Related papers: Gradient flows of $(K,N)$-convex functions with ne…

This paper studies the convexity properties of nonsmooth extended-real-valued weakly convex functions, a class of functions that is central to modern optimization and its applications. We establish new characterizations of convexity using…

We generalize the theory of gradient flows of semi-convex functions on CAT(0)-spaces, developed by Mayer and Ambrosio--Gigli--Savar\'e, to CAT(1)-spaces. The key tool is the so-called "commutativity" representing a Riemannian nature of the…

We consider the gradient flow of a one-homogeneous functional, whose dual involves the derivative of a constrained scalar function. We show in this case that the gradient flow is related to a weak, generalized formulation of the Hele-Shaw…

We study contractivity properties of gradient flows for functions on normed spaces or, more generally, on Finsler manifolds. Contractivity of the flows turns out to be equivalent to a new notion of convexity for the functions. This is…

Characterizations of all continuous, additive and $\mathrm{GL}(n)$-equivariant endomorphisms of the space of convex functions on a Euclidean space $\mathbb{R}^n$, of the subspace of convex functions that are finite in a neighborhood of the…

Let $V$ be a finite nonempty set. A transit function is a map $R:V\times V\rightarrow 2^V$ such that $R(u,u)=\{u\}$, $R(u,v)=R(v,u)$ and $u\in R(u,v)$ hold for every $u,v\in V$. A set $K\subseteq V$ is $R$-convex if $R(u,v)\subset K$ for…

We derive the system of differential equations for the gradient flow characterizing the training process of linear in-context learning in full generality. Next, we explore the geometric structure of the gradient flows in two instances,…

Motivated by a constrained minimization problem, it is studied the gradient flows with respect to Hessian Riemannian metrics induced by convex functions of Legendre type. The first result characterizes Hessian Riemannian structures on…

In this paper we study kinematic expansive flows on compact metric spaces, surfaces and general manifolds. Different variations of the definition are considered and its relationship with expansiveness in the sense of Bowen-Walters and…

In this paper we establish a rigorous gradient flow structure for one-dimensional Kimura equations with respect to some Wasserstein-Shahshahani optimal transport geometry. This is achieved by first conditioning the underlying stochastic…

We consider the evolution of sets by nonlocal mean curvature and we discuss the preservation along the flow of two geometric properties, which are the mean convexity and the outward minimality. The main tools in our analysis are the level…

In this paper, we study the asymptotic behavior of continuous- and discrete-time gradient flows of a ``lower-unbounded" convex function $f$ on a Hadamard manifold $M$, particularly, their convergence properties to the boundary $M^{\infty}$…

We study non-convex subgradient flows for training two-layer ReLU neural networks from a convex geometry and duality perspective. We characterize the implicit bias of unregularized non-convex gradient flow as convex regularization of an…

Eigenvalues inequalities involving (log) convex/concav functions and Hermitian matrices, positive unital maps are considered. Simple proofs of Bhatia-Kittaneh inequality and Naimark dilation theorem are given.

The main goal of this paper is to present results of existence and non-existence of convex functions on Riemannian manifolds and, in the case of the existence, we associate such functions to the geometry of the manifold. Precisely, we prove…

In this paper we consider the non local evolution equation $$ \frac{\partial u(x,t)}{\partial t} + u(x,t)= \int_{\mathbb{R}^{N}}J(x-y)f(u(y,t))\rho(y)dy+ h(x). %\,\,\, h \geq 0. $$ We show that this equation defines a continuous flow in…

We develop a geometric flow framework to investigate two classical shape functionals: the torsional rigidity and the first Dirichlet eigenvalue of the Laplacian. First, by constructing novel deformation paths governed by height-stretching…

Equipped with the L^2-distortion distance, the space "X" of all metric measure spaces (X,d,m) is proven to have nonnegative curvature in the sense of Alexandrov. Geodesics and tangent spaces are characterized in detail. Moreover, classes of…

We are interested in existence of gradient flows for shape functionals especially for first Laplacian eigenvalues. We introduce different techniques to prove existence and use different formulations for gradient flows. We apply a…

In our previous paper on this topic, we introduced the notion of k-Hessian measure associated with a continuous k-convex function in a domain \Om in Euclidean n-space, k=1,...,n, and proved a weak continuity result with respect to local…