Time series generation is widely used in real-world applications such as simulation, data augmentation, and hypothesis testing. Recently, diffusion models have emerged as the de facto approach to time series generation, enabling diverse synthesis scenarios. However, the fixed standard-Gaussian diffusion prior may be ill-suited for time series data, which exhibit properties such as temporal order and fixed time points. In this paper, we propose TimeBridge, a framework that flexibly synthesizes time series data by using diffusion bridges to learn paths between a chosen prior and the data distribution. We then explore several prior designs tailored to time series synthesis. Our framework covers (i) data- and time-dependent priors for unconditional generation and (ii) scale-preserving priors for conditional generation. Experiments show that our framework with data-driven priors outperforms standard diffusion models on time series generation.

In this study, we give a stability analysis of denoising autoencoder(DAE) from the novel perspective of dynamical systems when the input density is defined as a distribution on a manifold. We demonstrate the connection between the corrupted distribution and the learned reconstruction function of a nonlinear DAE, which motivates the use of a dynamic projection system (DPS) associated with the learned reconstruction function. Utilizing the constructed DPS, we prove that the high-density region of the corrupted data distribution asymptotically converges to the data manifold. Then, we show that the region is the attracting stable equilibrium manifold of the DPS which is completely stable. These results serve a theoretical basis of the DAE in recognizing the high-density region of the highly corrupted data with large deviations through the DPS. The effectiveness of this analysis is verified by conducting experiments on several toy examples and real image datasets with various types of noise.

Support-based clustering using kernels suffers from serious computational limitations inherent in many kernel methods when applied to very large-scale problems despite its ability to identify clusters with complex shapes. In this paper, we propose a novel clustering algorithm called Voronoi cell-based clustering to expedite support-based clustering using kernels. In contrast to previous studies, including the basin cell-based method, the proposed method achieves computational efficiency in both the training phase to construct a support estimate using sampled data to reduce the evaluation of kernels and the labeling phase to assign a cluster label on each data point nearest its representative point. The performance superiority of the proposed method over the other basin cell-based methods in terms of computational time and storage efficiency is verified by various experiments using benchmark sets and in real applications to image segmentation.

Clustering methods utilizing support estimates of a data distribution have recently attracted much attention because of their ability to generate cluster boundaries of arbitrary shape and to deal with outliers efficiently. In this paper, we propose a novel dissimilarity measure based on a dynamical system associated with support estimating functions. Theoretical foundations of the proposed measure are developed and applied to construct a clustering method that can effectively partition the whole data space. Simulation results demonstrate that clustering based on the proposed dissimilarity measure is robust to the choice of kernel parameters and able to control the number of clusters efficiently.