arXiv research

The study calculates Weyl entropy in spacetime regions and shows its monotonic behavior.

In this paper, we introduce a monotonicity formula for the mean curvature flow which is related to self-expanders. Then we use the monotonicity to study the asymptotic behavior of Type III mean curvature flow on noncompact hypersurfaces.

In this paper, we introduce a monotonicity formula for the mean curvature flow. We also apply this monotonicity formula to study the asymptotic behavior of eternal solutions.

Deep networks show less disorder and more monotonic behavior.

Study on biharmonic map heat flow with monotonicity formula.

New algorithms avoid non-monotonic risk curves in statistical learning.

Framework mitigates risk non-monotonicity in high-dimensional predictions.

Learning performance can show non-monotonic behavior. That is, more data does not necessarily lead to better models, even on average. We propose three algorithms that take a supervised learning model and make it perform more monotone. We prove consistency and monotonicity with high probability, and evaluate the algorit…

Investigates polar tangential angles of curves and their monotonicity.

Plotting a learner's average performance against the number of training samples results in a learning curve. Studying such curves on one or more data sets is a way to get to a better understanding of the generalization properties of this learner. The behavior of learning curves is, however, not very well understood and…

We study the qualitative behavior of nonlinear Dirac equations arising in quantum field theory on complete Riemannian manifolds. In particular, we derive monotonicity formulas and Liouville theorems for solutions of these equations. Finally, we extend our analysis to Dirac-harmonic maps with curvature term.

We prove a monotonicity formula for mean curvature flow with surgery. This formula differs from Huisken's monotonicity formula by an extra term involving the mean curvature. As a consequence, we show that a surgically modified flow which is sufficiently close to a smooth flow in the sense of geometric measure theory is…

The theory of monotone Riemannian metrics on the state space of a quantum system was established by Denes Petz in 1996. In a recent paper he argued that the scalar curvature of a statistically relevant - monotone - metric can be interpreted as an average statistical uncertainty. The present paper contributes to this su…

This paper benchmarks monotone-constrained models for credit PD across datasets and finds constraints are mostly costless.

In this paper, we establish a general monotonicity formula of the following elliptic system $$ Δu_i+f_i(u_1,...,u_m)=0 \quad {\rm in} Ω, \label{0.1} $$ where $Ω\subset\subset \mathbb{R}^n$ is a bounded domain, $(f_i(u_1,...,u_m))=\nabla F(\vec{u})$, and $F(\vec{u})$ is a given smooth function of $\vec{u}=(u_1,...,u_m)$…

Develops monotone tree-based GAMI models using XGBoost.

A number of machine learning (ML) methods have been proposed recently to maximize model predictive accuracy while enforcing notions of group parity or fairness across sub-populations. We propose a desirable property for these procedures, slack-consistency: For any individual, the predictions of the model should be mono…

Gradient descent on neural nets often operates at the Edge of Stability, where loss behavior is complex but loss decreases over time.

Empirical time series of inter-event or waiting times are investigated using a modified Multifractal Detrended Fluctuation Analysis operating on fluctuations of mean detrended dynamics. The core of the extended multifractal analysis is the non-monotonic behavior of the generalized Hurst exponent $h(q)$ -- the fundament…

We present a heuristic based algorithm to induce \textit{nonmonotonic} logic programs that will explain the behavior of XGBoost trained classifiers. We use the technique based on the LIME approach to locally select the most important features contributing to the classification decision. Then, in order to explain the mo…

Develops methods to analyze feature-outcome associations in subpopulations.

This paper classifies solutions for a specific geometric problem.

We present turnpike-type results for the risk tolerance function in an incomplete market setting under time-monotone forward performance criteria. We show that, contrary to the classical case, the temporal and spatial limits do not coincide. We also show that they depend directly on the left- and right-end of the suppo…

Develops a two-level monotonic multistage recommender system for better user-specific prediction.

The paper improves PCS approximation for ranking and selection under limited simulation budgets.

Efficient algorithms find optimal monotone transforms for calibration under strictly convex losses.

This paper studies how adding leaves to a tree affects its spectral properties.

The paper examines how the first Steklov-Dirichlet eigenvalue changes with the distance between two concentric circles.

Extends tracking guarantees for time-varying variational inequalities.

In foreign exchange markets monotonic rate changes can be observed in time scale of order of an hour on the days that governmental interventions took place. We estimate the starting time of an intervention using this characteristic behavior of the exchange rates. We find that big amount of interventions can shift the a…

This work explains how large neural networks generalize well despite overparameterization.

In this note we discuss how several results characterizing the qualitative behavior of solutions to the nonlinear Poisson equation can be generalized to harmonic maps with potential between complete Riemannian manifolds. This includes gradient estimates, monotonicity formulas and Liouville theorems under curvature and …

The paper extends the avoidance principle for mean curvature flows, proving new intersection dimension monotonicity results.

In this paper we define the torsion flow, a CR analogue of the Ricci flow. For homogeneous CR manifolds we give explicit solutions to the torsion flow illustrating various kinds of behavior. We also derive monotonicity formulas for CR entropy functionals. As an application, we classify torsion breathers.

In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized Ricci flow of model geometries. In every Bianchi class, we estimate the derivative of the eigenvalue. Then we construct monotonic quantities under the Ricci flow and obtain upper and lower bounds for…

The dilatation of a pseudo-Anosov braid is a conjugacy invariant. In this paper, we study the dilatation of a special family of pseudo-Anosov braids. We prove an inductive formula to compute their dilatation, a monotonicity and an asymptotic behavior of the dilatation for this family of braids. We also give an example …

Study the long-time behavior of Hermitian-Yang-Mills flow on non-Kähler manifolds.

The Cheeger constant increases under Ricci flow on spheres.

New findings show privacy affects generalization error in a non-monotonic way.

Estimates for VPMCF show ancient MCF solutions and finite-time behavior.

In this paper, by modifying the arguments in \cite{WY}, we get some rigidity theorems on compact manifolds with nonempty boundary. The results in this paper are similar with those in \cite{ST} and \cite{WY}. Like \cite{ST} and \cite{WY}, we still use quasi-spherical metrics introduced by \cite{Ba} to get monotonicity o…

Monotonic improvement in uncertainty estimation with Gaussian processes as dimension increases.

The extragradient method fails for hypomonotone variational inequalities.

BL learns interpretable optimization structures from data.

Exploiting low-rank structure of the user-item rating matrix has been the crux of many recommendation engines. However, existing recommendation engines force raters with heterogeneous behavior profiles to map their intrinsic rating scales to a common rating scale (e.g. 1-5). This non-linear transformation of the rating…

Deep learning models can have low bias and variance, contrary to classical theory.

Consider the following coupled elliptic system of equations \begin{equation*} \label{} (-Δ)^s u_i = (u^2_1+\cdots+u^2_m)^{\frac{p-1}{2}} u_i \quad \text{in} \ \ \mathbb{R}^n , \end{equation*} where $0<s\le 2$, $p>1$, $m\ge1$, $u=(u_i)_{i=1}^m$ and $u_i:\mathbb R^n\to \mathbb R$. The qualitative behavior of solutions of…

The paper proves existence and behavior of Lagrangian tori in complex projective plane.