arXiv research

Many statistical learning problems can be posed as minimization of a sum of two convex functions, one typically a composition of non-smooth and linear functions. Examples include regression under structured sparsity assumptions. Popular algorithms for solving such problems, e.g., ADMM, often involve non-trivial optimiz…

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.

This work proves the continuum limit of t-SNE for data visualization.

New scheme optimizes BMI through probabilistic and geometric shaping.

Study shows convergence rates for Cheeger cuts on data clouds.

Smooth knots can be embedded into a specific Menger continuum.

We prove the following result announced in Todorov and Valov: Any homogeneous, metric $ANR$-continuum is a $V^n_G$-continuum provided $\dim_GX=n\geq 1$ and $\check{H}^n(X;G)\neq 0$, where $G$ is a principal ideal domain. This implies that any homogeneous $n$-dimensional metric $ANR$-continuum with $\check{H}^n(X;G)\neq…

We introduce the continuum self-similar tree (CSST) and characterize it topologically. We apply this to answer a question of Curien about the topology of the continuum random tree (CRT). We also give a topological characterization of other trees with branch points of finite or infinite valences.

Continuum Dropout improves neural differential equations by preventing overfitting.

Dimension reduction of multivariate data supervised by auxiliary information is considered. A series of basis for dimension reduction is obtained as minimizers of a novel criterion. The proposed method is akin to continuum regression, and the resulting basis is called continuum directions. With a presence of binary sup…

Derives continuum model from discrete $\varepsilon$-graphs with connectivity functional.

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 $V^n$-continua to cohomological dimensions.

Proves continuum limits of Lipschitz learning using Γ-convergence.

This paper studies continuum-armed bandits under Besov smoothness conditions and derives minimax rates.

A mesh-free method solves continuum-marginal optimal transport problems.

Optimal reinsurance contracts designed for a continuum of risk types.

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.

Continuum transformers learn operators in context via gradient descent.

Given a trivalent graph in the 3-dimensional Euclidean space, we call it a discrete surface because it has a tangent space at each vertex determined by its neighbor vertices. To abstract a continuum object hidden in the discrete surface, we introduce a subdivision method by applying the Goldberg-Coxeter subdivision and…

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.

Continuum-wise hyperbolicity is exactly the pseudo-Anosov dynamics with spine singularities.

This paper establishes the consistency of spectral approaches to data clustering. We consider clustering of point clouds obtained as samples of a ground-truth measure. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. We investigate the…

A crucial challenge in image-based modeling of biomedical data is to identify trends and features that separate normality and pathology. In many cases, the morphology of the imaged object exhibits continuous change as it deviates from normality, and thus a generative model can be trained to model this morphological con…

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.

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…

We analyze convergence of Fermat distances and their application in clustering.

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…

New finite element method for complex forms in any dimension.

Framework learns physics-informed continuum models from molecular data.

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 propose a stochastic process driven by memory effect with novel distributions including both exponential and leptokurtic heavy-tailed distributions. A class of distribution is analytically derived from the continuum limit of the discrete binary process with the renormalized auto-correlation and the closed form momen…

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 $AC(X, x)$…

We overview some results on distributed learning with focus on a family of recently proposed algorithms known as non-Bayesian social learning. We consider different approaches to the distributed learning problem and its algorithmic solutions for the case of finitely many hypotheses. The original centralized problem is …

We present a rigorous homogenization theorem for distributed dislocations. We construct a sequence of locally-flat Riemannian manifolds with dislocation-type singularities. We show that this sequence converges, as the dislocations become denser, to a flat non-singular Weitzenböck manifold, i.e. a flat manifold endowed …

Paper presents a new approach to continuum mechanics using port-Hamiltonian framework.

Study of convergence of point-object configurations to a charged dust continuum.

We propose a stochastic process driven by the memory effect with novel distributions which include both exponential and leptokurtic heavy-tailed distributions. A class of the distributions is analytically derived from the continuum limit of the discrete binary process with the renormalized auto-correlation. The moment …

We develop a theory to represent dislocated single crystals at the mesoscopic scale by considering concentrated effects, governed by the distribution theory combined with multiple-valued kinematic fields. Our approach gives a new understanding of the continuum theory of defects as developed by Kroener (1980) and other …

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…

We present a homogenization theorem for isotropically-distributed point defects, by considering a sequence of manifolds with increasingly dense point defects. The loci of the defects are chosen randomly according to a weighted Poisson point process, making it a continuous version of the first passage percolation model.…

Derives Black-Scholes model without stochastic calculus or PDEs.

Optimal strategy proposed for maximizing cumulative reward in continuum-armed bandits.

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…

We consider a distributed learning setup where a network of agents sequentially access realizations of a set of random variables with unknown distributions. The network objective is to find a parametrized distribution that best describes their joint observations in the sense of the Kullback-Leibler divergence. Apart fr…

Brin-Thompson groups have new properties for n>=2.

Generalizes risk sharing models to a continuum of agents.