Constructs continuous families of minimal surfaces and holomorphic immersions.
problem Creating continuous families of minimal surfaces and holomorphic immersions.
method Continuous family of complex structures and conformal minimal immersions.
result Continuous families of proper Jb-conformal minimal immersions and holomorphic null immersions. Efficient method for learning continuous exponential families beyond Gaussian.
problem Learning continuous exponential families with unbounded support.
method Interaction Screening approach for scalable learning of continuous graphical models.
result Our estimator maintains similar accuracy and sample complexity scalings compared to alternative approaches, while improving run-time.
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.
Graph continuous operators become Riesz continuous after multiplication by unitary operators.
problem Characterizing Riesz continuity of graph continuous operators.
method Multiplication by unitary operators to transform graph continuity to Riesz continuity.
result The index of graph continuous families of Fredholm operators coincides with N. Ivanov's index.
Notes on continuity of discrete-spectrum Fredholm operators.
problem Continuity properties of discrete-spectrum families of Fredholm operators.
method Relates recent work on discrete-spectrum families to classical continuity properties.
result Establishes connections between new and classical concepts.
Continuous family of elliptic operators' projections maintain Cauchy data spaces.
problem Maintaining Cauchy data spaces for a continuous family of elliptic operators.
method Elementary tools and classical results applied to operator graphs, Sobolev spaces, and Green's formula.
result Orthogonalized Calderón projections form a continuous family of projections.
Paper introduces kernel deformed exponential families for sparse continuous attention.
problem Creating efficient attention mechanisms for sparse data.
method Developed kernel deformed exponential families, theoretically and experimentally.
result Kernel deformed exponential families can attend to multiple compact regions of data.
Paper extends sparse alternatives to softmax for continuous domains, enabling efficient attention mechanisms.
problem Efficiently assigning zero probability to irrelevant categories in continuous domains.
method Extend alpha-entmax to continuous domains, introducing continuous-domain attention mechanisms.
result Continuous attention allows attending to time intervals and compact regions, improving text classification, machine translation, and visual question answering.
We construct continuous families of scattering manifolds with the same scattering phase. The manifolds are compactly supported metric perturbations of Euclidean Rn for n≥8. The metric perturbation may have arbitrarily small support.
The study constructs families of nilpotent Lie groups with geodesic orbit metrics.
problem Finding geodesic orbit metrics on nilpotent Lie groups.
method Construction of continuous families of nilpotent Lie groups.
result Continuous families of non-isomorphic nilpotent Lie groups with geodesic orbit metrics.
We construct continuous families of Riemannian metrics on certain simply connected manifolds with the property that the resulting Riemannian manifolds are pairwise isospectral for the Laplace operator acting on functions. These are the first examples of simply connected Riemannian manifolds without boundary which are i…
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.
The purpose of this paper is to present the first continuous families of Riemannian manifolds isospectral on functions but not on 1-forms, and simultaneously, the first continuous families of Riemannian manifolds with the same marked length spectrum but not the same 1-form spectrum. The examples presented here are Riem…
This paper develops sparse alternatives to continuous distributions, including new types of Gaussians and attention mechanisms.
problem Creating flexible continuous distributions with varying support for machine learning applications.
method Defining Ω-regularized prediction maps and Fenchel-Young losses for arbitrary domains, and deriving new types of Gaussians and attention mechanisms. result Sparse alternatives to continuous distributions, including deformed exponential families and β-Gaussians, are introduced. The paper generalizes spectral section concepts to non-compact spaces.
problem Generalizing spectral sections to non-compact base spaces.
method Generalization to arbitrary base spaces, applications to cobordism theorems, investigation of Riesz continuity.
result If a family of operators has a spectral section, it is Riesz continuous.
The abstract discusses families of holomorphic maps to Oka manifolds with approximation theorems.
problem Approximating \(J_b\)-holomorphic maps to Oka manifolds.
method Constructing continuous or smooth families of \(J_b\)-holomorphic maps to Oka manifolds with approximation on compact Runge sets.
result Runge and Mergelyan approximation theorems and Weierstrass interpolation theorem for families of open Riemann surfaces.
Study continuity of Bergman kernels on degenerating varieties.
problem Continuity and uniform convergence of Bergman kernels on degenerating varieties.
method Introduced fiberwise Bergman kernel for flat families of polarized varieties, established continuity and uniform convergence results.
result Uniform convergence of Fubini-Study currents and continuity of fiberwise Bergman kernel on test configurations.
Novel simplex-valued distribution improves on existing models.
problem Limitations of existing simplex-valued distributions like Dirichlet.
method Introducing continuous categorical distribution.
result Continuous categorical resolves limitations of Dirichlet.
Improved inference for models with continuous latent variables.
problem Inference accuracy with traditional variational methods is limited.
method Reparameterized Variational Rejection Sampling (RVRS) using a proposal distribution with a reparameterized gradient estimator.
result RVRS offers a better trade-off between computational cost and inference fidelity.
The paper proves the existence of a continuous family of translating surfaces under a specific curvature flow.
problem Existence of convex translating surfaces under flow by α-th power of Gauss curvature. method Constructing a family of translating surfaces using Jacobi fields and analyzing their effective growth rates.
result The family of translating surfaces is a topological manifold with quantitative convergence rates.
New bounds for score matching in polynomial exponential families.
problem Understanding the sample complexity of score matching for polynomial exponential families.
method Non-asymptotic sample complexity analysis for score matching.
result First finite sample bounds for score matching in polynomial exponential families.
We characterize when the level sets of a continuous quasi-monotone functional defined on a suitable convex subset of a normed space can be uniquely represented by a family of bounded continuous functionals. Furthermore, we investigate how regularly these functionals depend on the parameterizing level. Finally, we show …
Researchers study the normalizing constant of a continuous categorical distribution.
problem Understanding the normalizing constant of the continuous categorical distribution.
method Characterize numerical behavior and present theoretical and methodological advances.
result The normalizing constant can be written in closed form using elementary functions.
The study finds new infinite dilogarithm identities related to number sequences and continued fractions.
problem Finding new infinite dilogarithm identities.
method Demonstrating families of identities associated with specific number sequences and continued fractions.
result New infinite dilogarithm identities related to Fibonacci, Lucas numbers, convergents of even period continued fractions, and recurrence relations.
Develops a new method for learning discrete distributions without embedding them in a continuous space.
problem Challenges in learning discrete distributions using current methodologies.
method Introduces a MAD invertible map and a mixed variational flow (MAD Mix) for discrete distributions.
result MAD Mix produces more reliable approximations than continuous-embedding flows.
In the current article we study complex cycles of higher multiplicity in a specific polynomial family of holomorphic foliations in the complex plane. The family in question is a perturbation of an exact polynomial one-form giving rise to a foliation by Riemann surfaces. In this setting, a complex cycle is defined as a …
Let M be a manifold, possibly with boundary. We show that the deRham differential from k-forms to exact (k+1)-forms has a continuous right inverse when both spaces are given the weak Whitney topology. This antidifferential operator is given a fairly explicit formula depending on the choice of a suitable good cover of M…
Study the spaces of flat connections for classical Lie groups using Chern-Weil theory.
problem Understanding the weak homotopy type of spaces of flat connections for classical Lie groups.
method Use Chern-Weil theory and relate to the functorial map involving continuous families of representations.
result Relate the spaces of flat connections to the weak homotopy type of the spaces of representations.
Analyzes Kähler-Einstein metrics on families of Fano varieties.
problem Establishing Kähler-Einstein metrics on Fano varieties in families.
method Analytic method to show unique Kähler-Einstein metrics on neighboring fibers.
result Uniform a priori estimates and continuous variation of Kähler-Einstein potentials.
In this paper we propose and study a family of continuous wavelets on general domains, and a corresponding stochastic discretization that we call Monte Carlo wavelets. First, using tools from the theory of reproducing kernel Hilbert spaces and associated integral operators, we define a family of continuous wavelets by …
New invariants detect Fano varieties' K-stability.
problem Detecting K-stability in Fano varieties.
method Introducing valuative invariants based on higher moments.
result Can detect K-stability of Fano varieties.
For a continuous curve of families of Dirac type operators we define a higher spectral flow as a K-group element. We show that this higher spectral flow can be computed analytically by $\heta$-forms, and is related to the family index in the same way as the spectral flow is related to the index. We introduce a notion…
We consider a continuous family (fs), s∈[0,1] of complex polynomials in two variables with isolated singularities, that are Newton non-degenerate. We suppose that the Euler characteristic of a generic fiber is constant (or equivalently the sum of the affine Milnor number and the Milnor number at infinity $μ(s)+λ…
Generalization bounds derived for neural ODEs and deep residual networks.
problem Understanding the generalization capability of neural ODEs and deep residual networks.
method Lipschitz-based argument and analogy with deep residual networks.
result A generalization bound involving the magnitude of weight matrix differences.
We define a family of kernels for mixed continuous/discrete hierarchical parameter spaces and show that they are positive definite.
The classical H surfaces of H. A. Schwarz form a 1-parameter family of triply periodic minimal surfaces (TPMS) that are usually described as close relatives to his more famous P surface. However, a crucial distinction between these surfaces is that the P surface belongs to a 5-dimensional smooth family of embedded TPMS…
We present a construction of complete self-dual Einstein metrics of negative scalar curvature on an uncountable family of manifolds of infinite topological type, which are enumerated by continued fraction expansions of irrational numbers. These manifolds may be regarded as limits of the resolutions of cyclic quotient s…
Two Riemannian manifolds are said to be isospectral if the associated Laplace-Belttrami operators have the same eigenvalue spectrum. If the manifolds have boundary, one specifies DIrichlet or Neumann isospectrality depending on the boundary conditions imposed on the eigenfunctions. We construct continuous families of (…
The article constructs Spin(7) metrics with Aloff--Wallach spaces as orbits.
problem Creating Spin(7) metrics with specific geometric properties.
method Continuous 1-parameter families of non-compact Spin(7) metrics with chiralities, focusing on Aloff--Wallach spaces.
result Construction of Spin(7) metrics with Aloff--Wallach spaces as principal orbits, including geometric transitions.
We generalize the circle bundle examples of ancient solutions of the Ricci flow discovered by Bakas, Kong, and Ni to a class of principal torus bundles over an arbitrary finite product of Fano Kähler-Einstein manifolds studied by Wang and Ziller in the context of Einstein geometry. As a result, continuous families of $…
Continuous time analysis of bubble formation in harmonic maps.
problem Understanding bubble formation in harmonic map heat flow.
method Continuous time approach to analyze bubbling sequences.
result Solutions approach multi-bubble configurations in continuous time.
Families of alternating knots (links) and tangles are studied using as building block the conway defined as the twisting of two strands. The regular representation of knots assumes the projection has the minimal number of overpassings, and the minimal number of conways. The continued fraction associated to rational kno…
New examples of degenerating metrics on R^4 found.
problem Degenerating Poincaré-Einstein metrics on R^4.
method Riemannian ansatz of Debever and Deimaniański, constructing families.
result Continuous families of metrics with cusps and unexpected degenerations.
New periodic solution found in 4-body problem, not part of expected geometrical family.
problem Finding new periodic solutions in the 4-body problem not fitting the expected geometrical family.
method Analytic continuation of a numerical solution to discover a new family of solutions.
result Existence of a non-planar periodic solution for any pair of masses and integer n.
In this paper we introduce a family of stochastic gradient estimation techniques based of the perturbative expansion around the mean of the sampling distribution. We characterize the bias and variance of the resulting Taylor-corrected estimators using the Lagrange error formula. Furthermore, we introduce a family of va…
New tensor network models learn continuous data effectively.
problem Tensor network models' limitations in handling continuous data.
method Developed a new family of tensor network generative models for continuous data.
result Models can approximate any reasonably smooth probability density function with arbitrary precision.
We prove a Frobenius-type theorem for singular distributions generated by a family of locally Lipschitz continuous vector fields satisfying almost everywhere a quantitative finite type condition.
Discrete analogues of ellipsoids with preserved circular cross sections.
problem Constructing discrete analogues of ellipsoids with preserved geometric properties.
method A novel discretization procedure to create discrete analogues of ellipsoids composed of planar quadrilaterals.
result Discrete analogues of ellipsoids have preserved circular cross sections and can be deformed.