomology.
result Establishes Central Limit Theorem and invariance principle for unbounded quasimorphisms.
Researchers factorize Dirac operators on toric noncommutative manifolds, finding curvature terms.
problem Factorizing Dirac operators on toric noncommutative manifolds.
method Using unbounded KK-theory and Kasparov modules, they show tensor sums of operators coincide with the Dirac operator on the manifold.
result There is a curvature term that arises as an obstruction for tensor sum decomposition in unbounded KK-theory.
Paper derives Li-Yau inequality for unbounded Laplacian on graphs.
problem Deriving Li-Yau inequality for unbounded Laplacian on graphs.
method Assumption of curvature-dimension inequality C D E ′ ( n , K ) CDE'(n,K) C D E ′ ( n , K ) and derivation of Li-Yau inequality. result First results on Li-Yau inequality for unbounded Laplacian on graphs.
Upper bounds on Wasserstein distance for empirical measures in unbounded functional spaces.
problem Analyzing convergence and concentration of empirical measures in unbounded functional spaces.
method Generalized upper bounds using Wasserstein distance, covering large dimensional Euclidean spaces and Gaussian processes.
result Rate-optimal upper bounds for functional data distributions with specific decay rates.
Using the unbounded picture of analytical K-homology, we associate a well-defined K-homology class to an unbounded symmetric operator satisfying certain mild technical conditions. We also establish an ``addition formula'' for the Dirac operator on the circle and for the Dolbeault operator on closed surfaces. Two proofs…
Golden L surface has unbounded bunching of saddle connections
problem Unbounded bunching of saddle connections on translation surfaces
method Translation surface with golden ratio
result Every positive integer K has a ball containing at least K saddle connection periods
The Einstein/Abelian-Yang-Mills Equations reduce in the stationary and axially symmetric case to a harmonic map with prescribed singularities $\p\colon\R^3\smΣ\to\H^{k+1}_\C$ into the ( k + 1 ) (k+1) ( k + 1 ) -dimensional complex hyperbolic space. In this paper, we prove the existence and uniqueness of harmonic maps with prescribed sing…
New manifolds show Riesz transform unbounded for p > 2.
problem Understanding Riesz transform behavior on manifolds.
method Constructing Riemannian manifolds with specific properties.
result Riesz transform unbounded on L p ( M ) L^p(M) L p ( M ) for all p > 2 p > 2 p > 2 . Metric transforms make Euclidean half lines hyperbolic, preserving geodesic properties.
problem Characterizing metric transforms that make Euclidean half lines hyperbolic.
method Characterization of metric transforms φ \varphi φ such that ( [ 0 , ∞ ) , ∣ ⋅ ∣ φ ) ([0,\infty),|\cdot|_\varphi) ([ 0 , ∞ ) , ∣ ⋅ ∣ φ ) is Gromov hyperbolic. result Metric transform rigidity for roughly geodesic Gromov hyperbolic spaces.
New inequalities for unbounded functions improve denoising score matching.
problem Statistical error bounds for denoising score matching with unbounded objective functions.
method Derive new concentration inequalities using McDiarmid's inequality and Rademacher complexity bounds.
result Improved statistical error bounds for denoising score matching.
It is shown that the compactly supported identity component of the diffeomorphism group of the 2-dimensional punctured torus T p 2 \mathbb T^2_p T p 2 is an unbounded group. It follows that the fragmentation norm of T p 2 \mathbb T^2_p T p 2 is unbounded.