In unsupervised causal representation learning for sequential data with time-delayed latent causal influences, strong identifiability results for the disentanglement of causally-related latent variables have been established in stationary settings by leveraging temporal structure. However, in nonstationary setting, existing work only partially addressed the problem by either utilizing observed auxiliary variables (e.g., class labels and/or domain indexes) as side-information or assuming simplified latent causal dynamics. Both constrain the method to a limited range of scenarios. In this study, we further explored the Markov Assumption under time-delayed causally related process in nonstationary setting and showed that under mild conditions, the independent latent components can be recovered from their nonlinear mixture up to a permutation and a component-wise transformation, without the observation of auxiliary variables. We then introduce NCTRL, a principled estimation framework, to reconstruct time-delayed latent causal variables and identify their relations from measured sequential data only. Empirical evaluations demonstrated the reliable identification of time-delayed latent causal influences, with our methodology substantially outperforming existing baselines that fail to exploit the nonstationarity adequately and then, consequently, cannot distinguish distribution shifts.

Following from Section 2.1, we review several relaxations of faithfulness, which give rise to different We provide further simulation results for the analysis of the SUCF assumption in Section 3.2 . Different number of nodes and expected degrees are considered. X-axes are visualized in log scale. With the above lemmas, we now provide the proofs of the main results. We proceed by contraposition in both parts of the proof.

However, in nonstationary setting, existing work only partially addressed the problem by either utilizing observed auxiliary variables (e.g., class labels and/or domain indexes) as side-information or assuming simplified latent causal dynamics. Both constrain the method to a limited range of scenarios.

Sliding window-factor graph optimization (SW-FGO) has gained more and more attention in navigation research due to its robust approximation to non-Gaussian noises and nonlinearity of measuring models. There are lots of works focusing on its application performance compared to extended Kalman filter (EKF) but there is still a myth at the theoretical relationship between the SW-FGO and EKF. In this paper, we find the necessarily fair condition to connect SW-FGO and Kalman filter variants (KFV) (e.g., EKF, iterative EKF (IEKF), robust EKF (REKF) and robust iterative EKF (RIEKF)). Based on the conditions, we propose a recursive FGO (Re-FGO) framework to represent KFV under SW-FGO formulation. Under explicit conditions (Markov assumption, Gaussian noise with L2 loss, and a one-state window), Re-FGO regenerates exactly to EKF/IEKF/REKF/RIEKF, while SW-FGO shows measurable benefits in nonlinear, non-Gaussian regimes at a predictable compute cost. Finally, after clarifying the connection between them, we highlight the unique advantages of SW-FGO in practical phases, especially on numerical estimation and deep learning integration. The code and data used in this work is open sourced at https://github.com/Baoshan-Song/KFV-FGO-Comparison.

In this section, we provide the derivations of the equations provided in section 2.

Reinforcement learning (RL) methods frequently assume that each new observation completely reflects the environment's state, thereby guaranteeing Markovian (one-step) transitions. In practice, partial observability or sensor/actuator noise often invalidates this assumption. This paper proposes a systematic methodology for detecting such violations, combining a partial correlation-based causal discovery process (PCMCI) with a novel Markov Violation score (MVS). The MVS measures multi-step dependencies that emerge when noise or incomplete state information disrupts the Markov property. Classic control tasks (CartPole, Pendulum, Acrobot) serve as examples to illustrate how targeted noise and dimension omissions affect both RL performance and measured Markov consistency. Surprisingly, even substantial observation noise sometimes fails to induce strong multi-lag dependencies in certain domains (e.g., Acrobot). In contrast, dimension-dropping investigations show that excluding some state variables (e.g., angular velocities in CartPole and Pendulum) significantly reduces returns and increases MVS, while removing other dimensions has minimal impact. These findings emphasize the importance of locating and safeguarding the most causally essential dimensions in order to preserve effective single-step learning. By integrating partial correlation tests with RL performance outcomes, the proposed approach precisely identifies when and where the Markov assumption is violated. This framework offers a principled mechanism for developing robust policies, informing representation learning, and addressing partial observability in real-world RL scenarios. All code and experimental logs are accessible for reproducibility (https://github.com/ucsb/markovianess).

Thus, the observed data can be summarized into a sequence of "observation-action-reward" triplets ( O t, A t, R t) t 0. It is worth noting that the observation O t at each time step is not equivalent to the environment's state S t. Indeed, the state can be viewed as a special observation with the Markov property, and we will elaborate on the difference between the two later. Policies: The goal of RL is to learn an optimal policy π based on the observation-action-reward triplets to maximize the agent's cumulative reward. Mathematically, a policy is defined as a conditional probability distribution function mapping the agent's observed data history to the action space. It specifies the probability of the agent taking different actions at each time step. Below, we introduce three types of policies (see Figure 1(b) for a visualization of their relationships): (1) History-dependent policy: This is the most general form of policy. At each time t, we define H t as the set containing the current observation O t and all prior historical information (O i, A i, R i) i

This paper studies how to test the stationarity of stochastic gradient with momentum using some advanced testing statistics that take the time correlations into accounts. Extensive experiments are run to demonstrate the advantage of the proposed method over existing approaches. Originality: The paper is based on extending a recent paper by Yaida. It does not seem that original to me but the authors do combine the condition by Yaida with some more advanced testing statistics in a new way. Overall I think the extension is quite natural, so the conceptual novelty is not that high.

The key idea in this paper is to maintain this property of discrete state space models while relaxing the stationary Markov assumption on the transition probabilities that we typically use to simplify inference. Although this idea is not new (e.g. The variational inference algorithm for this model also seems to be new. In practice, we can relax the "strict" Markov assumption (i.e. the state in year t 1 is conditionally independent of the past given the state at year t) by augmenting the state with the past h 1 years. This keeps the inference exact and relatively easy to implement.