Mathematical approach defines stability conditions for ML models.
problem Ensuring stability of machine learning models.
method Adopted topological and metric spaces theory to define stability.
result Stability of ML models depends on topological properties of classification sets.
This paper deals with stability in the numerical solution of the prominent Heston partial differential equation from mathematical finance. We study the well-known central second-order finite difference discretization, which leads to large semi-discrete systems with non-normal matrices A. By employing the logarithmic sp…
The paper proves stability of the positive mass theorem using intrinsic flat convergence.
problem Stability of the positive mass theorem in mathematical relativity.
method Intrinsic flat convergence of points and applications to stability.
result Revisits and strengthens the stability results for graphical hypersurfaces of Euclidean space.
CLIP controls neural network stability by bounding Lipschitz constants.
problem Neural networks lack mathematical guarantees of stability, especially to adversarial examples.
method Develops a variational regularization method (CLIP) to control the Lipschitz constant of neural networks.
result CLIP provides a tighter bound on the actual Lipschitz constant compared to layer-wise methods.
This paper provides a mathematical framework for time-delay reservoir computing.
problem Lack of rigorous mathematical foundations for reservoir computing properties.
method Control-theoretic framework, formal definitions of separation and fading memory, explicit lower bound derivation.
result Established formal definitions and connections to stability notions for time-delay systems.
Paper studies stability of generalized Ricci solitons with mathematical rigor.
problem Stability of generalized Ricci solitons under curvature assumptions.
method Computed second variation formula, proved stability conditions.
result Equivalence of dynamical and linear stability on steady gradient solitons.
Mathematical framework investigates fire sales amplification and stability.
problem Market instability caused by fire sales amplification.
method Developed a mathematical framework to investigate system characteristics and resilience.
result Characterized systems resilient to small shocks for financial stability assessment.
Proposes a RL method using simulators for stabilizing uncertain systems.
problem Limited experiences and potential dangerous actions during RL learning of real systems.
method Two-stage approach: virtual systems for Q-function learning, real system interactions for final policy.
result Proposed method improves RL performance in uncertain discrete-time systems.
New mathematical foundations for stable RKHSs improve system identification.
problem Improving stability tests and modeling of impulse responses.
method Providing new structural properties and stability conditions for stable RKHSs.
result Any stable kernel admits feature maps induced by orthogonal eigenvectors in l2.
We propose an investigation of stability of vacua in string theory by studying their stability with respect to a (suitable) world-sheet renormalization group (RG) flow. We prove geometric stability of (Euclidean) anti-de Sitter (AdS) space (i.e., Hn) with respect to the simplest RG flow in closed string the…
The paper examines how macroeconomic control tools lost effectiveness, leading to a 'dark ages' period.
problem Loss of effectiveness of control tools in macroeconomic stabilization policy.
method Historical analysis of macroeconomic stabilization policy from 1948 to 1993.
result The overstatement of the Lucas critique and Kydland and Prescott's time-inconsistency led to a period of ineffective stabilization policy.
Paper shows equivalence between MM and PH for n-D Morse functions.
problem Relationship between Mathematical Morphology and Persistent Homology.
method Examined pairing of extrema in Morse functions using dynamics and persistence.
result Equivalence proven between dynamics and persistence on n-D Morse functions.
In this paper, we introduce a new concept of stability for cross-validation, called the (β,ϖ)-stability, and use it as a new perspective to build the general theory for cross-validation. The (β,ϖ)-stability mathematically connects the generalization ability and the stability of…
Paper examines two methods for FX market volatility modeling.
problem FX market volatility modeling problem.
method Classical econometric GCH and mathematical approaches (SSA, dynamical systems stability analysis).
result Both mathematical tools show promising results in FX market volatility modeling.
Mathematical methods of population genetics and framework of exchangeability provide a Markov chain model for analysis and interpretation of stochastic behaviour of equity markets, explaining, in particular, market shape formation, statistical equilibrium and temporal stability of market weights.
New method stabilizes tensegrity structures suitable for engineering.
problem Tensegrity structures often have unstable modes unsuitable for engineering.
method Proposes a relationship between rods and strings for full-rank convexity.
result Designs a stable three-rod three-string tensegrity.
Deep convolutional neural networks have led to breakthrough results in practical feature extraction applications. The mathematical analysis of these networks was pioneered by Mallat, 2012. Specifically, Mallat considered so-called scattering networks based on identical semi-discrete wavelet frames in each network layer…
Study proves stability of big bang singularity in complex system.
problem Stability of Kasner solutions in Einstein-Maxwell-scalar field-Vlasov system.
method Detailed mathematical structures and new delicate arguments.
result Nonlinear stability with Kasner exponents in full strong sub-critical regime.
The paper examines stable CMC hypersurfaces with boundaries on parallel hyperplanes.
problem Stability of CMC hypersurfaces with free boundaries.
method Analysis and numerical computations.
result Equilibrium hypersurfaces are stable without self-intersection in all dimensions.
Study stability of non-Kähler Calabi-Yau metrics using critical points of generalized Einstein Hilbert action.
problem Stability of critical points of the generalized Einstein Hilbert action in non-Kähler Calabi-Yau theory.
method Analysis of Bismut Hermitian Einstein manifolds and Bismut flat pluriclosed steady solitons, proving stability conditions.
result All Bismut Hermitian Einstein manifolds are linearly stable, and all Bismut flat pluriclosed steady solitons with positive Ricci curvature are linearly strictly stable.
The OGY method is one of control methods for a chaotic system. In the method, we have to calculate a stabilizing periodic orbit embedded in its chaotic attractor. Thus, we cannot use this method in the case where a precise mathematical model of the chaotic system cannot be identified. In this case, the delayed feedback…
Weight decay stabilizes training dynamics by slowing progressive sharpening.
problem Understanding how weight decay affects training stability in deep learning models.
method Analyzing weight decay effects at the Edge of Stability, developing a mathematical framework.
result Weight decay dampens oscillations and stabilizes sharpness in CNNs, causing a phase transition in MLPs.
New findings show mini-batch SGD operates in a 'Edge of Stochastic Stability' regime.
problem Understanding the stability and convergence of mini-batch SGD.
method Analyzing the mini-batch Hessian and its directional curvature.
result Mini-batch SGD operates in a different stability regime (Edge of Stochastic Stability) compared to full-batch GD.
In this paper, simple mathematical models from Control Theory are applied to three very important economic paradigms, namely (a) minimum wages in self-regulating markets, (b) market-versus-true values and currency rates, and (c) government spending and taxation levels. Analytical solutions are provided in all three par…
Study proves existence, uniqueness, and stability for specific stochastic Volterra equations.
problem Analyzing existence, uniqueness, and stability of affine stochastic Volterra equations with L1-kernels. method Approximations with L2-kernels, stability result, duality argument, deterministic Riccati--Volterra integral equation. result Established weak uniqueness for the equations using Fourier--Laplace transform and a deterministic Riccati--Volterra integral equation.
Quantum GAN improves volatility modeling in finance.
problem Improving volatility modeling in finance using GANs.
method Developed a quantum GAN for volatility modeling.
result Quantum GAN provides exponential advantage over classical methods.
New risk measures improve portfolio diversification and stability.
problem Concentration risk in traditional portfolio optimization methods.
method Equal-correlation portfolio strategy with mathematical optimization.
result Improved risk diversification and stable returns.
New memory effect discovered in gravitational wave behavior.
problem Understanding gravitational wave behavior in spacetimes with angular momentum.
method Mathematical analysis of Minkowski spacetime and Kerr black holes.
result Angular momentum memory effect observed at future null infinity.
AOPU stabilizes NN training by approximating natural gradient, improving stability and convergence.
problem Stability and interpretability in online NN training for industrial soft sensors.
method AOPU truncates gradient backpropagation, optimizing trackable parameters, and approximating natural gradient.
result AOPU achieves stable convergence and superior performance on chemical process datasets.
Survey on stability of Minkowski spacetime in relativity.
problem Nonlinear stability of Minkowski spacetime in general relativity.
method Decay assumptions, geometric foliations, energy identities, and gauge choices.
result Understanding of decay, dispersion, and geometry-analysis interplay.
This paper argues that the fundamental principle of contemporary financial economics is balanced reciprocity, not the principle of utility maximisation that is important in economics more generally. The argument is developed by analysing the mathematical Fundamental Theory of Asset Pricing with reference to the emergen…
New insights link memory loss to system stability in dynamical systems.
problem Understanding the relationship between dynamics and computation, especially stability and memory loss.
method Analyzing driven dynamical systems responding to temporal inputs.
result Memory loss in driven systems leads to consistent responses to similar inputs and affects stability.
SmoothDARTS stabilizes DARTS-based architecture search by smoothing loss landscapes.
problem DARTS-based NAS methods suffer from instability, leading to deteriorating architectures.
method SmoothDARTS (SDARTS) uses perturbation-based regularization to smooth the loss landscape.
result SmoothDARTS improves the generalizability and performance of DARTS-based methods.
Recurring international financial crises have adverse socioeconomic effects and demand novel regulatory instruments or strategies for risk management and market stabilization. However, the complex web of market interactions often impedes rational decisions that would absolutely minimize the risk. Here we show that, for…
Math proves deep learning unstable, despite stable neural networks existing.
problem Unstable neural networks in deep learning despite stable ones existing.
method Mathematical proof showing instability of current training procedures.
result Proven existence of stable and accurate neural networks with variable dimensions, but current algorithms cannot compute them.
A new likelihood ratio metric for GANs training stability.
problem Training consistency and stability in GANs.
method Likelihood ratio approach for adversarial optimization.
result New metric for online convergence and stability assessment.
Recently, machine learning (ML) has established itself in various worldwide benchmarking competitions in computational biology, including Critical Assessment of Structure Prediction (CASP) and Drug Design Data Resource (D3R) Grand Challenges. However, the intricate structural complexity and high ML dimensionality of bi…
The Helfrich functional, denoted by H^{c_0}, is a mathematical expression proposed by Helfrich (1973) for the natural free energy carried by an elastic phospholipid bilayer. Helfrich theorises that idealised elastic phospholipid bilayers minimise H^{c_0} among all possible configurations. The functional integrates a sp…
OptiLIME improves LIME explanations by balancing stability and adherence.
problem LIME's instability and lack of reliability in explanations.
method OptiLIME uses a deterministic sampling approach and feature selection to maximize stability while retaining predefined adherence.
result OptiLIME provides more reliable and interpretable explanations.
Survey of mathematical foundations for reinforcement learning.
problem Design and analysis of modern reinforcement learning algorithms.
method Organizes mathematical structures from probability, optimization, and operator theory.
result Unified mathematical entry point for researchers in various fields.
Study examines crypto-backed stable derivatives in DeFi, focusing on DAI.
problem Stability of crypto-backed stablecoins in DeFi.
method Introduced a belief parameter to simulate DAI, proposed a mathematical model, analyzed risk factors.
result Belief parameter improves simulation of DAI price stability.
Lyapunov exponents help understand RNN stability.
problem Optimizing RNNs is sensitive to various parameters.
method Use Lyapunov exponents as dynamical system tools.
result Lyapunov spectrum measures training stability.
Mode connectivity is a surprising phenomenon in the loss landscape of deep nets. Optima -- at least those discovered by gradient-based optimization -- turn out to be connected by simple paths on which the loss function is almost constant. Often, these paths can be chosen to be piece-wise linear, with as few as two segm…
This paper studies the transition from disequilibrium to equilibrium in financial markets.
problem Modeling financial markets as disequilibrium models and analyzing their transition to equilibrium.
method Mathematical analysis using asymptotic limits and Tikhonov-Fenichel reduction.
result Stability of the reduced equilibrium model and economic implications are discussed.
We provide an introduction to the mathematics and physics of the deformed Hermitian-Yang-Mills equation, a fully nonlinear geometric PDE on Kahler manifolds which plays an important role in mirror symmetry. We discuss the physical origin of the equation, and some recent progress towards its solution. In dimension 3 we …
We stabilize MI-based losses by adding a regularization term, improving their performance and stability.
problem Instability of MI-based losses in machine learning.
method Added a novel regularization term to stabilize MI-based losses.
result Regularization stabilizes training and improves the performance of MI-based losses.
This work ensures stability in POD basis interpolation for pMOR in hyperelasticity.
problem Stability of POD basis interpolation on Grassmann manifolds for pMOR in hyperelasticity.
method Stability conditions derived from Grassmannian Exponential map and principal angles.
result Explicit stability conditions for practical pMOR applications and non-monotonic error behavior.
The paper examines fair pricing and hedging stability under small numéraire perturbations.
problem Fair pricing and hedging stability under numéraire perturbations.
method Reformulating the stochastic control problem to show stability and deriving asymptotic formulas.
result Fair price and hedging strategy are stable with small numéraire perturbations.