This paper is concerned with online time series forecasting, where unknown distribution shifts occur over time, i.e., latent variables influence the mapping from historical to future observations. To develop an automated way of online time series forecasting, we propose a Theoretical framework for Online Time-series forecasting (TOT in short) with theoretical guarantees. Specifically, we prove that supplying a forecaster with latent variables tightens the Bayes risk--the benefit endures under estimation uncertainty of latent variables and grows as the latent variables achieve a more precise identifiability. To better introduce latent variables into online forecasting algorithms, we further propose to identify latent variables with minimal adjacent observations. Based on these results, we devise a modelagnostic blueprint by employing a temporal decoder to match the distribution of observed variables and two independent noise estimators to model the causal inference of latent variables and mixing procedures of observed variables, respectively. Experiment results on synthetic data support our theoretical claims. Moreover, plugin implementations built on several baselines yield general improvement across multiple benchmarks, highlighting the effectiveness in real-world applications.

Computation in recurrent networks of neurons has been hypothesized to occur at the level of low-dimensional latent dynamics, both in artificial systems and in the brain. This hypothesis seems at odds with evidence from large-scale neuronal recordings in mice showing that neuronal population activity is high-dimensional. To demonstrate that low-dimensional latent dynamics and high-dimensional activity can be two sides of the same coin, we present an analytically solvable recurrent neural network (RNN) model whose dynamics can be exactly reduced to a lowdimensional dynamical system, but generates an activity manifold that has a high linear embedding dimension. This raises the question: Do low-dimensional latents explain the high-dimensional activity observed in mouse visual cortex? Spectral theory tells us that the covariance eigenspectrum alone does not allow us to recover the dimensionality of the latents, which can be low or high, when neurons are nonlinear. To address this indeterminacy, we develop Neural Cross-Encoder (NCE), an interpretable, nonlinear latent variable modeling method for neuronal recordings, and find that high-dimensional neuronal responses to drifting gratings and spontaneous activity in visual cortex can be reduced to low-dimensional latents, while the responses to natural images cannot. We conclude that the high-dimensional activity measured in certain conditions, such as in the absence of a stimulus, is explained by low-dimensional latents that are nonlinearly processed by individual neurons.

Traditional models of climate change use complex systems of coupled equations to simulate physical processes across the Earth system. These simulations are highly computationally expensive, limiting our predictions of climate change and analyses of its causes and effects. Machine learning has the potential to quickly emulate data from climate models, but current approaches are not able to incorporate physicallybased causal relationships. Here, we develop an interpretable climate model emulator based on causal representation learning. We derive a novel approach including a Bayesian filter for stable long-term autoregressive emulation. We demonstrate that our emulator learns accurate climate dynamics, and we show the importance of each one of its components on a realistic synthetic dataset and data from two widely deployed climate models.

While deep neural networks possess the capability to perform semantic segmentation, producing a single deterministic output limits reliability in safety-critical applications caused by uncertainty and annotation variability. To address this, stochastic segmentation models using Conditional Variational Autoencoders (CVAE), Bayesian networks, and diffusion have been explored. However, existing approaches suffer from limited latent expressiveness and interpretability. Furthermore, our experiments showed that models like Probabilistic U-Net rely excessively on high latent variance, leading to posterior collapse. This work propose a novel framework by integrating Gaussian Mixture Model (GMM) with Normalizing Flow (NF) in CVAE for stochastic segmentation. GMM structures the latent space into meaningful semantic clusters, while NF captures feature deformations with quantified uncertainty. Our method stabilizes latent distributions through constrained variance and mean ranges. Experiments on LIDC, Crack500, and Cityscapes datasets show that our approach outperformed state-of-the-art in curvilinear structure and medical image segmentation.

Causal representation learning (CRL) is the process of disentangling the latent low-dimensional causally-related generating factors underlying high-dimensional observable data. Extensive recent studies have characterized CRL identifiability and perfect recovery of the latent variables and their attendant causal graph. This paper introduces the notion of reward-oriented CRL, the purpose of which is to move away from perfectly learning the latent representation and instead learning it to the extent needed for optimizing a desired downstream task (reward). In reward-oriented CRL, perfectly learning the latent representation can be excessive; instead, it must be learned at the coarsest level sufficient for optimizing the desired task. Reward-oriented CRL is formalized as the optimization of a desired function of the observable data over the space of all possible interventions and focuses on linear causal and transformation models. To sequentially identify the optimal subset of interventions, an adaptive exploration algorithm is designed that learns the latent causal graph and the variables needed to identify the best intervention. It is shown that for an n-dimensional latent space and a d-dimensional observation space, over a horizon T the algorithm's regret scales as O(d

Mean-field variational inference (VI), despite its scalability, is limited by the independence assumption, making it unsuitable for scenarios with correlated data instances. Existing structured VI methods either focus on correlations among latent dimensions which lack scalability for modeling instance-level correlations, or are restricted to simple first-order dependencies, limiting their expressiveness. In this paper, we propose High-order Tree-structured Variational Inference (HoT-VI)2, that explicitly models k-order instance-level correlations among latent variables. By expressing the global posterior through overlapping k-dimensional local marginals, our method enables efficient parameterized sampling via a sequential procedure. To ensure the validity of these marginals, we introduce a conditional correlation parameterization method that guarantees positive definiteness of their correlation matrices. We further extend our method with a tree-structured backbone to capture more flexible dependency patterns. Extensive experiments on time-series and graphstructured datasets demonstrate that modeling higher-order correlations leads to significantly improved posterior approximations and better performance across various downstream tasks.

Reinforcement Learning (RL) has shown promising results in solving Job Scheduling Problems (JSPs), automatically deriving powerful dispatching rules from data without relying on expert knowledge. However, most RL-based methods train only a single decision-maker, which limits exploration capability and leaves significant room for performance improvement. Moreover, designing reward functions for different JSP variants remains a challenging and labor-intensive task. To address these limitations, we introduce a novel and generic learning framework that optimizes multiple policies sharing a common objective and a single neural network, while enabling each policy to learn specialized and diverse strategies. The model optimization process is fully guided by a self-labeling manner, eliminating the need for reward functions. In addition, we develop a training scheme that adaptively controls the imitation intensity to reflect the quality of self-labels. Experimental results show that our method effectively addresses the aforementioned challenges and significantly outperforms state-of-the-art RL methods across six JSP variants. Furthermore, our approach also demonstrates strong performance on other combinatorial optimization problems, highlighting its versatility beyond JSPs.

Modeling hierarchical latent dynamics behind time series data is critical for capturing temporal dependencies across multiple levels of abstraction in real-world tasks. However, existing temporal causal representation learning methods fail to capture such dynamics, as they fail to recover the joint distribution of hierarchical latent variables from single-timestep observed variables. Interestingly, we find that the joint distribution of hierarchical latent variables can be uniquely determined using three conditionally independent observations. Building on this insight, we propose a Causally Hierarchical Latent Dynamic (CHiLD) identification framework. Our approach first employs temporal contextual observed variables to identify the joint distribution of multi-layer latent variables. Sequentially, we exploit the natural sparsity of the hierarchical structure among latent variables to identify latent variables within each layer. Guided by the theoretical results, we develop a time series generative model grounded in variational inference. This model incorporates a contextual encoder to reconstruct multi-layer latent variables and normalize flowbased hierarchical prior networks to impose the independent noise condition of hierarchical latent dynamics. Empirical evaluations on both synthetic and realworld datasets validate our theoretical claims and demonstrate the effectiveness of CHiLD in modeling hierarchical latent dynamics.

We propose the Distance-informed Neural Process (DNP), a novel variant of Neural Processes that improves uncertainty estimation by combining global and distanceaware local latent structures. Standard Neural Processes (NPs) often rely on a global latent variable and struggle with uncertainty calibration and capturing local data dependencies. DNP addresses these limitations by introducing a global latent variable to model task-level variations and a local latent variable to capture input similarity within a distance-preserving latent space. This is achieved through bi-Lipschitz regularization, which bounds distortions in input relationships and encourages the preservation of relative distances in the latent space. This modeling approach allows DNP to produce better-calibrated uncertainty estimates and more effectively distinguish in-from out-of-distribution data. Empirical results demonstrate that DNP achieves strong predictive performance and improved uncertainty calibration across regression and classification tasks.

Modeling hierarchical latent dynamics behind time series data is critical for capturing temporal dependencies across multiple levels of abstraction in real-world tasks. However, existing temporal causal representation learning methods fail to capture such dynamics, as they fail to recover the joint distribution of hierarchical latent variables from \textit{single-timestep observed variables}. Interestingly, we find that the joint distribution of hierarchical latent variables can be uniquely determined using three conditionally independent observations. Building on this insight, we propose a Causally Hierarchical Latent Dynamic (CHiLD) identification framework. Our approach first employs temporal contextual observed variables to identify the joint distribution of multi-layer latent variables. Sequentially, we exploit the natural sparsity of the hierarchical structure among latent variables to identify latent variables within each layer. Guided by the theoretical results, we develop a time series generative model grounded in variational inference. This model incorporates a contextual encoder to reconstruct multi-layer latent variables and normalize flow-based hierarchical prior networks to impose the independent noise condition of hierarchical latent dynamics. Empirical evaluations on both synthetic and real-world datasets validate our theoretical claims and demonstrate the effectiveness of CHiLD in modeling hierarchical latent dynamics.