Deep QMC ansatzes improve variational QMC accuracy.
problem Improving variational QMC accuracy with neural network ansatzes.
method Analysis of deep neural network ansatzes PauliNet and FermiNet convergence to fixed-node limit.
result Deep QMC ansatzes can reach fixed-node limit with large network sizes.
Investment strategy optimized under wealth limits for exponential utility maximization.
problem Maximizing wealth under fixed upper and lower limits for exponential utility.
method Combining optimal investment strategy with options to handle constraints.
result Investment strategy distribution analyzed for change of quantiles.
Study gradient flow of phase transitions with fixed contact angle.
problem Understanding phase transitions with fixed contact angle.
method Gradient flow of the Allen-Cahn equation with fixed boundary contact angle.
result Established interior and boundary convergence properties for solutions and energy measures.
Study reveals three limiting regimes for neural network functionals.
problem Understanding the behavior of functionals of random neural networks.
method Central and non-central limit theorems, Hermite expansions, Diagram Formula, Stein-Malliavin techniques.
result Three distinct limiting regimes based on fixed points of covariance function.
In this paper we prove some general results on constant mean curvature lamination limits of certain sequences of compact surfaces Mn embedded in R3 with constant mean curvature Hn and fixed finite genus, when the boundaries of these surfaces tend to infinity. Two of these theorems generalize to the non…
Incorrect fixed point results in digital topology are corrected.
problem Incorrect fixed point results in digital topology.
method Analysis of recent papers in digital topology.
result Corrected incorrect conclusions in fixed point results.
Study describes limits of surfaces in a mathematical space.
problem Understanding limits of surfaces in mathematical spaces.
method Completely described Gromov-Hausdorff closure of surfaces.
result Completely described the closure of surfaces.
This note fixes a small gap in Kerckhoff's proof that the limit set of the handlebody set has measure zero.
Training of large-scale deep neural networks is often constrained by the available computational resources. We study the effect of limited precision data representation and computation on neural network training. Within the context of low-precision fixed-point computations, we observe the rounding scheme to play a cruc…
Characterizes extreme points in polygon limit sets.
problem Identifying boundary points in polygon limit sets.
method Characterization through affine dilations and polygon vertices.
result Characterizes which points lie on the boundary of convex hull.
The paper studies Lagrangian structures in Higgs bundle moduli spaces and their conformal limits.
problem Understanding Lagrangian structures in Hitchin moduli space.
method Analyzing semistable and polystable Higgs bundles, focusing on the intersection with Lagrangian sublocus.
result The conformal limit of stable Higgs bundles on a specific Lagrangian sublocus exists under certain conditions.
Study infinite-depth limits of neural networks with fixed width.
problem Understanding the behavior of neural networks as depth increases with fixed width.
method Analyzing finite-width residual networks with random Gaussian weights, focusing on the infinite-depth limit.
result The pre-activations converge to a zero-drift diffusion process, differing from the infinite-width limit.
The paper corrects and improves previous assertions in digital topology.
problem Incorrect or insufficiently proven assertions in digital topology.
method Review and correction of existing assertions.
result Improved and corrected assertions in digital topology.
The paper analyzes fixed step-size SA schemes on Riemannian manifolds.
problem Developing efficient algorithms for optimization on curved spaces.
method Fixed step-size stochastic approximation schemes in a Riemannian framework.
result The schemes converge to the solution as the step-size approaches zero.
Study of fixed points in large networks with random dependencies.
problem Systemic risk in large financial networks.
method Analysis of vector fixed point equations on random graphs, obtaining finite dimensional limits.
result Approximate solutions to random FP equations for large networks.
Classifies limits of groups of involutions in SL(2,F) over local fields.
problem Classifying limits of groups of involutions in SL(2,F) over local fields.
method Classifying involutions, proving polar decompositions, classifying limits.
result Classification of Chabauty limits of various groups of involutions.
We study super--replication of contingent claims in markets with fixed transaction costs. This can be viewed as a stochastic impulse control problem with a terminal state constraint. The first result in this paper reveals that in reasonable continuous time financial market models the super--replication price is prohibi…
The paper corrects and improves previous assertions in digital topology.
problem Incorrect or poorly proven assertions in digital topology.
method Review and correction of existing assertions.
result Improved and corrected assertions in digital topology.
We show that for a strongly convergent sequence of purely loxodromic finitely generated Kleinian groups with incompressible ends, Cannon-Thurston maps, viewed as maps from a fixed base limit set to the Riemann sphere, converge uniformly. For algebraically convergent sequences we show that there exist examples where eve…
Study transverse measures on infinite type hyperbolic surfaces.
problem Characterize the cone of transverse measures on infinite type hyperbolic surfaces.
method Use inverse limits and geodesic laminations to describe and construct cones of transverse measures.
result Explicit descriptions and bases of cones of transverse measures exist for many laminations.
The softmax content-based attention mechanism has proven to be very beneficial in many applications of recurrent neural networks. Nevertheless it suffers from two major computational limitations. First, its computations for an attention lookup scale linearly in the size of the attended sequence. Second, it does not enc…
We will study the blowup behavior of a surface sequence immersed in R2 with bounded Willmore functional and fixed genus.
Reconstructing polytopes with fixed facet directions from support function evaluations.
problem Reconstructing polytopes with known facet directions from limited data.
method Least-squares estimate via convex quadratic program, combinatorial characterization for uniqueness, algorithm convergence.
result The least-squares estimate for a fixed simplicial normal fan is a convex quadratic program, and the solution is unique under certain conditions.
Study Gromov-Hausdorff limits of K3 surface metrics via moduli compactification.
problem Understanding limits of K3 surface metrics.
method Moduli-theoretic framework for collapsing Ricci-flat Kahler metrics.
result Gromov-Hausdorff limits of hyperKahler metrics with fixed diameters.
The study limits the number of specific foliations with bounded geometry.
problem Bounding the number of isoparametric foliations with bounded geometry.
method Proving finitely many foliations with specific properties and constructing infinite families of non-diffeomorphic foliations.
result There are only finitely many isoparametrically foliated closed connected Riemannian manifolds with bounded geometry, up to foliated diffeomorphism.
This note refers to our previous paper "The emergence of torsion in the continuum limit of distributed edge-dislocations". It identifies and fixes an error in the notion of convergence of Weitzenböck manifolds defined in the paper, and in the proof of the well-definiteness of this notion of convergence.
Study on the geometric Dyson Brownian motion of non-square matrix products.
problem Understanding the spectrum of a product of non-square random matrices.
method Proportional depth-width limit followed by mean-field limit, solving Burgers equation.
result Free log-normal law is obtained in the identity-start case.
The study analyzes the Hessian spectrum of DNNs during training using the NTK.
problem Understanding the Hessian spectrum of DNNs during training.
method Analysis using the Neural Tangent Kernel (NTK) in both fixed and mean-field limits.
result Characterization of the asymptotics of the spectrum of the Hessian at initialization and during training.
Examines challenges and proposes new approaches in machine learning theory.
problem Challenges in machine learning as a function approximation and optimization.
method Mathematical analysis of gradient descent, fixed network limitations, and RNNs.
result New insights and mathematical approaches to improve machine learning.
New findings on complexity limits in fixed budget bandit identification.
problem Determining the best possible error rate for fixed budget bandit identification.
method Analyzing the best non-adaptive sampling procedures and showing the existence of complexities.
result No fixed complexity for certain bandit identification tasks.
Continuous MDS embeds sequences of dissimilarities in Euclidean space.
problem Embedding sequences of dissimilarities as n increases. method Continuous MDS reformulates MDS for sequences of dissimilarity matrices.
result Uniform convergence of interpolated embeddings.
Study a market with uncertain informed traders, finding price impact depends on both asset value and informed trader count distribution.
problem Uncertain participation of informed traders in a market with limit orders.
method Characterized equilibrium by a fixed point integral equation, analyzed large order asymptotics, solved numerically.
result Equilibrium price impact depends on both asset value and distribution of informed traders, not just expected number of informed traders.
Let L be an ample holomorphic line bundle over a compact complex Hermitian manifold X. Any fixed smooth Hermitian metric on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k:th tensor power of L. In this paper various convergence results are obtained for the correspond…
Study smooth linear statistics on random covers of hyperbolic surfaces, showing central limit and variance results.
problem Analyzing fluctuations and energy variance of random covers of compact hyperbolic surfaces.
method Examining fluctuations in a small energy window around a fixed energy level, considering the variance of a typical surface, using a double limit where n and L go to infinity. result Distribution of fluctuations tends to a Gaussian with variance of GOE/GUE, and energy variance of a typical random n-cover is that of GOE/GUE. Compactifies metrics on K3 surfaces with algebraic description.
problem Classify Gromov-Hausdorff limits of K3 surfaces with fixed structures or polarizations.
method Algebraic description of Gromov-Hausdorff compactification.
result Classification of Gromov-Hausdorff limits of K3 surfaces.
Machine learning finds a compact fixed point action for SU(3) gauge theory.
problem Finding accurate and compact parametrizations of fixed point actions for SU(3) gauge theory.
method Used machine learning, specifically a gauge equivariant convolutional neural network.
result Obtained a superior parametrization of a fixed point action for SU(3) gauge theory.
This work studies Gaussian geometry under entropy-regularized 2-Wasserstein distance.
problem Understanding Gaussian distributions in uncertainty quantification and diffusivity.
method Entropy-regularized 2-Wasserstein distance, closed-form solutions, fixed-point characterization.
result Closed-form expressions for the 2-Sinkhorn divergence and fixed-point barycenter.
Stochastic gradient descent converges to universal limits in high dimensions.
problem Statistical tasks in high dimensions with specific data projections.
method Stochastic gradient descent applied to mixture distributions, proving universality of limits.
result The ODE limits are universal for mixtures of arbitrary product distributions.
New methods for federated learning reduce communication costs.
problem Efficiently solving optimization problems in a distributed setting.
method Developed two strategies for achieving consensus in federated learning: fixed number of local steps and randomized computations.
result Convergence analysis and experiments show benefits of the proposed methods.
The paper develops bootstrap methods for ACD models with random durations.
problem Bootstrap inference for autoregressive duration models with random durations.
method Recursive schemes for fixed calendar span or realized event count.
result The bootstrap method reproduces the conditional Gaussian component for ACD models with 0<κ<1. We prove that convex hypersurfaces in Rn+1 contracting under the flow by any power α>n+21 of the Gauss curvature converge (after rescaling to fixed volume) to a limit which is a smooth, uniformly convex self-similar contracting solution of the flow. Under additional central symmetry of the ini…
In this paper, asymptotic results in a long-term growth rate portfolio optimization model under both fixed and proportional transaction costs are obtained. More precisely, the convergence of the model when the fixed costs tend to zero is investigated. A suitable limit model with purely proportional costs is introduced …
In this paper, we will study the limiting behavior of the Brown-York mass of the coordinate spheres in an asymptotically flat manifold. Limiting behaviors of volumes of regions related to coordinate spheres are also obtained, including a discussion on the isoperimetric mass introduced by Huisken \cite{Huisken}. We will…
Study on fixed points of random permutations with surface group constraints.
problem Understanding fixed points of random permutations with surface group constraints.
method Computing expected number of fixed points using word maps and surface group constraints.
result The expected number of fixed points is bounded by O(1/dimχ) for shortest representatives. Polynomial-time algorithm for optimal stopping with fixed accuracy.
problem High-dimensional path-dependent optimal stopping problems.
method Efficient simulator of underlying information process, polynomial-time algorithm based on novel expansion.
result Polynomial-time solution for epsilon-optimal stopping policies and values.
The paper considers a general semi-Markov model for Limit Order Books with two states, which incorporates price changes that are not fixed to one tick. Furthermore, we introduce an even more general case of the semi-Markov model for LimitOrder Books that incorporates an arbitrary number of states for the price changes.…
Scaling limits for super-replication costs in models with transient price impact.
problem Analyzing the cost of options in models with transient price impact.
method Proving a scaling limit theorem using a Cox--Ross--Rubinstein binomial model.
result The scaling limit coincides with PDE methods for purely temporary price impact models, expanding to path-dependent options.
We study the a.s. convergence of a sequence of random embeddings of a fixed manifold into Euclidean spaces of increasing dimensions. We show that the limit is deterministic. As a consequence, we show that many intrinsic functionals of the embedded manifolds also converge to deterministic limits. Particularly interestin…