arXiv research

Study of focal-entropy for class-imbalanced classification.

Study focal surfaces of wave fronts with unbounded curvatures.

Self-focal points on ellipsoids of dimension 3 or higher are rare.

Study of cuspidal edges on focal surfaces of regular surfaces.

Study on transnormal functions and their level sets on Finsler manifolds.

The focal submanifolds of isoparametric hypersurfaces in spheres are all minimal Willmore submanifolds, mostly being $\mathcal{A}$-manifolds in the sense of A.Gray but rarely Ricci-parallel (\cite{QTY},\cite{LY},\cite{TY3}). In this paper we study the geometry of the focal submanifolds via Simons formula. We show that …

This paper connects billiards in ellipses to focal billiards in ellipsoids.

Study on focal surfaces and evolutes of framed curves in hyperbolic 3-space using Legendrian duality.

Focal loss improves classification but not class-posterior probability estimation.

We prove that a submanifold with parallel focal structure, which is a generalization of isoparametric and equifocal submanifolds, induces a singular Riemannian foliation of the ambient space by its parallel and focal manifolds.

Study on evolutes and focal surfaces of pseudo-spherical framed immersions in anti-de Sitter space.

Here, we focus on focal surfaces of a tubular surface in Euclidean 3-space E^3: Firstly, we give the tubular surfaces with respect to Frenet and Darboux frames. Then, we define focal surfaces of these tubular surfaces. We get some results for these types of surfaces to become flat and we show that there is no minimal f…

Study on focal surfaces of lightcone framed surfaces in Lorentz-Minkowski 3-space.

Study of families of lines on spheres and their focal sets.

The {\em focal curve} of an immersed smooth curve $γ:s\mapsto γ(s)$, in Euclidean space $\R^{m+1}$, consists of the centres of its osculating hyperspheres. The focal curve may be parametrised in terms of the Frenet frame of $γ$ (${\bf t},{\bf n}_1, ...,{\bf n}_m$), as $C_γ(s)=(γ+c_1{\bf n}_1+c_2{\bf n}_2+...+c_m{\bf n}…

We characterize singularities of focal surfaces of wave fronts in terms of differential geometric properties of the initial wave fronts. Moreover, we study relationships between geometric properties of focal surfaces and geometric invariants of the initial wave fronts.

In this note, we consider the rigidity of the focal decomposition of closed hyperbolic surfaces. We show that, generically, the focal decomposition of a closed hyperbolic surface does not allow for non-trivial topological deformations, without changing the hyperbolic structure of the surface. By classical rigidity theo…

The paper bounds bandwidth and focal radius for manifolds with positive isotropic curvature.

Given a closed Riemannian manifold (M, g), there is a partition Σ_i of its tangent bundle TM called the focal decomposition. The sets Σ_i are closely associated to focusing of geodesics of (M, g), i.e. to the situation where there are exactly i geodesic arcs of the same length joining points p and q in M. In this note,…

We study the geometry of curves in the Minkowski space and in the de Sitter space, specially at points where the tangent direction is lightlike (i.e. has length zero) called lightlike points of the curve. We define the focal sets of these curves and study the metric structure of them. At the lightlike points, the focal…

Study on focal locus of submanifolds in Finsler manifolds, showing regularity and smoothness.

We prove that the focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal to the symmetry direction containing the source, while the focal surface is translation invariant. This is done by const…

Focal loss reduces model curvature for better calibration.

We characterize symmetric spaces without focal points by the equality case of general equalities between geometric quantities.

Survey of Cartan's work on isoparametric hypersurfaces in spheres.

Study calculates indices and nullities of focal manifolds in spheres.

In this note we show that a compact asymptotically harmonic manifold without focal points is either flat or a rank one locally symmetric space.

Characterizes submanifolds with minimum ratio of diameter to focal radius.

Method for generating new curves from plane curves on cylinders.

Pairs (Hamiltonian system, Lagrangian distribution), called dynamical Lagrangian distributions, appear naturally in Differential Geometry, Calculus of Variations and Rational Mechanics. The basic differential invariants of a dynamical Lagrangian distribution w.r.t. the action of the group of symplectomorphisms of the a…

The focal locus $Σ_X$ of an affine variety $X$ is roughly speaking the (projective) closure of the set of points $O$ for which there is a smooth point $x \in X$ and a circle with centre $O$ passing through $x$ which osculates $X$ in $x$. Algebraic geometry interprets the focal locus as the branching locus of the endpoi…

Miscalibration - a mismatch between a model's confidence and its correctness - of Deep Neural Networks (DNNs) makes their predictions hard to rely on. Ideally, we want networks to be accurate, calibrated and confident. We show that, as opposed to the standard cross-entropy loss, focal loss [Lin et. al., 2017] allows us…

An equifocal submanifold M of a symmetric space N of compact type induces a foliation with singular leaves on N. In this paper we will show how to reconstruct the equifocal foliation starting from one of the singular leaves, the so-called focal manifolds. To be more concrete: The equifocal submanifold is equal to a par…

Enhanced loss function boosts fraud detection in auto insurance claims.

Focalized GP improves Bayesian optimization for large datasets.

The space ${\Bbb{L}}$ of oriented lines, or rays, in ${\Bbb{R}}^3$ is a 4-dimensional space with an abundance of natural geometric structure. In particular, it boasts a neutral Kähler metric which is closely related to the Euclidean metric on ${\Bbb{R}}^3$. In this paper we explore the relationship between the focal se…

FOCaL meta-learner estimates functional treatment effects robustly.

In this article, we study the ergodicity of the geodesic flows on surfaces with no focal points. Let $M$ be a smooth connected and closed surface equipped with a $C^\infty$ Riemannian metric $g$, whose genus $\mathfrak{g} \geq 2$. Suppose that $(M,g)$ has no focal points. We prove that the geodesic flow on the unit tan…

This paper proposes a Residual Convolutional Neural Network (ResNet) based on speech features and trained under Focal Loss to recognize emotion in speech. Speech features such as Spectrogram and Mel-frequency Cepstral Coefficients (MFCCs) have shown the ability to characterize emotion better than just plain text. Furth…

We study the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces. In particular, we provide a characterization of the focal set in terms of its second fundamental form and determine the principal curvatures of the homogeneous hypersurfaces together with their multiplicities.

Sharp focal radius estimate for hypersurfaces in manifolds with positive curvature.

A focal representation of a generic regular curve γ in E^{m+1} consists of the centers of the osculating hyperplanes. A k-slant helix γ in E^{m+1} is a (generic) regular curve whose unit normal vector V_{k} makes a constant angle with a fixed direction U in E^{m+1}. In the present paper we proved that if γ is a k-slant…

The paper addresses selection bias in conformal prediction for focal units.

An isoparametric hypersurface in unit spheres has two focal submanifolds. Condition A plays a crucial role in the classification theory of isoparametric hypersurfaces in [CCJ07], [Chi16] and [Miy13]. This paper determines $C_A$, the set of points with Condition A in focal submanifolds. It turns out that the points in $…

Study on Gehring link problem and width of bands in curved manifolds.

This is a continuation of Tang and Yan, which investigated the first eigenvalues of minimal isoparametric hypersurfaces with $g=4$ distinct principal curvatures and focal submanifolds in unit spheres. For the focal submanifolds with $g=6$, the present paper obtains estimates on all the eigenvalues, among others, giving…

In this paper we consider on a complete Riemannian manifold $M$ an immersed totally geodesic hypersurface $\Si$ existing together with an immersed submanifold $N$ without focal points. No curvature condition is needed. We obtained several connectedness results relating the topologies of $M$ and $\Si$ which depend on th…

The study counts geodesics on special manifolds without focusing points.