Deep Koopman-layered Model with Universal Property Based on Toeplitz Matrices

• URL: http://arxiv.org/abs/2410.02199v1
• Date: Thu, 3 Oct 2024 04:27:46 GMT
• Title: Deep Koopman-layered Model with Universal Property Based on Toeplitz Matrices
• Authors: Yuka Hashimoto, Tomoharu Iwata,
• Abstract summary: The proposed model has both theoretical solidness and flexibility. The flexibility of the proposed model enables the model to fit time-series data coming from nonautonomous dynamical systems.
• Score: 26.96258010698567
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We propose deep Koopman-layered models with learnable parameters in the form of Toeplitz matrices for analyzing the dynamics of time-series data. The proposed model has both theoretical solidness and flexibility. By virtue of the universal property of Toeplitz matrices and the reproducing property underlined in the model, we can show its universality and the generalization property. In addition, the flexibility of the proposed model enables the model to fit time-series data coming from nonautonomous dynamical systems. When training the model, we apply Krylov subspace methods for efficient computations. In addition, the proposed model can be regarded as a neural ODE-based model. In this sense, the proposed model establishes a new connection among Koopman operators, neural ODEs, and numerical linear algebraic methods.

Related papers

• Latent Space Energy-based Neural ODEs [73.01344439786524]
  This paper introduces novel deep dynamical models designed to represent continuous-time sequences. We train the model using maximum likelihood estimation with Markov chain Monte Carlo. Experimental results on oscillating systems, videos and real-world state sequences (MuJoCo) demonstrate that our model with the learnable energy-based prior outperforms existing counterparts.
  arXiv  Detail & Related papers  (2024-09-05T18:14:22Z)
• Data-driven Nonlinear Model Reduction using Koopman Theory: Integrated Control Form and NMPC Case Study [56.283944756315066]
  We propose generic model structures combining delay-coordinate encoding of measurements and full-state decoding to integrate reduced Koopman modeling and state estimation. A case study demonstrates that our approach provides accurate control models and enables real-time capable nonlinear model predictive control of a high-purity cryogenic distillation column.
  arXiv  Detail & Related papers  (2024-01-09T11:54:54Z)
• Representer Point Selection for Explaining Regularized High-dimensional Models [105.75758452952357]
  We introduce a class of sample-based explanations we term high-dimensional representers. Our workhorse is a novel representer theorem for general regularized high-dimensional models. We study the empirical performance of our proposed methods on three real-world binary classification datasets and two recommender system datasets.
  arXiv  Detail & Related papers  (2023-05-31T16:23:58Z)
• Capturing dynamical correlations using implicit neural representations [85.66456606776552]
  We develop an artificial intelligence framework which combines a neural network trained to mimic simulated data from a model Hamiltonian with automatic differentiation to recover unknown parameters from experimental data. In doing so, we illustrate the ability to build and train a differentiable model only once, which then can be applied in real-time to multi-dimensional scattering data.
  arXiv  Detail & Related papers  (2023-04-08T07:55:36Z)
• Discovering Dynamic Patterns from Spatiotemporal Data with Time-Varying Low-Rank Autoregression [12.923271427789267]
  We develop a time-reduced-rank vector autoregression model whose coefficient are parameterized by low-rank tensor factorization. In the temporal context, the complex time-varying system behaviors can be revealed by the temporal modes in the proposed model.
  arXiv  Detail & Related papers  (2022-11-28T15:59:52Z)
• Gaussian Process Koopman Mode Decomposition [5.888646114353371]
  We propose a nonlinear probabilistic generative model of Koopman mode decomposition based on an unsupervised Gaussian process. Applying the proposed model to both synthetic data and a real-world epidemiological dataset, we show that various analyses are available using the estimated parameters.
  arXiv  Detail & Related papers  (2022-09-09T03:57:07Z)
• Learning and Inference in Sparse Coding Models with Langevin Dynamics [3.0600309122672726]
  We describe a system capable of inference and learning in a probabilistic latent variable model. We demonstrate this idea for a sparse coding model by deriving a continuous-time equation for inferring its latent variables via Langevin dynamics. We show that Langevin dynamics lead to an efficient procedure for sampling from the posterior distribution in the 'L0 sparse' regime, where latent variables are encouraged to be set to zero as opposed to having a small L1 norm.
  arXiv  Detail & Related papers  (2022-04-23T23:16:47Z)
• Bayesian Active Learning for Discrete Latent Variable Models [19.852463786440122]
  Active learning seeks to reduce the amount of data required to fit the parameters of a model. latent variable models play a vital role in neuroscience, psychology, and a variety of other engineering and scientific disciplines.
  arXiv  Detail & Related papers  (2022-02-27T19:07:12Z)
• MINIMALIST: Mutual INformatIon Maximization for Amortized Likelihood Inference from Sampled Trajectories [61.3299263929289]
  Simulation-based inference enables learning the parameters of a model even when its likelihood cannot be computed in practice. One class of methods uses data simulated with different parameters to infer an amortized estimator for the likelihood-to-evidence ratio. We show that this approach can be formulated in terms of mutual information between model parameters and simulated data.
  arXiv  Detail & Related papers  (2021-06-03T12:59:16Z)
• Anomaly Detection of Time Series with Smoothness-Inducing Sequential Variational Auto-Encoder [59.69303945834122]
  We present a Smoothness-Inducing Sequential Variational Auto-Encoder (SISVAE) model for robust estimation and anomaly detection of time series. Our model parameterizes mean and variance for each time-stamp with flexible neural networks. We show the effectiveness of our model on both synthetic datasets and public real-world benchmarks.
  arXiv  Detail & Related papers  (2021-02-02T06:15:15Z)
• Predicting Multidimensional Data via Tensor Learning [0.0]
  We develop a model that retains the intrinsic multidimensional structure of the dataset. To estimate the model parameters, an Alternating Least Squares algorithm is developed. The proposed model is able to outperform benchmark models present in the forecasting literature.
  arXiv  Detail & Related papers  (2020-02-11T11:57:07Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.