Develops an oblique projection technique to approximate a foliation for non-normal dynamics.
problem Modeling dynamics far from a primary Spectral Submanifold (SSM) in non-normal systems.
method Oblique projection technique based on experimental data.
result Approximates a stable invariant foliation for non-normal dynamics efficiently.
This paper develops methods to solve saddle-point problems on Riemannian manifolds with exponential stability.
problem Solving saddle-point problems on Riemannian manifolds with exponential stability.
method Developed a projected dynamical system on a Riemannian manifold to solve saddle-point problems, leveraging the strong monotonicity of the gradient of the Lagrangian function.
result Established exponential stability and convergence of the projected dynamical system to the unique saddle-point.
New method learns nonlinear projections for reduced-order modeling of complex dynamical systems.
problem Modeling transient dynamics near a manifold in nonlinear systems.
method Constrained autoencoder neural networks with invertible activation functions and biorthogonal weight matrices.
result Demonstrated effectiveness on a vortex shedding model, learning oblique fibers for fast dynamics.
By using a projective connection over the space of two-dimensional affine connections, we are able to show that the metric interaction of Polyakov 2D gravity with a coadjoint element arises naturally through the projective Ricci tensor. Through the curvature invariants of Thomas-Whitehead, we are able to define an acti…
Proposes PredVAR model for reduced-dimensional dynamics from noisy data.
problem Extracting low-dimensional dynamics from high-dimensional noisy data.
method Probabilistic reduced-dimensional vector autoregressive model with oblique projection.
result Iterative algorithm yields dynamic latent variables with rank-ordered predictability.
Data-driven model reduction captures non-Markovian dynamics using Koopman and Mori-Zwanzig formalisms.
problem Modeling complex, non-Markovian dynamics efficiently and understanding their underlying mechanisms.
method Formulates data-driven model reduction within Koopman and Mori-Zwanzig formalisms, deriving NARMAX models from dynamical systems.
result Shows how data-driven methods can represent non-Markovian dynamics using Koopman and Mori-Zwanzig formalisms.
Geometrically transforms nonconservative dynamics to linearize Kepler and Manev systems.
problem Regularizing and linearizing nonconservative central force dynamics.
method Projective transformation and conformal scaling in configuration and phase spaces.
result Full linearization of Kepler and Manev dynamics in any finite dimension.
PNDEs project neural dynamics onto constraint manifolds, improving accuracy and stability.
problem Learning dynamics from data without violating known constraints.
method Projecting the learned vector field onto the tangent space of the constraint manifold.
result PNDEs outperform existing methods in learning constrained dynamical systems.
Integrable dynamics explained via geometric maps and cluster algebras.
problem Integrable dynamics in projective geometry.
method Twisted triple crossing diagram maps and cluster integrable systems.
result Cross-ratio dynamics described by geometric R-matrices. Characterizes convex cocompact actions in projective space with dynamical properties.
problem Understanding convex cocompact group actions in projective space.
method Dynamical characterization and expansion property analysis.
result Equivalence of convex cocompactness to an expansion property in different Grassmannians.
Given a real vector space V of finite dimension, together with a particular homogeneous field of bivectors that we call a "field of projective forces", we define a law of dynamics such that the position of the particle is a "ray" i.e. a half-line drawn from the origin of V. The impulsion is a bivector whose support is …
We propose a projected gradient dynamical system as a model for a bargaining scheme for an asset for which the two interested agents have personal valuations which do not initially coincide. The personal valuations are formed using subjective beliefs concerning the future states of the world and the reservation prices …
A method models nonlinear dynamics from data using barycentric coordinates and memory.
problem Modeling complex dynamical systems from data.
method SPA for data projection, barycentric coordinates, delay-embedding theorem for memory.
result Stable models of chaotic dynamics and attractors are reproduced.
Regression learns Mori-Zwanzig operators for dynamical systems.
problem Learning Mori-Zwanzig operators for complex dynamical systems.
method Statistical regression to extract Markov and memory operators.
result Regression models improve learning of memory-dependent corrections.
The study examines model risk in real option valuation methods.
problem Model risk in real option valuation methods.
method A decision tree framework to value options to invest or divest in projects.
result Real option values can decrease with volatility and increase with investment costs, contrary to previous literature.
This manuscript is an introduction to the theory of holomorphic foliations on the complex projective plane. Historically the subject has emerged from the theory of ODEs in the complex domain and various attempts to solve Hilbert's 16th Problem, but with the introduction of complex algebraic geometry, foliation theory a…
The paper introduces a fast algorithm for learning and forecasting nonlinear dynamics from noisy time series data.
problem Challenges in capturing nonlinear dynamics from noisy time series data.
method A projected nonlinear state-space model with kernel functions applied to projected lines.
result The model effectively learns and forecasts complex nonlinear dynamics with computational efficiency.
A method for dynamic portfolio choice with uncertain parameters using Pontryagin projection.
problem Continuous-time CRRA portfolio choice in markets with estimated and uncertain coefficients.
method Simulation-based two-stage solver (DPO + Pontryagin projection) to maximize ex-ante objective.
result Projection stabilizes learning and accurately recovers analytic decisions, improving over model-free PPO.
New statistic κ-profile helps monitor weather, soundscapes, and dynamical systems.
problem Monitoring intrinsic dimensionality of large data sets.
method Optimization problem to find κ-profile, which is the norm of the shortest projected secant. result The κ-profile provides a useful statistic for understanding and monitoring large data sets. With the rapid increase of available data for complex systems, there is great interest in the extraction of physically relevant information from massive datasets. Recently, a framework called Sparse Identification of Nonlinear Dynamics (SINDy) has been introduced to identify the governing equations of dynamical systems…
Dynamic treatment effects estimated over time using covariate balancing.
problem Estimating treatment effects in panel data with dynamic treatments.
method Dynamic covariate balancing with potential local projections.
result Established inferential guarantees for the proposed method.
Framework for continuous-time network data representation learning.
problem Learning reliable representations of dynamic network interactions.
method Three-stage process: intensity estimation, projection learning, evolving node representation construction.
result Trajectories satisfy structural and temporal coherence, providing robust inference.
The paper characterizes strong Hamel functions using symmetries and proves their preservation properties.
problem Characterizing strong Hamel functions and their symmetries in Finsler spaces.
method Analyzing geodesic spray, strong dual symmetries, and strong dynamical symmetries.
result Strong Hamel functions can be characterized in terms of strong dual symmetries and strong dynamical symmetries.
The paper proposes a dynamic risk measure approach for evaluating defined-contribution pension funds.
problem Periodic evaluation of defined-contribution pension funds to manage risk and improve projections.
method Dynamic risk measure criterion, model-free reinforcement learning, Lee-Carter mortality model.
result Periodic evaluations lead to more risk-averse strategies, while mortality improvements encourage risk-seeking behaviors.
Efficient methods reduce projections in non-stationary online learning.
problem Optimizing dynamic and adaptive regret in non-stationary online learning environments.
method Presented efficient methods reducing the number of projections per round from O(logT) to 1. result Reduced number of projections per round from O(logT) to 1 for optimizing dynamic and adaptive regret. The paper studies geometric loci and their invariants in complex dynamics.
problem Analyzing geometric loci and their invariants in complex dynamics.
method Intersection theory and dynamical invariants on the flex and gothic loci.
result Determined the divisor class of the flex locus and various tautological intersection numbers on the gothic locus.
We study several new invariants associated to a holomorphic projective structure on a Riemann surface of finite analytic type: the Lyapunov exponent of its holonomy which is of probabilistic/dynamical nature and was introduced in our previous work; the degree which measures the asymptotic covering rate of the developin…
We model how Lipschitz continuity changes during neural network training.
problem Understanding how Lipschitz continuity evolves during training.
method We use a system of stochastic differential equations to capture the dynamics of Lipschitz continuity under SGD.
result We identify three factors driving the evolution of Lipschitz continuity: gradient flow projection, gradient noise, and Hessian projection.
The mesoscopic organization of complex systems, from financial markets to the brain, is an intermediate between the microscopic dynamics of individual units (stocks or neurons, in the mentioned cases), and the macroscopic dynamics of the system as a whole. The organization is determined by "communities" of units whose …
This paper solves the inversion problem for jump processes using Markovian projections.
problem Calibrating jump-diffusion models with both local and stochastic features.
method Inverting Markovian projections for pure jump processes.
result Constructs calibrated local stochastic intensity (LSI) models for credit risk applications.
The paper classifies and decomposes quaternionic projective transformations.
problem Classifying and decomposing elements of the projective linear group PSL(3,H). method Algebraic characterization of dynamical types using reversibility, decomposition of elements into simple elements.
result Offered a complete classification for elements of SL(3,R). Tool uses text mining to define innovative tech fields from abstracts.
problem Defining scope in dynamic, multidisciplinary tech projects.
method Text mining of Elsevier's Scopus abstracts.
result Tool provides crucial information for tech field definition.
New algorithm updates eigenvectors of evolving graphs efficiently.
problem Updating eigenvectors of dynamic graphs.
method Subspace projection based on Rayleigh-Ritz projections.
result Strong performance in eigenvector approximation and downstream tasks.
Next-gen reservoir computing models dynamical systems from time-series data.
problem Modeling dynamical systems from time-series data.
method Pseudorandom nonlinear projection of time-delay embedded inputs.
result Models remain stable over long rollouts and generalize beyond training data.
We prove dynamical stability and instability theorems for compact Einstein metrics under the Ricci flow. We give a nearly complete charactarization of dynamical stability and instability in terms of the conformal Yamabe invariant and the Laplace spectrum. In particular, we prove dynamical stability of some classes of E…
Random projections help in representing sparse graphs efficiently.
problem Efficiently representing sparse graphs of varying sizes and vertex sets.
method Random projection of adjacency matrices to retain graph functionality and properties.
result Random projections can accurately represent graphs of different sizes and vertex sets in the same space.
This work tackles phaseless subspace tracking, recovering time-varying signals from phaseless projections.
problem Recovering time-varying signals from phaseless linear projections under gradual subspace change.
method Dynamic subspace tracking approach, leveraging gradual subspace change over time.
result Demonstrates feasibility of phaseless subspace tracking with gradual subspace change.
Study variance-optimal hedging of forward curve derivatives under stochastic volatility.
problem Variance-optimal hedging of forward curve derivatives with stochastic volatility.
method Assumes HJM-Musiela dynamics modulated by stochastic covariance, uses Galtchouk-Kunita-Watanabe projection.
result Density of finite-maturity strategies, convergence of finite-rank projections, decomposition of hedging error.
The study examines how bias affects hypothesis formation in neural networks.
problem Characterizing the impact of bias on hypothesis formation in neural networks.
method An automated data-driven projection pursuit neural network to extract and select features for binary classification.
result The refinement of a working hypothesis converges to a robust multivariate perception of data.
We describe a general method to construct completely bounded idempotent mappings on operator spaces, starting from amenable semigroups of completely bounded mappings. We then explore several applications of that method to injective operator spaces, fixed points of completely contractive mappings, Toeplitz operators, dy…
A new method uses matrix sketches for efficient graph clustering in dynamic environments.
problem Efficiently clustering large, dynamic graphs in distributed memory systems.
method Inspired by spectral clustering, the approach uses random dimension-reducing projections to derive matrix sketches.
result The method produces embeddings that yield performant clustering results in a fully-dynamic stochastic block model stream.
AKOrN uses synchronized neurons to improve AI tasks.
problem Improving AI performance through better neural representations.
method AKOrN introduces synchronized neurons to replace threshold units.
result AKOrN improves performance across various AI tasks.
The paper proposes a new model for financial order books without assuming prices or quantities.
problem Understanding the geometry of financial order books without assuming prices or quantities.
method Modeling financial order books as an inflationary relational system without metric, temporal, or price coordinates. Observable quantities arise through spectral embeddings of the graph Laplacian.
result Projected supply and demand are constrained to gamma-like functional forms, which can be observed as integrated-gamma cumulative profiles in high-frequency data.
In this paper, I give a new construction of a Kähler-Einstein metrics on a smooth projective variety with ample canonical bundle. This result can be generalized to the construction of a singular Kähler-Einstein metric on a smooth projective variety of general type which gives an AZD of the canonical bundle. Also the va…
In this paper, we train a recurrent neural network to learn dynamics of a chaotic road environment and to project the future of the environment on an image. Future projection can be used to anticipate an unseen environment for example, in autonomous driving. Road environment is highly dynamic and complex due to the int…
Model analyzes corruption dynamics on an Ising lattice.
problem Tackles corruption dynamics on an Ising lattice.
method Formulated as an Ising lattice model with stochastic Markov process.
result Demonstrates different asymptotic states of corruption networks.
Proposes a method to learn stable invariant sets in dynamical systems.
problem Learning stable invariant sets in general dynamical systems.
method Generalizes Manek and Kolter's approach by introducing projection onto latent space shapes and using invertible neural networks.
result Validates the method and shows its usefulness for long-term prediction.
Bi-contact surgery operations can be applied to Anosov flows.
problem Characterizing Anosov flows and their properties.
method Metric and contact geometric characterizations, Liouville geometry, Reeb dynamics.
result Bi-contact surgery operations can be applied to Anosov flows.