arXiv research

Study calibrates high-dimensional binary classifiers using angle between estimator and true weights.

Investigates portfolio optimization with and without gearing constraints.

In a recently published paper [1], it is shown that deep neural networks (DNNs) with random Gaussian weights preserve the metric structure of the data, with the property that the distance shrinks more when the angle between the two data points is smaller. We agree that the random projection setup considered in [1] pres…

Generalizes results on Bieri-Neumann-Strebel-Renz invariants and tropical varieties.

We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperbolic polyhedra with …

A novel method optimizes variable-stiffness structures for better strength and weight.

Piecewise flat approximations for curvature in Euclidean and non-Euclidean spaces.

Counting HCMU sphere components using weighted trees.

The study connects minimal and maximal surfaces in 3D and 3-L space.

Geometric approach clusters intersecting manifolds with high probability.

Establishes convexity and coercivity of K-energy functional for complex tori.

In this paper we study the right-angled Coxeter groups that acts geometrically on the Salvetti complex of a certain right-angled Artin group, which we refer to as Croke-Kleiner spaces. We prove that any right-angled Coxeter group that acts geometrically on the Croke-Kleiner spaces acts with $π/2$ angles between reflect…

We investigate the relationship between stability and the existence of extremal Kähler metrics on certain toric surfaces. In particular, we consider how log stability depends on weights for toric surfaces whose moment polytope is a quadrilateral. We introduce a space of symplectic potentials for toric manifolds, which …

Surveying connections between graph combinatorics and algebraic right-angled Artin groups.

The study examines discrete curvature notions on Cayley graphs of certain groups.

The paper explores connections between perimeter, area, and visual angle of convex sets.

We give a new notion of angle in general metric spaces; more precisely, given a triple a points $p,x,q$ in a metric space $(X,d)$, we introduce the notion of angle cone ${\angle_{pxq}}$ as being an interval ${\angle_{pxq}}:=[\angle^-_{pxq},\angle^+_{pxq}]$, where the quantities $\angle^\pm_{pxq}$ are defined in terms o…

This paper explores historical and philosophical aspects of angles and solid angles, inspired by Euler's work.

We obtain the following version of Lidskii theorem. Let L, M, N be p-dimensional subspaces in R^n. Let ψ_j be the angles between L and M, let φ_j be the angles between M and N, and let θ_j be the angles between L and N. Consider the orbit of the vector ψwith respect to permutations of coordinates and inversions of axis…

Associated to any finite flag complex L there is a right-angled Coxeter group W_L and a contractible cubical complex Sigma_L (the Davis complex) on which W_L acts properly and cocompactly, and such that the link of each vertex is L. It follows that if L is a generalized homology sphere, then Sigma_L is a contractible h…

We first define a complex angle between two oriented spacelike planes in 4-dimensional Minkowski space, and then study the constant angle surfaces in that space, i.e. the oriented spacelike surfaces whose tangent planes form a constant complex angle with respect to a fixed spacelike plane. This notion is the natural Lo…

Smooth approximations bound dihedral angles of convex polytopes.

On a compact Riemannian manifold with boundary, the absolute and relative cohomology groups appear as certain subspaces of harmonic forms. DeTurck and Gluck showed that these concrete realizations of the cohomology groups decompose into orthogonal subspaces corresponding to cohomology coming from the interior and bound…

For a hyperbolic surface S of finite type we consider the set A(S) of angles between closed geodesics on S. Our main result is that there are only finitely many rational multiples of πin A(S).

The paper studies prescribed angle surfaces in Riemannian manifolds with torse-forming vector fields.

We survey the role of right-angled Artin groups in the theory of diffeomorphism groups of low dimensional manifolds. We first describe some of the subgroup structure of right-angled Artin groups. We then discuss the interplay between algebraic structure, compactness, and regularity for group actions on one--dimensional…

The goal of the paper is to study the angle between two curves in the framework of metric (and metric measure) spaces. More precisely, we give a new notion of angle between two curves in a metric space. Such a notion has a natural interplay with optimal transportation and is particularly well suited for metric measure …

We establish the short-time existence of the Ricci flow on surfaces with a finite number of conic points, all with cone angle between 0 and $2π$, where the cone angles remain fixed or change in some smooth prescribed way. For the angle-preserving flow we prove long-time existence and convergence. When the Troyanov angl…

We provide geometric conditions on a pair of hyperplanes of a CAT(0) cube complex that imply divergence bounds for the cube complex. As an application, we classify all right-angled Coxeter groups with quadratic divergence and show right-angled Coxeter groups cannot exhibit a divergence function between quadratic and cu…

The study introduces hyperbolic angles in Lorentzian spaces and characterizes curvature bounds.

The paper defines and proves the existence of curves in Riemannian manifolds with prescribed angles to torse-forming vector fields.

Dynamic Mode Decomposition (DMD) yields a linear, approximate model of a system's dynamics that is built from data. We seek to reduce the order of this model by identifying a reduced set of modes that best fit the output. We adopt a model selection algorithm from statistics and machine learning known as Least Angle Reg…

We discuss the existence of the angle between two curves in Teichmüller spaces and show that, in any infinite dimensional Teichmüller space, there exist infinitely many geodesic triangles each of which has the same three vertices and satisfies the property that its three sides have the same and arbitrarily given length…

Union of Subspaces (UoS) is a popular model to describe the underlying low-dimensional structure of data. The fine details of UoS structure can be described in terms of canonical angles (also known as principal angles) between subspaces, which is a well-known characterization for relative subspace positions. In this pa…

The paper examines how the angle between inputs in ReLU networks decreases with depth, impacting training.

For a convex domain $D$ that is enclosed by the hypersurface $\partial D$ of bounded normal curvature, we prove an angle comparison theorem for angles between $\partial D$ and geodesic rays starting from some fixed point in $D$, and the corresponding angles for hypersurfaces of constant normal curvature. Also, we obtai…

We prove the existence of a minimal diffeomorphism isotopic to the identity between two hyperbolic cone surfaces $(Σ,g_1)$ and $(Σ,g_2)$ when the cone angles of $g_1$ and $g_2$ are different and smaller than $π$. When the cone angles of $g_1$ are strictly smaller than the ones of $g_2$, this minimal diffeomorphism is u…

We develop an analogy between right-angled Artin groups and mapping class groups through the geometry of their actions on the extension graph and the curve graph respectively. The central result in this paper is the fact that each right-angled Artin group acts acylindrically on its extension graph. From this result we …

Subspace models play an important role in a wide range of signal processing tasks, and this paper explores how the pairwise geometry of subspaces influences the probability of misclassification. When the mismatch between the signal and the model is vanishingly small, the probability of misclassification is determined b…

Generalizes warped product submersion to conformal case.

Quantifying similarity between data objects is an important part of modern data science. Deciding what similarity measure to use is very application dependent. In this paper, we combine insights from systems theory and machine learning, and investigate the weighted cepstral distance, which was previously defined for si…

In this work, we study robust deep learning against abnormal training data from the perspective of example weighting built in empirical loss functions, i.e., gradient magnitude with respect to logits, an angle that is not thoroughly studied so far. Consequently, we have two key findings: (1) Mean Absolute Error (MAE) D…

LCW reduces activation shift in neural networks, improving training efficiency and generalization.

Enhances DNN robustness and accuracy with $L_{2,\infty}$ normalization.

We establish a connection between capillary floating in neutral equilibrium and the billiard ball problem. This allows us to reduce the question of floating in neutral equilibrium at any orientation with a prescribed contact angle for infinite homogeneous cylinders to a question about billiard caustics for their orthog…

We consider here a generalization of a well known discrete dynamical system produced by the bisection of reflection angles that are constructed recursively between two lines in the Euclidean plane. It is shown that similar properties of such systems are observed when the plane is replaced by a regular surface in ${\mat…

Constructs minimal capillary cones with specific symmetry and proves their existence and uniqueness.

Study of curves and surfaces in Riemannian spaces making a constant angle with a parallel transported direction.