Towards Identifiability of Hierarchical Temporal Causal Representation Learning
Authors: Zijian Li
Minghao Fu
Junxian Huang
Yifan Shen
Ruichu Cai
Yuewen Sun
Guangyi Chen
Kun Zhang
Venue: NeurIPS 2025 poster ()
Keywords: causal representation learning
Primary area: probabilistic_methods

Abstract: 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.

Towards Identifiability of Hierarchical Temporal Causal Representation Learning
Zijian Li
Minghao Fu Junxian Huang
Yifan Shen? Ruichu Cai
Yuewen Sun 1.2
Guangyi Chen
Kun Zhang! 12,
Carnegie Mellon University
2 Mohamed bin Zayed University of Artificial Intelligence
"Gunggdong University of Technology
University of California San Diego
Abstract
Modeling hierarchical latent dynamics behind time series data critical for captur- ing temporal dependencies across multiple levels of abstraction in real-world tasks. However, existing temporal causal representation learning methods fail capture such dynamics, as they fail to recover the joint distribution of hierarchical latent variables from joint distribution of hierarchical latent variables can be uniquely determined using three conditionally independent observations. Building on this insight, we propose Causally Hierarchical I. Latent Dynamic (CHiLD) identification framework. Our approach first employs temporal contextual observed variables to identify the joint distribution of multi-layer latent variables. Sequentially,
observed variables. Interestingly, we find that the
exploit the natural sparsity of the hierarchical structure among latent variables to identify latent vari- ables within each layer. Guided by the theoretical results, develop time series generative model grounded in variational inference. This model incorporates a contextual encoder reconstruct multi-layer latent variables and normalize flow- based hierarchical prior networks to impose the independent noise condition hierarchical latent dynamics. Empirical evaluations both synthetic and real- world datasets validate our theoretical claims and demonstrate the effectiveness CHiLD in modeling hierarchical latent dynamics.
1 Introduction
Hierarchical temporal structures are pervasive in time series data such weather records and stock prices. These structures arise from latent processes operating at multi-level abstraction-ffor example, seasonal, monthly, and daily variations in climate data. Understanding the hierarchical latent dynamics underlying time series remains a fundamental challenge [68 4 [7] 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 27 to identify latent variables. To extend to nonlinear scenarios, researchers
various assumptions, such as sufficient changes K77 72 47 777, ensure the independent varia-
tion of latent variables. Specifically, some approaches leverage auxiliary variables [[8] 19 28 29 33]
31 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
Eqqual contribution.
Corresponding authors.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Recent advances have further tackled the challenge of identifying latent dynamics with instantaneous dependencies under assump- tions like interventions [54] grouping observations [57. and the
2
z
4++1
sparse causal influence 48 Please refer to more discussion of the
related works and real-world implications of hierarchical latent dynamics in Appendix and C
respectively.
Despite recent advances, these methods predominantly assume single layer of latent variables and will fail to identify hierar- chical latent dynamics. As result, under a hierarchical struc- ture, using historical observation as surrogate to leverage the sufficient changes condition is not applicable anymore since the high-level variables will introduce noise. Consequently, methods
z
X++1
Figure I: Illustration of data gen- eration process with hierarchical temporal dynamics that consists of L-layer latent variables. The observed variables x, are gener- ated by x, =g(z), <f), where g and ef denote the nonlinear mix- ing function and noise, respec- tively. And z are influenced by its time-delayed and hierarchical parents zI_1 and z 2 1+,,,1 L-1, respectively.
under the single-layer assumption [28] 74 face challenges in re- covering 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. In- terestingly, 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 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 causal understanding of the process.
Based on this insight, we therefore establish 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 practical algorithm, we develop time series generative model based 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, enforce the independent noise condition of hierarchical latent dynamics, the normalizing Boseb hicrarchical 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 identifiabiliy 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.
2
Preliminaries
2.1 Hierarchical Data Generation Process
We begin with the time series generation process with a hierarchical latent causal process as shown in Figure 1 To facilitate clarity, we adopt the terminology widely used in ICA literature like observed variables Xt, latent variables 2,, and the mixing function g. In the context of hierarchical latent dynamics, we let 2, {z! ... zi.... zf{
where z! E RR" denotes the I-th layer latent variables.
Suppose that we have observed variables with discrete timestamps X {xI.- ".,.". X }. where X, E IR" is only generated from first layer latent variables z! Z C R (low-level latent
"llustration shows first-order Markov structure, method can extend to higher-order dependencies.

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variables) via a nonlinear mixing function g and independent noise e,. Mathematically, we have:
X = g(),
For the (-th layer, i-th dimension latent variable at t timestamp =i, 1 E {1...., L 11 is generated via the latent causal procedure, which is assumed to be related to lime-delayed parent Pau(21,) (2,,,) and the hierarchical parents Par, (z/,)) respectively. Formally, it can be written via structural equation model as follows
4 (Paa(E)))) Pan (a..)).), with & - Pel."
where ci,, is the temporally and spatially independent noise sampled from Pet, And the L-th layer latent variables of is only related to the time-delayed parent Par(=t,). which is formalized as follows:
+ = 1. (Pad C.).; with 2 ~ Pelyy
Note that the hierarchical structure among latent variables of data generation process in Figure [ and Equation 1 3 is naturally sparse, since there are no direct causal relations between z and 2
I+1 meaning that 2 is independent of z conditional on other latent variables.
Based on the generation process above, we further define the block-wiss identifiability and component- wise identifiability of latent variables in Definition and respectively.
Definition 1 (Block-wid wise Identifiability of Latent Variables zi (69). The block-wise identifuability of zi E R" means that for ground-truth z there exists 2 E II" and an invertible function h R
IR,, such that zi- = h(2)).
Definition (Component-wiss wise Identifiability of Latent Variables =
42 The component-wise
identifiability of z E IR" is that for each zi, i E {1,...,, .n), there exists a corresponding estimated component 3.3 J E (1,".-,,²] and an invertible function h² R + E, such that z = h,,(2
2.2 Why previous theoretical results can hardly identify the hierarchical latent variables?
- we provide detailed explanation of why previous methods fail to identify hierarchical latent variables. First, the previous methods for temporal causal representation learning (8 29 73 74 usually assume that the observed variables X₁ are generated from single-layer latent variables Z, via an invertible and deterministic mixing function g, e.g., X = (, implying that we can recover the distribution of latent variables from observed variables by matching the marginal distribution of observed variables. Sequentially, these methods can leverage the historical observation X as a surrogate for - to meet the sufficient change condition and further achieve identifiability.
Based on the data generation process described in Equations
However, when the data generation process follows hierarchical latent dynamics (we suppose there are two latent layers for convenience), since the low-level latent variables, e.g., z are generated from the high-level latent variables e-g., z?, and noise c, we cannot recover the joint distribution of 2: {zt z,) due to the additional noise c,. Therefore, previous methods cannot consider x,-1 as the surrogate of 2ı-1. and hence cannot achieve identification under the hierarchical latent dynamics.
3 Identifiability of Hierarchical Latent Variables
In this section, we show how to identify the hierarchical latent dynamics under temporal contextual observations. Specifically, we first establish the block-wise identifiability results of multi-layer latent variables z, = [z! ... } with 2L adjacent observed variables (Theorem Dp. Sequentially, we leverage the estimated multi-layer latent variables 2N as a surrogate of the true variables to meet the sufficient change condition. Then we leverage the connection between natural sparsity of the hierarchical structure among latent variables and cross derivative 52 to achieve biock-wise identifiability for latent variables within each layer (Theorem (2). Finally, we further exploit the previous results 74 to achieve component wise identifiability of latent variables z (Lemma)
3

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3.1 Identifiability of Multi-layer latent variables
In this subsection, we show the identifiability results of multi-layer latent variables zt. For better explanation, we first provide the definition of the linear operator [23] 25 as follows.
Definition 3 (Linear Operator [23] L.I. Consider two random variables a and b with support A and B, the linear operator Lbja is defined as a mapping from a probability function Pa in some
function space F(A) onto the probability function pt - Lbja o Pn in some function space F(B),
F(A) + F(B): Ph - Lbla o Pa -0 pb a (alaa)papaadd
(4)
Based on the definition of linear operator, we show that the multi-layer latent variables 2t (z).... 2t) can be block-wise identifiability as follows.
Theorem 1. (Block-wise Identifiability of Latent Variables z. in Hierarchical Latent Process.)
Suppose the observed and L-layer latent variables follow the data generation process in Figure
7 By matching the true joint distribution of 2L + 1 number of adjacent observed variables, i.e., XL-L:".. "xp'".,, ;Xt+)" we further make the following assumptions:
(i) The joint distribution of x, z, and their marginal and conditional densities are bounded and continuous.
(ii) The linear operators Ligetr... x1t1/a, and Lx--t.- -e-1 (x N.+1. are injective for bounded function space.
(iii) For all 24, z E Ze with 2, F z/g the set (x+ p(x)(2e) & p(xi(z))) has positive probability.
Suppose that we have learned (g, f-pe) ) to achieve Equations and (2. then the latent variables
2g =z) _²) are block-wise identifiable.
Intuition and Proof Sketch. Compared with previous works that leverage spectral decomposition to achieve identifiability [23] [4], we do not require the monotonicity and normalization assumption
(23 (see Appendix [B.4] and quantitatively formulate the relations between required observations and the latent layers. Intuitively, Theorem [ shows that at least 2L adjacent observations are the minimal information for the identification of under the data generation process described in Equations (3). This is because of the injectivity of the two operators, where an operator that maps some functional space onto lower dimensions cannot be injective by Definition 3 The proof can be summarized in the two steps. First, by using the fact that {xI-L.-.- x {x,}. and [x++1,"' ... X are conditionally independent given 2,, we can construct an eigenvalue- eigenfunction decomposition regarding the integral operator. Next, by leveraging the relationship between the uniqueness of spectral decomposition (Theorem XV.4.3.5 (II), latent variables are block-wise identifiable when the marginal distribution of observed variables is matched.
Implication. Theorem T demonstrates that temporal contextual observations can be used to identify the joint distribution of multi-layer latent variables, even when additional noises are introduced into the generation process from high-level latent variables to observations. As an additional benefit, the same identification result can be achieved even when the mixing process from latent observed variables is non non-deterministic and involves independent noise.
Discussion Assumptions. To elarify theoretical results, we provide an explanation of these assumptions and their relevance to real-world scenarios. 1) The assumption of the bounded and continuous conditional densities is standard in the literature on the identification of latent variables under measurement error 233 25), meaning that the observed and latent variables are bounded and change continuously. For example, in the financial markets example, the stock prices (observed variables) and the interest rates (latent variables) usually change continuously. And their values are usually within a certain reasonable range. 2) The injective linear operator assumption is also used in the literature on the identification of latent variables under measurement error [14]. (More examples of this condition can be found in Appendix B.3). Intuitively, an injective operator Luje implies that there is enough variation in the density of b for different distributions of a. Using the same financial markets example, the injectivity of Ix..,...... xx1 1,s, means that the profitability (i.e., 21) of a company has sufficient influence on the stock prices of that company (i.e., X(+1, .)+
And Lxx L... X; ||x 11.....- 12 means that the stock price of a company has a significant impact on its future stock price. Therefore, this assumption is easy to meet in practice. 3) The third assumption means that the density of x, can be different when the values of 2, are different, which is also easy to meet, For example, under different interest rates, the variance of the stock price is usually different.

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3.2 Identifiabiliy of Single-layer Latent Variables z!
Theorem 2. (Block-wise Identifiability of Latent Variables z, in any 1-th Layer.) For a series of observed variables X₁ E IR" and estimated latent variables 2, E IR" with the corresponding process fú, P()), B. suppose that the process subject to observational equivalence X, = [ E We let
C. _ 11)-1.Zz]= R2xLxx and that Mc, be the variable set of two consecutive timestamps and the corresponding Markov network respectively, and further employ following assumptions:
(i) (Smooth and Positive Density [74] 42)):
The conditional probability function of the latent
variables c₁ is smooth and positive.
(ii) (Sufficient Variability)
481: Denote |Me. as the number of edges Markov network Me.- Let
w(m)= = 10 p(cc) ,'
0 log pl ci cal(z-22)
8 log p(c)((1-22) ,' 02 log ((cc((212)))
det 2.m
10
0 logg((c,
e de,, 0 2.m (1)))
(5)
where denotes concatenation operation and (i.j) E E(M..) denotes all pairwise indice such that Cr, Cr. are adjacent in MM... For m E [1,-.., ,.]. there exist 4n + +MMel different values of 22 such that the 4n + [Me,l values of vector vector functions w(m) are linearly independent.
Then for latent variables zi at the 1-th layer, 2 is block-wise identifublee without permutation, i.e.
there exists 2) and an invertible function h' IR" + R,,, such that z! = h (22))
This proof is built upon the results of Theorem 1 since we cannot achieve the joint distribution of latent variables in the hierarchical struc- ture. By considering latent variables C, ( from two adjacent timestamps, the cross-layer la- tent variables are conditionally independent, e.g, z 1L 1 cc \zzi,s 1 Vi, E {1.... .n..
Intuition and Proof Sketch. Proof can be found in Appendix A3
And such conditional independence implics 8² pig.)..) = O by leveraging the connection between Ozt [+1
conditional independence and cross derivatives [52]. Moreover, since we have achieved the block- wise identifiability of 2g latent variables and further introduce the sufficient variability assumption [48). As a result, we can construct a full-rank linear system with the unique solution. By further exploiting the hicrarchical temporal structure of latent variables, we can show that the estimated 2 can be block-wise identifiable without permutation, i.e, there exists an invertible function ht R - R", such that 2 = (2.).
leverage the contextual observation the surrogates of historical
Compared with Previous Theoretical Results. Although both 488 and Theorem 2 use the sparsity influence of latent dynamics to achieve identifiability, yet our contribution goes substantially further. First, IDOL (8 assumes invertible mixing procedure, which begins from matching the marginal distribution of X₁. While our method allows that partial latent variables contribute to the observations it is built the block-wise identification from Theorem Second, IDOL shows the component-win wise identifiability with permutation. However, CHiLD shows the block-wise identifiable without permutation (please find it at Lemma A2 Appendix B.3). This result is stronger than that of IDOL. We further show that each component within each layer z/,, is component wise identifiable.
with a noise-contaminated mixing procedure,
Discussion on Assumptions. We further explain the assumptions and their real-world implications. First, the smooth and positive density assumption is commonly adopted assumption in nonlinear ICA studies [37 [74), implying that the latent variables can change continuously based on historical information. This assumption can be casily met in real-world scenarios like financial markets. For example, the exchange rates usually change smoothly over time.
Second, the sufficient variability assumption is widely employed in existing results on the identifiabil- ity of temporally causal representation learning [73] 74 5 48), reflecting the changeability of latent variables. Take stock price forecasting an example, the latent variables may represent quantitative factors such as the price-to-earnings ratio, trading volume spread at different time steps. The linear independence of the latent variables implies that changes in one factor, such as the price-to-earinngs ratio, cannot be linearly represented by trading volume. Moreover, while the sufficiency assumption forms the foundation of identifiability theory, it is not overly restrictive in practical applications.
5

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Table 1: Attributes of causal representation learning theories. A check denotes that method has attribute or can be applied to a setting, whereas a cross denotes the opposite.
Method
Hierarchical Structure
Noisy Mixing Procedure
Instantancous Dependency
Time Series Data
Stationarity
X
TDRL 74
CaRING 5
X
A AXX
X
General
X
IDOL 48
X
CHiLD
Lemma 1. (Component-wise Identifiability of Latent Variables 2/ in any 1-th Layer [74 42
For a series of observed variables x, IK" and estimated latent variables 2, RuxL with the corresponding process fú, P(ε), &. suppose that the process subject to observational equivalence X = 8(2,,2)) Besides the conditional independence assumption, i.e., log p(z]///.1 + E log p(2,( 1211112 we further assume the sufficient variability for each latent component. For any 2 2 C RD and 12/-1; 21+1) there exist 2n a-++1 different values of ú, m = 0. 0.... 2n,
such that these 2n vectors V1,m Vi.0 are linearly independent, where Vim is defined as:
82 P( Vi, - (02
02 In P In P( (02t )2
O PInP(zE ( Ozh
(6)
02h
Then for i-th latent variable zi,s at the 1-th th layer, zi, is component-wise identifiability. i.e., there exists 3 and an invertible function h4 RR R. such that Et, (2,,)
Discussion. Please refer to the proof in Appendix B.7, Compared with existing works [2 29 74
42 that also leverage similar assumptions (i.e., conditional independence and sufficient variability assumptions) to achieve component-wist -iis identification, the main difference that our method simultaneously leverage the time-delayed and hierarchical parents meet the sufficient variability condition instead of domain or historical information.
3.3 Comparison with Existing Results of Causal Representation Learning
Compared to recent methods for causal representation learning, the proposed CHiLD model better aligns with real-world scenarios from multiple perspectives, summarized in Table Specifically, TDRL 744) and PCL (29] struggle to handle time-series data with hierarchical latent d dynamics and noisy mixing processes. Although CaRiNG (5] also leverages temporal contextual information for identification, it remains limited as it fails to account for noisy mixing processes and cannot identify latent dynamics with instantancous dependencies. More recently, Zhang al. [77] and Li et [48] have employed sparse causal influences to identify latent variables with instantaneous dependencies. However, these methods neither address the challenges posed by noisy mixing processes nor identify latent variables within hierarchical structures. Another closely related work, Fu et al. [14), also utilizes unique eigeaaffutttnttttii decomposition to achieve identifiability. However, this method is specifically designed for causal discovery in climate data and does not account for hierarchical latent dynamics. In contrast, our model extends previous results by establishing the relationship between the minimal required temporal observations and the size of hierarchical latent variables.
4 Approach
In this section, we propose the VAE-based model [39 22 for time series generation as shown in Figure 2 Specifically, begin by estimating the hierarchical latent variables using a contextual encoder which processes a sequence of observations, X;-L:t+L [] and outputs the estimated latent variables 2. Sequentially, the reconstructed observations generated from the latent space via a step-wise decoder, which uses 2, to produce &
'The subscript denotes the observations from t L to L time steps.

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To capture the hierarchical latent dynamics, we em- play the Kullback-Leibler divergence between the posterior distribution of the estimated latent vari- ables and prior distribu- tion, which coincides with the natural sparsity of hier- archical structure in Theo- rem[land and the conditional independence condition in Lemma [ Moreover, devise a hierarchical prior network to estimate the prior distribution of latent variables, which based on the normalizing flow 61
Step-wise Decoder
82
Dec
Dec
Dec
4-1
1-1
Hierarchical Latent
Variabless
fz
f
x
Contextual Hierarchical Encoder
Hierarchical Prior Networks
Enc
Enc
Enc
X1
X2
XT
Figure 2: The overall framework of CHiLD, which incorporates contex- tual Hierarchical encoder (Enc), step-wise decoder (Dec), and hierarchi- cal prior networks.
63
and converts the prior distribution into for implementation details.
Gaussian noise like (74
Please refer to Appendix F.3
4.1 Contextual Hierarchical Encoder and Step-wise Decoder
Guided by Theorem I we the temporal context observations to identify multi-layer latent variables. Therefore, we devise a contextual hierarchical encoder for the inference process in Equation (7)
2 (2((((")
which Φ1 encodes the observations to the first latent layer, and adopts a similar form of ID convolu-
tional neural networks. Subsequently, each Φr encode the preceding latent layer, 2 2 1, to extract the deeper or higher-level components, yielding 2 L can be considered as the size of the receptive field. Accordingly, we employ neural architecture based on temporal convolutional neural networks (TCN) 44 45 as the contextual encoder. Given the estimated latent variables 2,, further develop a step-wise decoder that only takes the bottom-layee -ayer 2, as input to generate X, for each time-step within the receptive field, as shown in Equation (8]
x: (2.).
(
4.2 Hierarchical Prior Networks
To make the estimated latent variables capture the hierarchical property and the conditional indepen- dent property one straightforward solution is to minimize the Kullback-Leibler divergence between the posterior distribution and the prior distribution. However, the prior distribution is unknown and we estimate the prior density directly, arbitrary density functions may be generated, leading to an incorrect posterior distribution. To address this challenge, motivated by 77 5) who use normal- izing flow [6]] to model the causal procedure of latent variables by restricting the independence of estimated noise, we propose the hierarchical prior networks estimate the prior distribution of latent variables. Specifically, given time delay т, we let r {ri. ..,,..". .ri.. be set of learned inverse transition functions that take 2/, 2/_ and 2 as input and generates the noise term, i.e., t, r 2..T, 21+1). And r² is implemented by MLPs. Sequentially, we can devise transformation (b/ 2/+1 2/) - (b/.. 2/+1 2 with corresponding Jacobian
0
J.. 0
I
0
where denotes a matrix. By applying the change of variables
ari DE
diag
formula, we have:
p(2 y 2, .2,) = p(2),,,2/ [
(9)
By dividing 2 +111 both sides, the prior distribution can be derived as follows:
del
log 122 2(41) log logg(('/12 2(+1) + log |J,|= Σ (l0gple).) + log
(.
(10)
0E
7

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Table 2: Experimental results of MCC on simulation data
Datasct
CIGLD
IDOL
CaRING
TDRL
Beta- VAE
Slow VAE
IVAE
FacturVAB
PCL
TCL
0.8:20.003) 0.91(0043)
0.8230.0255 0.836(0.155)
0.2500.005)
0.774j0.022)
0.66/0.023)
0.50900003)
0.513/00027)
0,40500.021
054110.37)
0.558(0.004)
0.94100001)
0.942(0.076)
0.755010445
0.524/0.154) 0.3500065)
D417000055)
0.344(0011))
0.404:0.0K51
0.35/00012)
0.801c0.03
0290
0.797(0.118)
0.748/0.017)
0.742(0067)
0.61500023)
0.322/0.098)
0.329(0003)
0.356g0.012)
0.44119.016)
0.765(0.055)
0.788(00082)
0.22100.003
0.708(0.015)
0.738(0007)
0.710:0.050)
06790004)
0.45210004)
0.462000101
0.86210.026)
0.723/8.057)
(233)
0.702(0.032)
0.71600.024)
0.515(0007))
0.725/0.00))
0.741(0.043)
(/22
0.069/0.55)
0.806/0.022)
0.723:0.024) 0.687(0029)
0.676000788)
0.667(0.034)
0.685(0.066)
0.69100.335
0.63000.009
0.384(0.150)
0,31900.024)
0.77410.007)
0.60X0.075)
0.572(0004)
0.30500.996)
0.5250.017)
0.2990009))
0661100.055)
0.499X0.008)
0.442/00021)
0,345/0.059)
where p(ε),) follow Gaussian distributions, For the reconstruction likelihood LR we use the mean- squared error (MSE) to measure the discrepancy between the generated and original observations.
Please note that although CaRiNG [5] also leverages the contextual observation in VAE-based framework, the main difference between CaRiNG and CHiLD can be summarized into the following two folds. 1) The model design comes from different theoretical guidance. Motivated by the nonlinear ICA-based theoretical results, CaRiNG leverages the contextual observations to recover the lost latent information and address the challenge of a non-invertible mixing function. In contrast, our method guided by the uniqueness of spectral decomposition and overcomes the difficulties of hierarchical latent dynamics. 2) The prior distribution estimations of these two methods are also different. CaRiNG assumes single-layer latent variables and uses the time-delayed and current latent variables to estimate noise. Meanwhile, the CHiLD allows hierarchical latent dynamics and hence harnesses the current, time-delayed, as well hierarchical latent variables for noise estimation.
Further Discussion:
4.3
Optimization
Finally, we train the proposed model by optimizing the Evidence Lower Bound (ELBO) as follows
ELBO-
Eg(arr(xir) 1n (1 - DKs/u(aarr/x
T
= B
log p(pp(x/ + ( Bo(ar)Mrr)
- log ( -
CKL
(11)
where DK denotes the Kullback-Leible divergence and the detailed derivation can be found in Appendix F.2 For the reconstruction loss L. we use the mean-squared error (MSE) to measure the discrepancy between the generated and original observations. And we leverage the Equation (TO) to approximate the KL. divergence.
5
Experiment
5.1 Experiments on Synthetic Data
Data Generation. We generate synthetic time series data based fixed latent causal process, as defined in Equations [ 3 To evaluate the proposed theoretical results, we provide seven different datasets from A to G, with different numbers of latent variables, different complexities of generation processes, and different latent layers. Note that Dataset B consists of single-layer latent variables, while other datasets adhere to the assumptions underlying the proposed theoretical framework. Details of the data generation process and evaluation metrics can be found in Appendix G.T
Baselines. To evaluate the effectiveness of our method, we consider the following baselines. First, we consider the standard 8 VAE (20) and FactorVAE 38) which do not use historical information. Then consider TDRL (74), iVAE (37]. TCL 28 PCL 29 and SlowVAE [40], which use temporal information but do not contain instantancous dependency. We also consider CaRiNG [5] and IDOL [48], which leverage contextual observations and consider instantaneous dependency, respectively. We use mean correlation coefficient (MCC) as the evaluation metric.
Results and Discussion. Experiment results are shown in Table 2 According to the experiment results, we can find that: 1) the proposed CHiLD achieves the highest MCC performance in Datasets A, C. D, and E; 2) the CHiLD and the recently proposed methods like IDOL and CaRiNG achieve good performance in Dataset since it does not contain hierarchical latent dynamics. 3) In the more challenging Dataset F and G (datasets with large dimensions and latent layers), although the
8

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Table 3: Experiment results of real-world datasets for the time series generation.
ComseAt-FID
Weather
WeatherBench
CESM2
Box
Gestures
Throweasch
Discussion
Purchases
WalkDog
CHILD
0.507(0.042)
0.078:0.017)
0.01800.003)
0.092/0.011)
0.032/0.004)
0.022/0.003)
0.02910.005)
0.021:0.099)
0.010000005
KoVAR
2,131(0.637)
(0.02)
0.195/0.050
0.856:00809))
0.67910.025)
0.18340.0688
0.235/0.33))
0.149/0.053)
137/0.0554
Diflusion TS
3.82600.915)
1.71200 163)
0.407(0.003)
0.258:0.017)
0.088/0.006)
0.151:0.011
0.487/0. 166)
0.300x0.048)
0.334(0.057)
TmeGIAN
6.43411299)
1.56400.200 2003
3.255(0.252)
0.92990.087)
0.83010.081
3.50
3.76110.3933
L825K0.225)
388800.4601
moI
(495(0.485)
0.23000.000 (R0)
0. 0:142/0.017)
0.125 (26)
0.018(0.044)
0,025 (02)
0.082:0.035
0.025X0.0144
0.01(00022)
VAE
5.61(1.359)
0.60910.066)
0.238(0.011)
0.215:0.026)
0.329(0.051
0.254(0.255
0.919/0.132 (32)
114900.146)
0,56500.126)
TAmeVA
391000858)
(.221(0.029))
0.10800029)
0.20880.022)
0.42800.163)
0414(0.334)
0.06710.020)
0,07420.008)
0.82200.121)
Correlational
Weater
WeatherBench
CDSM2
Throwcasch
Discussion
Purchases
WalkDog
0.16510.004)
1.2000.009)
0.08910.003)
0.065(0001)
0.014:0.000)
0.100K0.000
0.12610.001
KoVAR
162000.012)
(0.384()182)
(D:087(8.017)
1.75860 OER)
10010003300
1.26900051)
4.00000 023)
2.6890.054)
3015(0.021)
Dillusion
1992(0.021)
63.291(06600)
16.659/0.231
3.4070 125)
1.72400.200)
2.101(0.243 243)
5,85100 (09)
3.513/0.157)
3.70100.077)
BaEGANN
3439000020)
37.473(1.213)
7.37330-937) 20.643(01082)
(0.912116600
(6.419(.334)
27257600485)
25.56211 2000
21. 21.84800.2501
moI
1. (195/00014)
34.773((-059)
9.12540.016)
1.84000 035)
0.634/0.008 (08)
0.550/0000)
1.205/0.0100
1.148,0.008)
2.97(00016)
ca VAE
0.584(0.013)
34.69000131)
6. 6.68960.022)
0.812/0.026>
0.92610.064)
0.755(0.025)
8.62510.032)
7.741x0.0555
6.36700,079)
TameAA
0.70700.005)
24.873(0091)
6,363(0,022)
0.891(0.013)
1.04610.047)
L003(0046))
0.31010.022
0,508/0.002)
0.664/0.DEE3
(a) CHiLD
(b) TimeVAE
(o) DiffusionTS
(d)TimeGAN
Figure 3: Interpolation visualization of different models. For each method, after training the model, only one latent variable is gradually changed while keeping the other variables fixed. The images of each method from left to right represent the gradual increase of the latent variable.
MCC performance of the proposed CHiLD is lower, it can still outperform the existing methods for temporal causal representation learning, like IDOL, which supports our theoretical claims.
5.2 Experiments on Real-world Data
5.2.1 Experiment Setup
Datasets. We consider the task of unconditional time series generation and use the following datasets: Stock, ETTh1, fMRI, MuloCo, and two human motion datasets, Human3.6M and HumanEva-I. For Human3.6M (32. we choose motions: Discussion, Purchases, and WalkingDog. For HumanEva-1 [65]. we choose Box, Gesture, and Throwcatch. We further consider three climate datasets: Weather, WeatherBench, and CESM2. Please refer Appendix G.2.I and for the dataset description and the connection between time series generation and modeling hierarchical temporal latent dynamics. Baselines. First, we consider the VAE-based methods like cwVVAE (64), KoVAE [58] and VAE [9. We also consider other types of generative models like Diffusion-11 TS (76) and TimeGAN [75]. Moreover, we consider the causal representation- based method like IDOL
We choose Context- Fréchet Inception Distance (Context-FII)) and the Correlational Score evaluation metrics. We repeat each experiment over 3 random seeds and publish the mean and standard deviation. Please refer Appendix G.2.2 for the introduction to the evaluation metric.
5.2.2 Quantitative Results
The experimental results in Table show that the proposed CHiLD method significantly outperforms all baselines across most datasets, such as Box and Discussion, highlighting its potential for time series generation. While VAE and Ko VAE also employ VAE-based architectures, they assume independent latent variables. contrast, CHiLD enforces a hierarchical latent structure, enabling it to model high-level dependence and hence generate more realistic time series data. Other time series generation baselines, such as Diffusion-T's and TimeGAN, fail to deliver optimal performance, particularly on human motion datasets. This can be attributed to the higher dimensionality of human motion data compared to datasets like ETTh1, which implies more complex latent dynamics. Please refer to Appendix G.3.2 and G.3. for more experiment results and the ablation studies.
5.2.3 Qualitative Results
We also provide the interpolation visualization results. Specifically, after training each model, we change one latent variable gradually and keep the other fixed.

---

Controllable Generation. To evaluate controllable genera- tion, we compare TAM, TimeGAN, and DiffusionTS in the Discussion motion generation dataset. As shown in Figure 3 TimeGAN performs the worst, the generated motion patterns are difficult recognize human shapes. TimeVA produces more structured movements but lacks coordination. While DiffusionTTS generates recognizable human shapes, the variations in motion with respect to the latent variable are limited, indicating weak controllability. In contrast, CHiLD method generates high-quality human motions and enables coordinated changes in limb movements through latent variable control. Please find more interpolation results in the form of GIF visualization from the supplementary material.
Changing Joints
Stable Joints
(a) Interpolation visualizations (b) Interpolation Visualizations
of High-Level Latent Variables of Low-Level Latent Variables
Figure 4: Interpolation visualization of high-level and low-level latent variables of the proposed CHiLD method. Interpolating high-level latent variables leads more significant changes ob- served data compared to interpolating low-level latent variables.
Hierarchical Generation. We interpolate both high-level and low-level latent variables CHiLD and generate corresponding motions. As shown in Figure 4 green represents stable joints, while red indicates joints that change with the latent variable. The results show that high-level latent variables influence broader range of joints compared to low-level ones, demonstrating that our method effectively captures hierarchical abstractions in time-series data.
Conclusion
This paper introduces an identification framework for temporal causal representation learning under hierarchical latent dynamics. The proposed method leverages temporal contextual observations to estimate the joint distributions of multi-layer latent variables and exploits the sparsity of hierarchical structures to achieve precise latent variable identification within each layer. A VAE-based generative model, incorporating a contextual encoder and normalizing flow-based hierarchical priors, enforces independent noise conditions while enhancing the capacity to generate realistic time series data. Empirical evaluations on both synthetic and real-world datasets validate the theoretical guarantees and demonstrate the effectiveness of our approach. Future works will extend this framework to more complex domains, including controllable video generation, as well as applications in neuroscience, finance, and climate modeling. Limitation: However, the component-wise identifiability result of our work also relies on other assumptions like layer-wise conditional independence. How to further relax these assumptions would be an interesting future direction.
7 Acknowledgment
The authors would like to thank the anonymous reviewers for helpful comments and suggestions during the reviewing process. The authors would like acknowledge the support from NSF Award No. 2229881, 1, AI Institute for Societal Decision Making (AI-SDM), the National Institutes of Health (NIH) under Contract RO1HLI 15905, and grants from Quris AI, Florin Court Capital, and MBZUAI-WIS Joint Program, and the Al Deira Causal Education project. Moreover, this research was supported in part by National Science and Technology Major Project (2021ZD011 11501, National Science Fund for Excellent Young Scholars (62122022). Natural Science Foundation of China (U24A20233, 62206064, 62206061, 62476163, 62406078), Guangdong Basic and Applied Basic Research Foundation (2024A 151501 1901, 223300417000
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Supplement to TToarrs Identifiability of Hierarchical Temporal Causal Representation Learning" "
Useful Definitions and Lemmas
24
Proof of Theoretical Results
25
B.
Notations
25
B.2
Proof of wisck-wise Identifiability of Latent Variables 2, Hierarchical Latent Process
25
B.3
Examples injective linear operators
28
B.4
Monotonicity and Normalization Assumption
28
B.5
Block-wise Identifiability of Latent Variables z, in any (-th Layer
28
Extension to Multiple Lags
32
B.7
Component-wise Identifiability of Latent Variables o any 1-th Layer
32
Real-world Implications of Modeling Time Series Data in Hierarchical Manner
33
Relationship between Time Series Generation and Hierarchical Dynamics
34
Related Works
34
Implementation Details
35
Prior Likelihood Derivation
35
Evident Lower Bound
36
Model Details
36
F.3.1
Reproducibility of Simulation Experiment
36
F33.2
Reproducibility of Real-world Experiment
36
Experiment Details
36
G.1
Simulation Experiment
36
G.I.I
Data Generation Process
36
Evaluation Metries
37
G.1.3
More Simulation Experiments of Different Length of Observations
38
G.2
Real-world Experiment
38
G.2.1
Dataset Description
38
G.2.2
Evaluation Metric
39
G.3
More Experiment Results
39
G.3.1
Experiment results on other datasets
39
G.3.2
Ablation Study
39
Broader Impacts
40
23

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A Useful Definitions and Lemmas
Definition (Diagonal Operator) Consider two random variable a and b, density functions Pe and are defined on some support A and B, respectively. The diagonal operator Dbja maps the density function Pa to another density function D o Pa defined by the pointwise multiplication of the function Pala at fixed point b:
ab P) = Doja Pa: where Dola = (b 1-).
(A1)
Definition 5. (Completeness) A family of distribution ()) is complete if the only solution p(a) to
JA A pla)pa da for all b B,
(A2)
is p(a) = 0. In other words, no matter the range of an operator is on finite or infinite, it is complete if its null spacd or kernel is zero set. Completeness is always used to phrase the sufficient and necessary condition for injective linear operator [59 6).
Theorem A1. (Theorem XV 4.5 in (TT) Part III) A bounded operator T a spectral operator if and only if it is the sum T = S + N of a bounded scalar type operator S and quasi-nilpotent operator N commuting with S. Furthermore, this decomposition is unique and T and have the same spectrum and the same resolution of the identity.
Lemma A1. (Lemma! I in (24)) Under Assumption densely defined over ((X) (for 9 C2, f.aa).
1 if Tuja is injective, then T exists and is
Properties of linear operator. We outline useful properties of the linear operator to facilitate understanding of our proof:
i. (Inverse) If linear operator: Laja exists a left-inverse Lij such Liju o Luja opo Pa for all E A. Analogously, if Laja exists a right-inverse Lnjo- such Loja L/o Pa = Pa for all A. If Lbja is bijective, there exists left-inverse and right-inverse which are the same.
ii (Injective) Luja is said to be injective linear operator if its L is defined over the range of the operator Lbje
If so, under assumption (ii ) L Laj exists and is densely defined over F(A).
(24].
iii (Composition) Given two linear operators Lejh F(B)- + F(C) and Laja F(A) + F(C). with the function space supports defined uniformly on the range of supports for the domain spaces as characterized by Taja- it follows that Leje = Lepp Laja- Furthermore, the properties of linearity and associativity are preserved in the operation of linear operators. However, is crucial to note the non-commutativity of these operators, i.e., Lo a F Loja Lop.. indicating the significance of the order of application.
Definition 6 (Markov Network). Markov network is undirected graph G (V, E) with a set of random variables XeeV, where any two non-ad adjacent variables like Da and Tob are conditionally independent given all other variables. That is.
Sa 1 xolxVMabb
V(a b) $ E.
(A3)
Markov Networks and Directed Acyclic Graphs (DAGs) both graphical models employed to represent joint distributions and to illustrate conditional independence properties. Based on the definition of Markov Network, we further define Vector-Nod: Markov Network as follows:
Definition 7 (Vector-Node Markov Network). A Markov network where each node represents vector, rather than scalar variable. In this structure, the joint distribution is factorized over a graph
G = (V, E), where V is the set of nodes, and each v E V corresponds to multidimensional vector X.. Given any two non-adjacent vectors Xa and Xb, any Xa,i E Xa and Xb.j E Xr are conditionally independent given all other variables. That is.
Xa.
Xb. X [V
V(a, b) & F.
(A4)
'The null space or kernel of an operator L to be the set of all vectors which L maps to the zero vector: null L {o V Le 0).
24

---

Definition 8 (Isomorphism of Markov networks). We let the V(-) be the vertical set of any graphs, an isomorphism of Markov networks M and M is bijection between the vertex sets of M and M f VV + V(M)
such that any vertices and of M adjacent G if and only f'() and [(o) adjacent M.
Definition (Definition of left and right inverse.). Let f : y and g y + be fanctions. We say that is a left inverse to f, and that f is right inverse of 9. if g o f idx, where id is the identity function.
B
Proof of Theoretical Results
B.1
Notations
This section collects the notations used in the theorem proof's for clarity and consistency. Table A4: List of notations, explanations, and corresponding values.
Index
Explanation
Support
T k,,
Number of variables
" n N+
Index of latent variables
i.j.k, S,00
Time index
E N+
L
Number of latent layers
L E N+ and L >
I
Index of latent layers
= 21 L}
Variable
X,
Support of observed variables in time-index
A C R"
Z,
Support of latent variables
Z, S R
X4
Observed variables in time-index t
X₁ E I"
Latent variables in indexindex t
2, R"
4t,1
The i-th, 1-th-layer latent variable at t-th step
2,, E R
Ca
{z,-1. }
C₁ E R2xnxL
G
Independent noise of mixing procedure Independent noise of the latent transition of x!
c Pep
t ~ Pel.
Function
Pa s. (b)
Density function of given b Joint density function of (a, b) given and Time-delayed parents
(a,
p
pap
Hierarchical parents Nonlinear mixing function
g())
IR - IR
f-)-)
Transition function of z
IR Pa. (w), 111 |Pan (x), + R
h())
Invertible transformation from z, to 2,
IR" T1
IR"
T()
Permutation function
IR IR"
Function space
Me.
Markov network over C₁
Contextual hierarchical encoder for z!
Step-wise Decoder
Noise estimator of éL.,.
Symbol
J
Jacobian matrix of r
B.2 Proof of Block-wise Identifiability of Latent Variables z, in Hierarchical Latent Process
Theorem A2. (Block-wise Identifiability of Latent Variables z, Hierarchical Latent Process.) Suppose the observed and L-layer latent variables follow the data generation process in Figure
25

---

1 By matching the true joint distribution of 2L + number of adjacent observed variables, i.e.,
I.- -xs ,x,+ , we further make the following assumptions:
(i) The joint distribution of x,Z, Z, and their marginal and conditional densities bounded and continuous.
(ûi The linear operators Lxc+t.." sett and Lixe-ry" :x- lxeety S.41. are injective for bounded function space.
(iii) For all E,, z Zze # z(), the set fxe P(xi (z)) o P(x)(z)) has positive probability.
Suppose that we have learned (g. f. P(2)) to achieve Equation and 2 then the latent variables
Z ....... .z(} are block-wise identifiable.
Proof. We first follow Hu et al. [24] framework prove that z, is block-wise identifiable given sufficient observation. Sequentially, we prove that we require at least 2L + adjacent observed variables to achieve block-wise identifiability.
Given time series data with timesteps = (x1,-"' X . xx and L-layer latent variables, we let x and Xy be {x1, x,-1) and (x++1, .x) respectively. Note that the length of X<t and X> are larger than L, i.e., ixcal > I. and [x>1] > L. Sequentially, according to the data generation process in Figure we have:
PX 2i) P(xx)/z)) P(x)/xx = P(x>
(A5)
Sequentially, the observed P(x:-1) and joint distribution P(x P(xst, Xx x(xx))), and we have:
X1, xex) directly indicates
P(x>n X [x(x))===
P(xst: Xx, Ze 2r(xx1)d22
P(x Z1, Xx) P(x,, (cce
2,
2,
Intogration over 2e
Factorization of joint conditional probability
P(x)elz))P(xx 24 2xx
Conditional Independence
Bayes Law
P(xxslz)) P(z, (x))ddz
(A6)
We further incorporate the integration over A'c follows:
/
frr Xex 2z
Xxx ((xxe)
(A7)
According to the definition of linear operator, have:
S(L P(x P(xxx (Ixpexel P|(x)))
f Xes
P(ziixet)
[L2ixxe Px(x))
(A8)
P(xsolzeddz Lxsiler
By combining Equation A7
and A8
we have:
TxxxxxxXxes P)(xxt) (Tixalee Dxilzs Izcixes P)(x)t),
(A9)
which implies the operator equivalence:
Lxgexelxer Lx Exs Dx, Lee
(A10)
26

---

Sequentially, we further integrate out and have:
JxL Ixxsuxelxa, dxi = S(c Txxsolee Dx.la, (xcxxxx
(A11)
and it results in:
Lxsilxee = Loxs is Lus bect. According to assumption (ii), the linear operator I Txxpele, injective, Equation (A12) can be rewritten
(A12)
as:
x> les Lx |ecc = Lxii
(A13)
By combining Equation A10 and
A13
we have
I xx xi xxct
T 22
Dxilz, xxte/e /xxtexeet
(A14)
By leveraging Lemma Al if Lx Lxclss is injective, then L x exists. Therefore, we have:
xxsex xx Lx D Dx xx
(A15)
Then leverage assumption (iii) and the linear operator is bounded. Consequently, Txxsxx xei I x is also bounded, which satisfies the condition of Theorem A1 and hence the the operator Tx Dx 'x>t/>; have a unique spectral decomposition, where T xxx1 and Dx, correspond to eigenfunctions and eigenvalues, respectively.
Since both the marginal and conditional distributions of the observed variables matched, the true model and the estimated model yield the same distribution over the observed variables. Therefore,
also have:
Tixss Xet x>||x< = T. &e "
(A16)
where the L.H.S corresponds the true model and the R.H.S corresponds the estimated model. Moreover, L 8//., &&ct L also have the unique decomposition, so the L.H.S of the Equation A16 can be written
Lx x. (xaee xx1|xx LLg.le.
D2.12 L L
(A17)
Integrating Equation A15 and Equation A17 and noting that their L.H.S. are identical, it follows that they share the same spectral decomposition. This yields
L Ch Dxi/21 P
(A (A18)
where C' is scalar accounting for scaling indeterminacy and P is a permutation on the order of elements in D as discussed in [IT. These forms of indeterminacy are analogous to those in eigendecomposition, which can be viewed as fniie-dimensional special case. P is mapping from distribution distribution
Since the normalizing condition
f
Px, dx;
(A19)
must hold for every 2t one only solution is to set C
Hence, D& la, and Dxclzs are identical up to a permutation on their repsective elements. We use unordered sets express this equivalence:
{p(x, zz)] = {p(x, 2,)}. for all z,, 2,
(A20)
Due to the set being unordered, the only way to match the R.H.S. with the L.H.S. in consistent order is to exchange the conditioning variables, that is,
{ol (p(xal((") p(x ....... (2(xx12 ), p(x, 12, (1(2))
(A21)
where superscript (.) denotes the index of conditioning variable, and π is reindexing the conditioning variables. We use relabeling map h to represent its corresponding value mapping:
p(xe|zz) p(xx ((x)),((;)), for all z,, 2,
(A22)
27

---

Since 22. x2. and Lx)x,, are continuous, h is continuous and differentiable. Moreover, by (2
leveraging Assumption (iii), different values of i.e., 2((t) 2t imply p(x:|2,") p( So
we can construct function : - p(x, Zz). and we have:
F(z
F(z (2)
(A23)
implying that is injective. Moreover, by using Equation A22 we have F(zi) = F(h(2,)), which implies Z, h(2)).
The aforementioned result leverage x Xt, and X>r as three different measurement of zes where [xcal >> Mel. X>+| > | and |xe| < [Ze]- It may imply that when the x, cannot provide enough information recover Zt, we can seek more information from X<t and X>*-
Sequentially, we further prove that when the observed and L-layer latent variables follow the data generation process in Equation ( and 2 we require at least 2L adjacent observed variables, i.e, (x Xz, follows.
X } to make block-wise identifiable. We prove it by contradiction as
Suppose we have 2L adjacent observations, which can be divided into two cases: I) (X--L+1;"" "t,".. , (
and (x--L,"'' ;,,, "t, x In the first case, suppose the di- mension of X2 and that of any layer of latent variables z are n, the dimension of X X (L- 1) X n and the dimension of z; is x n. conflicting with the assumption that Lx Xx. tereee injective. In the second case, the dimensions of Xt- ... X and X Xx+1-11 are n and (Z. 1) x respectively, conflicting with the assumption that 1xx--/ -x Xx-1 Xx-1|Xeet."" N++L in- jective. As result, require at least 2L adjacent observations, i.e., (,- Nty x++L) to make 2, block-wise identifiable.
B.3 Examples of injective linear operators
The assumption of the injectivity of linear operator is commonly employed in the nonparametric identification 24
3 26 Intuitively, it means that different input distributions of linear operator correspond different output distributions of that operator. For better understanding of this assumption, we provide several examples that describe the mapping from Pa - Pb. where a and b are random variables.
Example (Inverse Transformation). g(a). where g is an invertible function.
Example 2 (Additive Transformation). b a where p(e) must not vanish everywhere after the Fourier transform (Theorem 2.1 in (55)).
Example b = g(a) + 6, where the same conditions from Examples and are required.
(o)((a))e)) +é), a post-nonlinear model with invertible Example (Post-linear Transformation). b b gi (g2(a)
nonlinear functions 91, 92, combining the assumptions in Examples
Example (Nonlinear Transformation with Exponential Family). b = g(a, c), where the joint distribution p(a,) b) follows an exponential family.
Example (General Nonlinear Transformation). b g(a, c). general nonlinear formulation.
Certain deviations from the nonlinear additive model (Example 3). e.g., polynomial perturbations, can still be tractable.
B.4 Monotonicity and Normalization Assumption
Assumption 1 (Monotonicity and Normalization Assumption (26)). For any Xt E Xr. there exists a known functional such that G [Px.n 2. (-1x), 2c)] is monotonic in 24- We normalize 2; G 1 (-1xe, zr)].
B.5 Block-wise Identifiability of Latent Variables z! any 1-th Layer
Theorem A3. (Block-wise Identifiability of Latent Variables z in any 1-th Layer.) For a series of observed variables x, RR and estimated latent variables 2, E RR" with the corresponding process fi, P(), P()). g, suppose that the process subject observational equivalence =g(ž,, (t)...)). We
28

---

let C; - {z B2x1xxn and that Mc. be the variable set of two consecutive timestamps and the corresponding node-vector Markov network respectively, and further employ following assumptions:
probaabbiyy (i) (Smooth and Positive Density); The conditional probability function the latent variables
C₁ is smooth and positive, i.e., p(c, (L-2) is third-order differentiable and p(c, (Zt-2 > over IR2xXxx
(ii) (Sufficient Variability): Denote |Ma, as the number of edges in Markov network Me. Let
= 8 logple
O log (ugp(cil(e))))
ae(m)
00f, 221-2,
2eh 024-2.
@ log
8 plgp(cc (2-22
8 logp(ce (LS)EE(MO))
0cc,102-22.
de 22 2,m
de, ( (A24)
where @ denotes concatenation operation and (i, E(Me.) denotes all pairwise indice such that Cr,s, Ctjj are adjacent Mo. For m [1,... nl, there exist 4n + [Me,l different values of 2,m such that the 4n + [Mael values of vector functions w(m) are linearly independent.
Then latent variables z, at the 1-th layer, z; is block-wise identifiable, i.e., there exists i and an invertible function K1, IR" + IR",, such that z, = h1 (z.).
Proof. By reusing Theorem A2 with more observed variables, {z,-2, Ze-1, ze} and {z--1, ze) are also block-wise identifiable, implying that there exists invertible functions hg and h2, such that 2 2 2: ha (,, Z zı) and 2 2, h2(2t-1, Zz). So we have:
P(2 2.)) P(2, Ze- 11, Jzzl = 2( = P(z--1, 2()2112))
P(e, 22-2) P(c, (cc
In P(c)(2,-2) In p(c)(2r-2) In Jhaal
(A25)
And then we partition the latent variables into two parts
Let 2-1 and 2/2 be two variables that denote the k-th variable of 2/_, and o-th variable of 2/2 respectively, where 4 > f2. According the data generation process, is not hard to find that 2/L1. and 2/c, not adjacent in the estimated Markov network Me, over è, {2,-1, 2. the first-order derivative w.r.t.
We conduct
2/11, and have
log plc.
2L
Cogp(cc/2r-1 Ocl,
04,, Ølog [Jaal
(A26)
02 1,K
02.-1. 1,K
82 1,
We further conduct the second-order derivative w.r.t. 2/- and 2 then we have:
8 log pog.p(c/222. (2)2)
21
2L
8
acy,
deg
02,-1,0001
Dct dci
02
02
1=1
(A27)
25
Ologp(c, 2(-2)
02,
82 log [
+
Dct,
0z 1 02%
Diti_ 1, 02/
Since 2/1 1. é, lde, G. Utilizing the fact that conditional independence can lead to zero cross derivative (53 for each value of 2,-2, we have:
1, and 2/:, not adjacent in MM,, 2/1 2 1. and 2/c are conditionally independent given
8a log p(crp((/.eaa 22)
8 log pesp(ee,
1/ 1,21
2/2 2/ 2(-2)
0² Plogple,,1e M2 \2 1, 2 2(-2)
+
02,1 02 02/3
02-11
02/3
02"11,001
2 log p(c,)
12
1
20,01
2/
1 (2(22)
0.
02
02/2
4-
(A28)
29

---

Bring in Equation
A28 Equation A2
can be further derived as:
21
8² log pl à 2)
ach,
dch, 02
O(c
)
02
02 log p(c)(21-2)
och.
det
Σ
02
02,,0
1 i=1 s(e.))E((Me)
(i))e,, 61 adjacent Mor
(A29)
21 a
E Σ
"
0² log log ((c))(122)
Jch,
deg
dc, Oct
02
02
(a j=1
(iiiie, and or,y not adjacent in Mo:
24
2 log p(ce 22-2)
+ Olog J
dct
, 1-1 02
O2
where (j.i) E {Me,)) denotes that Ct and a, are adjacent in Me. Similar to Equation (A28). we have ppp(c_le__a)
when Ct,i, Ctj are not adjacent Me.. Thus, Equation A29 can be rewritten
000,)
2L
a 8 logp(c,,2.... 2logp(c.1112)
22)
ad,
2L n
n
02 log (lg)(cc)
Oct,
Ocq
Σ
O(ch,1)
oz
Oz
dct ,ct,00t,
02 o2
1=1
(()
26
log ple 22)
pd,
Olog [Jn.e]
dc,,
0z 1 02l2
02
(A30)
Then for each m 1.2,...
nL and each value of é. 2,m conduct partial derivative on both
sides of Equation A30 and have:
2L
3 log (c.
oct, dct
O(c,, )20,))200 20
0 1,k 0212
24
log logp(c, (2-22
Oc,,
det (A31)
ΣΣ
Σ
Σ
Jct, dc,
22r-2.m 2.m
02
27.0
I=1 i=l s(s.))E((M))
j=1
21.
0² log p(cc (2-2)
82 c4
dc,, 021-2.m 2,m
02 1,k 0213
Finally, we have
2L
8 log pcppp(ll122
det, act,
2L
82 log p(c, 24-2)
8
DiC,) 200
02
gct, 02,
02 1, 02
1=1==1
y3 log (2)))
acts
Oct
dct
ac,,
Σ
+
(s(()))EE(M.) t=1 ae, 0cg, Uhr 2.m
02 02
azk.
02
02
(A32)
According Assumption A2, we can construct 4nL |Mel n² different equations with different values of 21-2.m and the coefficients of the equation system they form are linearly independent. To ensure that the right- hand side of the equations always O, the only solution is
ach.2
Ock,
-0,
(A33)
0
1
02.2
30

---

dess +
deg
Dey,
=0,
(A34)
22 1-1k 1,k
2z,
02 1-1k
02
0(C, ,(C,,)
=0.
(A35)
0z 1, 22
Bringing Eq A33 into Eq That is,
A34 at least one product must be zero, and the other must zero as well.
dct,
dct =0. 0z
(A36)
0z 1-1
According the aforementioned results, 2/L,, and z/:o be two variables that denote the k-th variable of 2 and o-th variable of z/², respectively, where 1 l2, draw the following conclusions. (i) Equation A33)) implies that, each ground-truth latent variable c,, in I-th block is function of most one of c and ez which are in 41-th and 12-th layer, respectively
(ii) Equation (A36) implies that, for each pair of ground-truth latent variables ct,s and o7, in I-th and I-th blocks, respectively, that are adjacent in Mc. over C₁2 they can not be a function of d and ala respectively.
According to the data generation process, restrict the independent noises among blocks, and hence the estimated node-vector Markov network isomorphic to the ground-truth node-vector Markov network.
Sequentially, we further give the proof that under the same permutation z block 2 only function of 2, (+) Since the permutation happens on each timestamp respectively, the cross-timestamp block-wise identifiability presented clearly.
Suppose there exists pair of indices I, s E {1,-... ... L.. Since h2 is invertible, there exists a permuted version of the estimated blocks, denoted c/, such that:
Ozh 2.j
# O, 1= L, and i.j j 1,..... n,
(A37)
Ox # and
we have case.
#
Let us discuss case by case, We can discuss the following
Ban
If z not adjacent to zi, we have 2, (() is not adjacent to 2, (x), Using Equation (A33). we
have dct del, O, which leads to ox! O.
02 03-"..
82
If block z! is adjacent to block zi, we have 2, adjacent to 2, TG) The intimate neighbor set of z is empty, there exists at least one index & such that z is adjacent to z but not adjacent to z/. Similarly, we have the same structure on the estimated Markov network.
which means that (7((k) is adjacent (7(() but not adjacent to 2: Using Equation A36
we have act aci 02.-1. 0z.7
O, which leads az 0.
a2,
In conclusion, we always have 0z = 0, meaning that there exists a permutation æ the estimated
02
blocks, such that zł and z, (()
is one-to-one corresponding.
Sequentially, we further leverage Lemma A2 to show that there is exist an invertible function h', such that 2/ W (z)).
Lemma A2. (Hierarchical Structure Resolves Layer- Wise Permutation Indeterminacy) If the estimated latent causal graph represents hierarchical structure, the layer-wise components zi 23 ..... z are block-wise identifiable without permutation.
31

---

Proof. Let A, represent the true causal adjacency matrix of the latent causal graph at time t in layer-wise level, and let A, represent its estimation. By definition, we have
Uu iz 0,
k=1+1,
k k=1+1,
A
0z 0,
k 11=1,... I...,..., L.
(A38)
otherwise,
otherwise,
Consider the nonzero elements of the adjacency matrices, which occur at positions where 1. If a layer-wise permutation is applied, the rows and columns of the A, are both permuted according to the same permutation. Specifically, we have
A4t1. = D1+1.1
(A39)
where D141,1 is a diagonal matrix representing the scaling indeterminacy, and x {l...., ......,.J.... L} + {1..... L} is permutation function. The subscripts of Å and A above indicate the following equation:
<1 + = +1))
(A40)
The recursive formula in Equation (A40)
implies:
(1) = x11 + (1-
(A41)
For (() to be valid permutation covering all values {L...., L}. it is necessary that x 1. (1) & 1, the sequence ([() = T1) + (1- would either exceed L or fail to include 1, violating the bijective property of Hence, conclude that
(1) -
(A42)
indicating that the layer-wise components remain unpermuted.
B.6 Extension to Multiple Lags
For the sake of simplicity, we consider only one special case with 1 in Theorem 2 Our theoretical results can actually be extended to arbitrary lags and subsequences easily. For any given time lag т, and future horizons which is centered at Z, with historical and future Ty steps, i.e. 24 Met-,,""" ",,,... . + In this case, the vector function w(m) in the
Sufficient Variability assumption should be modified as
I log p(e, Zt-TA- -1,.
8 log p(c: Ze- -1. vn
w(m)= =
, 1,m
de, 2n O2c- -1,m
8 log p(c: 21- 12,
8 log p(c: 2e
(A43)
0c,,
8c, U2c 0ee-n -1
80 8log log B -1.10
dc,, Uc,, Ü2e- 1,m
(L.))
Besides, 2 x n x (Th + T + 1) + [Me, values of linearly independent vector functions in 2 .m for I' E (L- Th - -,,... , Tb. - and m E (1..... .t.....,77 , are required as well. The rest of the theorem remains the same, and the proof can be easily extended in such setting.
B.7 wisponent-wise Identifiability of Latent Variables z in any 1-th Layer
Lemma A3. (Component-wise Identifiability of Latent Variables 2i in any 1-th Layer) For series observed variables E R. and estimated latent variables 2, 12" with the corresponding process fi, P(č), P()), &, suppose that the process subject observational equivalence x; gg(2,, (p) We employ sufficient variability assumptions as follows:
(i) (sufficient variability) For any of E Z C R and ú (21-122,11 there exist 2n + 1 different values of û, m = -..... 2n, such that these 2n vectors Vt, Vio are linearly independent, where Vi,m is defined as:
Vi.m (0P) In p(z
8 In P( Il
())
0l P(2,(()
Oln Plm,(z), dah
(8%) ((82))))
(azl, }2 Ozh ,
(A44)
32

---

Then for i-th latent variable zi, at the 1-th layer, z is component-wise identifiability. i.e., there exists a and an invertible function h4 IR R, such that 2/. 11 (2).))
Proof. According to Theorem A3 we have achieved the block- block-ideniffibiiityy of z by letting 2 and 2 1 be the temporal and hierarchical parents, we further have
P( 2 P(h 2i-1: Ze 1) P(z 12 1
- P( = In P (i) + In J
In P( (ū) = Σ In p() /i) + In [J
2=
(A45)
where Jng is the Jacobian matrix of the transformation associated with 14, Sequentially, we dif- ferentiate both sides of the Equation (455) w.r.t 2 and Ei,,> where j, k {1,... ,n} and + yields
J2 In P(z (oz),)²
Oz Oz 02
Oln P(z,
8z ,
J ,
(A46)
oz k
Oz
02 1,k
Therefore, for u Ug, 1 U2n. we have 2n + such equations. Subtracting each equation corresponding to u1 U2n with the equation corresponding to up results in 2n equations:
n 0 In ((zi,,(mm) (3
0 P(z (uo))
Oz Oz ozl
Σ
(2
(A47)
Oln (z), Oln P(z), (úu))
Øzzi,
dz
Oz
where m = 1,,... 2n. As a result, under the sufficient variability assumption, the linear system is a 2n 2n full-rank system, and the only solution is
02 Jz. and 0221.e 02 22 3z
And Oz dz
0
implies that for each i = 1,....
f 0 for at most one element (1,... n). Therefore,
there only at most one non-zero entry in each row indexed by i in the Jacobian J Jag. Moreover, the
invertibility of b necessitates Jng to be full-rank which implies that there is exactly one non-zero
component in each row of JR implying that the z is component wise identifiable.
C Real-world Implications of Modeling Time Series Data in a Hierarchical Manner
Real-world scenarios are usually governed by hierarchical layers of temporal latent dynamics instead of single one. Capturing the hierarchical temporal latent dynamics plays critical role in modeling time series data. Here is two real-world scenarios.
Human Motion Modeling.
Human locomotion unfolds over multiple temporal layers 16
60
2
Higher-level latent variables:
Capture coarse movement categories- e.g. walking versus running or resting-capturing the switch-like changes in overall gait dynamics.
Lower-level latent variables: Resolve fine- ine-grained kinematics such joint angles, stride frequency, and instantaneous speed.
This decomposition isolates semantically meaningful factors: the high-level code tells what activity is occurring, while the low-level code details how it is executed. Such structure supports down- stream tasks, including activity recognition, early anomaly detection (e.g., pathological gaits), and personalized coaching systems that adapt exercises to an individual's movement profile.
33

---

which is Climate Modeling. shown as follows:
Atmospheric processes also exhibit a natural hierarchy 34
71,
Top-level latent variables capture large-scale regimes like monsoon cycles or transitions between seasons, governing the dominant energy balance of a region.
Middle-level latent variables reflect regional or intra-seasonal patterns-e.g.. the twelve traditional solar terms in East Asia or monthly precipitation phases that modulate local ecosystems.
Bottom-level latent variables track short-term fluctuations in temperature, humidity, and wind that drive daily weather variability.
Modeling these tiers jointly enables scientists disentangle long-term trends from transient dis- turbances, improving extended-range forecasts, pinpointing abnormal events (heat-wave onsets, unseasonal cold snaps), and facilitating attribution studies of climate change.
D Relationship between Time Series Generation and Hierarchical Dynamics
The goal of time series generation is to identify the joint distribution of observed variables of different time steps, i.e., p(x1.'''' ... Xr). To achieve this goal, we should simultaneously capture long-range structure spanning hundreds of steps and fine-grained, Collapsing every timescale into single latent layer forces one hidden state to shoulder both roles. This entangles long-term context with short-term detail, squanders model capacity resolving incompatible dependencies, and obscures the semantics the latent dimensions dimensions-making directed manipulation counterfactual sampling nearly impossible.
cooment-to-momen variability.
By contrast, an explicit hierarchical latent structure allocates dedicated timescale each layer: slow-clock top tier encodes global intent or seasonality. intermediate tiers refine meso-level patterns, and fast-elock bottom tier captures instantancous noise. This separation prevents semantic mixing. lets the model reuse parameters efficiently, and provides clearly interpretable handles for controllable generation (e.g., editing only high-level intent while preserving fine detail).
E Related Works
Time Series Generation Generating realistic time series data (66
33
13 80 is important in
numerous fields, including finance, healthcare, and engineering, where access to sufficient data can be challenging. Traditional generative methods, such Generative Adversarial Networks (GANs) [17]
56
62 35 and Variational Autoencoders (VAEs) 39 46 71. have been widely explored for time series generation. TimeGAN (75), GAN-based framework, integrates adversarial and supervised losses to effectively capture temporal dynamics, demonstrating superior performance over previous methods in terms of similarity and predictive performance. However, such GAN-based models often face challenges such as training instability and mode collapse. To overcome these limitations, VAFs robust alternative. VAE [TO] incorporates domain knowledge to
have been explored model temporal patterns like trends and seasonalities, improving both interpretability and training efficiency. Recent advances have also introduced diffusion-basee models [4] 15 211, such as Diffusion-TS (76), which leverage disentangled temporal representations and Fourier-based training objectives generate high-quality and interpretable time series while mitigating common limitations of autoregressive approaches. Furthermore, Koopman VAEs (KoVAE) [58] leverage linear latent dynamics inspired by Koopman theory. enabling the integration of domain knowledge and stability analysis. These methods collectively illustrate the evolving landscape of time series generation, with a focus on balancing realism, interpretability, and computational efficiency. Other relevant works include [36,, which synthesize time series by manipulating existing time series.
Identifiability Causal Representation Learning Independent Component Analysis (ICA) has been widely used to identify latent variables with identifiability guarantees, traditionally assuming a linear mixing process
30 27 However, this linearity constraint restricts its applicability in more complex scenarios. To handle nonlinear scenarios, researchers have proposed nonlinear
8
ICA by introducing additional assumptions, such as auxiliary variables
or structural
sparsity 77, 49). Specifically, methods based on auxiliary variables [28 29 50 typically assume
34

---

that latent sources are conditionally independent given auxiliary information, such as domain indices or temporal segments, thereby enabling identifiability. For unsupervised approaches. structural sparsity in the generative process has been exploited to recover causal latent factors without relying on auxiliary variables [79
78 67 This approach ensures identifiability by enforcing sparsity constraints on the Jacobian of the mixing function, allowing for the identification up component- wise transformations and permutations.
Temporal observations introduce unique challenges to identifiability, especially in the presence of instantaneous dependencies and dynamic latent structures. 28 achieves identifiability for stationary data using permutation- based contrastive learning, while 29 extends this approach to nonstationary time series by leveraging variability data segments. Recent methodss [73] 74) assume conditional
independence of latent variables given their time-delayed parent to identify latent causal relations.
However, these assumptions often fail in real-world scenarios where instantaneous dependencies are prevalent, such as in human motion data, making them of limited applicability in such contexts. Recently, 48 proposed an identification framework for instantaneous latent dynamics, introducing sparse influence constraint to enforce sparsity in both time-delayed and instantancous causal relationships among latent processes. While these advances enhance identifiability, addressing hierarchical latent dependencies remains key challenge for further improving the identifiability of causal structures in temporal data.
Implementation Details
F.1 Prior Likelihood Derivation
We first consider the prior of In p(z14). We start with an illustrative example of stationary latent causal processes with two time-delay latent variables, i.e. 2, [2r,1, Z,,2 with maximum time lag = 1, i.e., Rt.s = 7, with mutually independent noises. Then we write this latent process as transformation map f (note that we overload the notation f for transition functions and for the transformation map):
1t-1,1
žt- 1,1
2t-1,2
=f
2t-1,2
2t,1
6r,1
21,2
6,
By applying the change of variables formula the map f, evaluate the joint distribution of the latent variables p(2--1,1, 24-1,2, 21,1 21,2) as
2t,1; p p(2-1111 1.1; 2t-1,2; 21,1 2t,2) p(2 1,1, 2 C1,1,
(A48)
det Jrl
where Jr is the Jacobian matrix of the map f. where the instantaneous dependencies are assumed to be a low-triangular matrix:
0
0
0
Jr
0
02x.1
02s.1
0
02y-1,1
8.y-1.2
051,2
0sc-1.1
Uze-1.2
Oea,1
Dec.2
Given that this Jacobian is triangular, we can efficiently compute its determinant II. 8a., Fur-
ex.
thermore, because the noise terms are mutually independent, and hence G, 1 Ctjj for j * and 1. 2t-1,
with the RHS of Equation (A48) follows
p(21-111 1,2, 2t,1, s1.2) p(2 1,1, 12) x p(Er.1, 66,2) p( x p(er,) [Je]
(A49)
[Jel
Finally, we generalize this example and derive the prior likelihood below. Let fry (r1),,1,,,3.-- be a set of leamned inverse transition functions that take the estimated latent causal variables, and output the noise terms, i.e., é,, = (2(,,, (2. Then we design a transformation - B with low-triangular Jacobian follows:
In,xa XL
[. L. 3 26]
mapped to " 2 1. with JA-B
diag
Dra,1 DE
B
(A50)
35

---

Similar to Equation A49 we can obtain the joint distribution of the estimated dynamics subspace
as:
log log p0gp(2 22-1)+
log p(e,,) (og((ee((]])
(A51)
Because of mitually independent moise assumptiion
Finally, we have:
the
Ōr:
log
Σ p(e,,)
log
(AS2)
Since the prior of (( 11: 0(2:|2;-1 with the assumption of first-order Markov assumption, we can estimate p 1(2++1:7 (1:))) in similar way.
F.2 Evident Lower Bound
In this subsection, we show the evident lower bound. We first factorize the conditional distribution according to the Bayes theorem.
In In P(X1.T. Z1.T)
= Bg In p(XI:T: Zi.T (Zz.T X1:7)
p(2u. XI.T)
p(z X1T (z(Z X1.T)
Bggarrben (e)) In (xp bir)) - DKK(g(Zz (ir)lp(()r))) ELBO. (A53)
LELD
F.3 Model Details
F.3.1
Reproducibility of Simulation Experiment
For the implementation of baseline models, we utilized publicly released code for TDRL, CaRiNG. Since the source code of IDOL is not available, we implemented based on TDRL with the sparsity constraint. And the proposed CHiLD is also modified based on the code of TDRL, and shared hyperparameters remain the same.
F.3.2 Reproducibility of Real-world Experiment
We follow the setting of (76.. The model details of the proposed method are shown in Table A5 As for other baselines like KoVAE, Diffusion-TS, TimeGAN, and TAEE we employ the official implementations and use the default hyperparameters for fair comparison. To achieve the results from the well-fit baselines, we employed the default hyperparameters and tried different values of learning rate for the best models. And the number of latent variables each real-world dataset is shown in Table|A6
G Experiment Details
G.1 Simulation Experiment
G.1.1 Data Generation Process
As for the temporally latent processes, we use MLPs with the activation function of LeakyReLU to model the sparse time-delayed. That is:
- (L 2.21, Ver. 2ÿ,0)) + +
(A554
2
Lea -2, 0.2) 0.2) + E.
where W is the i-th row W and Vei,i is the first i i- columns in the i-th row of V. Moreover, each independent noise ε,;; is sampled from the distribution of normal distribution. We further let the data generation process from latent variables to observed variables be MLPs with the LeakyReLU units. And the generation procedure can be formulated as follows:
X; (eeckyyBeeLULLLLaeeeeeeeee 0.2)) + 6 0.2),
(A55)
36

---

Table A5; Architecture details. T, length of time series. [x/]: imput dimension. ne: latent dimension.
LeakyReLLU: Leaky Rectified Linear Unit. Tanh: Hyperbolic tangent function.
Configuration
Description
Output
Latent Variable Encoder
In
Observed time series
Batch Sixe x dimension
Convolution neural networks
[x,] neurons
Batch Sizextx X
Concat zero
concatenation
Batch Sizex Tx [
Dense
neurons
Batch Size Txn
Decoder
Input:2:.T
Latent Variable
Batch Size Txn Batch Size Tx [4]
Dense
[
neurons, Tanh
Modular Prior Networks
Input: Z1:T"
Latent Variable
Batch Sizex(n+1) (n+1)x122
Dense
128 neurons,,LeakRReLU
Dense
128 neurons,LeakyRRen ReLU
128 x 128
Dense
128 Leurons,LeeakRRLLU
128 128
Dense
1 neuron
Batch Size
Jacobian Compute
Compute log(det(J))
Batch Size
Table A6: Number of latent variables on each real-world dataset
ETTh
Stocks
MuJoco
IMRI
Box
Gestures
Throwaatch
Discussion
Purchases
WalkDog
Sample Size
10:000
10000
1889
74481
10087
Dimension of High level Layer Dimension of Low-level Layer Number Layer
20
15
15
25
25
10
30
30
30
30
26
26
26
where the dimension of is 1.
We generate six different synthetic datasets. For dataset A, the first layer contains 4 latent variables and the second layer contains latent variable. For dataset B, all the latent variables are in the first layer, since there is only one latent layer. For dataset C, the first layer contains latent variables and the second layer contains 2 latent variables. For dataset D, we consider the second-order latent markov process, and the first and second layers contain 4 and 1 latent variables, respectively. For dataset E, we consider a more complex generation procedure. Specifically, each causal relation and generation using a 2-layer MLP with LeakyReLU activation, which are shown as follows:
> (2 +
z/, = 0.5 M (M (x)-)) 0.5 0.5 (M(M((2 +el, X,, = (M (z))) + +
(A56)
where M is linear layer followed by Leaky activation. For dataset F, we consider three latent layers with 1, 2, and 4 latent variables. For dataset G, we consider the case with large number of latent layers and dimensions, which contains three latent layers with 8, 8, and latent variables. respectively. The total size of the dataset is 100,000, with 1,024 samples designated as the validation set. The remaining samples are the training set.
G.1.2 Evaluation Metrics
To evaluate the identifiability performance of our method under instantaneous dependencies, we employ the Mean Correlation Coefficient (MCC) between the ground-truth and the estimated é, higher MCC denotes better identification performance the model can achieve. In addition, we also draw the estimated latent causal process to validate our method. Since the estimated transition function will be a transformation of the ground truth, we do not compare their exact values, but only the activated entries.
37

---

G.1.3 More Simulation Experiments of Different Length of Observations
To evaluate our theoretical results that L-layer latent variables require 2L+1 consecutive observations, we designed an experiment on synthetic two-layer model with 8 latent variables each layer. In our experimental setting, we vary the number of consecutive observations, i.e., 1,3,5, denoted by settings J1. J3, and J5, respectively. The results are shown in Table
Table A7: MCC performance under different numbers of consecutive observations.
Setting
MCC (mean std)
J1
0.6859 + 0.024
J3
0.7001 0.008
J5
0.7622 + 0.004
According to the experiment results, we can find that we observe that as the length of the consecutive observation window increases, the recovery performance of the latent variables improves. In particular, when the length is 2T 1, i.e., 5, the identifiability result is optimal.
G.2 Real-world Experiment
G.2.1 Dataset Description
The detailed descriptions of the datasets are shown as follows:
Stock is the Google stock price data from 2004 to 2019. Each observation represents one day and has 6 features.
ETTh dataset contains the data collected from electricity transformers, including load and oil temperature that is recorded every 15 minutes between July 2016 and July 2018.
fMRI is benchmark for causal discovery, which consists of realistic simulations of blood- (BOLD) time series.
MuloCo (multi-joint dynamics with contact) is time series dataset generated by the physics engine.
Huaman 3.6 is collected over 3.6 million different human poses, viewed from 4 different angles, using an accurate human motion capture system. The motions were executed by 11 professional actors, and covered diverse set of everyday scenarios including conversations, eating, greeting, talking on the phone, posing, sitting, smoking, taking photos, waiting, and walking in various non-typical scenarios. We randomly use four motions, i.e., Gestures (Ge), Jog (J), Throw (CT), and Walking (W).
HVa-nEVA-L comprises subjects, each performing several action categories. Each pose has 15 joints with three axis. We choose 6 motions, i.e., Discussion (D), Greeting (Gr), Purchases (P). SittingDown (SD), Walking (W), and WalkTogether (WT) for the task of human motion forecasting. Specifically, the ground truth motion of the body was captured using a commercial motion capture (MoCap) system from ViconPeak 5 The system uses reflective markers and six IM-pixel cameras to recover the 3D position of the markers and thereby estimate the 3D articulated pose of the body. We consider the joints as latent variables and the signals recorded from the system as observations.
Weather dataset offers 10-minute summaries from an automated rooftop station at the Max Planck Institute for Biogeochemistry in Jena, Germany.
WeatherBench is benchmark dataset for data-driven medium- range weather forecasting, It repackages forty years (1979-2018) ERA5 global reanalysis into srrchinelearning-eee ready NetCDF tensors sampled every six hours. Fourteen core surface and pressure-level geriables-geopoteentaa height, temperature, humidity, wind components, vorticity, po- tential vorticity, cloud cover, precipitation, solar radiation, and more- are provided on -aitude-lonnituuee grids of 0.25%, 1.406 2.812%, and 5.625". Static fields such land-sea mask, soil type, orography, and grid coordinates are included.
is
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CESM2 9 delivers 100 fully coupled Earth system simulations at 1 resolution spanning 1850-2100 under CMIP6 historical and SSP3-7.0 forcing. Ensemble spread arises from distinct arean-atmosphere initial states: ten members seeded from evenly spaced low-drift periods; eighty members created by tiny 10-14 atmospheric perturbations applied to four AMOC-phased control years; and ten MOAR members useful for regional downscaling.
G.2.2 Evaluation Metric
Context-Freecee Inception Distance (Context-FID) score 35 quantifies the quality of the synthetic time series samples by computing the difference between representations of time series that fit into the local context.
Correlational score [51] uses the absolute error between cross-correlation matrices by real data and synthetic data assess the temporal dependency.
G.3 More Experiment Results
G.3.1 Experiment results other datasets
We would like to clarify that while some absolute improvements may seem small, e-g., 0.079 (ours) vs. 0.098 (IDOL) in the fmri dataset, it represents a 19% relative improvement. Focusing on percentage improvements, we observe significant gains of 34.07% and 23.91% % two metrics, showing the effectiveness of CHiLD.
Table A8: Experiment results on other datasets.
ETTh
Stocks
Mujoco
fmri
CHiLD
0.111(0.022)
0.001(0.000)
0.238(0.055)
0.025(0.001)
KoVAE
0.120(0.009)
0.095(0.013)
0.024(0.009)
1.086(0. 142)
Diffusion TS
0.116(0.010)
0.147(0.025)
0.013(0.001)
0.105(0.006)
Context-FID Score
TimeGAN
0.300(0.013)
0.103(0.013)
0.563(0.052)
1.292(0.218)
IDOL
0.077(0.013)
0.022(0.005 (02))
0.062(0.012)
0.065(0.002)
CwVAE
0.892(0.059)
0,807(0.252)
1.010(0.195)
0.896 103)
TAE
0.805(0.186)
0.215(0.035)
0.251(0.015)
(0.449(0.969)
CHiLD
0.008(0.000)
0.001(0.000) (.001(0.000)
0.079 (0.00)
KoVAE
0.045(0.006)
0.007(0.002)
0.203(0.031)
(1.832(0.007)
Diffusion-
0.049(0.008)
0.004(0.001)
0.193(0.027)
(0.01()
Correlational
TimeGAN
0.210K0.006)
0.063(0.005)
0.886(0.039)
23.502(0.039)
Score
IDOL
0.002(0.000)
0,006(0.000)
0.002(0.000)
0.098(0.003)
cw VAE
0.070(0.001)
0.053(0.001)
0.050(0.001)
18.434(0.030)
VAE
0.111(0.020)
0.095(0.008)
0.388(0.041)
17.296(0.526)
G.3.2 Ablation Study
To evaluate the effectiveness of each module, we further devise two variants of the proposed CHiLD as follows. 1) CHiLD-KL: we remove the K L divergence restriction of prior estimation. 2) CHiLD-C: we do not use the context information for the proposed CHiLD. Experiment results are shown in A9
find that incorporating the contextual observation and KI divergence has a positive impact on the overall performance of the model.
Table A9: Experiment results of different model variants on the Humaneva-Box and Humaneva- Throweatch datasets.
Box
Throwcatch
Model
Context-FID Score
Correlational Score
Context-FID Score
Correlational Score
CHiLD CHiLD-KL 2.038(0.003)
0.007(0.001)
(.021(0.002)
0.063(0.056)
0.662(0.007)
(.637(0.006)
0.077(0.004))
0.974(0.0039)
CHiLD-C
0.017(0.0034)
0.036(0.0015)
0.093(0.011)
1.077(0.0098)
"aapss//elimatedat.ii wtt.aaiii.......aaaaeaeeeeeeeee. kr/dattaccesm22Iens
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H Broader Impacts
The proposed CHiLD method offers a significant advancement in identifying hierarchical latent dynamics, which is crucial for generating realistic and interpretable time series data across wide range of domains such as healtheare, finance, climate science, and autonomous systems. By enabling the generation of time series that are both data-driven and causally grounded, our method not only enhances the credibility of the generated sequences also improves the performance of downstream tasks like forecasting, anomaly detection, and decision-making These improvements foster increased trust and adoption machine learning models in critical applications, particularly where reliable data-driven insights of paramount importance.
Furthermore, our work contributes the broader field of generative modeling by providing principled framework for hierarchical latent dynamics. This framework mitigates issues like mode collapse, a common problem in generative models, and enhances the model's generalizability across various datasets. As result, CHiLD stands powerful tool for both researchers and practitioners looking to explore complex time series data and apply generative modeling techniques in real- world scenarios. This method paves the way for future research in more complex domains, such as controllable video generation, neuroscience, and predictive analytics in environmental systems, thereby further extending its applicability.
40


Reviews:
[{"review_id": "zUnf4RepUu", "reviewer": "Reviewer_nYtx", "summary": "This paper investigates the problem of hiearchical temporal CRL, which investigates a dynamic latent Bayesian network where the observations are generated by the last layer of latent variables, which itself is generated by its 1-level higher and 1-step previous blocks of latent variables. They show that under regularity conditions, identifying the entire latent space with $L$ layers is possible using the joint distribution of $2L+1$ consecutive observation values. They then apply existing ideas to further refine this result to per-block and per-component recovery.", "questions": "- Even if it is obvious, can the authors include a formal equivalence, consistency or \"closeness\" statement for the implementation method?\n- In the spirit of Appendix G.1.3, can you show that adding more than necessary information does not significantly increase or reduce performance?\n- Please add a clarification in experiments section about (i) the synthetic experiments showing ID performance and (ii) real-world experiments showing data synthesis performance; this knowledge shouldn't be expected from a reader but written explicitly. I suggest slightly renaming the subsections to reflect this difference.\n- I did not find the compute resources necessary to run the experiments. Can you add it to the appendix in writing? Even if it is negligible -- I know it is not -- simply say so.", "limitations": "yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "The main departure of this work from existing temporal CRL papers is the observation model: The observed variables is NOT generated by an invertible transformation of the entire latent space, but, only the final layer of it. As far as I am aware, this way of inducing a lack of invertibility -- not due to a measurement noise, but from actual missing information in latent space -- is new, and perhaps promising in the context of real-world applications.\n\nWhile I appreciate the extensive study done for the experimentation, I find the overall workflow of converting identifiability results to theory-inspired VAE implementations slightly backward. Even if it is obvious, including a formal equivalence, consistency or \"closeness\" statement for the implementation method the authors describe would make the contribution -- and the overall flow of the paper -- more concrete.", "was_revised": true, "final_justification": "The authors have satisfactorily answered all my concerns in the rebuttal/discussion phases. I believe the paper is solid except for motivation/setting. However, I think that the extensive work done in other sections (e.g. experiments) strengthens the motivation enough for acceptance.", "rebuttal": "Dear Reviewer nYtx, we sincerely appreciate your encouraging and insightful feedback, which has significantly helped us enhance the completeness of our theoretical results and soundness of our experimental results. We have added a consistency result for the estimation and additional experimental results with different numbers of consecutive observations. Please find our point-by-point responses below.\n\n>Q1&Q2 (include a formal equivalence, consistency, or \"closeness\" statement for the implementation method): While I appreciate the extensive study done for the experimentation, I find the overall workflow of converting identifiability results to theory-inspired VAE implementations slightly backward. Even if it is obvious, including a formal equivalence, consistency, or \"closeness\" statement for the implementation method the authors describe would make the contribution -- and the overall flow of the paper -- more concrete.  \n\n\nA1 and A2: Thank you for your thoughtful feedback. In light of your suggestion, we have provided a consistency theoretical result for our implementation in the updated version, which is shown as follows:\n\nTheorem (**Consistency of Estimation**) Suppose the following assumptions hold:\n1) The variational posterior distribution $q(z_t|x_{t-L:t+L})$ is expressive enough to contain the prior $p(z_t|z_{t-1:t-\\tau})$.\n2) The ELBO is optimized jointly with respect to both the generative parameters and inference parameters.\n\nThen, in the limit of infinite data, the estimated latent variables are identified up to permutation and component-wise invertible transformations.\n\nProof: The ELBO can be written as follows:\n\n$ELBO=\\mathbb{E}_ {q(z_ {1:T}|x_ {1:T})}\\sum_ {t=1}^T\\log p(x_ t|z_ t) + \\mathbb{E}_ {q(z_{1:T}|x_{1:T})}[\\sum_ {t=1}^T(\\log p(z_ t|z_ {t-1:t-\\tau}))-\\log q(z_ t|x_ {t-L:t+L})]$\n\nUnder assumption (1), the true conditional prior lies within the variational family, so the KL divergence term can be minimized to zero as the model capacity and data size grow. In this case, optimizing the ELBO is equivalent to maximizing the log-likelihood, and the VAE inherits the consistency properties of maximum likelihood estimation. Moreover, since our identifiability results (Theorems 1–2) guarantee that the latent variables are identifiable up to an invertible transformation, the estimated latent variables converge to an equivalent representation of the true latent variables in the limit of infinite data. \n\n\n>Q3: In the spirit of Appendix G.1.3, can you show that adding more than necessary information does not significantly increase or reduce performance?\n\nA3: Thank you for this insightful suggestion, which helps strengthen the robustness of our experimental validation. To assess the impact of including more than the necessary amount of information, we conducted additional experiments on a synthetic dataset with three-layer hierarchical latent dynamics. Specifically, we varied the number of consecutive observations: 1, 3, 5, 7, and 9 time steps. The corresponding results in terms of MCC are reported below:\n\n| Length of Observations | 1 | 3 | 5 | 7 | 9 |\n|---|---:|---:|---:|---:|---:|\n| MCC | 0.6859(0.024) | 0.7001(0.008) | 0.8042(0.004) | 0.8241(0.015) | 0.8266(0.023)|\n\nThis indicates that once a sufficient amount of information is available to recover the latent structure, adding more data does not significantly affect the identification performance, which aligns with the intuition that excessive redundancy contributes little additional benefit.\n\n>Q4: Please add a clarification in the experiments section about (i) the synthetic experiments showing ID performance and (ii) real-world experiments showing data synthesis performance; this knowledge shouldn't be expected from a reader but written explicitly. I suggest slightly renaming the subsections to reflect this difference.\n\nA4: Thank you so much for your insights! In light of your suggestions, we have explicitly added a clarification in Section 5. Specifically, the synthetic experiments are designed to evaluate the identifiability performance. i.e., whether the learned latent variables align with the ground-truth latent variables. And the real-world experiments focus on data synthesis performance, demonstrating that the learned latent representations can generate realistic and consistent time series data. \n\nMoreover, we have renamed the subsection titles to better reflect these objectives:\n\n“Experiments on Synthetic Data” → “Synthetic Experiments: Evaluation of Identifiability\"\n\n“Experiments on Real-World Data” → “Real-World Experiments: Evaluation of Time Series Generation”\n\n>Q5: I did not find the computing resources necessary to run the experiments. Can you add it to the appendix in writing? Even if it is negligible -- I know it is not -- simply say so.\n\nA5: Thanks a lot! We have provided the compute resources in the updated version. Specifically, we conduct all the experiments on a single NVIDIA GTX 3090 24GB GPU and an i7-7700k CPU with 64Gb memory."}, {"review_id": "8yf8HD3YOV", "reviewer": "Reviewer_9SP8", "summary": "This submission addresses the problem of performing inference in a hierarchical latent causal process given time series data. The authors establish the component-wise identifiability of the latent variables under such a model, and develop a variational autoencoder architecture that produces the correct conditional independencies in the generation model and can accurately recover the latent posterior at inference time. The resulting generative model achieves state-of-the-art performance, while utilizing the hierarchical structure to disentangle levels of abstraction at generation time. This separation allows for controllable generation.", "questions": "Does performance degrade when the number of layers is increased above 2?", "limitations": "yes", "rating": 5, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 2, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\n\n- The paper provides extensive theoretical contributions that establish the identifiability of latent variables under a hierarchical causal process.\n\n- The experiments are promising, and show widespread state-of-the-art performance on simulated and real data.\n\nWeaknesses:\n\n1. I found some parts of the submission to lack clarity, although I acknowledge that these issues might be a result of a lack of confidence in this specific setting. For example, Definitions 1 & 2 seem to be missing key conditions: $\\hat{z}_t^l$ needs to be defined as the estimated latent variable from observations, or else that statement \"there exists an invertible function $h$ and value $\\hat{z}$ such that $z=f(\\hat{z})$\" is trivially true. \n\n2. On real world data, only $L=2$ is explored, reducing the impact of working with a hierarchical latent process.", "was_revised": true, "final_justification": "See discussion. The added experiment further demonstrates the benefits of their model's hierarchical structure. I am keeping my score at 5 and believe this submission should be accepted.", "rebuttal": "Dear Reviewer 9SP8, we are very grateful for your valuable comments, helpful suggestions, and encouraging feedback. They have greatly helped us improve the readability of our paper and enhance the completeness of our experiments. As per your valuable suggestions, we have provided a clearer definition and added more real-world experiments to better evaluate the different layers of latent variables.\n\n>Q1: I found some parts of the submission to lack clarity, although I acknowledge that these issues might be a result of a lack of confidence in this specific setting. For example, Definitions 1 & 2 seem to be missing key conditions: $\\hat{z}_t^1$ needs to be defined as the estimated latent variable from observations, or else that statement \"there exists an invertible function $h$ and value $\\hat{z}_t$ such that $z_t=f(\\hat{z}_t)$\" is trivially true.\n\nA1: Thank you for pointing this out. In light of the suggestion, we have revised Definitions 1 and 2 to explicitly state that $\\hat{z}_t$ denotes the estimated latent variables inferred from observations. \n\n>Q2: On real-world data, only $L=2$ is explored, reducing the impact of working with a hierarchical latent process.\n\nA2: Thank you for this insightful comment. In light of your suggestions, we have conducted additional experiments with 3 and 4 latent layers on five real-world datasets: Weather, WeatherBench, Throwcatch, Gestures, and CESM2. Experiment results are shown as follows.\n\n| Dataset | Weather | WeatherBench | Throwcatch | Gestures | CESM2 |\n|:---:|:---:|:---:|:---:|:---:|:---:|\n| # of Levels | Context-FID |  |  |  |  |\n| 1 | 0.676(0.103) | 0.088(0.014) | 0.025(0.004) | 0.041(0.006) | 0.126(0.023) |\n| 2 | 0.507(0.042) |  0.078(0.017) |  0.022(0.003) | 0.032(0.004) | 0.018(0.003) |\n| 3 | 0.486(0.199) | 0.068(0.013) | 0.021(0.002) | 0.026(0.005) | 0.034(0.004) |\n| 4 | 0.270(0.035) | 0.063(0.019) | 0.029(0.005) | 0.044(0.007) | 0.042(0.010) |\n| # of Levels | Correlational Score |  |  |  |  |\n| 1 | 0.169(0.003) | 5.476(0.025) | 0.093(0.009) | 0.112(0.005) | 1.520(0.025) |\n| 2 | 0.165(0.004) |  5.190(0.016) | 0.065(0.001) |  0.089(0.003) | 1.250(0.009)  |\n| 3 | 0.155(0.007) | 4.952(0.017) | 0.080(0.003) | 0.082(0.004) | 1.521(0.023) |\n| 4 | 0.158(0.005) | 4.770(0.015) | 0.082(0.002) | 0.095(0.009) | 1.638(0.049) |"}, {"review_id": "o332UW0PRZ", "reviewer": "Reviewer_nGAz", "summary": "# Summary\n\nThe paper establishes identifiability results for hierarchical temporal causal representations. It presents two main theorems and one lemma:\n\n1. **Theorem 1** : Proves block-wise identifiability of the latent vector $z_t$.  \n2. **Theorem 2** : Extends block-wise identifiability to each hierarchical layer.  \n3. **Lemma 1** : Shows component-wise identifiability of individual latent variables $z^l_{i,t}$.\n\nAdditionally, the authors propose an algorithm to reconstruct the multilayer latent variables.", "questions": "- I remain uncertain about the generative process chosen for the model. Could the authors clarify its motivation? Specifically, what underpins the assumptions that (i) variables in layer $l+2$ cannot influence those in layer $l$ or layer 1, (ii) only a single temporal lag is considered, and (iii) neither $z^{l-1}_{t-1}$ nor $z^{l-2}_t$ can be causal for $z^{l}_t$?\n\n- Several modelling assumptions appear tailored to meet the identifiability conditions rather than arising from domain considerations. Could the authors explain how these assumptions are justified independently of the identifiability proofs, in other words are there any clear real world application of such structure?\n\n- The framework assumes that each observation $x_t$ is generated solely by the first latent layer. Why are contributions from higher layers ruled out, and how realistic is this restriction in practical settings?\n\n- Every latent layer is specified to have dimensionality $n$, matching that of the observed data. What motivates this choice, and how sensitive are the identifiability results to relaxing the dimensionality constraint?", "limitations": "The authors mentioned one limitation related to their assumptions like layer wise conditional independence. However, as we mentioned, we still have concerns regarding the hierarchical structure assumed by this paper as well as the identifiability techniques used to prove the results.", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "# Strengths\n\n- **Novel hierarchical generative process:** The paper introduces a new multi-layer generative model in which each layer injects additional noise, making the identifiability problem notably more challenging.\n\n- **Methodological contribution:** It thoughtfully adapts techniques from the measurement-error literature to causal latent-variable identifiability.\n\n- **Clear intuition for assumptions:** For every theorem, the authors provide intuition that clarifies each assumption, helping readers in understanding the results.\n\n# Weaknesses and Questions:\n\n- **Clarity and motivation:** The introduction does not yet offer a clear motivation for the specific hierarchical structure used in the paper or its real-world relevance. Concrete examples or use cases would greatly improve the writing and the storytelling of this paper.\n\n- **Originality of exposition:** Portions of the introduction closely resemble text from the IDOL paper, though paraphrased. Clearer attribution or more distinctive framing would enhance transparency and underscore the paper’s unique contributions.\n\n- **Premature technical detail:** Technical discussion of the single-layer assumption appears on the first page without sufficient context. Deferring such material until after the problem statement and motivating examples would aid readability.\n\n- **Generative process assumptions:** Some modeling choices, not allowing influences from layer \\(l+2\\) to \\(l\\), limiting interactions to a single lag, and excluding links such as $z^{l-1}_{t-1}$, $z^{l}_{t}$ seem unrealistic. Several assumptions feel tailored to produce identifiability rather than arising from domain knowledge. Stronger justification would help substantiate these assumptions.\n \n- **Observation model:** The framework assumes that only the first latent layer generates \\(x_t\\). Explaining why higher layers do not contribute, or exploring a more flexible observation mechanism, would improve realism.\n\n- **Layer dimensionality:** Each latent layer is set to have the same dimension \\(n\\) as the observations. Discussing scenarios where this design choice is appropriate and motivate it\n\n- **Overlap with prior proofs:**   *Theorem 1* closely follows Theorem 1 in “Instrumental Variable Treatment of Nonclassical Measurement Error Models.”  *Theorem 2* adapts the proof strategy of Theorem 1 in the IDOL paper with only minor modifications. \n\n- **Minor errors:** There are typographical issues in Eq(3), Appendix Eq(A19), and the sentence around line 1076 that should be corrected.\n- Please correct me if I’m mistaken, identifiability results in CRL identify latent variables $z_t$ and try to prove the existence of an invertible mapping between $\\hat{z}_t$ and $z_t$. Given that the observation $x_t$ are observed and fixed can you explain why $\\hat{x}_t$ in your proofs ? Using a different symbol for the estimated model Eq(A16) might make the notation clearer. \n\n- **Operator definition (EqA8):** It would be more rigorous to define the operator $L_{x_{>t}\\mid z_t}$ explicitly, as done in *Instrumental Variable Treatment of Nonclassical Measurement Error Models* proof in page 17.\n\n- **Fairness of the IDOL comparison:** The manuscript claims to achieve identifiability without a permutation step, unlike IDOL. However, IDOL’s component-wise identifiability also holds up to permutation, while the present work avoids only layer-index permutation due to its hierarchical design.", "was_revised": true, "final_justification": ".", "rebuttal": "Dear Reviewer nGAz, we highly appreciate the valuable comments and helpful suggestions on our paper and the time dedicated to reviewing it. Below, please see our point-to-point responses.\n\n>Q1: \"Clarity and motivation\"\n\nA1: Thank you for your comment. We actually put a lot of effort into the motivation and the clarity, as you see **Page 1, Lines 34–35 in introduction**. Moreover, we have provided motivations and real-world examples in **Line 1243-1268 in Appendix C**. Please kindly let us know if you cannot find it. For information below the summary of the two real-world examples. \n1. Human Motion Modeling: The higher-level latent variables correspond to coarse movement categories (e.g., walking, running), while the lower-level latent variables capture fine-grained kinematics.\n2. Climate Data: We consider a three-layer hierarchical structure where the latent variables represent seasonal, monthly, and daily factors that influence weather patterns.\n\n>Q2: \"Originality of exposition\"\n\nA2: These two papers are in the same line of research, aiming to identify the latent variables of time series data. And therefore, we agree that to some level, the motivations are related. Moreover, we would like to respectfully highlight that the clear difference between these two works is as follows:\n1. **Different research focus**. IDOL is motivated by identifying latent causal processes with instantaneous dependencies under sparse causal influences. CHiLD specifically targets the identifiability of hierarchical latent dynamics, which has not been addressed previously. The introduction directly starts from hierarchical temporal structures and discusses why existing single-layer methods fail in this setting.\n2. **Distinct motivation and framing**. IDOL emphasizes relaxing strong assumptions, such as interventions or grouping of observations, to achieve identifiability in the single-layer case. In contrast, we highlight that the commonly used “sufficient changes” condition breaks down under hierarchical structures, thereby motivating the need for a new framework to recover the joint distribution of latent variables.\n3. **Independent theoretical contributions**. IDOL establishes component-wise identifiability based on a sparse latent process. While CHiLD proposes another block-wise identifiability, showing that the joint distribution of latent variables can be uniquely determined with certain adjacent observations, and further achieves component-wise identifiability within each layer. This theoretical insight is entirely different. \n4. **Clearer attribution**. While the two works share a general background on causal representation learning, we will revise the introduction to explicitly acknowledge IDOL as a single-layer baseline and further emphasize how CHiLD goes beyond it by addressing hierarchical structures.\n\nIn light of your comment, we have also included a clear distinction in the revised version.\n\n>Q3: \"Premature technical detail\"\n\nA3: We kindly remind you that the technical discussion has been provided in **Section 2.2 (Page 3)** of the original submission with the title of “Why previous theoretical results can hardly identify the hierarchical latent variables?”. We believe this placement aligns with your suggestion to defer technical content until after sufficient context is provided.\n\n>Q4&11&12: As for \"Generative process assumptions\".\n\nA4: Thank you for noticing this. Kindly let us explain why the model is actually flexible as follows:\n\n1. Not allowing influences from layer l+2 to l. Although the generative process in Fig. 1 illustrates a first-order Markov structure, we explicitly state on **Line 69 and in the footnote on Page 2** that “the illustration shows a first-order Markov structure; however, our method can extend to higher-order dependencies.” Moreover, as shown in Equ.(2), $z_{t,i}^l=f_i(Pa_e(z_{t,i}^l),Pa_d(z_{t,i}^l), \\epsilon_{t,i}^l)$, we does not imply the first-order dependencies, and cross-layer interactions e.g., from layer (l+2) to (l), can be naturally incorporated. Furthermore, when influences from layer (l+2) to (l) exist, we can consider layers (l) to (l+2) as a merged layer, which is a special case of the 1-order Markov structure.\n2. Limiting interactions to a single lag. As our model is not limited to a first-order Markov structure, we have considered scenarios with multiple temporal lags, and the corresponding proof has been provided in **Appendix B.6**.\n3. Excluding links such as $z_{t-1}^{l-1} \\to z_t^l$. If there is a link from $z_{t-1}^{l-1}$, we can only achieve the block-wise identifiability results, where $z_{t-1}^{l-1}$ and $z_t^l$ compose a block. However, we should make such an assumption to establish very strong component-wise identifiability results. If that's not the case, the result is still non-trivial. However, it is not as strong as the present one. In light of your reminder, we have also added a simulation experiment where there are links from $z_{t-1}^{l-1}$ to $z_t^{l}$, the experiment results are shown as follows:\n\n||MCC|\n|-|-|\n|w. $z_{t-1}^{t-1}\\to z_t^l$ |0.811(0.021)|\n|w.o $z_{t-1}^{t-1}\\to z_t^l$ |0.631(0.019)|\n\nExperiment results show that the component-wise identification is hard to achieve when there is $z_{t-1}^{l-1}\\to z_t^l$. To summarize, strong identifiability results rely on appropriate assumptions. However, we have discussed the consequences of the violation of this particular assumption in the revised version.\n\nIn light of your reminder, we have highlighted the aforementioned cases in the updated version. \n\n>Q5&13: \"Observation model\"\n\nA5: Thank you for this valuable comment. We would like to clarify that the assumption that only the first latent layer directly generates the observations $x_t$ is motivated by hierarchical latent processes often encountered in real-world scenarios, such as climate data. In such settings, high-level latent variables govern abstract temporal dynamics, which are manifest in the observed data via the bottom latent variables. \nMoreover, this case is more challenging, as some latent variables do not directly contribute to $x_t$. Motivated by the real problems, this constraint allows us to establish the identifiability of the joint distribution, even in the presence of noise, because the latent variable cannot be recovered as determining functions.\n\n>Q6&14: \"Layer dimensionality\"\n\nQ6: Thanks! First, our theoretical results do not require the dimensionality of the latent variables at each layer to match that of the observations. The same dimensionality was chosen in our work primarily for convenience, as it allows us to explicitly analyze how the relationship between latent dimensionality and observed dimensionality affects identifiability. As stated in Lines 1094–1101, the injectivity is ensured when the total observed dimensionality in $x_{t-\\tau},\\cdots,x_{t-1}$ is greater than or equal to the total latent dimensionality, which is also been verified in our synthetic experiments. We have added a further discussion in the revised version to explicitly emphasize that the latent dimensionality is flexible in practice.\n\n>Q7: \"Overlap with prior proofs\"\n\nA7: Thank you very much! As mentioned above, we aim to address a real problem in machine learning as you can see the motivation example in Appendix C. And this problem has not been handled in previous works like IDOL. These prior results do not account for the hierarchical temporal structure or the form of non-invertibility and noisy mixing structure in our setting.\n\nTo achieve identifiability in new cases, we have to formulate the specific model and then tailor the proof and incorporate further constraints to achieve our identifiability results. We hope you will find this work serves as an essential step in dealing with important problems in representation learning.\n\n>Q8: \"Minor errors\"\n\nA8: Thanks! We would like to clarify the following:\n1. Eq(3): There is no typographical error here. Eq(3) is correct as it explicitly expresses that the top-layer latent variables are only influenced by time-delayed variables, consistent with the generative process assumptions.\n2. Appendix Equation (A19): You are correct, and we have revised it in the updated version to $\\int_ {\\hat{X}_ t}p_ {\\hat{x}_ t|\\hat{z}_ t}d\\hat{x}_ t$.\n3. Sentence around Line 1076. We guess you are confused with $D_ {\\hat{x}_ t|\\hat{z}_ t}$ 和 $D_ {x_ t|z_ t}$, but they are correct since they represent the estimated conditional distribution and the true conditional distribution, respectively. \n\n>Q9: The explaination of $\\hat{x}$.\n\nA9: Thank you for your question. We use the “^” notation to denote estimated variables, which is a convention widely adopted in related literature[1]. Specifically, $\\hat{x}_t$ in our proofs refers to the estimated observed variables from the learned generative model. In light of your suggestion, we have added a clarification in the notation table of the updated version to explicitly state this point.\n\n[1] Score-based Causal Representation Learning: Linear and General Transformations\n\n>Q10: Operator definition (EqA8)...\n\nA10: Thank you for your helpful suggestion. In the original version, we expressed the linear operator in terms of its probabilistic form for better intuitive understanding. We have now explicitly defined the operator $L_{x>t|z_t}$ in the updated version.\n\n>Q11: \"Fairness of the IDOL comparison\"\n\nA11: Thank you for pointing out the connection. Our results without layer-wise permutation benefit from the hierarchical prior. This prior reflects the natural organization of many real-world temporal processes and allows us to distinguish layers without permutation. Furthermore, compared with IDOL, our model allows for a noisy mixing procedure where the latent variables are not deterministic functions of the observations, and some latent variables may not directly contribute to $x_t$. This setting is more aligned with many real-world problems, where the generation process involves uncertainty."}, {"review_id": "PBmuocSoaw", "reviewer": "Reviewer_yDBF", "summary": "This work considers a novel setting that combines hierarchical causal latent structure with temporal dynamics. Under strengthened assumptions on the latent generative model, it proposed a hierarchy of increasingly refined identifiability results based on observations from time steps t-L to t+L, where L is the number of hierarchical layers. The theoretical contributions include i) the block-identifiability of all the variables in time t, ii) block-identifiability of the variables in time t and layer l, and finally iii)element-wise identifiability. To empirically validate the theoretical results, the authors develop a VAE-based model and evaluate it on both synthetic data and nine real-world datasets, including three climate datasets and two human motion datasets. Overall, the proposed method outperforms the other baseline models, including VAE-based methods, generative models, and causal representation learning methods.", "questions": "1.\tAbout linear operators, does it exist for arbitrary two distributions? If the answer is no, then assumption (ii) made in Theorem 1, does it implicitly assume that the existence of a linear transformation between the distributions of Lxt+1,··· ,xt+L|zt and Lxt−L,··· ,xt−1|xt+1,··· ,xt+L? The example 4 in appendix B.3, is it a typo? Post-linear in the bold part but non-linear later? \n\n2.\tIn the hierarchical data generation process, does it assume that the dimensionality of the latent variables at each layer matches that of the observations (since only $n$ is used in the Sec2.1? While from the experimental setting it seems no? Or this is some assumption for theorem only? Does it assume the dimension of latent is known as pior? If no, how to estimate it in the experiment for both real datasets and synthetic?", "limitations": "Some missing points are listed in the weaknesses above", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths\n\n1.\tThe setting, considering hierarchical causal latent structure with temporal dynamics, is novel and well-motivated. The structure of the series of identifiability results is clear and easy to follow.\n\n2.\tUnlike many existing causal representation learning approaches that assume a deterministic mixing process, this work allows for noise in the mixing functions and also cross-layer connections. \n\n3.\tThe experiments on real datasets are sufficient and well-explained. It matches the setting in the theorem well, showing strong evidence about the efficiency of the effectiveness of the proposed method.\n\nWeaknesses\n\n1.\tThe theoretical contributions, while clearly presented, offer limited novelty. Both the core assumptions and the proof strategy closely follow prior work.\n\n2.\tThe assumptions made for achieving identifiability might be restrictive, especially the conditional independence assumption in the final theorem (as the authors also clearly mentioned in the discussion section) and the linear operator at the beginning, which might not be true for many cases. \n\n3.\tIn the synthetic dataset, the mixing function between x and z is only a simple 2-layer of leakyReLU. I am concerned about whether the setup is sufficiently expressive to convincingly demonstrate the method’s ability to handle complex nonlinear mixing functions. \n\n4.  Although increasing the number of layers in the latent space adds complexity to the identifiability problem, the authors ease this by using a broader temporal window (from t-L to t+L), which introduces a tradeoff and could be impractical in certain applications.", "was_revised": true, "final_justification": "For completeness", "rebuttal": "Dear Reviewer yDBF, we sincerely appreciate your informative feedback and helpful suggestions that helped clarify our contributions, improve the readability of our theories, and the completeness of our experiments. Please see our point-to-point response below. \n\n>Q1: The theoretical contributions, while clearly presented, offer limited novelty. Both the core assumptions and the proof strategy closely follow prior work.\n\nA1: Thank you for your thoughtful comment, which helps us highlight the contribution of our work. We would like to emphasize that our contribution lies in addressing a fundamentally different and practically motivated problem. Specifically, as illustrated by the motivating example in Appendix C, we aim to address a real-world challenge in representation learning: recovering hierarchical temporal latent structures where only a subset of the latent variables contributes directly to the observations, and where the latent-to-observed mapping is non-invertible and noisy. This setting is not covered by existing results such as those in IDOL, which rely on stronger assumptions (e.g., full invertibility, flat latent structures) and are not applicable in the presence of hierarchical dependencies and partial observability.\n\nTo tackle this more realistic yet theoretically challenging problem, we developed a specific model formulation and carefully tailored the proof strategy, introducing additional hierarchical constraints. This allows us to establish identifiability results under broader and more practical conditions. In light of your suggestion, we have highlighted the contribution of our work in the updated version.\n\n>Q2: The assumptions made for achieving identifiability might be restrictive, especially the conditional independence assumption in the final theorem (as the authors also clearly mentioned in the discussion section) and the linear operator at the beginning, which might not be true for many cases.\n\nA2: Thank you for your insightful comment, which helps us further clarify the intuition behind our assumptions and improve the presentation of the paper. First, we would like to clarify that the conditional independence assumption [1,3] and the invertible linear operator [2] assumption are common in the literature of causal representation learning. Below, we elaborate on why these assumptions are not overly restrictive and how they can be further relaxed:\n\n1. As for the conditional independence assumption, we assume conditional independence within each layer of latent variables, while allowing instantaneous effects across layers. This assumption is primarily made for clarity in presenting the identifiability of latent variables within each layer. Even the the conditional independence assumption does not hold in practice, the latent variables from each layer can still achieve the block-wise identifiability according to Theorem 2, facilitating the hierarchically controllable generation. Moreover, this assumption can be relaxed by leveraging the results of [4], identifiability can still be established by exploiting the sparsity of latent dynamics even in the presence of instantaneous effects. \n\n2. As for the linear operator assumption, I guess you mean that the linear operator assumption denotes a type of linear transformation. However, the linear operator describes a mapping between any two arbitrary distributions. And the assumption of an injective linear operator, e.g., $L_{x_{t+1},\\cdots,x_{t+L}|z_t}$, intuitively means that the information contained in the observations is equivalent to that in the latent variables. They are commonly assumed when we try to recover the joint distribution in the presence of latent variables. Furthermore, it is also a natural assumption for real-world scenarios, since if the temporal observations $x_{1:T}$ do not encode the information of $z_t$, the latent variables cannot be recovered from the observations. In this paper, we explicitly state this assumption to clarify the sufficient conditions under which hierarchical latent variables are identifiable. Additionally, we have provided several **practical examples of injective linear operators in Appendix B.3** to support the plausibility of this assumption.\n\n[1] Hyvarinen, Aapo, and Hiroshi Morioka. \"Unsupervised feature extraction by time-contrastive learning and nonlinear ica.\" NeurIPS2016  \n[2] Fu, Minghao, et al. \"Identification of Nonparametric Dynamic Causal Structure and Latent Process in Climate System.  \n[3] Yao, Weiran, Guangyi Chen, and Kun Zhang. Temporally disentangled representation learning.  \n[4] Li, et al. \"On the identification of temporally causal representation with instantaneous dependence.\" (ICLR2025) \n\n>Q3: In the synthetic dataset, the mixing function between x and z is only a simple 2-layer of leakyReLU. I am concerned about whether the setup is sufficiently expressive to convincingly demonstrate the method’s ability to handle complex nonlinear mixing functions.\n\nA3: Thank you for this constructive comment, which improves the soundness of our experiment results. In light of your suggestions, we have provided additional synthetic experiments with different nonlinear mixing functions, including 3- and 4-layer MLPs with Tanh and leakyReLU activation, respectively. Experiment results are shown as follows. \n\n|Mixing Function| MCC |\n|---|---|\n| 3-layer Leakly_ReLU | 0.8024 (0.0291) |\n| 4-layer Leakly_ReLU | 0.7564 (0.0123) |\n| 3-layer Tanh | 0.8118 (0.0217) |\n\n\n>Q4: Although increasing the number of layers in the latent space adds complexity to the identifiability problem, the authors ease this by using a broader temporal window (from t-L to t+L), which introduces a tradeoff and could be impractical in certain applications.\n\n A4: Thank you for this insightful comment.  We agree that in certain practical applications, obtaining a sufficiently long temporal window may be challenging. This naturally leads to a trade-off between the length of the observation window and the extra assumption required for identifiability.\n\nTo illustrate this, let us consider a simplified case with only two layers of latent variables. Instead of requiring 5 consecutive observations as in our main setting, the latent variables can be identified with only 3 time steps of observations under the sparsity assumption. When each latent variable has a sufficient number of pure children (e.g., 2 conditional independent observed variables), we can first identify the lowest-layer latent variables by leveraging the results in [5]. Then, based on the recovered lower-layer representations, the higher-layer latent variables can be identified iteratively. This significantly relaxes the requirement on the temporal window length. \n\n[5] Kong, et al. \"Identification of nonlinear latent hierarchical models.\" NeurIPS2023\n\n>Q5: About linear operators, does it exist for arbitrary two distributions? If the answer is no, then assumption (ii) made in Theorem 1 does it implicitly assume that the existence of a linear transformation between the distributions of Lxt+1,···,xt+L|zt and Lxt−L,···,xt−1|xt+1,···,xt+L? Example 4 in Appendix B.3, is it a typo? Post-linear in the bold part, but non-linear later?\n\nA5: Thanks for your good question. Let us answer it point by point. First, the linear operators exist for arbitrary two distributions as mentioned in [6]. We also thank you for pointing out the typos; the \"Post-linear\" should be \"Post-nonlinear\".\n\n[6] Nelson Dunford and Jacob T. Schwartz. Linear Operators. John Wiley & Sons, New York, 1971\n\n>Q6: In the hierarchical data generation process, does it assume that the dimensionality of the latent variables at each layer matches that of the observations (since only n is used in the Sec2.1? While from the experimental setting it seems no? Or is this some assumption for the theorem only? Does it assume the dimension of the latent is known a priori? If no, how to estimate it in the experiment for both real datasets and synthetic?\n\nA6: Thank you for raising this important question, which helps us clarify the assumptions on the dimensionality of latent variables. \n\nFirst, our theoretical results do not assume that the dimensionality of the latent variables at each layer matches that of the observations. In our experiments, we chose equal dimensionality primarily for simplicity, as it allows us to clearly examine how the relative size of the latent space and observation space affects identifiability. As stated in Lines 1094–1101, the mapping from latent to observed variables is injective as long as the total dimensionality of the observed variables in $x_{t-\\tau},\\cdots,x_{t-1}$ is greater than or equal to the total dimensionality of the latent variables. This result has also been empirically supported by our synthetic experiments. Following your suggestion, we have added a corollary in the revised version to explicitly state that the latent dimensionality is not restricted in theory and can vary in practice.\n\nSecond, we assume the dimensionality of latent variables to be known in our theoretical analysis. For real-world scenarios where the true dimensionality is unknown, we treat the number of latent variables and the number of layers as hyperparameters. In practice, these can be estimated via standard model selection techniques, such as evaluating predictive accuracy or log-likelihood under different configurations, to select the optimal values."}]

Author rebuttal:

Meta-review: ## My review of the paper

This paper shows that hierarchical latent variables in temporal settings are identifiable by leveraging past, $x_{<t}$ and future $x_{>t}$ observations as context. They begin by showing blockwise identifiability of the L layers of hierarchical latents by assuming that one matches the joint over adjacent variables under an injective linear operator assumption. On the surface, block-identification of the layers doesn't tell you much---one hasn't yet disentangled the hierarchical relationship between latents---but this sets up their later results, and I thought that the linear operator assumption & proof technique was interesting. I've seen similar assumptions being used in the non-parametric instrumental variable literature (this connection is cited), but I'm not aware of prior work using these assumptions being used for causal representation learning. They then proceed to show that the sparsity of the hierarchical structure (where variables only depend on hierarchical parents and time lagged hierarchical) gives block identifiability of the latent layers, and finally conditional independence gives component-wise identifiability. These identifiability results are turned into an algorithm via a variational autoencoder that enforces the appropriate dependency structure. They evaluated their approach on both simulated data, where they achieved decent MCC results (though it is hard to say how well that would transfer to real settings), and a variety of real datasets. The real data experiments were impressively diverse (covering weather, stocks, human motion, etc.) but just focused on evaluating the observable variables (FID & correlation scores), with one qualitative experiment on the human motion dataset.

#### Strengths & weaknesses
 - I found the theory section very interesting - I was surprised at how much can be identified if you are prepared to assume injectivity of linear operators & the hierarchical structure. For this reason, I think the paper should be accepted.
 - While I appreciated the breadth of the experiments, I think that they would have been far more convincing if there was more effort put into trying to understand what hierarchical latent structure was estimated in practice. Causal representation learning has made a lot of theoretical progress, but the algorithms remain very difficult to apply on real problems because not enough effort has been put into this task of understanding the resulting latents (and critiquing the model).

Presentation issues & minor comments:

I felt that Section 2 / 3 could be improved by explicitly introducing a running example to illustrate the hierarchical data generating process that the authors have in mind. The section already uses examples from finance to illustrate the assumptions, so introducing that example earlier to provide examples of what you hope to identify & what makes it hard, would make these sections more accessible.

## Metareview

My opinion of the paper (captured in the review above) is fairly consistent with what the reviewers noted. I agree with Reviewer nYtx that the connection between the identifiability results and the algorithm should be formalized by including the outcomes of the discussion from the rebuttal in the text. There should also be a more explicit discussion on the limitations of applying these methods in practice (it is hard to know whether your assumptions are correct, and interpreting the latents is a challenge). But as I said in my review, I think the theoretical contributions are interesting, and the paper is worthy of acceptance.
Decision: Accept (poster)