We provide the first convergence guarantee for the Consistency Models (CMs), a newly emerging type of one-step generative models that is capable of generating comparable samples to those sampled from state-of-the-art Diffusion Models. Our main result is that, under the basic assumptions on score-matching errors, consistency errors, and smoothness of the data distribution, CMs can efficiently generate samples in one step with small $W_2$ error to any real data distribution. Our results (1) hold for $L^2$-accurate assumptions on both score and consistency functions (rather than $L^\infty$-accurate assumptions); (2) do not require strong assumptions on the data distribution such as log-Sobelev conditions; (3) scale polynomially in all parameters; and (4) match the state-of-the-art convergence guarantee for score-based generative models. We also show that the Multi-step Consistency Sampling procedure can further reduce the error comparing to one step sampling, which supports the original statement from Song Yang’s work. Our result can be generalized to arbitrary bounded data distributions that may be supported on some low-dimensional sub-manifolds. Our results further imply TV error guarantees when making some Langevin-based modifications to the output distributions.

In the real world, multivariate time series from the dynamical system are correlated with deterministic relationships. Analyzing them dividedly instead of utilizing the shared-pattern of the dynamical system is time consuming and cumbersome. Multitask learning (MTL) is an effective inductive bias method to utilize latent shared features and discover the structural relationships from related tasks. Base on this concept, we propose a novel MTL model for multivariate chaotic time-series prediction, which could learn both dynamic-shared and dynamic-specific patterns. We implement the dynamic analysis of multiple time series through a special network structure design. The model could disentangle the complex relationships among multivariate chaotic time series and derive the common evolutionary trend of the multivariate chaotic dynamical system by inductive bias. We also develop an efficient Crank-Nicolson-like curvilinear update algorithm based on the alternating direction method of multipliers (ADMM) for the nonconvex nonsmooth Stiefel optimization problem. Simulation results and analysis demonstrate the effectiveness on dynamic-shared pattern discovery and prediction performance.

How to make full use of the evolution information of chaotic systems for time-series prediction is a difficult issue in dynamical system modeling. In this article, we propose a maximum information exploitation broad learning system (MIE-BLS) for extreme information utilization of large-scale chaotic time-series modeling. An improved leaky integrator dynamical reservoir is introduced in order to capture the linear information of chaotic systems effectively. It can not only capture the information of the current state but also achieve the compromise with historical states in the dynamical system. Furthermore, the feature is mapped to the enhancement layer by nonlinear random mapping to exploit nonlinear information. The cascading mechanism promotes the information propagation and achieves feature reactivation in dynamical modeling. Discussions about maximum information exploration and the comparisons with ResNet, DenseNet, and HighwayNet are presented in this article. Simulation results on four large-scale data sets illustrate that MIE-BLS could achieve better performance of information exploration in large-scale dynamical system modeling.

High-dimensional and large-scale time series processing has aroused considerable research interests during decades. It is difficult for traditional methods to reveal the evolution state in dynamical systems and discover the relationship among variables automatically. In this paper, we propose a unified framework for nonuniform embedding, dynamical system revealing, and time series prediction, termed as Structured Manifold Broad Learning System (SM-BLS). The structured manifold learning is introduced for nonuniform embedding and unsupervised manifold learning simultaneously. Graph embedding and feature selection are both considered to depict the intrinsic structure connections between chaotic time series and its low-dimensional manifold. Compared with traditional methods, the proposed framework could discover potential deterministic evolution information of dynamical systems and make the modeling more interpretable. It provides us a homogeneous way to recover the chaotic attractor from multivariate and heterogeneous time series. Simulation analysis and results show that SM-BLS has advantages in dynamic discovery and feature extraction of large-scale chaotic time series prediction.