Survey on statistical learning theory for control, focusing on linear systems.
problem Applying machine learning techniques to control systems, especially linear ones.
method Adapting tools from modern high-dimensional statistics and learning theory.
result Recent advances in statistical learning theory for control, particularly for linear systems.
Introduces linear K-systems for Hamiltonian Floer theory.
problem Constructing Floer cohomology for Hamiltonian systems on compact manifolds.
method Geometric realization of linear K-systems via pseudo-holomorphic curves and inductive construction of Kuranishi structures.
result Construction of Floer cohomology and isomorphism with singular cohomology.
Study on neural scaling laws for solving linear systems in-context.
problem Theoretical guarantees for solving linear systems using a linear transformer architecture.
method Neural scaling laws and task diversity for in-domain and out-of-domain generalization.
result Novel notion of task diversity for necessary and sufficient condition of generalization under task shifts.
Symmetries in shell theory lead to multiple deformation possibilities.
problem Understanding symmetries in thin shell deformation theory.
method Analyzing symmetries in the context of linear theory of thin shells.
result Infinitely many deformations without shear strains and twisting.
Learning theory for linear systems with compositional inputs.
problem Training linear system operators with unknown variables constrained to non-negativity and unity.
method Bayesian inversion method for inferring unknown variable from noisy linear system output.
result Quantified uncertainty in trained operator and convergence rates for various cases.
The paper explores when linear system identification is hard or easy, especially for under-actuated systems.
problem Statistical hardness of learning linear systems, especially under-actuated or under-excited systems.
method Using tools from minimax theory and recent statistical tools for finite sample analysis of system identification.
result The controllability index of linear systems affects the sample complexity of identification, making some systems hard to learn.
Modeling financial systemic risk with optimal control theory for stability.
problem Analyzing and stabilizing systemic risk in interconnected financial entities.
method Developed a theoretical model using optimal control theory, including steps for synthesizing stabilizing controllers.
result The model ensures that the H ∞ H^{\infty} H ∞ norms of the mappings from disturbance to output are less than a predefined constant, stabilizing the system. Tutorial on using concentration inequalities for linear system identification.
problem Learning state-space parameters of linear systems.
method Large-deviations and self-normalized martingales.
result Data-dependent and independent bounds on learning rate.
New insights into cascade feedback linearization of control systems.
problem Obtaining a cascade feedback linearization for invariant control systems.
method Introducing truncated versions of operators from the calculus of variations to prove new theorems.
result Established new geometry and foundational theorems for future work.
New algorithm learns linear dynamical systems from measurements.
problem Learning system dynamics from linear measurements efficiently and accurately.
method Method of moments estimator to directly estimate Markov parameters.
result First polynomial time algorithm for learning linear dynamical systems.
The study sets limits on how well systems can be controlled adaptively.
problem Learning to control unknown linear Gaussian systems with quadratic costs.
method Combining ideas from experiment design, estimation theory, and perturbation bounds of information matrices.
result Regret lower bounds of the order of T \sqrt{T} T in the time horizon T T T accurately capture control-theoretic parameters. A tutorial on non-asymptotic system identification methods.
problem Identifying system parameters in linear models.
method Covering technique, Hanson-Wright Inequality, method of self-normalized martingales.
result Streamlined proofs of least-squares based estimator performance.
Regularization leads to balancedness in deep linear networks.
problem Balancedness in deep linear networks.
method Geometric invariant theory and Riemannian geometry of fibers.
result Balancing flows converge to the balanced manifold at a uniform exponential rate.
Methods from learning theory are used in the state space of linear dynamical and control systems in order to estimate the system matrices. An application to stabilization via algebraic Riccati equations is included. The approach is illustrated via a series of numerical examples.
Unified analysis of TD learning using MJLS theory for linear function approximators.
problem Characterizing the exact behaviors of TD learning algorithms with linear function approximators.
method Exploiting connections to Markov jump linear systems (MJLS) theory to analyze TD learning algorithms.
result Closed-form expressions for mean and covariance matrix of TD estimation error at any time step.
Geometric framework for dynamic feedback linearization of control systems with symmetry.
problem Dynamic feedback linearization of control systems with symmetry.
method Geometric framework based on Lie symmetry, systematic procedure for all smooth, generic system trajectories.
result Sufficient condition for dynamic feedback linearizability obtained.
Reduces Hamilton-Jacobi theory for nonholonomic systems with symmetries.
problem Hamilton-Jacobi theory for nonholonomic systems with symmetries.
method Reduction procedure for Hamilton-Jacobi equation.
result Reconstructs solutions in the unreduced picture from reduced equations.
We propose a geometric correspondence between (a) linearly degenerate systems of conservation laws with rectilinear rarefaction curves and (b) congruences of lines in projective space whose developable surfaces are planar pencils of lines. We prove that in projective 4-space such congruences are necessarily linear. Bas…
The Kosambi-Cartan-Chern (KCC) theory represents a powerful mathematical method for the investigation of the properties of dynamical systems. The KCC theory introduces a geometric description of the time evolution of a dynamical system, with the solution curves of the dynamical system described by methods inspired by t…
A powerful mathematical method for the investigation of the properties of dynamical systems is represented by the Kosambi-Cartan-Chern (KCC) theory. In this approach the time evolution of a dynamical system is described in geometric terms, treating the solution curves of a dynamical system by geometrical methods inspir…
Investigates financial and economic systems using statistical mechanics and information theory.
problem Complexity, asymmetry, stochasticity, and non-linearity in financial and economic systems.
method Model-based and empirical analyses using statistical mechanics and information theory.
result Derives probability distribution functions for better understanding of financial and economic dynamics.
New method identifies key genes affecting phenotypes in biological systems.
problem Identifying genes that drive specific phenotypes in complex biological systems.
method Data-driven observability decomposition using Koopman operators.
result Koopman operator representation identifies genes that drive phenotypes.
We learn linear models from nonlinear systems using multiple trajectories and regularization.
problem Identifying linear models from data when the underlying dynamics are nonlinear.
method Multiple trajectories data acquisition followed by regularized least squares.
result Learn linearized dynamics with arbitrarily small error given enough samples.
We study the eleven dimensional supergravity equations which describe a low energy approximation to string theories and are related to M-theory under the AdS/CFT correspondence. These equations take the form of a non-linear differential system, on B 7 × S 4 \mathbb{B}^7\times\mathbb{S}^4 B 7 × S 4 with the characteristic degeneracy at t…
Develops inverse EKF for non-linear systems with stability guarantees and learning unknown dynamics.
problem Estimating adversary's Kalman-filtered estimates in highly non-linear systems.
method Proposes inverse extended Kalman filter (I-EKF) for second-order, Gaussian sum, and dithered forward models. Uses reproducing kernel Hilbert space for learning unknown dynamics.
result Derives theoretical stability guarantees for inverse second-order EKF.
The paper studies optimal transport in linear quadratic systems and derives interpolation inequalities.
problem Optimal transport problem in Linear Quadratic optimal control systems.
method Well-posedness of the Monge problem, regularity of optimal transport map, displacement interpolation of measures.
result Derivation of general interpolation inequalities for entropy functionals.
We study the interplay between the differential Galois group and the Lie algebra of infinitesimal symmetries of systems of linear differential equations. We show that some symmetries can be seen as solutions of a hierarchy of linear differential systems. We show that the existence of rational symmetries constrains the …
We identify linear models from nonlinear systems with initialization constraints.
problem Identifying linear models from nonlinear systems with initialization constraints.
method Multiple trajectories-based deterministic data acquisition algorithm followed by regularized least squares.
result We provide a finite sample error bound on the learned linearized dynamics.
New systems of linear PDEs discovered in 3D contact manifolds.
problem Investigating linear PDEs of s l 3 \mathfrak{sl}_3 sl 3 -type. method Complete local classification using extrinsic geometry.
result 7 new systems of second-order linear PDEs with 8-dimensional solution spaces.
Efficient algorithm controls unknown systems with adversarial perturbations.
problem Controlling unknown linear systems with adversarial perturbations and convex losses.
method Measures regret against an optimal linear policy, provides efficient algorithm with sublinear regret bound.
result First efficient algorithm with sublinear regret bound of T^{2/3}.
The paper tackles joint learning of linear systems, improving accuracy with pooled data.
problem Estimating transition matrices of multiple related linear systems more accurately.
method Developed novel techniques to bound estimation errors and establish high probability bounds for singular values.
result Significant gains in accuracy achieved by pooling data across systems.
Safe control of systems with unknown dynamics using persistent excitation.
problem Tension between safety and exploration in data-driven control.
method System identification through persistent excitation, robust constraint satisfaction, and synthesis of feedback controllers.
result Non-asymptotic guarantees on estimation and controller performance.
This paper studies the Yang--Mills ASD equation over the cylinder as a non-linear evolution equation. We consider a dynamical system consisting of bounded orbits of this evolution equation. This system contains many chaotic orbits, and moreover it becomes an infinite dimensional and infinite entropy system. We study th…
Study derivative-free methods for linear policies in linear-quadratic systems.
problem Optimizing policies in linear-quadratic systems with limited derivative information.
method Derivative-free methods applied to linear policies over various noise and reward feedback settings.
result These methods converge to near-optimal policies with a polynomial number of zero-order evaluations.
We show that the theory of isothermic surfaces in $\E^3$ -- one of the oldest branches of differential geometry -- can be reformulated within the modern theory of completely integrable (soliton) systems. This enables one to study the geometry of isothermic surfaces in $\E^3$ by means of powerful spectral methods availa…
New theory explains how Normalizing Flows represent data.
problem Lack of theoretical foundation in Normalizing Flows.
method Linear systems theory applied to Normalizing Flows.
result Optimal flows learn to represent local covariance.
New approach learns mixtures of linear dynamical systems without separation conditions.
problem Learning mixtures of linear dynamical systems with better fit or understanding.
method Tensor decompositions to learn mixtures of linear dynamical systems.
result Algorithm succeeds without strong separation conditions and can compete with Bayes optimal clustering.
Paper tackles robust control policy learning for uncertain systems.
problem Learning control policies for an unknown linear dynamical system with quadratic cost.
method Convex optimization method balancing exploitation and exploration.
result Minimizes worst-case cost by reducing uncertainty in model parameters.
Signature tensors uniquely identify ODE solutions.
problem Identifying ODE solutions from signature tensors.
method Geometric theory of nonlinear systems of ODEs.
result Necessary and sufficient algebraic conditions for signature tensors to represent ODE solutions.
In this paper we investigate overdetermined systems of scalar PDEs on the plane with one common characteristic, whose general solution depends on 1 function of 1 variable. We describe linearization of such systems and their integration via Laplace transformation, relating this to Lie's integration theorem and formal th…
This study provides a new mathematical structure for Koopman eigenfunctions.
problem Understanding and representing nonlinear dynamics as linear.
method Theoretical, analytical, and numerical approaches to Koopman eigenfunction space.
result Equivalence of minimal generating set and maximal independent set, defining conditions for independence.
New proof of generalized Chow-Rashevskii theorem for non-linear systems.
problem Generalized Chow-Rashevskii Theorem for non-linear systems.
method Independent proof structure allowing generalizations to orbits of compositions of flows.
result Proof structure applicable to applications in Control Theory and controllability criteria.
Unified method for solving extrinsic geometry problems.
problem General equivalence problem of extrinsic geometry.
method Formulation of osculating maps and algorithm for invariants.
result Categorical isomorphism between extrinsic geometries and involutive systems of linear differential equations.
Complex frequency generalizes eigenvalues in LTI systems.
problem Characterizing dynamics of signals with complex values.
method Geometric frequency interpretation and transformation analysis.
result Complex frequencies in LTI systems match eigenvalues.
Differential Galois theory connects connections with parameters to isomonodromic deformations.
problem Understanding the Galois group of connections with parameters.
method Geometric setting and classical results on differential algebraic groups and Lie algebra bundles.
result Galois groups of connections with parameters are determined by isomonodromic deformations.
The aim of this paper is to prove that a control affine system on a manifold is equivalent by diffeomorphism to a linear system on a Lie group or a homogeneous space if and only the vector fields of the system are complete and generate a finite dimensional Lie algebra. A vector field on a connected Lie group is linear …
SKOLR uses linear RNNs to approximate Koopman operators for time-series forecasting.
problem Nonlinear dynamical system analysis and time-series forecasting with infinite-dimensional Koopman operators.
method Established a connection between Koopman operator approximation and linear RNNs, integrating learnable spectral decomposition and MLP.
result SKOLR delivers exceptional performance in various forecasting benchmarks and dynamical systems.
We show that classical Wilczynski--Se-ashi invariants of linear systems of ordinary differential equations are generalized in a natural way to contact invariants of non-linear ODEs. We explore geometric structures associated with equations that have vanishing generalized Wilczynski invariants and establish relationship…