Lower bound found for Perron-Frobenius degrees of certain complex numbers.
problem Finding lower bounds for the Perron-Frobenius degree of complex numbers.
method Using Doug Lind's idea, proving results for both cubic and biPerron numbers.
result Arbitrary large Perron-Frobenius degrees for certain complex numbers.
Deep learning framework for kernel methods using RKHM and Perron-Frobenius operators.
problem Kernel methods in deep learning with potential overfitting issues.
method Combining RKHM and Perron-Frobenius operator to derive a new Rademacher bound and analyze deep kernel methods.
result Theoretical interpretation of benign overfitting and milder dependency on output dimension.
A pseudo-Anosov surface automorphism φ has associated to it an algebraic unit λφ called the dilatation of φ. It is known that in many cases λφ appears as the spectral radius of a Perron-Frobenius matrix preserving a symplectic form L. We investigate what algebraic units could potentially appear as dilatatio…
Develops a metric for comparing nonlinear dynamical systems.
problem Developing a metric for nonlinear dynamical systems.
method Using Perron-Frobenius operators in reproducing kernel Hilbert spaces.
result Includes existing fundamental metrics as special cases.
Groups grow faster than their subgroups, proven using special tools.
problem Growth rates of subgroups within hyperbolic groups.
method Automatic structures, Perron-Frobenius theory, growth tightness, rotating families.
result Non-elementary hyperbolic groups grow exponentially faster than their quasiconvex subgroups.
Study of endperiodic maps on infinite graphs, proving homotopy and eigenvalue properties.
problem Understanding endperiodic maps on infinite graphs with finitely many ends.
method Adapting relative train track maps and combinatorial techniques to infinite type setting.
result Any generalized endperiodic map is homotopic to a relative train track map.
Existence and uniqueness of discrete Einstein metrics on trees proven.
problem Existence and uniqueness of discrete Einstein metrics on trees.
method Using Perron-Frobenius theory and Lin-Lu-Yau Ricci curvature.
result Existence and uniqueness of discrete Einstein metrics on trees established.
Study on self-consuming generative models with diverse human curation, focusing on convergence and stability.
problem Analyzing self-consuming generative models with heterogeneous human curation.
method Investigates the asymptotic behavior of retraining dynamics using nonlinear Perron--Frobenius theory and Banach contraction mapping.
result Improves convergence results and provides stability and non-stability analyses for the model.
IGNN captures long-range graph dependencies using fixed-point equations.
problem Limited GNN ability to capture long-range graph dependencies.
method Fixed-point equilibrium equations involving implicitly defined state vectors, leveraging Perron-Frobenius theory and projected gradient descent.
result IGNN consistently captures long-range dependencies and outperforms state-of-the-art GNNs.
New methods for clustering graphs using spectral analysis.
problem Graph clustering for complex systems.
method Transfer operators and spectral properties.
result Spectral clustering can be interpreted using Koopman operators.
Paper introduces RKHM for more explicit variable structures analysis.
problem Explicitly analyzing structures among variables.
method Orthonormal systems in Hilbert C∗-modules, RKHM. result Theoretical and practical procedures for RKHM orthonormalization.
Stochastic discount factor (SDF) processes in dynamic economies admit a permanent-transitory decomposition in which the permanent component characterizes pricing over long investment horizons. This paper introduces an empirical framework to analyze the permanent-transitory decomposition of SDF processes. Specifically, …
GradientDICE improves offline estimation for reinforcement learning policies.
problem Estimating density ratios between target policy and sampling distributions in reinforcement learning.
method GradientDICE reparameterizes the optimization problem to avoid nonlinearity, ensuring convergence and consistency.
result GradientDICE is provably convergent and eliminates the need for nonlinearity in parameterization.
This paper extends transfer operator theory to McKean-Vlasov equations.
problem Analyzing the behavior of complex dynamical systems using transfer operators.
method Extended dynamic mode decomposition and Galerkin projection.
result Finite-dimensional approximations of transfer operators computed.
Fixed points of nonnegative neural networks are analyzed using fixed point theory.
problem Analyzing fixed points in nonnegative neural networks.
method Fixed point theory, nonlinear Perron-Frobenius theory, monotonic and scalable mappings.
result Conditions for the existence of fixed points in nonnegative neural networks are provided.
Paper develops metrics for random dynamical systems using vector-valued RKHSs.
problem Creating metrics for random nonlinear dynamical systems.
method Develops metrics on random dynamical systems using Perron-Frobenius operators in vector-valued reproducing kernel Hilbert spaces (vvRKHSs). Uses operator-valued kernels and time-wise independence criteria.
result Extends existing metrics for deterministic systems and introduces kernel maximal mean discrepancy for random processes.
The k-dimensional Dehn (or isoperimetric) function of a group bounds the volume of efficient ball-fillings of k-spheres mapped into k-connected spaces on which the group acts properly and cocompactly; the bound is given as a function of the volume of the sphere. We advance significantly the observed range of behavior f…
Develops a Krylov subspace method for estimating nonlinear systems with random noise.
problem Estimating nonlinear dynamical systems with random noise.
method Lifted representation of nonlinear dynamical systems using transfer operators, extended Arnoldi method, and shift-invert Arnoldi method.
result Empirical validation of methods on synthetic and real-world healthcare data.
Asset prices contain information about the probability distribution of future states and the stochastic discounting of those states as used by investors. To better understand the challenge in distinguishing investors' beliefs from risk-adjusted discounting, we use Perron-Frobenius Theory to isolate a positive martingal…
If (M,g) is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes of order o(λ) saturating sup-norm estimates. In particular, it gives optimal conditions for existence of eigenfunctions satisfying maximal sup norm bounds. The condition is …
In 1981 Masur proved the existence of a dense geodesic in the moduli space for a Teichmüller space. We prove an analogue theorem for reduced Outer Space endowed with the Lipschitz metric. We also prove two results possibly of independent interest: we show Brun's unordered algorithm weakly converges and from this prove …
Kernel operators help detect patterns in complex data.
problem Detecting long-lived coherent patterns in high-dimensional time-series data.
method Dominant eigenfunctions of kernel transfer operators combined with gradient-based optimization.
result Effective detection of long-lived coherent patterns in high-dimensional time-series data.
This paper provides a functional analytic foundation for singular value decomposition of RKHS operators.
problem Singular value decomposition of operators on RKHSs.
method Functional analytic approach, extending matrix eigenvalue problems to RKHS operators.
result Solid foundation and extension of singular value decomposition to RKHS operators.
The paper estimates key metrics for linear models with Markov or hidden Markov sources.
problem Estimating free energy, mutual information, and MMSE for linear models with specific signal priors.
method Replica analysis in statistical physics, focusing on Markov and hidden Markov sources.
result The linear model with Markov or hidden Markov sources can be simplified into decoupled AWGN channels.
In this paper we develop the metric theory for the outer space of a free product of groups. This generalizes the theory of the outer space of a free group, and includes its relative versions. The outer space of a free product is made of G-trees with possibly non-trivial vertex stabilisers. The strategies are the same…
Kernel-based methods extend transfer operator theory to new domains.
problem Analyzing complex dynamical systems and extracting meaningful information.
method Eigendecompositions in reproducing kernel Hilbert spaces.
result Kernel-based methods can be applied to any domain with a kernel similarity measure.
RaNNDy uses randomized neural networks to learn transfer operators efficiently.
problem Efficiently learning transfer operators from data.
method Randomized neural network approach with randomly initialized hidden layers and trained output layer.
result Significant reduction in training time and resources with improved stability.
Data-driven methods link graphon limits to random walks and spectral clustering.
problem Clustering signals evolving over time with graphon limits.
method Transfer operators, Koopman and Perron-Frobenius, for estimating graphon from signal data.
result Spectral clustering can be extended to graphons, reconstructing transition densities and graphons.
Study of particle systems with singular interaction through hitting times, revealing new phenomena and equilibrium strategies.
problem Understanding and predicting times of fragility in particle systems with strategic connections.
method General driving processes, inhomogeneous connection structures, strategic particle connections, max-plus algebra.
result Characterization of times of fragility and system regularization in equilibrium.
New methods improve stability of Sinkhorn algorithm in machine learning.
problem Stability of Sinkhorn semigroups in high-dimensional settings.
method Semigroup analysis based on contraction coefficients and Lyapunov-type operator-theoretic techniques.
result Unified and simplified arguments in Sinkhorn algorithm stability.
New guarantees for uniquely identifying transport maps and vector fields from finite measure-valued data.
problem Unique recovery of transport maps and vector fields from finite measure-valued data.
method Use of Whitney and Takens embedding theorems to establish conditions for unique identification.
result New metric for comparing diffeomorphisms and analogous results in infinitesimal settings.
New algorithms compute Koopman operators on RKHSs efficiently and accurately.
problem Data-driven spectral analysis of Koopman operators on RKHSs.
method General, provably convergent algorithms for RKHSs.
result Optimal algorithms with error control and spectral measures.