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.
Develops a dynamical method to prove the sharp Berezin-Li-Yau inequality.
problem Proving the sharp Berezin-Li-Yau inequality for convex domains.
method Volume-preserving mean curvature flow and a new monotonicity principle.
result Shows the sharp Berezin-Li-Yau bound for every smooth convex domain.
We propose a new framework for imposing monotonicity constraints in a Bayesian nonparametric setting based on numerical solutions of stochastic differential equations. We derive a nonparametric model of monotonic functions that allows for interpretable priors and principled quantification of hierarchical uncertainty. W…
FLOWGEM generates complete datasets from incomplete data with non-monotone MAR missingness.
problem Dealing with non-monotone Missing at Random (MAR) missingness in data.
method Iterative particle evolution of Wasserstein Gradient Flow, approximated by local linear estimators of density ratio.
result FLOWGEM achieves state-of-the-art performance across various settings, including non-monotone MAR mechanisms.
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.
We derive some restrictions on the topology of a monotone Lagrangian submanifold L⊂Cn by making observations about the topology of the moduli space of Maslov 2 holomorphic discs with boundary on L and then using Damian's theorem which gives conditions under which the evaluation map from this moduli …
Researchers propose a non-monotone quantum natural gradient for quantum systems.
problem Applying natural gradient methods to quantum systems without monotonicity.
method Introducing a non-monotone quantum natural gradient (QNG) and demonstrating its superiority over conventional QNG.
result Non-monotone QNG outperforms conventional QNG in terms of convergence speed.
Both analytic and geometric forms of an optimal monotone principle for Lp-integral of the Green function of a simply-connected planar domain Ω with rectifiable simple curve as boundary are established through a sharp one-dimensional power integral estimate of Riemann-Stieltjes type and the Huber analytic and geome…
The paper evaluates the importance of monotonicity in AI fairness across various fields.
problem Ensuring fairness in AI applications across criminology, education, health care, and finance.
method Theoretical reasoning, simulation, and extensive empirical analysis of monotonic neural additive models (MNAMs).
result Monotonicity is essential for fairness in AI ethics and society, especially in criminology, education, health care, and finance.
The paper tackles online learning problems with monotone arm sequences, achieving optimal or near-optimal regret bounds.
problem Online learning problems with ordinal and monotone arm sequences, such as dynamic pricing and clinical trials.
method Proposes algorithms for continuum-armed bandit problems with monotone arm sequences, achieving optimal or near-optimal regret bounds.
result Achieves optimal or near-optimal regret bounds for monotone arm sequences, differing from the continuous-armed bandit literature.
Anisotropic minimal graphs over half-spaces are flat.
problem Characterizing minimal graphs over half-spaces.
method Maximum principle and fully nonlinear PDE theory.
result Anisotropic minimal graphs over half-spaces are flat.
Paper shows how to replicate payoffs without oracles in CFMMs.
problem Replicating payoffs without oracles in CFMMs.
method Using liquidity provider shares in CFMMs to match any monotonic payoff.
result Explicit method and formula for trading functions and earnings.
A pricing principle is introduced for non-attainable claims in incomplete markets.
problem Pricing non-attainable contingent claims in incomplete markets.
method Distorted Radon-Nikodym derivative and Tsallis relative entropy over a family of equivalent martingale measures.
result The pricing principle is closely related to backward stochastic differential equations and is arbitrage-free and time-consistent.
Monotonicity of normalized implied-volatility coordinates under no-arbitrage
problem Monotonicity of normalized implied-volatility coordinates under no-arbitrage
method Elementary discrete no-arbitrage proof
result Monotonicity principle extended to Bachelier implied volatility
New formula shows how causal vectors relate to mass-minimizing data.
problem Understanding mass-minimizing initial data sets and their geometry.
method Developed a new monotonicity formula for causal Killing vectors.
result Established strong maximum principles for the Lorentzian length.
We consider Liouville-type and partial regularity results for the nonlinear fourth-order problem $$ Δ^2 u=|u|^{p-1}u\ \{in} \ \R^n,$$ where p>1 and n≥1. We give a complete classification of stable and finite Morse index solutions (whether positive or sign changing), in the full exponent range. We also compute an…
Curve shortening flow shrinks curves to points.
problem The behavior of curves under curve shortening flow.
method Nonlinear partial differential equations, maximum principle, monotonicity formulas, Harnack inequalities, blowup analysis.
result The curve shortening flow shrinks any closed embedded curve in the plane to a round point.
New energy functional and fields for Yang-Mills theory, proving monotonicity and vanishing theorems.
problem Developing new mathematical tools for Yang-Mills theory.
method Introducing normalized exponential Yang-Mills energy functional, deriving monotonicity formula and vanishing theorem.
result Monotonicity and vanishing theorems for exponential Yang-Mills fields.
Study tackles nonlinear factor models with unknown monotone links from incomplete and noisy data.
problem Learning nonlinear factor models with unknown monotone links from incomplete and noisy data.
method Formulated as joint recovery of low-rank factors, loadings, and nonlinear link function; proposed BCD algorithm with regularization.
result Established convergence guarantees and sublinear regret bounds for link-function updates.
The main result of the paper is a version of the fundamental theorem of asset pricing (FTAP) for large financial markets based on an asymptotic concept of no market free lunch for monotone concave preferences. The proof uses methods from the theory of Orlicz spaces. Moreover, various notions of no asymptotic arbitrage …
DRCD identifies causal direction between continuous and discrete variables using density ratio monotonicity.
problem Inferring causal direction between continuous and discrete variables from observational data.
method Density Ratio-based Causal Discovery (DRCD) method.
result DRCD identifies causal direction between continuous and discrete variables using density ratio monotonicity.
A new approach to group fairness treats it as a bargaining problem.
problem Fairness in deploying predictors across subpopulations.
method Interpreting fairness as a bargaining problem and proposing relative improvement.
result Relative improvement provides axiomatic justification and finite-sample convergence guarantees.
The Ekeland variational principle implies what can be regarded as a strong version, in the C1 category, of the Yau minimum principle: under the appropriate hypotheses {\it every} minimizing sequence admits a {\it good shadow}, a second minimizing sequence that has good properties and is asymptotic to the original on…
Stochastic control problems in finance often involve complex controls at discrete times. As a result numerically solving such problems, for example using methods based on partial differential or integro-differential equations, inevitably give rise to low order accuracy, usually at most second order. In many cases one c…
We demonstrate how easy it is for modern machine-learned systems to violate common deontological ethical principles and social norms such as "favor the less fortunate," and "do not penalize good attributes." We propose that in some cases such ethical principles can be incorporated into a machine-learned model by adding…
Optimizes AI learning with limited human feedback budgets.
problem Optimizing allocation of a fixed annotation budget for AI learning.
method Preference-Calibrated Active Learning (PCAL) using semi-parametric inference.
result Proves asymptotic optimality and robustness of the PCAL estimator.
We consider a convex Euclidean hypersurface that evolves by a volume or area preserving flow with speed given by a general nonhomogeneous function of the mean curvature. For a broad class of possible speed functions, we show that any closed convex hypersurface converges to a round sphere. The proof is based on the mono…
We study the optimal transport between two probability measures on the real line, where the transport plans are laws of one-step martingales. A quasi-sure formulation of the dual problem is introduced and shown to yield a complete duality theory for general marginals and measurable reward (cost) functions: absence of a…
New scaling framework for MoE architectures ensures stability and optimal performance at scale.
problem Lack of principled understanding of how hyperparameters should scale in MoE architectures.
method Developed a novel Dynamical Mean Field Theory (DMFT) for three scaling regimes of MoE architectures.
result Derived Maximally Scale-Stable Parameterization (MSSP) for SGD and Adam, providing robust learning rate transfer and monotonic improvement with scale.
We investigate a class of quadratic-exponential growth BSDEs with jumps. The quadratic structure introduced by Barrieu & El Karoui (2013) yields the universal bounds on the possible solutions. With local Lipschitz continuity and the so-called A_gamma-condition for the comparison principle to hold, we prove the existenc…
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.
Boosted GFlowNets improve exploration by sequentially training GFlowNets with residual rewards.
problem GFlowNets struggle to evenly explore reward landscapes, leading to poor coverage of high-reward areas.
method Sequential training of an ensemble of GFlowNets, each optimizing a residual reward.
result Boosted GFlowNets achieve better exploration and sample diversity on multimodal benchmarks and peptide design tasks.
Null-Calibrated Conformal Selection via Target-Membership Scores
problem Identifying test candidates whose unknown responses fall in a target region while controlling the false discovery rate
method Membership-score-based conformal selection
result Finite-sample valid null p-values
Study curve shortening flows on specific surfaces, proving properties and existence.
problem Analyzing curve shortening flows on rotational surfaces with negative Gauss curvatures.
method Assume negative Gauss curvatures and conditions on Gauss curvature and curve curvature. Prove curve remains a graph and establish flow properties.
result Prove the curve remains a graph over parallels and establish long-time existence of the flow.
Probit Monotone BART estimates binary outcomes using monotonic functions.
problem Estimating conditional mean functions for binary outcomes with monotonicity constraints.
method Proposes a new BART variant that incorporates monotonicity constraints for binary outcomes.
result Allows for more precise estimation of monotonic functions in binary outcome models.
This paper focuses on martingale optimal transport problems when the martingales are assumed to have bounded quadratic variation. First, we give a result that characterizes the existence of a probability measure satisfying some convex transport constraints in addition to having given initial and terminal marginals. Sev…
Monotone neural networks can approximate and interpolate functions efficiently.
problem Understanding the efficiency and expressiveness of monotone neural networks.
method Solving the monotone interpolation problem using depth-4 networks and comparing size bounds with arbitrary networks.
result Monotone neural networks can approximate and interpolate functions efficiently, but may require exponential size in high dimensions.
We study flows of hypersurfaces in Riemannian manifolds with specific curvature speeds.
problem Understanding the evolution of hypersurfaces in Riemannian manifolds under Weingarten conditions.
method Investigating Weingarten flows with a Weingarten function that is homogeneous, monotonic, and positive.
result Existence and embedding preserving properties of Weingarten flows with isoparametric initial data.
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.
Study examines explainable machine learning for monotonic models, finding Integrated gradients better for strong monotonicity.
problem Applying explainable machine learning to science-informed models.
method Proposed axioms for monotonicity, tested Shapley value and Integrated gradients methods.
result Integrated gradients provides better explanations for strong monotonicity.
We consider the heat flow of corotational harmonic maps from R3 to the three-sphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel, systematic, robust, and constructive approach to the stability analysis of self…
Formula proves monotonicity for anisotropic minimal hypersurfaces.
problem Understanding anisotropic minimal hypersurfaces.
method Proved a monotonicity formula under a sign assumption on the Minkowski norm.
result Monotonicity formula for anisotropic minimal hypersurfaces.
New example of manifolds with monotonic heat kernels found.
problem Understanding monotonicity of heat kernels on manifolds.
method Analyzing new examples and classifying flat tori.
result Generic metrics fail monotonicity at large times.
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…
The paper develops algorithms to restore monotonicity in non-monotone functions.
problem Non-monotone solutions from heuristic algorithms need to be corrected.
method Develops algorithms to restore monotonicity with limited queries.
result Restores monotonicity while degrading the function value by at most ε.
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…
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…
The number of trees T in the random forest (RF) algorithm for supervised learning has to be set by the user. It is controversial whether T should simply be set to the largest computationally manageable value or whether a smaller T may in some cases be better. While the principle underlying bagging is that "more trees a…