arXiv research

GD monotonically decreases GFS sharpness in neural networks and scalar models.

Paper introduces algorithms for explaining monotonic classifiers.

GradaGrad adapts learning rate non-monotonically, overcoming AdaGrad's step size decrease.

New example of manifolds with monotonic heat kernels found.

In this paper, we study the relation of the monotonicity of Hawking Mass and geometric flow problems. We show that along the Hamilton-DeTurck flow with bounded curvature coupled with the modified mean curvature flow, the Hawking mass of the hypersphere with a sufficiently large radius in Schwarzschild spaces is monoton…

Recent reports have described that the equivalent sample size (ESS) in a Dirichlet prior plays an important role in learning Bayesian networks. This paper provides an asymptotic analysis of the marginal likelihood score for a Bayesian network. Results show that the ratio of the ESS and sample size determine the penalty…

Bartnik mass is positive and non-decreasing for black holes

Large GD stepsizes improve margins and speed up training for non-homogeneous networks.

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

The paper analyzes high-dimensional kernel regression, showing different risk curves based on data and regularization.

Nonnegative matrix factorization (NMF) factorizes a non-negative matrix into product of two non-negative matrices, namely a signal matrix and a mixing matrix. NMF suffers from the scale and ordering ambiguities. Often, the source signals can be monotonous in nature. For example, in source separation problem, the source…

The paper proves the monotonicity of a modified Perelman's W-entropy for mean curvature flow.

In this paper we introduce a new logarithmic entropy functional for the linear heat equation on complete Riemannian manifolds and prove that it is monotone decreasing on complete Riemannian manifolds with nonnegative Ricci curvature. Our results are simpler version, without Ricci flow, of R.-G. Ye's recent result (arXi…

Paper investigates monotonicity issues in AI preference learning.

New quantity helps map homotopy classes in complex spaces.

New models ensure monotonicity in preference learning, improving accuracy especially with limited data.

Proves flows of two-convex Lagrangians are regular, global, and converge.

ATSM are widely applied for pricing of bonds and interest rate derivatives but the consistency of ATSM when the short rate, r, is unbounded from below remains essentially an open question. First, the standard approach to ATSM uses the Feynman-Kac theorem which is easily applicable only when r is bounded from below. Sec…

The paper proves a transformation theorem under a monotone property of almost Euclidean factors of geodesic balls.

We give a locally minimal, but not globally minimal bridge position of a knot, that is, an unstabilized, nonminimal bridge position of a knot. It implies that a bridge position cannot always be simplified so that the bridge number monotonically decreases to the minimal.

Numerical observations on martingale couplings are confirmed under certain conditions.

In this article, we introduce a mass-decreasing flow for asymptotically flat three-manifolds with nonnegative scalar curvature. This flow is defined by iterating a suitable Ricci flow with surgery and conformal rescalings and has a number of nice properties. In particular, wormholes pinch off and nontrivial spherical s…

Topological entropy decreases strictly along Ricci flow near hyperbolic metrics.

Monotonic Linear Interpolation property in neural networks persists despite non-convexity.

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

Brenier isotonic regression extends multi-output isotonic regression using optimal transport.

In 2004, Manning showed that the topological entropy of the geodesic flow for a surface of negative curvature decreases as the metric evolves under the normalised Ricci flow. It is an interesting open problem, also due to Manning, to determine to what extent such behaviour persists for higher dimensional manifolds. In …

As surrogate functions of $L_0$-norm, many nonconvex penalty functions have been proposed to enhance the sparse vector recovery. It is easy to extend these nonconvex penalty functions on singular values of a matrix to enhance low-rank matrix recovery. However, different from convex optimization, solving the nonconvex l…

The Cheeger constant increases under Ricci flow on spheres.

We consider a closed manifold M with a Riemannian metric g(t) evolving in direction -2S(t) where S(t) is a symmetric two-tensor on (M,g(t)). We prove that if S satisfies a certain tensor inequality, then one can construct a forwards and a backwards reduced volume quantity, the former being non-increasing, the latter be…

Develops a dynamical method to prove the sharp Berezin-Li-Yau inequality.

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

Monotonicity of normalized implied-volatility coordinates under no-arbitrage

Study non-monotonic loss functions in CRC, achieving valid risk control with large calibration samples.

In this article, we continue the work in \cite{GL} and study a normalized hypersurface flow in the more general ambient setting of warped product spaces. This flow preserves the volume of the bounded domain enclosed by a graphical hypersurface, and monotonically decreases the hypersurface area. As an application, the i…

Researchers compute quasi-local mass of Kerr black hole horizon.

Improved analysis of extragradient methods for structured VIPs.

We study convexity and monotonicity properties for prices of bonds and bond options when the short rate is modeled by a diffusion process. We provide conditions under which convexity of the price in the short rate is guaranteed. Under these conditions the price is decreasing in the drift and increasing in the volatilit…

We extend short-time existence and stability of the Dirichlet energy flow as proven in a previous paper by the authors to a broader class of energy functionals. Furthermore, we derive some monotonely decreasing quantities for the Dirichlet energy flow and investigate an equation of soliton type. In particular, we show …

In financial time series there are periods in which the value increases or decreases monotonically. We call those periods elemental trends and study the probability distribution of their duration for the indices DJIA, NASDAQ and IPC. It is found that the trend duration distribution often differs from the one expected u…

Study eigenvalues of surfaces with collapsing handles or cross caps.

In this paper, we investigate the attractive properties of the proximal gradient algorithm with inertia. Notably, we show that using alternated inertia yields monotonically decreasing functional values, which contrasts with usual accelerated proximal gradient methods. We also provide convergence rates for the algorithm…

FISAR uses neural networks to optimize safe reinforcement learning with forward-invariant constraints.

In this paper, we extend the $β$-CNMF to two dimensions and derive exact multiplicative updates for its factors. The new updates generalize and correct the nonnegative matrix factor deconvolution previously proposed by Schmidt and Mørup. We show by simulation that the updates lead to a monotonically decreasing $β$-dive…

The paper tackles rested bandits with non-decreasing and concave rewards, deriving lower bounds and an efficient algorithm.

Two new methods improve monotonic constraint enforcement in regression and classification trees.

New metric shows how different regularization methods affect deep linear networks.

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…