The Koopman operator is a mathematical tool that allows for a linear description of non-linear systems, but working in infinite dimensional spaces. Dynamic Mode Decomposition and Extended Dynamic Mode Decomposition are amongst the most popular finite dimensional approximation. In this paper we capture their core essence as a dual version of the same framework, incorporating them into the Kernel framework. To do so, we leverage the RKHS as a suitable space for learning the Koopman dynamics, thanks to its intrinsic finite-dimensional nature, shaped by data. We finally establish a strong link between kernel methods and Koopman operators, leading to the estimation of the latter through Kernel functions. We provide also simulations for comparison with standard procedures.
Nonparametric Sparse Online Learning of the Koopman Operator
Coordinated Science Laboratory, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校协调科学实验室）；Department of Mathematics and Computer Science, Royal Military College of Canada（加拿大皇家军事学院数学与计算机科学系）；Department of Electrical and Computer Engineering, University at Albany, SUNY（纽约州立大学水牛城大学电气与计算机工程系）；Applied Physics Labs, Johns Hopkins University（约翰霍普金斯大学应用物理实验室）；Department of Electrical and Computer Engineering, Coordinated Science Laboratory, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校电气与计算机工程系，协调科学实验室）
Chair of Information-oriented Control, TU München（信息导向控制系，慕尼黑技术大学）；Munich Data Science Institute（慕尼黑数据科学研究所）；Munich Institute of Robotics and Machine Intelligence（慕尼黑机器人与机器智能研究所）；Control Systems Group, TU/e（控制系统组，埃因霍温理工大学）；Systems and Control Laboratory, HUN-REN Institute for Computer Science and Control（系统与控制实验室，HUN-REN计算机科学与控制研究所）
Verifiable Regularity Criterion for Conditional Expectation Operators and Conditional Mean Embeddings with Applications to Nonparametric Regression, Bayesian Inverse Problems, and Koopman Operators
Technische Universität Ilmenau（伊尔梅瑙工业大学）；Freie Universität Berlin（柏林自由大学）；Chemnitz University of Technology（开姆尼茨工业大学）
Graduate University of Advanced Studies (SOKENDAI)（综合研究大学院大学（SOKENDAI））；Institute of Statistical Mathematics（统计数学研究所）；RIKEN AIP（理化学研究所人工智能研究中心）
In this paper, we propose a nonlinear probabilistic generative model of Koopman mode decomposition based on an unsupervised Gaussian process. Existing data-driven methods for Koopman mode decomposition have focused on estimating the quantities specified by Koopman mode decomposition, namely, eigenvalues, eigenfunctions, and modes. Our model enables the simultaneous estimation of these quantities and latent variables governed by an unknown dynamical system. Furthermore, we introduce an efficient strategy to estimate the parameters of our model by low-rank approximations of covariance matrices. …
We study the problem of estimating the continuous response over time to interventions using observational time series---a retrospective dataset where the policy by which the data are generated is unknown to the learner. We are motivated by applications where response varies by individuals and therefore, estimating responses at the individual-level is valuable for personalizing decision-making. We refer to this as the problem of estimating individualized treatment response (ITR) curves. In statistics, G-computation formula (Robins, 1986) has been commonly used for estimating treatment responses…
Memorial Sloan Kettering Cancer Center（纪念斯隆凯特琳癌症中心）
We present a Bayesian hierarchical multi-view mixture model termed Symphony that simultaneously learns clusters of cells representing cell types and their underlying gene regulatory networks by integrating data from two views: single-cell gene expression data and paired epigenetic data, which is informative of gene-gene interactions. This model improves interpretation of clusters as cell types with similar expression patterns as well as regulatory networks driving expression, by explaining gene-gene covariances with the biological machinery regulating gene expression. We show the theoretical a…
Robust tracking of a target in a clutter environment is an important and challenging task. In recent years, the nearest neighbor methods and probabilistic data association filters were proposed. However, the performance of these methods diminishes as the number of measurements increases. In this paper, we propose a robust generative approach to effectively model multiple sensor measurements for tracking a moving target in an environment with high clutter. We assume a time-dependent number of measurements that include sensor observations with unknown origin, some of which may only contain clutt…
Non-parametric supervised learning algorithms represent a succinct class of supervised learning algorithms where the learning parameters are highly flexible and whose values are directly dependent on the size of the training data. In this paper, we comparatively study the properties of four nonparametric algorithms, K-Nearest Neighbours (KNNs), Support Vector Machines (SVMs), Decision trees and Random forests. The supervised learning task is a regression estimate of the time-lapse in medical insurance reimbursement. Our study is concerned precisely with how well each of the nonparametric regre…
Some challenging problems in tracking multiple objects include the time-dependent cardinality, unordered measurements and object parameter labeling. In this paper, we employ Bayesian Bayesian nonparametric methods to address these challenges. In particular, we propose modeling the multiple object parameter state prior using the dependent Dirichlet and Pitman-Yor processes. These nonparametric models have been shown to be more flexible and robust, when compared to existing methods, for estimating the time-varying number of objects, providing object labeling and identifying measurement to object…
Estimating the score, i.e., the gradient of log density function, from a set of samples generated by an unknown distribution is a fundamental task in inference and learning of probabilistic models that involve flexible yet intractable densities. Kernel estimators based on Stein's methods or score matching have shown promise, however their theoretical properties and relationships have not been fully-understood. We provide a unifying view of these estimators under the framework of regularized nonparametric regression. It allows us to analyse existing estimators and construct new ones with desira…
Technische Universität Berlin（柏林工业大学）；Machine Learning Group（机器学习组）
While nonparametric density estimators often perform well on low dimensional data, their performance can suffer when applied to higher dimensional data, owing presumably to the curse of dimensionality. One technique for avoiding this is to assume no dependence between features and that the data are sampled from a separable density. This allows one to estimate each marginal distribution independently thereby avoiding the slow rates associated with estimating the full joint density. This is a strategy employed in naive Bayes models and is analogous to estimating a rank-one tensor. In this paper …
Nagoya University（名古屋大学）；RIKEN（理化学研究所）；Kyushu University（九州大学）
Understanding nonlinear dynamical systems (NLDSs) is challenging in a variety of engineering and scientific fields. Dynamic mode decomposition (DMD), which is a numerical algorithm for the spectral analysis of Koopman operators, has been attracting attention as a way of obtaining global modal descriptions of NLDSs without requiring explicit prior knowledge. However, since existing DMD algorithms are in principle formulated based on the concatenation of scalar observables, it is not directly applicable to data with dependent structures among observables, which take, for example, the form of a s…
