arXiv research

The study shows how to measure translation surfaces with short saddle connections.

Polynomial Duistermaat-Heckman measure on symplectic groupoid quotients.

Fix a translation surface $X$, and consider the measures on $X$ coming from averaging the uniform measures on all the saddle connections of length at most $R$. Then as $R\to\infty$, the weak limit of these measures exists and is equal to the Lebesgue measure on $X$. We also show that any weak limit of a subsequence of …

Starting with the work of Preiss on the geometry of measures, the classification of uniform measures in $\mathbb R^d$ has remained open, except for $d=1$ and for compactly supported measures in $d=2$, and for codimension $1$. In this paper we study $1$-dimensional measures in $\mathbb R^d$ for all $d$ and classify unif…

A new method is proposed to compute connectivity measures on multivariate time series with gaps. Rather than removing or filling the gaps, the rows of the joint data matrix containing empty entries are removed and the calculations are done on the remainder matrix. The method, called measure adapted gap removal (MAGR), …

The paper extends Laplacian spectra approximations to vector bundles.

Study harmonic measures and rigidity in Seifert 3-manifolds using $S^1$-connections.

Flatness of the loss curve is conjectured to be connected to the generalization ability of machine learning models, in particular neural networks. While it has been empirically observed that flatness measures consistently correlate strongly with generalization, it is still an open theoretical problem why and under whic…

Proposes a new risk measurement method for risk-averse stochastic optimization.

In this article we consider the continuity of the eigenvalues of the connection Laplacian of $G$-connections on vector bundles over Riemannian manifolds. To show it, we introduce the notion of the asymptotically $G$-equivariant measured Gromov-Hausdorff topology on the space of metric measure spaces with isometric $G$-…

We compute the Riemannian volume on the moduli space of flat connections on a nonorientable 2-manifold, for a natural class of metrics. We also show that Witten's volume formula for these moduli spaces may be derived using Haar measure, and we give a new proof of Witten's volume formula for the moduli space of flat con…

This study describes the Fisher-Rao metric on Gaussian measures in infinite-dimensional spaces.

In this paper, we study the asymptotic behavior of the volume of spheres in metric measure spaces. We first introduce a general setting adapted to the study of asymptotic isoperimetry in a general class of metric measure spaces. We then introduce a notion of "being asymptotically isoperimetric" for a family of finite a…

Study rigidity of PSU(1,1) actions on circle via harmonic measures.

We introduce a new measure of performance of investment strategies, the monotone Sharpe ratio. We study its properties, establish a connection with coherent risk measures, and obtain an efficient representation for using in applications.

This paper connects monetary and star-shaped risk measures by showing their equivalence under certain conditions.

The paper connects higher order risk measures and stochastic dominance, showing their equivalence and integrating them with optimization.

One often finds in the literature connections between measures of fairness and measures of feature importance employed to interpret trained classifiers. However, there seems to be no study that compares fairness measures and feature importance measures. In this paper we propose ways to evaluate and compare such measure…

The paper proves a convergence theorem for Wiener measures on holonomy groups.

We characterize when a convex risk measure associated to a law-invariant acceptance set in $L^\infty$ can be extended to $L^p$, $1\leq p<\infty$, preserving finiteness and continuity. This problem is strongly connected to the statistical robustness of the corresponding risk measures. Special attention is paid to concre…

Consumers with low demand, like households, are generally supplied single-phase power by connecting their service mains to one of the phases of a distribution transformer. The distribution companies face the problem of keeping a record of consumer connectivity to a phase due to uninformed changes that happen. The exact…

In this work, we identify the most general measure of arbitrage for any market model governed by Itô processes. We show that our arbitrage measure is invariant under changes of numéraire and equivalent probability. Moreover, such measure has a geometrical interpretation as a gauge connection. The connection has zero cu…

New control methods improve dynamic measure transport paths.

Improves time series classification with forest proximities.

We consider the problem of identifying a unitary Yang-Mills connection $\nabla$ on a Hermitian vector bundle from the Dirichlet-to-Neumann (DN) map of the connection Laplacian $\nabla^*\nabla$ over compact Riemannian manifolds with boundary. We establish uniqueness of the connection up to a gauge equivalence in the cas…

A new kernel measures brain network similarities, improving disease classification.

The paper defines surface area for graphs and derives spectral estimates.

Paper explores properties of slice-matching operators for measure transfer.

The paper introduces submodular information measures for machine learning applications.

The aim of this article is to study rational parallelisms of algebraic varieties by means of the transcendence of their symmetries. The nature of this transcendence is measured by a Galois group built from the Picard-Vessiot theory of principal connections.

New concept of partial law invariance connects decision theory and financial risk management.

Study shows observability from a measurable set for Gevrey functions.

Study finds a measure for sponge components of Lalley-Gatzouras type.

The author suggests using non-Euclidean geometry for psychometric models.

We simplify information measure computation using learned features.

We define the Ricci curvature, as a measure, for certain singular torsion-free connections on the tangent bundle of a manifold. The definition uses an integral formula and vector-valued half-densities. We give relevant examples in which the Ricci measure can be computed. In the time dependent setting, we give a weak no…

Forré introduces a new conditional independence notion for mixed variables.

LOCUS separates brain network connectivity matrices efficiently.

Paper introduces a new measure combining entropy and Gini index.

The paper confirms a conjecture linking link bipyramid volume and Mahler measure.

Handlebody groups are rigid under measure equivalence.

Proposes variational Wasserstein barycenters for geometric clustering.

Partial connections are (singular) differential systems generalizing classical connections on principal bundles, yielding analogous decompositions for manifolds with nonfree group actions. Connection forms are interpreted as maps determining projections of the tangent bundle onto the partial connection; this approach e…

Paper connects risk consistency to L_p consistency for broader loss functions.

Study connections on Lie and Courant algebroids, defining basic curvature and Atiyah cocycle.

The existence of kinematic formulas for area measures with respect to any connected, closed subgroup of the orthogonal group acting transitively on the unit sphere is established. In particular, the kinematic operator for area measures is shown to have the structure of a co-product. In the case of the unitary group the…

Proposes QGC to distinguish between lower and upper tail connectivity in financial networks.

Let Q be a connected component of a stratum in the space of quadratic differentials for a non-exceptional Riemann surface of finite type. We show that the probability measure on Q in the Lebesgue measure class which is invariant under the Teichmueller flow is obtained by Bowen's construction.