arXiv research

One of the most important problems in Finsler geometry is to classify Finsler metrics of scalar flag curvature. In this paper, we study the classification problem of Randers metrics of scalar flag curvature. Under the condition that $β$ is a Killing 1-form, we obtain some important necessary conditions for Randers metr…

Paper introduces new importance metrics for machine learning models, linking them to CATE.

This paper evaluates and improves metrics for identifying important features in machine learning models.

Defines a new metric to measure importance of predictors in complex machine learning models.

iLOCO measures feature interactions without assumptions, providing statistical inference.

In this paper, we study an important class of Finsler metrics--square metrics. We give two expressions of such metrics in terms of a Riemannian metric and a 1-form. We show that Einstein square metrics can be classified up to the classification of Einstein Riemannian metrics.

The current paper deals with some new classes of Finsler metrics with reversible geodesics. We construct weighted quasi-metrics associated with these metrics. Further, we investigate some important geometric properties of weighted quasi-metric space. Finally, we discuss the embedding of quasi-metric spaces with general…

Bayesian approach improves AdaLoRA's performance and efficiency.

Spherically symmetric metrics form a rich and important class of metrics. Many well-known Finsler metrics of constant flag curvature can be locally expressed as a spherically symmetric metric on R^n. In this paper, we study spherically symmetric metrics with constant Ricci curvature and constant flag curvature.

Square metrics is an important class of Finsler metrics. Recently, we introduced a special class of non-regular Finsler metrics called singular square metrics. The main purpose of this paper is to provide a necessary and sufficient condition for singular square metrics to be of constant Ricci or flag curvature when dim…

Interpretability is an important area of research for safe deployment of machine learning systems. One particular type of interpretability method attributes model decisions to input features. Despite active development, quantitative evaluation of feature attribution methods remains difficult due to the lack of ground t…

Proposes a new method for variable importance using targeted learning.

We define the notion of weak Minkowski metric and prove some basic properties of such metrics. We also highlight some of the important analogies between Minkowski geometry and the Funk and Hilbert geometries.

Douglas metrics are metrics with vanishing Douglas curvature which is an important projective invariant in Finsler geometry. To find more Douglas metrics, in this paper we consider a class of Finsler metrics called general $(α,β)$-metrics, which are defined by a Riemannian metric $α=\sqrt{a_{ij}(x)y^iy^j}$ and a $1$-fo…

The problem of explaining deep learning models, and model predictions generally, has attracted intensive interest recently. Many successful approaches forgo global approximations in order to provide more faithful local interpretations of the model's behavior. LIME develops multiple interpretable models, each approximat…

An important tool in the study of conformal geometry, and the AdS/CFT correspondence in physics, is the Fefferman-Graham expansion of conformally compact Einstein metrics. We show that conformally compact metrics satisfying a generalization of the Einstein equation, Poincare-Lovelock metrics, also have Fefferman-Graham…

The paper studies weighted Ricci curvatures and characterizes Randers metrics.

Let $Σ_g$ be a closed hyperbolic surface of genus $g$ and let $Ham(Σ_g)$ be the group of Hamiltonian diffeomorphisms of $Σ_g$. The most natural word metric on this group is the autonomous metric. It has many interesting properties, most important of which is the bi-invariance of this metric. In this work we show that $…

We survey recent results on the existence of Kähler-Einstein metrics on certain smoothable Fano varieties, focusing on the importance of such metrics in the construction of compact algebraic moduli spaces of K-polystable Fano varieties. Moreover, we give some applications and we discuss some natural problems which dese…

Two new methods assess feature importance for fairness in machine learning models.

In this paper, we use a Killing form on a Riemannian manifold to construct a class of Finsler metrics. We find equations that characterize Einstein metrics among this class. In particular, we construct a family of Einstein metrics on $S^3$ with ${\rm Ric} = 2 F^2$, ${\rm Ric}=0$ and ${\rm Ric}=- 2 F^2$, respectively. T…

Igarashi introduce the concept of $(α, β)$-metric in Cartan space $\ell^{n}$ analogously to one in Finsler space and obtained the basic important geometric properties and also investigate the special class of the space with $(α, β)$-metric in $\ell^{n}$ in terms of $'invariants'$. In the present paper we determine th…

According to [8] if the stationary Schroedinger equation on n-dim. Riemann space admits R-separation of variables (i.e. separation of variables with a factor R), then the underlying metric is necessarily isothermic. An important sub-class of isothermic metrics are the so called binary metrics. In this paper we study co…

A new method uses asymmetric Shapley values to assess gene importance in clinical prediction models.

Several authors have pointed out the connection between Barbilian's metric introduced in 1934 and the recent study of Apollonian metrics. We provide examples of various distances that can be obtained by Barbilian's metrization procedure and we discuss the relation between this metrization procedure and important Rieman…

Graphs are versatile tools for representing structured data. As a result, a variety of machine learning methods have been studied for graph data analysis. Although many such learning methods depend on the measurement of differences between input graphs, defining an appropriate distance metric for graphs remains a contr…

The norm of Cartan torsion plays an important role for studying of immersion theory in Finsler geometry. Indeed, Finsler manifold with unbounded Cartan torsion can not be isometrically imbedded into any Minkowski space. In this paper, we find two subclasses of (?, ?)-metrics which have bounded Cartan torsion. Then, we …

We introduce a notion of metric on a Lie groupoid, compatible with multiplication, and we study its properties. We show that many families of Lie groupoids admit such metrics, including the important class of proper Lie groupoids. The exponential map of these metrics allow us to establish a Linearization Theorem for Ri…

Natural metric structures on the tangent bundle and tangent sphere bundles $S_rM$ of a Riemannian manifold $M$ with radius function $r$ enclose many important unsolved problems. Admitting metric connections on $M$ with torsion, we deduce the equations of induced metric connections on those bundles. Then the equations o…

Study bi-Hermitian metrics on complex surfaces and solve geometric PDEs.

Around 2007, A. Chang, J. Qing, and P. Yang proved a conformal gap theorem for Bach-flat metrics with round sphere as the model case. In this article, we extend this result to prove conformally invariant gap theorems for Bach-flat $4$-manifolds with $(\mathbb{CP}^2, g_{FS})$ and $(\mathbb{S}^2\times\mathbb{S}^2,g_{prod…

Study on completeness of Sobolev metrics on manifold-valued curves.

A Riemannian metric is of constant curvature if and only if it is locally projectively flat. There are infinitely many locally projectively flat Finsler metrics of constant curvature, that are special solutions to the Hilbert's Fourth Problem. In this paper, we use the technique in the paper titled "Finsler metrics wit…

This paper tackles label-efficient evaluation in extreme class imbalance.

Artificial neural networks (NN) are instrumental in realizing highly-automated driving functionality. An overarching challenge is to identify best safety engineering practices for NN and other learning-enabled components. In particular, there is an urgent need for an adequate set of metrics for measuring all-important …

For a finite family of 3-dimensional almost contact metric manifolds with closed the structure form $η$ is described a construction of an almost contact metric manifold, where the members of the family are building blocks - cells. Obtained manifold share many properties of cells. One of the more important are nullity c…

We study paracontact metric $(κ,μ)$-spaces with $κ=-1$, equivalent to $h^2=0$ but not $h=0$. In particular, we will give an alternative proof of Theorem 3.2 of [11] and present examples of paracontact metric $(-1,2)$-spaces and $(-1,0)$-spaces of arbitrary dimension with tensor $h$ of every possible constant rank. We w…

Study on Kähler-Einstein metrics on quasi-projective manifolds.

The isotropic almost complex structures induce a Riemannian metric $g_{δ,σ}$ on TM, which are the generalized type of Sasakian metric. In this paper, the Levi-Civita connection of $g_{δ,σ}$ is calculated and the harmonicity of unit vector fields from $(M, g)$ to $(S(M), i*g_{δ,0})$ is investigated, where $i*g_{δ,0}$ is…

Deep single-index Fréchet regression for metric space-valued outputs

Delisle's projection explained by Euler in 18th century.

It is an important problem in differential geometry to find non-naturally reductive homogeneous Einstein metrics on homogeneous manifolds. In this paper, we consider this problem for some coset spaces of compact simple Lie groups. A new method to construct invariant non-naturally reductive Einstein metrics on normal ho…

The study of the $k$-th elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric, the so called $σ_k$ curvature, has produced many fruitful results in conformal geometry in recent years. In these studies, the deforming conformal factor is considered to be a solution of a fully nonlinea…

Paper proposes a method to estimate variance reduction in DNN training using importance sampling.

Study degenerations of Kähler-Einstein metrics on surfaces.

Captum library simplifies model interpretability for PyTorch.

New metrics on curve spaces improve shape analysis.

Framework improves gradient estimation for faster training convergence.