This work's purpose is to understand the dynamics of some social systems whose properties can be captured by certain iterated function systems. To achieve this intension, we start from the theory of iterated function systems, and then we study two specific economic models on random utility function and optimal stochast…
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48 results for “iterative function systems”
Recurrent iterated function systems (RIFSs) are improvements of iterated function systems (IFSs) using elements of the theory of Marcovian stochastic processes which can produce more natural looking images. We construct new RIFSs consisting substantially of a vertical contraction factor function and nonlinear transform…
Study on self-similar sets on Riemannian manifolds with new separation conditions.
problem Analyzing self-similar sets on Riemannian manifolds with new separation conditions.
method Formulated weak separation and finite type conditions for conformal iterated function systems on Riemannian manifolds.
result Obtained formulas for Hausdorff dimensions of self-similar and graph self-similar sets.
The two main theorems of this paper provide a characterization of hyperbolic affine iterated function systems defined on Rm. Atsushi Kameyama (Distances on Topological Self-Similar Sets, Proceedings of Symposia in Pure Mathematics, Volume 72.1, 2004) asked the following fundamental question: given a topological self-si…
Adaptive optimal control of nonlinear dynamic systems with deterministic and known dynamics under a known undiscounted infinite-horizon cost function is investigated. Policy iteration scheme initiated using a stabilizing initial control is analyzed in solving the problem. The convergence of the iterations and the optim…
Stability Analysis of Optimal Adaptive Control using Value Iteration with Approximation Errorsmath.OC
The paper analyzes stability in adaptive control systems with value iteration, considering approximation errors.
problem Stability of adaptive control systems during learning with approximation errors.
method Theoretical analysis of stability using value iteration, considering both constant and time-varying control policies.
result Estimations of the region of attraction to ensure function approximation remains valid.
Paper introduces Ricci iteration for studying Kähler-Einstein metrics.
problem Study of coupled Kähler-Einstein metrics.
method Coupled Ricci iteration, proving smooth convergence and coercivity.
result Smooth convergence of iteration for both negative and positive first Chern classes.
Meta-learning control algorithm with finite-time guarantees for unknown systems.
problem Online control of unknown linear systems with constraints.
method Provable regret guarantees for an iterative control algorithm.
result Regret bounds of for controller cost and constraint violation.
Let be the limit set of a conformal dynamical system, i.e. a Kleinian group acting on either finite- or infinite-dimensional real Hilbert space, a conformal iterated function system, or a rational function. We give an easily expressible sufficient condition, requiring that the limit set is not too much bigger than …
In this paper, two Q-learning (QL) methods are proposed and their convergence theories are established for addressing the model-free optimal control problem of general nonlinear continuous-time systems. By introducing the Q-function for continuous-time systems, policy iteration based QL (PIQL) and value iteration based…
Adaptive neural network approximates stochastic system densities.
problem Approximating high-dimensional stochastic dynamical systems.
method Temporal KRnet (tKRnet) trained with adaptive collocation points and temporal decomposition.
result Improves density approximation for stochastic systems without curse of dimensionality.
Efficiently differentiate functions of large matrices using new adjoint systems.
problem Differentiating functions of large matrices in scientific and probabilistic machine learning models.
method Deriving and implementing new adjoint systems for Lanczos and Arnoldi iterations in JAX.
result Efficient differentiation of PDEs, Gaussian process models, and Bayesian neural networks.
An iterated function system consisting of contractive similarity mappings has a unique attractor which is invariant under the action of the system, as was shown by Hutchinson [Hut]. This paper shows how the action of the function system naturally produces a tiling of the con…
Deep networks analyzed using IFS theory for stability and generalization.
problem Stability and generalization of deep neural networks.
method Viewing deep architectures as place-dependent IFS and applying results from random dynamical systems.
result Derivation of a Wasserstein generalization bound and a new training objective.
This paper addresses the model-free nonlinear optimal problem with generalized cost functional, and a data-based reinforcement learning technique is developed. It is known that the nonlinear optimal control problem relies on the solution of the Hamilton-Jacobi-Bellman (HJB) equation, which is a nonlinear partial differ…
This dissertation advances scalable Gaussian processes using iterative methods and pathwise conditioning.
problem The classical Gaussian process formulation is not scalable for large datasets and modern hardware.
method Combining iterative methods and pathwise conditioning to improve scalability.
result Significantly reduced memory requirements and facilitated application to larger datasets.
Paper proposes a pre-conditioning method to speed up gradient descent in multi-agent optimization.
problem Speed up convergence of gradient descent in multi-agent optimization problems.
method Iterative pre-conditioning approach to mitigate the effect of problem conditioning.
result Significant improvement in convergence speed of gradient descent method.
Adaptive optimal control using value iteration (VI) initiated from a stabilizing policy is theoretically analyzed in various aspects including the continuity of the result, the stability of the system operated using any single/constant resulting control policy, the stability of the system operated using the evolving/ti…
AI-driven framework optimizes MCMC-based preconditioners for faster linear system solving.
problem Slow convergence of Krylov subspace solvers for ill-conditioned matrices.
method Graph neural surrogate and Bayesian optimization for AI-tuned MCMC parameters.
result 50% reduction in iterations to convergence on unseen system.
Iterative method learns unknown constraints for MPC control.
problem Learning to satisfy unknown polyhedral state constraints in iterative MPC.
method Collects and improves estimates of unknown constraints using collected data, designs an MPC controller to satisfy the estimated constraints.
result Robust and probabilistic guarantees of constraint satisfaction as a function of task iterations.
Exponential distribution is ubiquitous in the framework of multi-agent systems. Usually, it appears as an equilibrium state in the asymptotic time evolution of statistical systems. It has been explained from very different perspectives. In statistical physics, it is obtained from the principle of maximum entropy. In th…
Asynchronous framework improves distributed learning performance.
problem Heterogeneous computing machines hinder synchronous learning strategies.
method Asynchronous distributed framework with parameter exchanges.
result Convergence of consistency in distributed asynchronous methods for gradient iterations.
iDEM generates samples from Boltzmann densities without data.
problem Generating statistically independent samples from unnormalized distributions.
method Iterative algorithm using energy and gradient for diffusion-based sampler training.
result iDEM achieves state-of-the-art performance and trains faster than existing methods.
Symbolic regression constructs smooth value functions for reinforcement learning.
problem Function approximators in reinforcement learning are black-box models with hyper-parameter tuning.
method Symbolic regression methods for constructing smooth value functions in the form of analytic expressions.
result Symbolic regression methods yield well-performing policies and are compact and mathematically tractable.
New model-free algorithm for LQ control with sublinear regret.
problem Model-free reinforcement learning for adaptive control of linear quadratic systems.
method Reduction to expert prediction problem, policy iteration with forced exploration.
result Algorithm achieves sublinear regret and polynomial computation cost.
Proposes a privacy-preserving sign selection method for distributed systems.
problem Sign selection in distributed differentially private settings.
method Iterative peeling of stability function combined with exponential mechanism.
result Recovery of support and signs with optimal signal-to-noise ratio.
Extreme points in limit setsmath.DS
Characterizes extreme points in polygon limit sets.
problem Identifying boundary points in polygon limit sets.
method Characterization through affine dilations and polygon vertices.
result Characterizes which points lie on the boundary of convex hull.
Paper learns Koopman operator from sparse data, escaping function space constraints.
problem Learning Koopman operator from non-closed function spaces.
method Operator stochastic approximation algorithm using conditional mean embeddings (CME).
result Online sparse learning algorithm with trajectory-based sampling guarantees.
Theoretical model for iterative user discovery in recommender systems.
problem Iterative feedback loops in recommender systems and their biases.
method Theoretical framework to model system evolution and convergence properties.
result Theoretical bounds and convergence properties on user discovery and blind spots.
The paper studies Lipschitz equivalence of self-similar sets and their augmented trees.
problem Lipschitz equivalence of self-similar sets and their boundaries.
method Introducing simple augmented trees and using combinatorial devices to show Lipschitz equivalence.
result Lipschitz equivalence of self-similar sets and their boundaries.
Neural Lyapunov Redesigneess.SY
Combines Lyapunov functions with controller synthesis for safe control policies.
problem Ensuring safety in controller design for nonlinear systems.
method Iterative algorithm combining Lyapunov function estimation and controller synthesis.
result Effective control policies with large safe regions are derived.
Index iteration theory for symplectic paths with applications to nonlinear Hamiltonian systemsmath.DG
In recent years, we have established the iteration theory of the index for symplectic matrix paths and applied it to periodic solution problems of nonlinear Hamiltonian systems. This paper is a survey on these results.
Theoretical and Numerical Analysis of Approximate Dynamic Programming with Approximation Errorseess.SY
This study is aimed at answering the famous question of how the approximation errors at each iteration of Approximate Dynamic Programming (ADP) affect the quality of the final results considering the fact that errors at each iteration affect the next iteration. To this goal, convergence of Value Iteration scheme of ADP…
A large class of semi-Hamiltonian systems of hydrodynamic type is interpreted as the equations governing families of critical points of functions obeying the classical linear Darboux equations for conjugate nets.The distinguished role of the Euler-Poisson-Darboux equations and associated Lauricella-type functions is em…
DNNs can approximate fractal functions with exponential linear regions.
problem Understanding neural network approximations of complex functions.
method Using Iterated Function Systems (IFS) and neural networks to generate fractal functions.
result DNNs can generate fractal functions with a number of linear regions exponential in the number of parameters.
Several new mutation-periodic quivers of period higher than 1 are introduced as well as the associated discrete dynamical systems. The reduction of these systems is developed using either a presymplectic or a Poisson approach. The presymplectic approach leads to a reduced system whose iteration map is symplectic with r…
A method to construct fractal surfaces by recurrent fractal curves is provided. First we construct fractal interpolation curves using a recurrent iterated functions system(RIFS) with function scaling factors and estimate their box-counting dimension. Then we present a method of construction of wider class of fractal su…
We analyze deep neural networks in the large size and iteration limit, revealing a deterministic system of equations.
problem Understanding the behavior of deep neural networks in the asymptotic regime of large network sizes and iterations.
method Sequential limit of each hidden layer and characterization of parameter evolution, using weak convergence and stochastic analysis.
result The limit neural network recovers a global minimum with zero loss for the objective function.
New framework reduces fault tolerance costs in machine learning.
problem Fault tolerance in iterative-convergent machine learning algorithms.
method Developed a general framework to quantify and design strategies for checkpoint-based fault tolerance.
result SCAR reduces iteration cost of partial failures by 78% - 95%.
Convergence and Dynamical Behavior of the ADAM Algorithm for Non-Convex Stochastic Optimizationstat.ML
Adam optimizes non-convex functions, converging to critical points.
problem Finding local minima of non-convex functions.
method Continuous-time model of Adam, ODE approximation, decreasing stepsize.
result Adam converges to critical points of non-convex functions.
Addressing RL's agent-environment boundary issues, a novel analysis ensures optimal value functions are invariant.
problem Fundamental RL concepts like value functions are not uniquely defined due to the agent-environment boundary.
method A boundary-invariant analysis of Fitted Q-Iteration, ensuring optimality guarantees are independent of the boundary choice.
result Theoretical analyses of RL algorithms, including Fitted Q-Iteration, are made invariant to the boundary choice.
New study shows faster convergence of SGD and Kaczmarz methods.
problem Improving convergence rates of iterative linear system solvers.
method Last-iterate convergence analysis of SGD with greedy step size over smooth quadratics.
result The -th iterate attains an convergence rate.
This paper speeds up iterative GP inference with warm starting.
problem Improving scalability of Gaussian process inference.
method Warm starting sequential posteriors using known solutions.
result Significant speed-ups and improved Bayesian optimisation performance.
Refines neural network predictions using background knowledge for improved accuracy.
problem Compensate for lack of labeled data in neural networks.
method Introduces differentiable refinement functions and Iterative Local Refinement (ILR) algorithm to refine predictions efficiently and accurately.
result ILR finds competitive results in MNIST addition task and refines predictions on complex SAT formulas.
Rotates MFVI for better Gaussian approximations.
problem Improving variational approximations for complex distributions.
method Rotated coordinate system, PCA-based rotation, iterative Gaussianization.
result Significantly more accurate approximations with lower computational cost.
Warm starts improve Gaussian process regression by up to 16x.
problem Optimizing hyperparameters for Gaussian processes.
method Iterative Gaussian processes with warm start optimization.
result Warm starts achieve the same results as conventional methods but significantly speed up computations.
Algorithm optimizes system design and control for better rewards.
problem Optimizing system design and control for maximum rewards.
method Deep reinforcement learning combining policy gradient and model-based optimization.
result DEPS algorithm outperforms state-of-the-art methods in various environments.
We present a general theory of fractal transformations and show how it leads to a new type of method for filtering and transforming digital images. This work substantially generalizes earlier work on fractal tops. The approach involves fractal geometry, chaotic dynamics, and an interplay between discrete and continuous…