arXiv research

Optimal reinsurance contracts designed for a continuum of risk types.

Generalizes risk sharing models to a continuum of agents.

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

Flocking refers to collective behavior of a large number of interacting entities, where the interactions between discrete individuals produce collective motion on the large scale. We employ an agent-based model to describe the microscopic dynamics of each individual in a flock, and use a fractional PDE to model the evo…

We formulate a stochastic game of mean field type where the agents solve optimal stopping problems and interact through the proportion of players that have already stopped. Working with a continuum of agents, typical equilibria become functions of the common noise that all agents are exposed to, whereas idiosyncratic r…

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…

Study optimal investment in large populations of competitive, heterogeneous agents.

Optimal risk sharing without convex preferences using aggregate convexity.

In this paper we study the continuum time dynamics of a stock in a market where agents behavior is modeled by a Minority Game and a Grand Canonical Minority Game. The dynamics derived is a generalized geometric Brownian motion; from the Black & Scholes formula the calibration of both the Minority Game and the Grand Can…

Smooth knots can be embedded into a specific Menger continuum.

Graphon game model simplifies stochastic interactions among agents.

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.

Study optimal portfolios for many players in a market model with random coefficients.

Expands MFGs to handle real-world asymmetric multi-agent games efficiently.

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.

The paper analyzes optimal investment strategies in a game with jump risk, deriving mean field equilibria.

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…

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

Gradient flows on graphons converge to curves on graphon space.

In this paper the problem of optimal derivative design, profit maximization and risk minimization under adverse selection when multiple agencies compete for the business of a continuum of heterogenous agents is studied. The presence of ties in the agents' best-response correspondences yields discontinuous payoff functi…

Paper studies optimal tracking portfolio in mean field game of large fund competition.

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…

Continuum transformers learn operators in context via gradient descent.

Study many-player investment-consumption games with power FPPs, finding market-risk preference affects consumption.

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…

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 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.

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

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…