In [S. Basu, A. Gabrielov, N. Vorobjov, Semi-monotone sets. arXiv:1004.5047v2 (2011)] we defined semi-monotone sets, as open bounded sets, definable in an o-minimal structure over the reals, and having connected intersections with all translated coordinate cones in R^n. In this paper we develop this theory further by d…
New PDE systems generalize Hawking mass monotonicity.
problem Generalizing Hawking mass monotonicity to initial data sets.
method Introduced new systems of PDE on initial data sets ( M , g , k ) (M,g,k) ( M , g , k ) . result Generalized Geroch's monotonicity formula to initial data sets.
A coordinate cone in R^n is an intersection of some coordinate hyperplanes and open coordinate half-spaces. A semi-monotone set is a defnable in an o-minimal structure over the reals, open bounded subset of R^n such that its intersection with any translation of any coordinate cone is connected. This can be viewed as a …
The paper proves learning-curve monotonicity for maximum likelihood estimators in various parametric settings.
problem Establishing monotonicity guarantees for maximum likelihood estimators.
method Variants of GPT-5.2 Pro were used to derive the results.
result The paper proves monotonicity for maximum likelihood estimators in Gaussian and Gamma variables.
New formulas limit minimal submanifolds' area in curved spaces.
problem Bounding minimal submanifolds' area in curved spaces.
method Developed new monotonicity formulae involving energy-like integrals over non-geodesic sets.
result Imply sharp area bounds for minimal submanifolds through a prescribed point.
Improves k-NN for monotonic data with robustness against noise.
problem Class noise in real-life data violates monotonic constraints in k-NN.
method Monotonic Fuzzy k-NN (MonFkNN) with new fuzzy membership calculation.
result Significant accuracy improvements and robustness against monotonic noise.
Unified view of monotonicity formulas for inverse mean curvature flow and p p p -capacitary potentials.
problem Understanding monotonicity formulas for various geometric flows and potentials.
method Refined analysis of p p p -capacitary potentials and their level sets. result Strong convergence of p p p -capacitary potentials to inverse mean curvature flow and curvature varifolds. In this paper we generalize the monotonicity formulas of [C] for manifolds with nonnegative Ricci curvature. Monotone quantities play a key role in analysis and geometry; see, e.g., [A], [CM1] and [GL] for applications of monotonicity to uniqueness. Among the applications here is that level sets of Green's function on …
Study non-monotonic loss functions in CRC, achieving valid risk control with large calibration samples.
problem Non-monotonic loss functions in CRC, violating existing theory's monotonicity assumption.
method Finite grid selection, calibration sample size analysis, Lipschitz continuity, monotonicity, distribution shift.
result Valid CRC achieved with large calibration samples, optimal excess risk rate of log ( m ) / n \sqrt{\log(m)/n} log ( m ) / n . Proposes a Bayesian nonparametric model for monotonic functions.
problem Imposing monotonicity constraints in Bayesian nonparametric models.
method Numerical solutions of stochastic differential equations for nonparametric model of monotonic functions.
result Demonstrates competitive results on benchmark functions and utility in temporal alignment of time-series data.
Empirical risk minimizers can be non-monotonic in learning curves.
problem Understanding the behavior of learning curves for empirical risk minimizers.
method Introducing risk monotonicity and analyzing its implications for various learners.
result Standard learners that minimize empirical risk can be non-monotonic regardless of training sample size.
Paper tackles non-monotone DR-submodular maximization with approximation and regret guarantees.
problem Maximizing non-monotone DR-submodular functions over specific sets.
method Frank-Wolfe algorithm for general convex sets, Stochastic Gradient Ascent for down-closed convex sets.
result First approximation guarantees for both offline and online settings.
We propose learning deep models that are monotonic with respect to a user-specified set of inputs by alternating layers of linear embeddings, ensembles of lattices, and calibrators (piecewise linear functions), with appropriate constraints for monotonicity, and jointly training the resulting network. We implement the l…
Researchers extend monotonicity formulas for harmonic functions in RCD(0,N) spaces.
problem Generalizing monotonicity formulas for harmonic functions in m R C D ( 0 , N ) {
m RCD}(0,N) m R C D ( 0 , N ) spaces. method New estimates for harmonic functions and a functional version of the outer volume cone theorem.
result Proven rigidity and almost rigidity statements for harmonic functions in m R C D ( 0 , N ) {
m RCD}(0,N) m R C D ( 0 , N ) spaces. We show that if P P P is a quadratic polynomial with a fixed Cremer point and Julia set J J J , then for any monotone map $\ph:J\to A$ from J J J onto a locally connected continuum A A A , A A A is a single point.
Study finds non-monotonic Value of Information in dynamic multi-market monopoly.
problem Investigates non-monotonicity in Value of Information for a price-setting monopolist.
method Uses a Bayesian inverse problem with Kalman-Bucy-Stratonovich filter in a dynamic discrete model.
result Non-monotonic relationship between signal variance and Value of Information.
Paper finds formulas for minimizing perimeter in special spaces, proving key dimensions and existence.
problem Understanding the structure of perimeter minimizing sets in specific metric spaces.
method Established a monotonicity formula and proved rigidity for perimeter minimizers in RCD(0,N) spaces.
result Sharp Hausdorff dimension estimates for singular strata and existence of blow-down cones.
GD monotonically decreases GFS sharpness in neural networks and scalar models.
problem Oscillatory behavior of loss in GD training.
method Analysis of GFS sharpness and empirical validation.
result GFS sharpness decreases monotonically during GD training.
We construct an infinitely exchangeable process on the set $\cate$ of subsets of the power set of the natural numbers N \mathbb{N} N via a Poisson point process with mean measure Λ Λ Λ on the power set of N \mathbb{N} N . Each $E\in\cate$ has a least monotone cover in $\catf$ , the collection of monotone subsets of $\cate$ , an…
New proof of Positive Mass Theorem using Green's function and monotonicity formula.
problem Proving the Positive Mass Theorem in Riemannian geometry.
method Established through a newly discovered monotonicity formula for Green's function.
result New proof of the Positive Mass Theorem and Riemannian Penrose Inequality.
Study online monotone density estimation with expert aggregation and log-optimal calibration.
problem Online monotone density estimation and log-optimal calibration.
method Proposed two online estimators: Grenander estimator and expert aggregation estimator.
result Online estimators achieve O ( n 1 / 3 ) O(n^{1/3}) O ( n 1/3 ) cumulative log-likelihood gap and n log n \sqrt{n\log{n}} n log n pathwise regret bound. Constructs flow lines connecting unstable to stable self-expanders.
problem Existence of monotone Morse flow lines for expander functionals.
method Constructs a singular Morse flow line connecting unstable to stable self-expanders.
result Constructs a monotone flow line with a small singular set.
Study noncommutative Sobolev inequalities using quantum state metrics.
problem Establishing Sobolev inequalities in noncommutative settings.
method Generalizing monotone metrics in quantum states.
result Developed new matrix-valued Beckner inequalities.
New method tackles adversarial sign-corrupted isotonic regression, estimating monotonic signals under heavy dependence.
problem Estimating monotonic signals when responses are sign-corrupted and adversarially designed to violate monotonicity.
method Developed ASCIFIT, a three-step estimation procedure using PAVA with pre- and post-processing corrections.
result Theoretical guarantees of sharp high probability upper bounds and minimax lower bounds for ASCIFIT.
The paper bounds and identifies joint probabilities in causal inference with monotonicity assumptions.
problem Bounding and identifying joint probabilities of potential outcomes and observed variables under monotonicity assumptions.
method Proposes new families of monotonicity assumptions, formulates bounding problem as linear programming, introduces new monotonicity assumption for identification.
result Validated methods through numerical experiments and applied to real-world datasets.
This paper introduces a novel monotone curve estimation framework based on convex duality.
problem Estimating smooth, continuous, and monotonic curves in data.
method Convex duality and optimal transport theories.
result Established statistical guarantees for monotone curve estimates.
Bayesian optimization with preference learning using monotonic neural networks.
problem Optimizing complex systems with multiple conflicting objectives.
method Proposes a neural network ensemble for utility surrogate modeling, leveraging monotonicity.
result Demonstrates superior performance compared to existing methods.
New models ensure monotonicity in preference learning, improving accuracy especially with limited data.
problem Failure of widely used preference learning models to maintain monotonicity.
method Proposed Linear Generalized Bradley-Terry models with Diffusion Priors.
result New models improve accuracy, especially with limited data.
Study on p p p -Green functions on specific manifolds, proving monotonicity.
problem Monotonicity of p p p -Green functions on certain 3D manifolds. method Sharp monotonicity formula for p p p -Green functions along level sets. result Established monotonicity for 1 < p < 3 1<p<3 1 < p < 3 on specific manifolds. New formulas derived for scalar curvature in generalized Ricci flow.
problem Scalar curvature in generalized Ricci flow.
method Derivation of weighted scalar curvature monotonicity formulas and Perelman-type energy/entropy formulas.
result New convex Nash entropies and pseudolocality principles.
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 ⊂ ⊂ R n Ω\subset\subset \mathbb{R}^n Ω ⊂⊂ R n is a bounded domain, ( f i ( u 1 , . . . , u m ) ) = ∇ F ( u ⃗ ) (f_i(u_1,...,u_m))=\nabla F(\vec{u}) ( f i ( u 1 , ... , u m )) = ∇ F ( u ) , and F ( u ⃗ ) F(\vec{u}) F ( u ) is a given smooth function of u ⃗ = ( u 1 , . . . , u m ) \vec{u}=(u_1,...,u_m) u = ( u 1 , ... , u m ) …
Study on MMV in jump-diffusion models resolves MV's non-monotonicity issues.
problem Non-monotonicity and free cash flow stream problems in MV preferences.
method Explicit solution for MMV preferences in jump-diffusion models, proving non-negative potential measures.
result MMV resolves MV's non-monotonicity and free cash flow stream issues.
Paper characterizes monotonic mean-deviation risk measures.
problem Developing consistent risk measures from mean-deviation models.
method Applying a risk-weighting function to the deviation part of a mean-deviation model.
result Characterizes monotonic mean-deviation measures as consistent risk measures.
This paper bounds the volume of singular and critical sets for elliptic equations with Hölder coefficients.
problem Bounding the volume of singular and critical sets for elliptic equations with Hölder coefficients.
method Proves explicit bounds for ( n − 2 ) (n-2) ( n − 2 ) -dimensional Minkowski estimates of singular and critical sets using Hölder continuity and new almost monotonicity formula. result Optimal improvement on Cheeger-Naber-Valtorta's volume estimates on each quantitative stratum.
Bartnik mass is positive and non-decreasing for black holes
problem Quasilocal mass for black holes
method Defining a Bartnik mass and proving positivity and monotonicity
result Positive and non-decreasing Bartnik mass for black holes
Study optimal portfolio allocation in a general semimartingale model.
problem Dynamic optimal portfolio allocation under monotone mean-variance preferences.
method New results in semimartingale theory applied to Sharpe ratio analysis.
result Characterization of circumstances for improving mean-variance efficiency.
Monotone adversarial corruptions degrade optimal learning algorithms.
problem Optimal learning algorithms' reliance on exchangeability and independence is challenged.
method Introduces a monotone adversarial corruption model where an adversary adds monotone corruptions to a clean dataset.
result Optimal learning algorithms achieve suboptimal expected error on new test points.
Constructs families of monotone Lagrangians in Brieskorn-Pham hypersurfaces.
problem Constructing compact monotone Lagrangians in Brieskorn-Pham hypersurfaces.
method Inspired by monodromy considerations, techniques for controlling homology, Maslov class, and monotonicity constant.
result Infinite families of monotone Lagrangian S 1 i m e s Σ g S^1 imes Σ_g S 1 im es Σ g in C 3 \mathbb{C}^3 C 3 for g ≥ 2 g \geq 2 g ≥ 2 . Efficient algorithms find optimal monotone transforms for calibration under strictly convex losses.
problem Calibrating estimations to improve performance with monotone transforms.
method Proposed linear-time and space algorithm for finding optimal monotone transforms for specific loss functions. Also proposed an anytime algorithm with linear space and pseudo-linearithmic time complexity.
result Optimal monotone transforms are unique and can be found efficiently for various strictly convex loss functions.
The paper extends the avoidance principle for mean curvature flows, proving new intersection dimension monotonicity results.
problem Understanding the behavior of intersections in mean curvature flows.
method Proving new intersection dimension monotonicity results for mean curvature flows, Brakke flows, and level set flows.
result The dimension of the intersection of mean curvature flows is non-increasing over time.
New algorithms avoid non-monotonic risk curves in statistical learning.
problem Non-monotonic behavior of risk curves in statistical learning.
method Derive risk-monotonic algorithms under weak assumptions.
result Risk monotonicity does not necessarily lead to worse excess risk rates.
The paper studies nodal sets of solutions to parabolic equations, proving finiteness and monotonicity properties.
problem Analyzing nodal sets of solutions to parabolic equations with general coefficients.
method Generalized methods to handle time-dependent and Lipschitz continuous coefficients.
result Finiteness and monotonicity properties of the ( n − 1 ) (n-1) ( n − 1 ) -dimensional Hausdorff measure of nodal sets. The study examines distortion in specific homeomorphisms of Cantor sets.
problem Distortion in homeomorphisms of Cantor sets.
method Analyzes equivalence of conditions related to discontinuities and conjugacy.
result Elements are distorted if they satisfy certain conditions.
Cube category simplifies set modeling.
problem Modeling set operations efficiently.
method Introducing interval-preserving monotone functions between finite Boolean lattices.
result Cube category facilitates model structures equivalent to simplicial sets.
The paper addresses monotonicity in machine learning models for fairness and accountability.
problem Ensuring fairness and accountability in transparent machine learning models.
method Study of three types of monotonicity (individual, weak pairwise, strong pairwise) and propose monotonic groves of neural additive models.
result Monotonic groves of neural additive models maintain transparency, accountability, and fairness.
Study solves optimal portfolio selection using HJB equation.
problem Optimal portfolio selection problem.
method Maximal monotone operator method, Banach fixed-point theorem, Fourier transform, monotone operators technique.
result Existence and uniqueness of solution to HJB equation.
Paper tackles online DR-submodular maximization with various convex sets.
problem Maximizing DR-submodular functions online over different convex sets.
method Develops online algorithms with approximation guarantees for various convex sets.
result Achieves 1 / e 1/e 1/ e -approximation ratio with O ( T 2 / 3 ) O(T^{2/3}) O ( T 2/3 ) regret for down-closed sets. Monotonic Linear Interpolation property in neural networks persists despite non-convexity.
problem Understanding the geometric properties of neural network loss landscapes.
method Tools from differential geometry to analyze the monotonicity of neural network weights.
result Sufficient conditions for the Monotonic Linear Interpolation property under mean squared error.