A novel method for supervised dimension reduction using continuum directions.
problem Dimension reduction of multivariate data with auxiliary information.
method Minimizes a novel criterion to obtain continuum directions, bridging unsupervised to fully supervised methods.
result Sample continuum directions are inconsistent but have good classification performance.
The paper connects fluid mechanics, elasticity, and geometry to study wrinkled solutions.
problem Existence of wrinkled solutions in nonlinear partial differential equations.
method Develops connections between fluids, elasticity, and isometric embeddings, mapping mechanical equations into geometric frameworks.
result Geometric theory provides a method for addressing admissibility criteria in nonlinear conservation laws.
Continuum mechanics theory describes skin's complex anisotropic behavior.
problem Modeling the anisotropic tearing of skin.
method Finsler geometry fiber bundle approach, variational method, phase-field mechanics.
result Analytical solutions capture experimental data on skin tearing.
Introduces CSST and characterizes its topology.
problem Characterize the topology of the continuum random tree.
method Introduce continuum self-similar tree (CSST) and apply it.
result Characterizes the topology of CSST and other trees.
Smooth knots can be embedded into a specific Menger continuum.
problem Embedding smooth knots into a specific type of continuum.
method Explicit construction using cubical models and self-similarity of the Menger continuum.
result Every smooth knot can be isotoped into the Menger continuum.
We prove the following result announced in Todorov and Valov: Any homogeneous, metric ANR-continuum is a VGn-continuum provided dimGX=n≥1 and Hˇn(X;G)=0, where G is a principal ideal domain. This implies that any homogeneous n-dimensional metric ANR-continuum with $\check{H}^n(X;G)\neq…
New method converts video of dye plumes into PDEs for better understanding.
problem Inferring continuum models from uncalibrated video data.
method Develops a pipeline to convert grayscale recordings into scalar fields, isolates drift, and identifies transport laws.
result Selected reduced model outperforms advection-diffusion baselines and retains structural interpretability.
Generative model for morphological continuum of normal and pathological states.
problem Identifying trends and features that separate normality and pathology in biomedical images.
method Wasserstein Auto-encoder with HSIC regularization for latent features.
result Model generates a continuum of morphological changes corresponding to side information.
We describe the combinatorial stochastic process underlying a sequence of conditionally independent Bernoulli processes with a shared beta process hazard measure. As shown by Thibaux and Jordan [TJ07], in the special case when the underlying beta process has a constant concentration function and a finite and nonatomic …
Continuum Dropout improves neural differential equations by preventing overfitting.
problem Overfitting in Neural Differential Equations (NDEs).
method Introduces Continuum Dropout, a regularization technique based on alternating renewal processes.
result Continuum Dropout outperforms existing methods in various tasks, improving generalization and uncertainty quantification.
Derives continuum model from discrete ε-graphs with connectivity functional.
problem Modeling diffusion in networks with varying connectivity.
method Energy-based continuum limit derivation, neural-network reconstruction of connectivity.
result Error between discrete and continuum energies is O(ε), valid even with fluctuations. Overview of manifolds of mappings for continuum mechanics.
problem Understanding smooth mappings between manifolds.
method Presentation of manifolds of mappings and their properties.
result Smooth convenient manifold C∞(M,N) of mappings between manifolds. Optimizes CNNs by directing gradients along output channels.
problem Improving generalization error in CNNs.
method Output-channel directed re-weighted L2 or Sobolev metrics.
result Improves generalization error by optimizing gradients.
Study transforms discrete graph surfaces into smooth continua through iterative subdivision.
problem Abstracting a smooth continuum from a discrete graph surface.
method Iterative Goldberg-Coxeter subdivision method to converge discrete surfaces into a continuum.
result The limit set forms a continuum geometric object from the discrete surface.
An important question that discrete approaches to quantum gravity must address is how continuum features of spacetime can be recovered from the discrete substructure. Here, we examine this question within the causal set approach to quantum gravity, where the substructure replacing the spacetime continuum is a locally f…
Generalizes Alexandroff's Vn-continua to cohomological dimensions.
problem Extending Alexandroff's concept of Vn-continua to cohomological dimensions. method Proves that strongly locally homogeneous generalized continua with cohomological dimension n are generalized Vn-spaces. result Every strongly locally homogeneous continuum of covering dimension n is a Vn-continuum in the sense of Alexandroff. Proves continuum limits of Lipschitz learning using Γ-convergence.
problem Semi-supervised learning with graph-based methods and continuum limits of p-Laplacian learning. method Proves continuum limits of Lipschitz learning using Γ-convergence.
result Proves Γ-convergence in the L∞-topology to the supremum norm of the gradient. This paper studies continuum-armed bandits under Besov smoothness conditions and derives minimax rates.
problem Optimizing an unknown function with limited evaluations.
method Studies continuum-armed bandits under Besov smoothness conditions and derives minimax rates.
result Minimax rates over Besov spaces are identical to those over the smallest Hölder space into which Besov spaces embed.
The paper proves spectral clustering consistency for point cloud data.
problem Consistency of spectral clustering for point cloud data.
method Variational convergence approach to graph Laplacians.
result Sharp conditions for spectral convergence with respect to sample size.
A mesh-free method solves continuum-marginal optimal transport problems.
problem Recovering minimum-energy velocity fields from time-continuous probability marginals.
method Embeds weak continuity equation in a reproducing kernel Hilbert space, optimizing with mini-batch stochastic methods.
result Accurately recovers drift and maintains marginal consistency in synthetic experiments.
Optimal reinsurance contracts designed for a continuum of risk types.
problem Designing optimal reinsurance contracts with a continuum of risk types.
method Principal-agent model, VaR at risk tolerance level, change of variables, univariate approach.
result Optimal reinsurance contracts are in stop-loss form, classifying agents into high and low risk groups.
This work proves the continuum limit of t-SNE for data visualization.
problem Understanding the theoretical basis of t-SNE from a continuum limit perspective.
method Proving the Kullback-Leibler divergence consistency as no∞ for t-SNE. result The continuum variational problem involving non-convex gradient regularization and penalty on probability density function magnitude.
We characterize those planar Peano continua that are homotopy equivalent to 1-dimensional sets. While many planar Peano continua are not homotopically 1-dimensional, we prove that each has fundamental group that embeds in the fundamental group of a 1-dimensional planar Peano continuum. We leave open the following quest…
Gradient flows on graphons converge to curves on graphon space.
problem Optimizing functions on large, exchangeable graphs.
method Euclidean gradient flow on edge weights converges to a curve on graphon space.
result Gradient flows on graphons can be described as curves of maximal slope on graphon space.
We analyze directed, unweighted graphs obtained from xi∈Rd by connecting vertex i to j iff ∣xi−xj∣<ε(xi). Examples of such graphs include k-nearest neighbor graphs, where ε(xi) varies from point to point, and, arguably, many real world graphs such as co-purchasing graphs. We ask whethe…
Continuum transformers learn operators in context via gradient descent.
problem Generalizing transformers to handle infinite-dimensional inputs for in-context learning.
method Gradient descent in an operator RKHS, leveraging generalized representer theorems and gradient flows.
result Operator learned in context is Bayes Optimal Predictor in infinite depth limit.
Continuum-wise hyperbolicity is exactly the pseudo-Anosov dynamics with spine singularities.
problem Classification of continuum-wise hyperbolic surface homeomorphisms
method Proving a complete structural classification
result Every cwF-hyperbolic homeomorphism is pseudo-Anosov with spine singularities We show how to associate an R-tree to the set of cut points of a continuum. If X is a continuum without cut points we show how to associate an R-tree to the set of cut pairs of X.
Using the topologist sine curve we present a new functorial construction of cone-like spaces, starting in the category of all path-connected topological spaces with a base point and continuous maps, and ending in the subcategory of all simply connected spaces. If one starts by a noncontractible n-dimensional Peano cont…
It has been known for a long time that the fundamental group of the quotient of $\RR ^3$ by the Case-Chamberlin continuum is nontrivial. In the present paper we prove that this group is in fact, uncountable.
We analyze convergence of Fermat distances and their application in clustering.
problem Understanding convergence properties of Fermat distances on Riemannian manifolds.
method Geometric and statistical arguments in percolation theory, leveraging novel arguments for non-uniform densities and curved domains.
result Discrete, sample-based Fermat distances converge to their continuum analogues with a precise rate dependent on intrinsic dimensionality.
Unified algorithms for structured sparsity problems with optimal convergence rates.
problem Solving minimization problems with structured sparsity assumptions.
method Unified continuum of preconditioned forward-backward operator splitting algorithms and accelerated algorithms.
result The continuum of algorithms attains the theoretically optimal rate of convergence.
We introduce a concept of tree-graded metric space and we use it to show quasi-isometry invariance of certain classes of relatively hyperbolic groups, to obtain a characterization of relatively hyperbolic groups in terms of their asymptotic cones, to find geometric properties of Cayley graphs of relatively hyperbolic g…
Framework learns physics-informed continuum models from molecular data.
problem Discovering accurate and robust data-driven continuum models from molecular simulation data.
method Operator regression framework using neural networks in modal space with physical inductive biases.
result Learned operators generalize to unseen system characteristics.
We propose a continuous time model for financial markets with proportional transactions costs and a continuum of risky assets. This is motivated by bond markets in which the continuum of assets corresponds to the continuum of possible maturities. Our framework is well adapted to the study of no-arbitrage properties and…
We construct a functor AC(−,−) from the category of path connected spaces X with a base point x to the category of simply connected spaces. The following are the main results of the paper: (i) If X is a Peano continuum then AC(X,x) is a cell-like Peano continuum; (ii) If X is n−dimensional then AC(X,x)…
Paper presents a new approach to continuum mechanics using port-Hamiltonian framework.
problem Geometric formulation of solid and fluid mechanics.
method Port-Hamiltonian framework, Dirac structures, Hamiltonian reduction theory.
result Systematic derivation of port-Hamiltonian models for solid and fluid mechanics.
Study of convergence of point-object configurations to a charged dust continuum.
problem Understanding the convergence of discretized point-object configurations to a charged dust continuum.
method Establishing existence and uniqueness of horizons/minimal surfaces, studying geometries of regions exterior to minimal surfaces, and discussing limits.
result Examples of scalar curvature jumps upon taking Gromov-Hausdorff and intrinsic flat limits.
Hamilton's Ricci flow (RF) equations were recently expressed in terms of the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry, for d greater than or equal to 2. The structure of the simplicial Ricci flow (SRF) equations are dimensionally agnostic. These SRF equations were tested numerically and…
Proposes a variational approach to shallow neural networks, bypassing optimization.
problem Theoretical understanding and optimization of shallow neural networks.
method Replaces discrete training with a continuum variational surrogate, proving global well-posedness and regularity.
result Optimal parameter density can be obtained by solving a single linear system, achieving O(1/N) generalization error. Material properties linked to Lie groupoids and algebroids in continuum mechanics.
problem Understanding material properties through groupoids and algebroids.
method Intuitive treatment of material groupoids and their algebroids.
result Material groupoids and algebroids are linked to material properties and their deformations.
Optimal strategy proposed for maximizing cumulative reward in continuum-armed bandits.
problem Maximizing cumulative reward in a scenario with limited resources and unknown stochastic rewards.
method Proposed an optimal strategy for a nonparametric setting with side information on actions.
result Optimal regret scales as \(O(T^{1/3})\) up to poly-logarithmic factors when \(T\) is proportional to \(N\).
Derives Black-Scholes model without stochastic calculus or PDEs.
problem Deriving the Black-Scholes model without advanced math.
method Continuum limit of Binomial tree approach.
result Derives Black-Scholes model and exchange-option generalization.
Abstract: Necessary condition for critical sets in 3D space.
problem Identifying critical sets in 3D space.
method Provided a necessary condition for closed subsets to be critical sets of smooth functions.
result Examples like the Whitehead continuum and p-adic solenoid are not critical sets.
We prove non-metricity in a continuum limit of randomly-distributed defects.
problem Emergence of non-metricity in continuum limit of point defects.
method Homogenization theorem applied to isotropically-distributed point defects modeled as a weighted Poisson point process.
result Non-metricity tensor emerges in the continuum limit of point defects.
We derive a continuum model from discrete elastic models on smooth manifolds.
problem Modeling stress-free configurations in geometrically-incompatible elastic systems.
method Variational convergence of discrete models to a continuum model.
result No stress-free configurations unless the manifold is flat.
We prove that every homomorphism from the fundamental group of a planar Peano continuum to the fundamental group of a planar or one-dimensional Peano continuum is induced by a continuous map up to conjugation. This is then used to provide a family of uncountable many planar Peano continua with pairwise non-isomorphic f…
Generalizes risk sharing models to a continuum of agents.
problem Risk sharing among a large number of heterogeneous agents.
method Modeling agents as points in a measure space, using risk measures on a probability space, and deriving dual representations.
result Explicit formulas for specific risk measures (entropic and expected shortfall) and applications to Pareto efficiency.