PAPER: IntroductionHierarchical temporal structures are pervasive in time series data such as weather records and stock prices. These structures arise from latent processes operating at multi-level abstraction-for example, seasonal, monthly, and daily variations in climate data. Understanding the hierarchical latent dynamics underlying time series remains a fundamental challenge [68,4,70] and has received growing attention. The core of this problem is the identification of latent processes evolving across different levels of abstraction from observed data.To identify the temporal latent process, several methods have considered Independent Component Analysis (ICA) [30,8,27] to identify latent variables. To extend to nonlinear scenarios, researchers use various assumptions, such as sufficient changes [37,72,47,77], to ensure the independent variation of latent variables. Specifically, some approaches leverage auxiliary variables [18,19,28,29,37,31] to achieve strong identifiability of latent variables. In the context of time-series data, others utilize historical observations as surrogates for historical latent variables to induce sufficient changes [73,29].Figure 1: Illustration of data generation process with hierarchical temporal dynamics that consists of L-layer latent variables. The observed variables x t are generated by x t = g(z 1 t , ϵ 0 t ), where g and ϵ 0t denote the nonlinear mixing function and noise, respectively. And z l t are influenced by its time-delayed and hierarchical parents z l t-1 and z l+1 t , l ≤ L-1, respectively.Recent advances have further tackled the challenge of identifying latent dynamics with instantaneous dependencies under assumptions like interventions [54], grouping observations [57], and the sparse causal influence [48]. Please refer to more discussion of the related works and real-world implications of hierarchical latent dynamics in Appendix E and C, respectively. Despite recent advances, these methods predominantly assume a single layer of latent variables and will fail to identify hierarchical latent dynamics. As a result, under a hierarchical structure, using historical observation as a surrogate to leverage the sufficient changes condition is not applicable anymore since the high-level variables will introduce noise. Consequently, methods under the single-layer assumption [28,74] face challenges in recovering the joint distribution of hierarchical latent variables from single-timestep observations, particularly when the observations are derived from lower-level variables. This limitation prevents these methods from using historical observations as surrogates to meet the sufficient changes condition, resulting in suboptimal identifiability.The point above highlights the urgent need for temporal causal representation learning with hierarchical latent dynamics is to recover the joint distribution of hierarchical latent variables. Interestingly, we find that the distribution of latent variables can be uniquely determined with the help of three temporally adjacent observations [23,25]. Furthermore, we demonstrate that even with hierarchical structures in the latent processes, the joint distribution of multi-layer latent variables can be identified by incorporating additional temporal contextual observations. Therefore, we can derive a hierarchical dynamic process underlying the observed data via the natural sparsity of the hierarchical latent structure, facilitating time series generation grounded in a causal understanding of the process.Based on this insight, we therefore establish a Causally Hierarchical Latent Dynamic (CHiLD) identification framework by harnessing the temporal contextual observations and natural sparsity of the hierarchical structure among latent variables. Additionally, to connect the theoretical results with a practical algorithm, we develop a time series generative model based on a variational autoencoder with a contextual encoder and normalizing flow-based hierarchical prior networks. Specifically, to identify the joint distribution of multi-layer latent variables, the contextual encoder transforms the historical, current, and future observations into the hierarchical latent variables at the current timestamp. Moreover, to enforce the independent noise condition of hierarchical latent dynamics, the normalizing flow-based hierarchical prior networks determine the prior distribution of hierarchical latent variables. These two specialized modules together boost the identification of hierarchical temporal causal representation learning. We validate our approach through simulation experiments to demonstrate identifiability and evaluate its performance on eleven time-series generation benchmarks for both generation quality and controllability. The impressive quantitative and qualitative results underscore the effectiveness of our method.
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REVIEW
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# Summary Of The Paper

The paper presents **CHiLD (Causally Hierarchical Latent Dynamic)**, a novel method for **temporally causal representation learning** in time series data with **hierarchical latent dynamics**. The central idea is to recover the **joint distribution of multi-layer latent variables** by utilizing **temporal contextual observations** and leveraging the **natural sparsity** inherent in hierarchical structures. The approach is grounded in **nonlinear ICA and causal representation learning**, and builds on theoretical guarantees for **identifiability** under certain assumptions (such as injectivity of linear operators, smoothness, and sufficient variability).

The method employs a **variational autoencoder (VAE)** augmented with a **contextual encoder** that processes a sliding window of past, present, and future observations, and a **normalizing-flow-based hierarchical prior network** to enforce **independent noise conditions** across layers. The paper provides **formal theorems** (Theorem 1 and Theorem 2) that establish **block-wise and component-wise identifiability** of latent variables in hierarchical structures. 

Empirically, the method is validated on **both synthetic and real-world datasets**, including **stock prices, fMRI data, human motion (Human3.6M, HumanEva-I), and climate data (Weather, CESM2)**. Quantitative comparisons with **state-of-the-art methods** (e.g., IDOL, TDRL, TimeVAE, Diffusion-TS) demonstrate competitive performance, and **qualitative results** (via interpolation visualizations) highlight the ability of CHiLD to generate **coordinated and controllable motion** in human motion data.

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# Strengths

1. **Novel Theoretical Framework:** The paper introduces a comprehensive **theoretical foundation** for identifying **multi-layer latent variables** in time series with hierarchical structures. Theorems 1 and 2 rigorously justify **block-wise and component-wise identifiability** under explicit assumptions (smoothness, injectivity, sufficient variability), distinguishing it from previous work that focuses on **single-layer latent variables**.

2. **Robust Empirical Validation:** The method is extensively evaluated on **diverse datasets**, including **synthetic, biomedical (fMRI), human motion, and climate data**, covering a broad range of temporal and spatial complexities. Both **quantitative metrics** (MCC, Context-FID, Correlational Score) and **qualitative analyses** (interpolation visualizations) are used to support the claims.

3. **Architectural Innovation:** The proposed **VAE-based framework** integrates a **contextual encoder** that leverages **past, present, and future observations**, along with a **normalizing-flow-based hierarchical prior network** that enforces **independence across latent layers**. This combination addresses the **challenges of hierarchical latent dynamics** in a principled manner.

4. **Clear Contributions to Causal Representation Learning:** The paper advances the **causal representation learning** literature by extending **nonlinear ICA and identifiability theory** to **multi-layer latent structures**, offering a more flexible and expressive model than previous approaches that assume flat or single-layer latent variables.

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# Weaknesses

1. **Insufficient Exploration of Scalability and Generalization (High Severity):**
   - There is **limited discussion** on how the method scales with increasing **depth of the hierarchy** (i.e., number of latent layers) or **increasing dimensionality** of the latent variables. Without this, it is unclear whether the method is suitable for **large-scale real-world applications** involving deep hierarchies or high-dimensional data.
   - The **theoretical bounds** on the number of required observations (**2L + 1**) for identifiability raise concerns about **computational feasibility** for larger L. However, **no analysis** is provided on the trade-off between **accuracy and computational cost** as L grows.

2. **Incomplete Abstraction of Model Components (Moderate Severity):**
   - The **ablation study** is limited to only two variants: **CHiLD-KL** (without KL divergence) and **CHiLD-C** (without context information). These are insufficient to isolate the **individual contributions** of the contextual encoder and the hierarchical prior network.
   - The **role of the hierarchical prior network** in enforcing **independent noise** is **not clearly explained** in relation to the **VAE objective**. It is unclear whether the **normalizing flow** is essential or merely an architectural choice.

3. **Lack of Statistical Significance Testing (Moderate Severity):**
   - While the paper reports **mean and standard deviation** of performance metrics, it **does not provide statistical significance testing** (e.g., paired t-test, Wilcoxon signed-rank test) to confirm that the differences between CHiLD and other methods are **statistically significant**.
   - The **claim of superiority** over other methods (e.g., IDOL, Diffusion-TS) is based solely on **point estimates**, which limits the strength of the empirical validation.

4. **Code Availability Issues (Low Severity):**
   - The paper mentions that **code is available in the supplementary material**, but **no public repository link** is provided. This reduces **reproducibility** and **accessibility** for the broader research community.
   - Without a **publicly accessible implementation**, it is difficult to independently **verify the results** or build upon the method.

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# Questions For The Authors

1. In **Section 3.1**, the authors assert that **2L + 1 adjacent observations** are required for identifiability. Can this requirement be **relaxed** under **alternative assumptions**, such as **higher sparsity** or **additional constraints** on the hierarchical structure? Please provide a **counterexample or justification**.

2. **Theorem 1** relies heavily on the **injectivity of linear operators**. What is the **empirical criterion** for checking injectivity in real-world data? Have the authors tested this assumption on the **real-world datasets** used in the experiments?

3. In **Appendix B.4**, the **monotonicity and normalization assumption** is used for identifiability. What is the **impact of violating this assumption**? Could the method still be valid under **milder conditions**?

4. In **Equation (10)**, the **Jacobian** is treated as **block-diagonal**. Why is this approximation valid, and what is the **effect of ignoring off-diagonal elements** on the accuracy of the **prior estimation**?

5. In **Section 3.2**, the authors claim that **component-wise identifiability** is achieved **without permutation**. What is the **fundamental difference** between the hierarchical structure modeled here and the **single-layer models** in IDOL [48]? How does the **layered structure** avoid the need for **permutation**?

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# Limitations Not Addressed By The Authors

1. **Scalability and Complexity Trade-offs:** The paper does not analyze how the method behaves as the **number of latent layers increases** or as the **dimensionality of latent variables becomes very large**. This raises concerns about the **applicability** of the method to **deep hierarchical structures** or **high-dimensional time series**.

2. **Robustness to Violations of Key Assumptions:** The **assumptions** (e.g., injectivity, smoothness, sufficient variability) are not evaluated for **failure modes**. The authors do not investigate how the method performs when these assumptions are **violated**, which is crucial for assessing **robustness** in real-world applications.

3. **Interpretability of Learned Representations:** While the paper emphasizes **identifiability**, it does not explore the **interpretability** of the recovered latent variables. Are the **learned latent variables aligned with semantic or physical concepts** in the real-world datasets?

4. **Evaluation Against Stronger Baselines:** Some **comparisons** (e.g., with Diffusion-TS) are **one-sided** and lack **direct comparison** on shared metrics. The **baselines** selected are **not always representative** of the latest developments in **generative time series modeling**.

5. **Reproducibility Concerns:** The **absence of a public codebase** and **minimal documentation** in the supplementary material limits the **reproducibility** of the results. This makes it difficult for other researchers to **build upon or critique** the method.

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# Soundness: 3 (Good)

The paper presents a **well-defined theoretical framework** with **clear assumptions and formal proofs**. The **empirical results** are extensive and span multiple **domains**, suggesting **generalizability**. However, the **scalability and robustness** of the method under **various conditions** remain **unclear**, and **some assumptions** are **not critically examined** for violation.

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# Contribution: 3 (Good)

The paper makes a **meaningful contribution** to the **field of causal representation learning** by extending **nonlinear ICA and identifiability theory** to **multi-layer latent structures**. The **theoretical results** are **novel**, and the **empirical validation** is **extensive**. However, the **contribution is somewhat incremental**, as similar ideas appear in **related works** (e.g., Fu et al. [14], Zhang et al. [77]).

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# Confidence: 4 (Very Confident)

The **technical content** is **rigorous**, and the **experimental setup** is **thorough**. The **claims are well-supported**, though some **limitations** remain. The **paper is well-written**, and the **reviewers can confidently evaluate** its merits and shortcomings.

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# Rating: 8 (Accept)

The paper presents a **novel and theoretically sound approach** to **hierarchically structured time series modeling**. The **empirical results** are **convincing**, and the **methodology is well-motivated**. While there are **areas for improvement** (especially in **scalability and reproducibility**), the **overall contribution is substantial** and **justifies acceptance**.

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# Brief Justification For Rating

The **theoretical framework** is **original and well-founded**, and the **empirical results** demonstrate **solid performance** across **multiple domains**. The **methodological innovations** (contextual encoder, hierarchical prior network) are **relevant and impactful**. While the **limitations** (scalability, robustness, reproducibility) are **important**, they do not detract from the **validity of the core contribution**. Hence, the paper is **worthy of publication**.

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