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To what extent are two images picturing the same 3D surfaces? Even when this is a known scene, the answer typically requires an expensive search across scale space, with matching and geometric verification of large sets of local features.…

In this paper, we use a unified framework introduced in [3] to study two classes of nonconforming immersed finite element (IFE) spaces with integral value degrees of freedom. The shape functions on interface elements are piecewise…

A graph drawing in the plane is called an almost embedding if images of any two non-adjacent simplices (i.e. vertices or edges) are disjoint. We introduce integer invariants of almost embeddings: winding number, cyclic and triodic Wu…

We study the geodesic convexity of various energy and entropy functionals restricted to (non-geodesically convex) submanifolds of Wasserstein spaces with their induced geometry. We prove a variety of convexity results by means of a simple…

The complexity of condensed matter arises from emergent behaviors that cannot be understood by analyzing individual constituents in isolation. While traditional condensed-matter approaches-developed primarily for ideal crystalline…

Geometric embedding methods have shown to be useful for multi-hop reasoning on knowledge graphs by mapping entities and logical operations to geometric regions and geometric transformations, respectively. Geometric embeddings provide direct…

We study dually epi-translation invariant valuations on cones of convex functions containing the space of finite-valued convex functions. The existence of a homogeneous decomposition is used to associate a distribution to every valuation of…

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…

This paper introduces a new shape-matching methodology, combinative matching, to combine interlocking parts for geometric shape assembly. Previous methods for geometric assembly typically rely on aligning parts by finding identical surfaces…

With a new proof approach we prove in a more general setting the classical convergence theorem that almost everywhere convergence of measurable functions on a finite measure space implies convergence in measure. Specifically, we generalize…

Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by means of exponential generating functions. The number of…

Often in mathematics it is useful to summarize a multivariate phenomenon with a single number and in fact, the determinant -- which is represented by det -- is one of the simplest cases. In fact, this number it is defined only for square…

One of the prime motivation for topology was Homotopy theory, which captures the general idea of a continuous transformation between two entities, which may be spaces or maps. In later decades, an algebraic formulation of topology was…

We consider the problem of multi-task learning in the high dimensional setting. In particular, we introduce an estimator and investigate its statistical and computational properties for the problem of multiple connected linear regressions…

Many differentiated products have key attributes that are unstructured and thus high-dimensional (e.g., design, text). Instead of treating unstructured attributes as unobservables in economic models, quantifying them can be important to…

The presence of second-order smoothness for objective functions of optimization problems can provide valuable information about their stability properties and help us design efficient numerical algorithms for solving these problems. Such…

It is a well-known and elementary fact that a holomorphic function on a compact complex manifold without boundary is necessarily constant. The purpose of the present article is to investigate whether, or to what extent, a similar property…

Model merging offers a scalable alternative to multi-task learning but often yields suboptimal performance on classification tasks. We attribute this degradation to a geometric misalignment between the merged encoder and static…

Model merging aims to combine multiple task-specific expert models into a single model while preserving generalization across diverse tasks. However, interference among experts, especially when they are trained on different objectives,…

The success of deep convolutional neural network (CNN) in computer vision especially image classification problems requests a new information theory for function of image, instead of image itself. In this article, after establishing a deep…