Identifying latent state transitions in non-linear dynamical systems
Authors: Çağlar Hızlı
Çağatay Yıldız
Matthias Bethge
S. T. John
Pekka Marttinen
Venue: ICLR 2025 Poster (2025)
Keywords: nonlinear ica
identifiability
disentanglement
dynamical systems
Primary area: learning on time series and dynamical systems
TL;DR: We introduce a new framework that identifies unknown transition functions solely from high-dimensional observed sequences.

Abstract: This work aims to recover the underlying states and their time evolution in a latent dynamical system from high-dimensional sensory measurements. Previous works on identifiable representation learning in dynamical systems focused on identifying the latent states, often with linear transition approximations. As such, they cannot identify nonlinear transition dynamics, and hence fail to reliably predict complex future behavior. Inspired by the advances in nonlinear ICA, we propose a state-space modeling framework in which we can identify not just the latent states but also the unknown transition function that maps the past states to the present. Our identifiability theory relies on two key assumptions: (i) sufficient variability in the latent noise, and (ii) the bijectivity of the augmented transition function. Drawing from this theory, we introduce a practical algorithm based on variational auto-encoders. We empirically demonstrate that it improves generalization and interpretability of target dynamical systems by (i) recovering latent state dynamics with high accuracy, (ii) correspondingly achieving high future prediction accuracy, and (iii) adapting fast to new environments. Additionally, for complex real-world dynamics, (iv) it produces state-of the-art future prediction results for long horizons, highlighting its usefulness for practical scenarios.

Published as a conference paper at ICLR 2025
IDENTIFYING LATENT STATE TRANSITIONS IN DYNAMICAL SYSTEMS
Çaglar Hizh
Çagatay Yildiz
Matthias Bethge University of Tübingen Tübingen AI Center
Aalto University caglar www.liaalloo...oocoooa.a fi
University of Tübingen Tübingen AI Center
ST John Aalto University
Pekka Marttinen Aalto University
ABSTRACT
This work aims recover the underlying states and their time evolution in a latent dynamical system from digeesimension sensory measurements, Previous works on identifiable representation learning in dynamical systems focused on identifying the latent states, often with linear transition approximations. As such, they cannot identify nonlinear transition dynamics, and henee fail to reliably predict complex future behavior. Inspired by the advances in nonlinear ICA, we propose a state-space modeling framework which we can identify not just the latent states but also the unknown transition function that maps the past states to the present. Our identifiability theory relies on two key assumptions: (i) sufficient variability in the latent noise, and (ii) the bijectivity of the augmented transition function. Drawing from this theory, we introduce a practical algorithm based on variational auto-encoders. We empirically demonstrate that it improves generalization and interpretability of target dynamical systems by (i) recovering latent state dynamics with high accuracy, (ii) correspondingly achieving high future prediction accuracy. and (iii) adapting fast to new environments. Additionally, for complex real-world dynamics, (iv) produces sate--o-thhe-ee future prediction results for long horizons, highlighting its usefulness for practical scenarios.
INTRODUCTION
We focus on the problem of understanding the underlying states of a dynamical system from its high-dimensional sensory measurements. This task is prevalent across various fields, including reinforcement learning (Hafner et al. 2019a) and robotics (Levine et al. 2016). For example, consider cartpole, where instead of directly observing the underlying states (cart position and velocity, pole angle and angular velocity; Fig. (a)),, we observe video stream of its behavior (Fig. (b)). Our main objective is to learn latent representations and state transition functions from such high-dimensional sequences, which are useful for downstream tasks such as optimal control of the observed system.
Due to the partially observed nature of the problem, learning dynamics in the data space (e.g., pixel space) is not feasible, and previous works often focus on learning latent dynamical systems (Hafner al. 2019b). However, such latent models commonly are not guaranteed recover the true under- lying states and transitions tnon-identifiabllig resulting in entangled representations, lack of gen- eralization across new domains, and poor interpretability (Schmidhuber 1992 Bengio al. 2013).
Identifiable representation learning aims to address these challenges by learning the underlying fac- tors of variation in the true generative model. To achieve this, our approach builds on the classical ideas of ICA., namely, the assumption of independent components. In the case of a linear emission function, this independence assumption has been used to identify states in dynamical systems (Ciaramella et al.) 2006; Kerschen et al. 2007). For the nonlinear case, |Hvvärinen & Morioka (2016) 2017 7) introduced an identifiability framework using two main assumptions: (i) latent states show non-stationarity or autocorrelation driven by auxiliary variable (this is referred to as 'sufficient variability"), ) and (ii)
given the auxiliary variable, they are conditionally independent. However, this framework assumes
mutually independent latent states that do not affect each other (Hyvärinen & Morioka 2016 2017

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(a)
Theorem 2
(c)
Unobserved Physical System
Our
Dovs Pros
Identified transition function
Identifiblee
Dynamical
System
Theorem
0 Reax Rost
Ideattfied lazent states
((6)
(b) Raw Observed Signal (e.g.. video)
X1.7
Time
Figure 1: Sketch of our method and main theoretical contribution. (a) We assume an underlying unobserved dynamical system, e.g., cartpole, where the full state Z, is composed of cart position and velocity and angle and angular velocity of the pole: [x, R. 0, 6). (b) We partially observe the system as a sequence of video frames X1:7, which are used as input to our method.
(c) We learn an inverse emission function g x, + 2, that maps the raw observation signals to the estimated latent state variables, as well as a transition function 2, + 2.+1 that maps the past latent states to the present latent state, Identifiability of the latent states is ensured by Theorem 1, In addition to this, our main contribution is the identifiability of the transition function ensured by Theorem 2.
Hyvärinen et al., 2019 Khemakhem et al. 2020). This mutual independence assumption is unrealistic for dynamical systems, as the present state of the system depends on the past states, i.e., the transition function propagates the system state by nonlinearly mixing the past state components.
Identifiability in dynamical systems. Recently, Yao et al. [202 2022) extended identifiability theory to dynamical systems by replacing the assumption of mutually- independent states with temporally- mixed states, conforming to the same non-stationarity and autocorrelation conditions. Under the refor- mulated assumptions, they showed that it is possible to identify recover the true latent states in dy- namical system (up to element- tiise transformations). Using similar assumptions for an autoregressive process in the observation space, Moorioaa Maric,,Morioka al. (2021) identified the 'innovations'
process noise, which represents the stochastic impulses fed to the system. See Tab. 3 (App. A for a comparison.
Although these approaches yield provably identifiable representations of the latent states, their theoretical frameworks fall short in identifying the underlying dynamics. To resolve this, inspired by nonlinear ICA, we introduce additional stochastic variables that are conditionally independent of each other given auxiliary variables. This conditional independence assumption implicitly encodes new, external information that is sufficiently independent of the observations (Hyvärinen al. 2019, App. E). Interestingly, the idea of conditioning on auxiliary variables has been proposed in other contexts, e-g., Packard et al. (1980) showed that time-delayed auxiliary variables help resolve underlying data generating dynamics of chaotic systems from the residual independent noise.
Our contributions. We present the first framework that allows for the identification of the unknown transition function alongside latent states and the emission functior"] (see Fig.1 Following previous works YYoo et al., 2021 2022), we first establish the identifiability of the latent states (Fig | Theorem I). We show that identifying the transition function is ensured by further restricting the spaces of the process noise prior and the transition function (Fig. T Theorem 2). These restrictions form our main assumptions for identifiability: the process noise shows non- -on-stationaritt autocorrelation driven by an auxiliary variable, and the augmented transition function is bijective.
Next, we introduce where we build the identifiability conditions into the space of prior distributions of the latent states and the process noise. Our framework ensures that the estimated forward transition function provably converges to the true unknown transition function under our identifiability assumptions.
theoretically-inforr
learning algorithm based variational auto-encoders,
Herre the term "identification" encapsulates not only the estimation but also a theoretical framework showing in what circumstances the estimated model recovers the ground-truth generative model as in the nonlinear ICA literature (Hyvärinen al. 2023
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Our findings show that our framework recovers latent states almost perfectly (Sec. 4. predicts the future states of three different dynamical systems much more accurately than state-of-the-ae art baselines (Secs, 4.1 4 and be quickly adapted to new environments (Sec. 4.3), and enforcing iden- tifiability assumptions consistently improves identification and future prediction accuracy (App. F.2)
2
AN IDENTIFIABLE DYNAMICAL SYSTEM FRAMEWORK
This section starts with the notation and our generative dynamical model that leads to identifiable variables and functions under certain assumptions. Sec. 2.2 we state the assumptions to achieve identifiability, followed by our main theoretical contribution in Sec. 2.3
2.1 DATA GENERATING PROCESS
We are interested in inferring latent dynamical systems from high-dimensional observations X where x, E R.P, with the time point. As usual, we assume that the observations are generated from sequence of latent states 217, with 2, E RRK through an emission function g R RP::
x; = gZ,
Without loss of generality, the latent states 1 evolve as first-order Markov process
f s1)
where f 22K + RK is an auto-regressive transition function and the process noise E R represents additional variables influencing the state evolution. e.g., random or unmodeled external variations in the process, not captured by deterministic dynamics. For the cartpole example, this would correspond the strength and direction of the wind at any time.
Challenges in identifying the latent transition function. The transition function f(2--1, 8;) 2t maps a higher-dimensional space R2K to a lower-dimensional space R.K rendering it inherently non-bijective. Consequently, multiple transition functions can correspond the same sequence of latent variables, resulting in non-identifiabiiit.
Augmented dynamics. We solve this challenge by augmenting the transition function and the emission function g with previous time steps, resulting in the following generative process:
Zo Pao (Zu),
# initial state
(1)
St Paju (s:/u) = II Psu (Skc ())
Vt .....T. # process noise
(2)
))
((2)-1,81)
St
faug
Vt I, # state transition (3)
Z4-1
[24-
2t-1
g(z,) g (2,,-1)
X+-1]
gaug
[2-1
VE 2..,.... # observation
where is auxiliary variable modulating the noise distribution Psju that implies non-i.i.d structure in the data generating process. For the auxiliary variable u, we consider two practical use cases:
Setting u to an observed regime label r 1..... leads to nonstationary process noise similar to Hovärinne & Morioka (2016), where the data is observed under R distinct regimes. For the cartpole system, this would imply several observation regimes under different wind conditions, for which their distributions of process noise are sufficiently different.
Setting to the unobserved noise S;-1 at previous time step implies an autocorrelated noise process similar to HHvärinen & Morioka (2017). For the cartpole system, this would imply a single, windy environment where the wind speed or direction changes continuously.
II practice we observe multiple sequences, but we drop the sequence index in notation for clarity. The model definition trivially generalizes to multiple sequences under exchangeability. Higher-order processes he represented first-order ones augmentation.
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We build on Morioka et al. (2021), whose approach suffers from two key shortcomings: First, their approach does not learn the transitions f. Second and more importantly, their theory is constrained to autoregressive transitions between pairs of observational data (x,-1.X) X.). This becomes impractical when modeling high-dimensional raw signals, such as videos, which requires modeling latent dynamics (Hafner al., 20T9b). To tackle these, propose new latent dynamical framework in which we apply the augmentation within the latent space, rather than in the observational space. This requires augmenting the compositional function fo rather than just a single function f. complicating the analysis. Finally, identifiability of the latent transition function follows when assuming that the augmented dynamics function bijective (in addition other identifiability assumptions in Sec. 2.2)
2.2 IDENTIFIABILITY THEORY
Let M (faugs Saugo Ps|u) denote the ground-truth model. By fitting the observed sequences,
we learn a model M = Baug., such that the ground-truth and the learned densities are observationally equivalent: Pau ((x))==) Baugg,BuugrPon (1x))==)) Vx, E X for t E
1. T. We make the following assumptions:
(Al) Injectivity and bijectivity. The emission functions g and g are injective, implying the same for the augmented functions gaugr Saug. The augmented transition functions faug: faug are bijective.
(A2) Conditionally independent noise. A single latent variable 2ht is influenced by a single process noise variable 8ks Zki ( 8ka). Let qe (Skt, u) = 1(gp( [1) denote the conditional log-density of the noise variable 8kt- Let (2ka, u) log p(=ke 1 u) denote the conditional log-density of the state variable 2kz. Conditioned on the auxiliary variable u, the log densities decompose for all E I::
K
log ((s
Σ log Dlgg(ss:lig k=1
Σ Th (Ske, u),
(5)
2k (
K
K mi (2ke, u).
logp(ze(ze-1,
log p 2ms 2t-
(6)
(A3) Sufficient variability of latent state Z₁- For any Zz there exist some 2K values of u: U1...., U2K, such that the 2K vectors
vi u1), Vi u2k)
are linearly independent for some index of the auxiliary variable u, where
vi(z,, u) = (8' (f ( - dzi1dm
d (2KI, Im u) Ōzk:du
O I' dz Ou:
(8)
0z7,0m
(A4) Sufficient variability of process noise Sy, For any s,, there exist some 2K values of u: Ui...., U2K, such that the 2K vectors
Wr(Se; uI), Wi(Se, U2x)
are linearly independent for some index of the auxiliary variable u, where
wr(s: u) = (8 (8gi( gi (81e, (dsncu -
(g u) 0sk:0u 0s7,0m
(' u) 0 ( (10)
0sk,00
Discussion of Assumptions. In Assumption (A7), the bijectivity of the augmented latent transition function captures the functional dependence between a latent pair (z,-1, 2,) and the process noise s,. This differs from Morioka et al. (222)) who use augmented transition model on observations (x4-1xx)) x2). Returning to the cartpole example, the transition function would map a latent state z, (1)
(1) s??) differ. This as- to two different future states 2 z(?), if the corresponding wind variables 9/41.5
sumption would not hold the controller or the transition function has intrinsic stochasticity, breaking
the bijectivity. Assumptions (A3, A4) imply that the latent states and the process noise sufficiently
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different across different environments, similar to the autocorrelation and non-stationarity conditions in Hyvärinen et al. (2019 2023). Assumption (A3) generalizes the sufficient variability condition defined for regime label in Yao et al. 20 to general auxiliary variable u. For (A3) to hold, the changes in the wind distribution should sufficiently change the distribution of latent states, i.e., 21) positions and velocities. For (A4) hold, need to observe the cartpole system in sufficiently many environments with sufficiently different wind (noise) conditions. These assumptions are violated, e.g., if the wind constant across the observed environments. For more details, see App.
2.3 MAIN THEORETICAL CONTRIBUTION
In this section, we state our main theoretical contribution, that is, the novel identifiability result for a latent transition function f (Theorem 2). We start with a theorem on the identifiability of the conditionally independent latent states Z, 11 (Theorem 1), which is an extension of the identifiability result established for the nonstationary noise case in Yao et al. (2021). The proofs detailed in Apps. C.I and C.2
Theorem (Identifiability of latent states Z1:T, based on Yao et al. (2021)). Under assumptions (AI, A2, A3). latent states Z, are identifiable up to a permutation and element-wise invertible transformation, i.e., there exists function h TRKK - IR.K,, such that = h(22). where h = o T= a composition of a permutation a. : [K] - [K] and an element-wise invertible transformation T, R RKK Equivalently, the same holds for the emission function, i.e., g g h.
Theorem 2 (Identifiability of the transition function f). Under assumptions (Al, A2, A3, A4), the process noise St is identifible up to a permutation and element-wise invertible transformation, iee. there exists function k: TRKK - RKK such that s; k($)), where k = ns O is composition of a permutation n. [K] + [K] and an element-wise invertible transformation Ts BRKK + RR Equivalently, the same holds for the transition function faug h-1 o fang Kaug since augmented functions aug and Kaug decompose into block-wise functions: h haug [h h and kaug = |K, h).
such that k and h are already shown to be compositions of a permutation and an element-wiss invertible transformation.
Proof sketch. Latent states z, and Zt-1 are identifiable under sufficient variability and conditional independence assumptions states z. Intuitively, the idea is also to identify the noise variables st. which leads to the identifiability of the transition function as 2, = St))
In the proof in App. C.2 first show that the function Kaug RR2K -> R2K that maps the learned k ($ 2 (2 2
21-1 1)
pair (S., 1 to the ground-truth pair (st, 21-),
St Z
Kang
is bijective. Next, we show that the function components ki, k2 R2K RK that take the pair (Se, 11) as input are equivalent to invertible element-wise functions k, h RRK - R.K such that (i) k only depends
$t. i.e., St (S,, 2,) = k(s;) and (ii) h only depends 2, 2,-1, ie.. 2--1 = (8,, 24-1) = h(2 For (ii), we use the result of Theorem For (i), we use the assumption (A4), the sufficient variability in the process noise.
3 PRACTICAL IMPLEMENTATION USING VARIATIONAL. INFERENCE
We turn our theoretical framework into practical implementation using variational inference, which approximates the true posterior over the noise and the latent states given the observations. In contrast to previous works that approximate the inverse transition function (Yao et al.) 2021 2022). we directly estimate the forward transition function. X₁ to process noise St. from which then obtain the latent states Z, similar to Franceschi al. (2020). Different from Franceschi et al. (2020), our process noise prior p (s)u) does not depend on the previous state 24-1, this dependence would make the model non-identifiable. In the implementation and the experiments, we set the auxiliary variable u to a regime label where the data is observed under distinct regimes.
achieve this, we first map the observation
implementation, we build our assumptions into the model as follows. We enforce (i) conditionally independent process noise by using independent 1D conditional flows for
For a
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Po(skr [)), and (ii) conditionally independent latent states by modeling each output & of the transition function f& (Z(-1, Skt) as a separate MLP. We provide the learning algorithm in Algorithm 1. For space considerations, provide the implementation details in Fig. 6ad Apps. D and F3
Sequential prediction S1:7 and Zo:r. Each observation x; is mapped to an intermediate embedding x, via MLP CNN backbone depending on the input modality. To infer the initial latent state, an encoder (MLP) takes the embeddings I.T,, up to time Tie as input. It outputs the parameters of the initial state posterior, qo [zo] (1:T;)) from which the initial state Zo is sampled: Zo
(s (x:TT..
Once the initial state is sampled, the model predicts the noise 8, and the latent state 2, iteratively for each time step 1 ..... T.
At each step, the embeddings up time X1:,, along with the previously sampled state Z;-1, are used to infer the process noise. A forward sequential model (RNN + MLP) takes [ z[-1] as input to compute the parameters of the noise posterior, qs [5,[X]i:> Zt-1). from which the noise is sampled: S; Go [x 2;-1). Using the sampled noise and the previous state 2; the next state z, is predicted through the transition function Z₁ f(zz-1,8;) Specifically, the components of z, are computed as Zµe fx 5kx). for k 1, ,.... ... K. Due to the sequential computation, the time complexity of a forward-pass is linear in the input sequence length O(T).
input and outputs the parameters of the observation likelihood, P& (x, (l.))) )). For the EL.BO computation, we weigh the KL term with the hyperparameter (Higgins et al.) 2017), which is selected through validation. We assume a standard Gaussian prior for the initial state, p(zo) (N( I). We use the auxiliary variable in implementing the prior for the noise variables St, (S |u) II po (s(ss)l)), where each (o($kt () is modeled as an independent, trainable ID conditional flow. To allow for modall
ELBO computation and priors. The decoder takes the latent state
prior distributions of the ID noise variables Sks, we employ neural spline flows (Durkan et al. 2019).
Algorithm 1 Practical learning algorithm
Requires: Variational posterior networks (ICEncoder and Noi seEsodee)))) and Decoder
1. Encode initial condition parameters: Mzo: log o2. ICEncoder(Xi.nn)
Sample initial condition: zo N(moo 2
3. For & E I,...,::
(a) Encode noise parameters: Pa, log o NoiseEnccoeer(xuu....
(b) Sample noise: 8; ~ [N (
(c) Compute the next latent state: z; 2()).
(d) Decode: = (Decoder(z.)
4. Compute ELBO: L = LR 3LkI. Samples (SL:T, z are used to approximate LXL:
T LR Σ t=1 E By .... (logooxx:
LKL. Dxi (1Po())) E E Bog Der [u)]]
5. Update the parameters {0, o}.
4 EXPERIMENTS
In this section, we evaluate our method's ability to (i) recover true system dynamics in controlled setups validate our theory (Sec.
(ii) accurately forecast long horizons in complex synthetic and real-world datasets (Secs, 4.1 and 4.2). and (iii) adapt to unseen environments efficiently (Sec. (S3c. In addition, we compare joint vs. tto-stage dynamics training in Sec. 4.4 present an ablation study and additional figures in App. E. and further details in App. F Our implementation to reproduce the study can be found at https:::.. htt///gttttu wig..u........... com/ wwt.c//cccccaaaccccc
4.1
Baselines. For latent dynamics identification, we compare with ofate-of-the-art identifiable representa- tion learning (IRL) methods: 3-VAE (Higgins et al., (2017). LEAP (linear: LEAP-LIN, non-parametric: LEAP-NP: Yao et al.) 202 and TDRL (Yao et al.) 2022). For sample-efficient adaptation, we compare with the best persorreffoming IRL method, LEAP- LIN, For future prediction, we compare with
state-of-the-art neural stat-space,, ODE and SDE methods: KALMANVAE (Fraccaro al. 2017).
CRU (Schirmer et al. 2022). NODE (Chen et all 2018). Latent NODE (Rubanova et al. 2019). 2DE? VAE (Yildiz et al. 2019) MONODE (Auzina et al. 2024), and Latent SDE (Li et al. 2020).

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Table 1: Results for synthetic and cartpole datasets (mean + std.dev. across seeds). Rows are marked (N/A) for methods unable to predict the future the process noise. LEAP-NP and TDRL cannot generate future predictions as they only approximate the inverse transition function. Mcc: mean correlation coefficient; higher is better 1. MSE[X [tutur/r]] denotes mean squared error over future time points; lower is better 4. Best bolded based on Welch's (-test with p < 0.01.
MODELS
DATASET
METRICS
8 S-VAE
VALMANVAE
CRU
LEAP-LIN
LEAP-NP
TDRL
OURS
Mex Papo
0.60 anis
0.64 anas
0.47
0.68 2010
0.89
0.73
0.94
Siypt
N/A
soc4
SYNTH.
MSE Ramm
N/A
1.27
8.85 20.1
0.22
N/A
N/A
0.06
MSE
N/A
1.32 20.27
8.73 20.5
0.18 2010
N/A
N/A
0.08 20
MSE
N/A
1.72
9.08
0.59
NIA
N/A
0.20
Mec Rapa
0.68
0.65
0.85
0.81
0.76
0.56
0.95
CARTPOLE
Mox Panrate)
N/A
0.73
anis
0.70 2015
0.78
N/A
N/A
0.91
Cartpole
Velecity
LEAP- Lin
(eture) Mco|/uarr
0.06
0.79
0.02
0.00
0.30
0:54
0.93
0.73
(Sortad)
0.99
0.20
0.12
0.23
CRU
0.12
0.79
0.52
0:86
0.49
0:34
0.62
0.77
0.82
Estim. a (Sortad)
0.99
0.24
0.13
0.20
OURS
MeclFneare
0.89
0.19
0.28
0.95
0.50
0.16
0.67
0.67
0.95
Figure 2: Correlation plots for first input points. For illustration purposes, we use the data from the seed with median performance. Cartpole system has 4 true states: cart position and velocity á, pole angle 0 and angular velocity θ. Left column: The correlation matrix between ground-truth states and latent estimations for best- pess-performing 3 models. Columns correspond to ground-truth states z, à 0, Ó, while rows correspond to most highly-correlatee estimated latent dimensions. Darker colors on diagonal imply better one-to-one mapping. Next 4 columns: 4 scatter plots between each ground- truth state dimension and its aligned latent. Here, a diagonal line implies better one-to-one mapping. An extended version, also covering the last Timure time steps, is available in Fig. 10 App. E
Metrics. For latent dynamics identification, we use mean correlation coefficient (MCC) between the ground-trume -tunh states and the estimated latents, which is the standard metric in the nonlinear ICA literature (Hyvärinen et al. 2019). MCC McC(Auupu and MCC[$Smpue] denote McC on latent states and process noise for the states and the mean squared error MSE [ on the future observations Xiuture-
imput steps. For future prediction, we use McC (iuure on future latent
Datasets. In all experiments, the data observed under It distinct regimes, and we set the auxiliary variable to the one-hot encoded regime label Synthetic: To validate theory, generate synthetic dataset satisfying our identifiability assumptions as commonly done theoretical works on IRL. We follow the setups in Yao et al. (2021 2022). We have R = 20 regimes with nonstationary noise: the (conditional) distribution of noise s varies between regimes. To evaluate the long- long-horizon forecast performance, Tayn = 4 observations for training the dynamical model and T Tanure
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observations for evaluating future predictions. Cartpole: We use the setup as described in Yao et al. (2022). with one modification: Yao et al. only use fixed action sequences, i.e., only always go-right or always go-left, which is not representative of real-world scenarios. Instead, implement continuous actions to obtain a more realistic setup. We have R = 6 regimes (5 for training, for testing), with distinct gravity values. The underlying dynamics are deterministic; the noise accounts for the variability due to unknown initial conditions and different gravity values between the domains. Mocap: To evaluate future predictions on long horizons with complex real-world dynamics, we use three CMU motion capture (Mocap) datasets from the dynamical systems literature. The different regimes correspond to different persons walking. MOCAP-SINGIA (Yildiz et al.) 2019) ei al. (020) contains walking sequences of a single subject (Rt = 1), and MOCAP-MULTI and MOCAP-SHIFT (Auzina et al. 2024) consist of sequences of different subjects (R- 6); in MOCAP-SHIFT, one subject is left out for testing. Here, we interpret the process noise as modeling the external variations in the observed data, which might show nonstationary behavior due to subject-specific characteristics
such limb lengths, joint angles, sensor positioning, sensor noise, or initial conditions.
4.1 OUR MODEL RECOVERS TRUE SYSTEM DYNAMICS BETTER IN CONTROLLED SETUPS
We present MCC and MSE results for the synthetic and cartpole datasets in Table 1. In the synthetic dataset, all IRL methods (LEAP-LIN, LEAP-NP (Yao et al.) 2021); TDRL (Yao et al.. 2022); OURS) estimate latent states with high correlation to the true states, as indicated by MCC (Zinpun Our
; and
method has the highest Mcc[Zuupud and MCC[SSmpu]] scores among all methods.
High values of McC Zinport and McC[Simpue] indicate a close alignment between the estimated and true latent transition functions, as stated in Theorem 2. This is validated by our method's superior future prediction accuracy, with lower MSE MSE[Xsuttre] across 2-, 4-, and 8-step horizons. It is important to note that the IRL. methods LEAP-NP and TDRL cannot generate future predictions (N/A) since they approximate only the inverse transition function. As the prediction horizon increases e.g., to 8 steps, twice as much as observed during training- the performance gap between our method and the baselines widens further. See predicted trajectories with calibrated uncertainties in Fig. 1 in App. E
For cartpole, methods that model complex temporal dependencies (LEAP-LIN, LEAP-NP Yao et al., I0 CRU (Schirmer et al. 2022); and OURS) outperform others in McC Zinpul and McC Ziuture For the top-periorming models (LEAP-LIN (Yao et al., 2021). CRU (Schirmer et al. 2022). ) and OURS), we present MCC correlation matrices and scatter plots comparing the ground-truth states with the 4 most correlated latent dimensions for the first mmpu time steps in Fig. 2 An extended version of this figure, also covering the last Timuue time steps, is available in Fig. 10 in App.
E LEAP-LIN (Yao) eet al. (021) struggles to estimate well-aligned latents for both the first 8 input steps and the subsequent 8 future steps. CRU (Schirmer et al. (022)) recovers well-aligned latents for cart position and angle
0 during the first 8 steps but loses accuracy in future predictions. In contrast, our method consistently recovers the most highly correlated latents, with the performance gap increasing for future predictions.
Summary. Our method recovers the true latent dynamics more accurately than baselines in both synthetic and cartpole setups, shown by (i) higher McC for input latents and (ii) more accurate future predictions, with the performance gap widening as the prediction horizon increases.
4.2 OUR MODEL PREDICTS LONG-HORIZON REAL-WORLD DYNAMICS MORE ACCURATELY
We present the MSE results in Tab. for
Table 2: TEST-MSE()) for MOCAP-SINGLE, MOCAP-
all Mocap datasets. These datasets exhibit complex wolld dynamics with pre- diction horizons of 75-300 steps. Across all datasets, NODE (Chen et al.] 2018) performs the worst, highlighting the need to model latent dynamics. As an addi- tional strong baseline, we trained CRU (Schirmer et al. 2022) on Mocap datasets. Although this approach based Kalman filters fits the training data well, it reverts to its linear prior when extrapolating into
MULTI and MOCAP-SHIFT. *Results from Li et al. 2020 +tResults from Auzina al. (0024).
MODELS
MOCAP-SINGLE
MULII-MULTI
MOCAP-SHIFT
LEAP-LIN
17.08 23:4
57.9 slos
51.1 2s
NODE
22.49 soss
72.2 auatt
61.6 tt
ODE
8.09 =n.1
Latent NODE
5.98 20.2
Latent SDF
4.03 2
MONODE
57.7
58.0 tt
OURS
3.82
19.6 23.2
36.0 a33
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Lobert Trijecary
Mee Jae ve
Oconcatee Space (ORA
Tome.
Figure 3: Left: Latent trajectories and skeletons for OURS model in MOCAP-S DOCAP-SINGLE (observation trajectories Fig. (). Right: Test trajectory predictions for LEAP-LIN and OURS MOCAP-MULTH
At test time, first 75 time points are given input, and the full prediction horizon is T 150 steps.
the (unseen) future. Consequently, fails to predict the unseen time points, and we have omitted its results from this section.
MOCAP-SINGLE consists of sequences from single subject, which violates our identifiability assumptions. Despite this model misspecification, our method achieves the lowest MSE. In contrast, MOCAP-MULLTI and MOCAP-SHII show non-i.i.d. structures, with multiple subjects treated as distinct environments. Our method is particularly well-suited for these datascts, significantly out- performing other baselines, including MONODE (Auzina et al., 2024), which is designed to handle non-i.idd structures by predicting environment-speen speciffc dynamic variables.
In Fig. 3 (Right), we present the predictions of our model alongside those of LEAP-LIN for six dimensions within the data space that representative of complex dynamics (all other dimensions are included in Fig. 12 in App. E Both models exhibit high accuracy for the initial half of the sequence, provided as imput. However, into the future, indicative of a superior approximation of the underlying system. In Fig. (Teff)) we show estimated latent trajectories, along with the generated skeletons for time MOCAP-SINGO The latent trajectories exhibit smooth, cyclic patterns with minimal differences between trials, reflecting the cyclic nature of walking. The estimated trajectories for the joint angles closely match the actual data trajectories, with calibrated uncertainties.
model demonstrates significantly improved extrapolation
300 on
Summary. Our model achieves state-of-the-art performance across all datasets, outperforming strong neural ODE/SDE baselines that were specifically designed for the task. Notably, our ap- proach performs best on MOCAP- that violates our assumptions, suggesting that our method maintains strong real-world applicability, even in scenarios that challenge its core assumptions.
4.3 IDENTIPYING DYNAMICS IS THE KEY TO SUCCESSFUL ADAPTATION
In this section, we explore whether for dynamical model adapt- ability it is more important to identify the underlying dynamics than just the latent states. We utilize our modified cartpole dataset, consisting of five source domains with varying gravity levels (5. 10, 20, 30, 40} and a target domain with = 90. When tested on the target domain, the Mcc Zrauue] of both LEAP-LIN Yao et al. 202T) and our method drops significantly. To address this, we fine-tune the models on adaptation datasets with N {20, 50, 100, 1000} sequences. As a strong baseline, we also train our model on an extended dataset that includes the data from the source and target domains.
035
0.00
MCC| miture
0.55
0.70
0.70
4 Adaatation Samples (log scale)
Duss (Train All Duta)
Duns (Fincuun)
(Finetune)
As shown in Fig. fine-tuning on just 50 sequences allows model to recover the McC (raure] performance of the strong baseline trained on the complete dataset. Fine-tuning our model more data makes it better expert the adapted do-
Figure 4: MCC Zrauore] with vary- ing number of adaptation samples.

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main. In contrast, the McC score of LEAP-LIN Yao et al. 2021 does not consistently improve, regardless of dataset size.
The sample-efficient adaptation performance of our method suggests that identifying dynamics plays key role in successful dynamics adaptation, while identifying only the latent states (LEAP-LIN, Yao et al. 2021) does not guarantee successful transfer to new domains.
Summary.
4.4 JOINTLY LEARNING LATENTS AND THEIR DYNAMICS IMPROVES FUTURE PREDICTION ACCURACY
In principle, one can use an off-the-shelf identifiable representation learning method, such LEAP-NP (ao et al., 2021), identify the ground-truth factors, and fit state transition function the inferred latent codes. In this section, we demonstrate that such a two- stage approach approximates an unknown transition function less well than our joint learning procedure that identifies the transition and emission functions. We compare the two on our synthetic dataset. To learn the unknown transitions, we train (i) three- layer MLP (1-MLP) and (ii) K three-layer MLPs (K- MLP), with leaky-ReLU activations on the latent state sequence Zuain inferred by LEAP-NP (aao et al.) 202I). We unroll the learned transition function for an additional Tiuure time steps, decode all the latent states, and compute the MSE in the data space. We repeat this three times and compare the average error with the eπror of our approach in Fig. [ Our approach achieves smaller errors at all time points, and the gap widens as we predict for longer horizons. This indicates that identifying the emission and transition functions jointly improves future prediction accuracy.
Msn(Xx ( future]
LEAP-ee-ffinettaniii (1-Mar)
tarstion (K Mir)
801
OURS
0.03
# Prediction Steps
Figure 5: Comparison of MSE achieved by
our approach vs. two-stage LEAP training.
DISCUSSION
We have presented the first latent dynamical system that allows for the identification of the unknown transition function, and theoretically proved its identifiability, based standard assumptions. We evaluated our approach on synthetic data, the cartpole environment and real-world Mocap datasets, and showed that (i) the estimated latent states correlated strongly with the ground truth, (ii) our method had the highest future prediction accuracy with calibrated uncertainties, (iii) could adapt to environments using handful of data, and (iv) it produces state-of-the-art future prediction results on complex real-world dynamics for long horizons, highlighting its usefulness for practical scenarios. The main limitation stems from the identifiability assumptions, which are further discussed in detail in App. C For future work, would be intriguing to examine how their violation might influence the final model's performance. Finally, demonstrating improved downstream performance from our method, e.g. in model- based policy learning, would be of interest.
A Acknowledgements Çağatay Yildiz and Matthias Bethge are members of the Machine Learn- ing Cluster of Excellence, funded by the Deutsche Forschungsgemminschatt (DFG, German Re- search Foundation) under Germany's Excellence Strategy EXC number 2064/1 - Project number 390727645. Matthia Bethge acknowledges financial support via the Open Philanthropy Foundation funded by the Good Ventures Foundation.
REFERENCES
Ilze Amanda Auzina, Çağatay Yildız, Sara Magliacane, Matthias Bethge, and Efstratios Gavves. Modulated neural ODEs. Advances Neural Information Processing Systems, 36, 2024.
Yoshua Bengio, Aaron Courville, and Pascal Vincent. Representation learning: A review and new perspectives. IEEE transactions on pattern analysis and machine intelligence, 35(8): 1828, 2013.
10

---

Published as a conference paper at ICLR 2025
Greg Brockman, Vicki Cheung. Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba. OpenAl gym. arXiv preprint N.Kiv:1606601544 2016.
Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations. Advances in Neural Information Processing Systems, 31, 2018.
Angelo Ciaramella, Enza De Lauro, Salvatore De Martino, Mariarosaria Falanga, and Roberto Tagliaferri. ICA based identification of dynamical systems generating synthetic and real world time series. Soft Computing, 10:587-606, 2006.
Conor Durkan, Artur Bekasov, lain Murray, and George Papamakarios. Neural spline flows. Advances in Neural Information Processing Systems, 32, 2019.
Marco Fraccaro, Simon Kamronn, Ulrich Paquet, and Ole Winther. A disentangled recognition and nonlinear dynamics model for unsupervised learning. Advances in Neural Information Processing Systems, 30, 2017.
Jean- Yves Franceschi, Edouard Delasalles, Mickaël Chen, Sylvain Lamprier, and Patrick Gallinari. Stochastic latent residual video prediction. In International Conference Machine Learning, pp. PMLR, 2020.
Danijar Hafner, Timothy Lillicrap. Jimmy Ba, and Mohammad Norouzi. Dream to control: Learning behaviors by latent imagination. arXiv preprint /92iv:/92..01600 2019a.
Danijar Hafner, Timothy Lillicrap, Ian Fischer, Ruben Villegas, David Ha, Honglak Lee, and James Davidson. Learning latent dynamics for planning from pixels. In International Conference on Machine Learning, pp.
PMLR, 2019b.
Irina Higgins, Loic Matthey, Arka Pal, Christopher P Burgess, Xavier Glorot, Matthew M Botvinick, Shakir Mohamed, and Alexander Lerchner. 3-VAE: Learning basic visual concepts with a constrained variational framework. ICLR (Poster), 2017.
Biwei Huang, Fan Feng, Chaochao Lu, Sara Magliacane, and Kun Zhang. Adarl: What, where, and how to adapt in transfer reinforcement learning. arXiv preprint arXiv:2107.02729, 2021.
Aapo Hyvärinen and Hiroshi Morioka. Unsupervised feature extraction by time-contrastive learning and nonlinear ICA. Advances in Neural Information Processing Systems, 29, 2016.
Aapo Hyvärinen and Hiroshi Morioka. Nonlinear ICA of temporally dependent stationary sources. In Artificial Intelligence and Statistics, pp. 4690 PMLR, 2017.
Aapo Hyvärinen and Petteri Pajunen. Nonlinear independent component analysis: Existence and uniqueness results. Neural networks, 12(3):429-433 1999.
Aapo Hyvärinen, Hiroaki Sasaki, and Richard Turner. Nonlinear ICA using auxiliary variables and generalized contrastive learning. In The 22nd International Conference on Artificial Intelligence and Statistics, pp. 859-868 868. PMLR, 2019.
Aapo Hyvärinen, llyes Khemakhem. and Hiroshi Morioka. Nonlinear independent component analysis for principled disentanglement in unsupervised deep learning. Patterns, 4(10), 2023.
Gaätan Kerschen, Fabien Poncelet, and J-C Golinval. Physical interpretation of independent compo- nent analysis in structural dynamics. Mechanical Systems and Signal Processing, 1164):5561-1555, 2007.
Ilyes Khemakhem, Diederik Kingma, Ricardo Monti, and Aapo Hyvärinen. Variational autoencoders and nonlinear ICA: A unifying framework. In International Conference Artificial Intelligence and Statistics, pp. 2207-2217. PMLR, 2020.
David Klindt, Lukas Schott, Yash Sharma, Ivan Ustyuzhaninov, Wieland Brendel, Matthias Bethge, and Dylan Paiton. Towards nonlinear disentanglement in natural data with temporal sparse coding. arXiv preprint arXiv:2007. 2020.
Sergey Levine, Chelsea Finn, Trevor Darrell, and Pieter Abbeel. End-to-end training of deep visuomotor policies. Journal of Machine Learning Research, 17(39):1-40, 2016.
11

---

Published as a conference paper at ICLR 2025
Xuechen Li, Ting- Kam Leonard Wong, Ricky TQ Chen, and David Duvenaud. Scalable gradients for stochastic differential equations. In International Conference on Artificial Intelligence and Statistics, pp. 3870-3882 PMLR, 2020.
Francesco Locatello, Stefan Bauer, Mario Lucic, Gunnar Raetsch, Sylvain Gelly, Bernhard Schölkopf, and Olivier Bachem. Challenging common assumptions in the unsupervised learning of disentan- gled representations. In International Conference on Machine Learning. pp. 41 14 4124. PMLR, 2019.
Hiroshi Morioka, Hermanni Hälvä, and Aapo Hyvärinen. Independent innovation analysis for nonlinear vector autoregressive process. In International Conference on Artificial Intelligence and Statistics, pp. 1549-15577 PMLR, 2021.
Norman H Packard, James P Crutchfield, J Doyne Farmer, and Robert S Shaw. Geometry from time series. Physical review letters, 45(9):712, 1980.
Yulia Rubanova, Ricky TQ Chen, and David K Duvenaud. Latent ordinary differential equations for irregularlyssmppled time series. Advances in Neural Information Processing Systems, 32, 2019.
Mona Schirmer, Mazin Eltayeb, Stefan Lessmann, and Maja Rudolph. Modeling irregular time series with continuous recurrent units. In International Conference on Machine Learning, pp. 19388-19405. PMLR, 2022.
Jürgen Schmidhuber: Learning factorial codes by predictability minimization. Neural computation, (6):863-879, 1992.
Xiangchen Song. Weiran Yao, Yewen Fan, Xinshuai Dong, Guangyi Chen, Juan Carlos Niebles, Eric Xing, and Kun Zhang. Temporally disentangled representation learning under unknown nonstationarity. arXiv preprint arXiv:2310. 18615, 2023.
Vincent Stimper, David Liu, Andrew Campbell, Vincent Berenz, Lukas Ryll, Bernhard Schölkopf, and José Miguel Memááneez-Lobate normflows: A PyTorch package for normalizing flows. arXiv preprint arXiv:2302. 300.:2214..1111. 2023.
Jack M Wang, David Fleet, and Aaron Hertzmann. Gaussian process dynamical models for human motion. IEEE transactions on pattern analysis and machine intelligence, 30(2):283-298, 2007.
Quanhan Xi and Benjamin Bloem-Reddy Indeterminacy in generative models: Characterization and strong identifiability. In International Conference on Artificial Intelligence and Statistics, pp. 6912-6939, PMLR, 2023.
Weiran Yao, Yuewen Sun, Alex Ho, Changyin Sun, and Kun Zhang. Learning temporally causal latent processes from general temporal data. arXiv preprint arXiv:21 2021.
Weiran Yao, Guangyi Chen, and Kun Zhang. Temporally disentangled representation learning. Advances in Neural Information Processing Systems, 35:26492-26503, 2022.
Cagatay Yildiz, Markus Heinonen, and Harri Lahdesmaki. VAE Deep generative second order ODEs with Bayesian neural networks. Advances Neural Information Processing Systems, 32, 2019.
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A RELATED WORK
Table 3: A comparison of our approach and the previous work, highlighting that our approach the first predict future states of dynamical system by utilizing transition function that identifies the underlying, true dynamics.
Identifies
Models
Can predict future
Identifies
latent states
latent dynamics
latent dynamics
SLOWWAF, Klindt ct al.
2020
X
X
IIA,
Morioka al.
2021
X
X
LEAP,
Yao et al
2021
XIV
tt
X
TDRL.
Yao et al.
(0022
X
X
NODE, Chen et al.
(2018)
x
X
KALMAN VAE. Fraccaro et al.
2017
Latent NODE, Rubanova ct al.
2019)
ODE²VAE,
Yildiz ct al. (2019
x
x
Latent SDE.
Li al. 2020)
CRU, Secirmehh el al. 2022
MONODE, Auzına el al.
2024
OURS
only the process noise is identified,
t only for linear approximation (L.AA--LIN)
B DDENTIFIABILITY THEORY
In this section, we discuss the identifiability of the latent states and the transition function, and provide the detailed proofs.
We assume latent dynamical system which is viewed as high-dimensional sensory observations X1.7, where t is the time point and R We assume a sequence of latent states Z1:7, with Z E K are instantaneously mapped observations via an emission function g R R.
x = g (Z:)).
(11)
The latent states Z1.7 evolve according Markovian dynamics:
2, f(z)-1,8:)
(12)
where f: 22K > R is an auto-regressive transition function and sy E R corresponds to process noise.
Our aim to jointly identify the latent states 7i the dynamics function f, and the process noise $1:7- We remind that previous works (Klindt et al., 2020 Yao al. 2021 2022 Song et al., 2023) have concentrated on identifying the latent states 1 possibly with linear transition approximations, but a general transition function f. Yet, without general f, the methods can estimate the underlying states only when corresponding observations are provided or provide simplistic approximations in their absence. Hence, they cannot predict complex future behavior reliably.
Notice that learning a provably identifiable transition function f: 222K + RK is not straightforward, since the transition function is not injective. naive solution can be to simply use a plug-in method (Yao et al. 2 2022) for identifying the latents and then fitting transition function f on the estimated latents, however, we show empirically our experiments that it leads to poor prediction accuracy for the future behavior.
B. NONLINEAR ICA
The nonlinear ICA assumes that the data is generated from independent latent variables z with a nonlinear emission function g. following Eq. (11). It is well-known be non-identifiblle for i.i.d. data
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(Hyvärinen & Pajunen 1999 Locatello et al. 2019). Recent seminal works (Hyvärinen & Morioka 2016; 2017: Hyvärinen et al. 2019) showed that autocorrelation and nonstationarity existent in non-i.1.d. data can be exploited to identify latent variables in an unsupervised way. Compared to the vanilla ICA that considers independence only along latent dimensions, the idea of these works is to introduce additional independence constraints reflecting the existent structure in the data. These additional constraints are formulated mathematically as identifiability assumptions, which restrict the space of the emission function g and the space of the latent prior Pz (Hyvärinen et al. 2023 Xi & Bloem- Reddyy (023). The key insight is that, after sufficiently constraining the latent prior Pz using such assumptions, identifying the latent variables Z, and identifying the injective emission function g become equivalent tasks (Xi & Bloem-Reddy 2023).
B.2 AUGMENTED DYNAMICS FOR IDENTIFIABLE SYSTEMS
To identify the transition function f such that z, = f(z 1, Sr), we will use the same insight: After sufficiently constraining the noise prior Psi given identifiable latent pair (Zz-1, Zi), identifying the noise variables 8; and identifying the bijective dynamics function f should be equivalent. Hence, in addition the identifiability assumptions restricting the space of the emission function g and the space of the latent prior Px. we will further restrict the space of the dynamics function f, and the space of the noise prior Ps-
First, let us note that the identifiability of the process noise 8; is not trivial since the dynamics function f 1R2K + RKK is an injective function and hence it does have an inverse. Following the independent innovation analysis (IIA) framework Morioka et al. (2021), we trivially augment the image space of the transition function and denote the bijective augmented function by faug R2K R2K:
1)
f (2) 2
Ze
faug
St Z
(13)
Z:-
Contrary to Morioka al. (2021), which use an augmented autoregressive model on observations (X--1, xt), our formulation captures the functional dependence between a latent pair (Zi-1, Zi) and the process noise St.
Next, we make the standard assumption in the temporal identifiability literature (Klindt et al.
2020
Yao al. 2021; 2022; Song et al., 2023) that each dimension of the transition function {.fx k=
influenced by single process noise variable 8yt- The output is a single latent variable 25::
2kt fz ska), for is 1,..., K and t 1.....
(14)
Notice that this does not impose limitation on the generative model, it just creates a segmentation between noise variables and latent variables. For example, if this assumption is violated and there exists noise variable Ske that affects both Z and Zje. with + j. then the noise variable Skr can instead be modeled latent variable Skt-
We re-state the full generative model for completeness:
2o Pzn (zo),
# initial state (15)
S, ~ Pe)uu (u)
II Peu
(Bkz/ (u),
VI I.......,
process noise
(16)
faug
St
f(Zz-1,88)
VIE I..., # state transition (17)
=
Z(-1
Zt-1
Z1-1
g(zi) g(z,-1)
gaug
2, # observation mapping
X1-1
|z--
(18)
where u is an auxiliary variable, which modulates the noise distribution Patu
C DISCUSSION OF ASSUMPTIONS
In this section, we compare our assumptions with the related work while discussing their theoretical and practical implications. To make better connection with real-world dynamical sytem example,
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we discuss autonomously controlled drone. Here, the observations X1:7 would be video stream of the flying drone instead of the system state 21:T containing absolute position, velocity, and acceleration in 3D. The process noise S1.7 might represent additional variables influencing the state evolution, e.g., the strength and direction of the wind at any time or drone motor torque set by the controller. In the next paragraph, we specifically consider the latent state 2, as the 3D position (location and orientation) of the drone at time t, and the external noise S as wind affecting the drone.
Here, we refer the observational equivalence of the models Assumption (A0), i.e., the ground-truth and the learned densities are observationally equivalent: PaansMaur:Pjj ((x:)==))
Pan )Tx)) Vx, E X for & E I.
Assumptions (A0,A1,A2) are standard assumptions in the nonlinear ICA literature (Klindt et al. 2020; Morioka et al. 2021; Yao et al.) 2021: 2022). Assumption (A0) ensures that the model M is sufficiently flexible that it learns the correct distribution in the limit of infinite data, e.g., the learned model components are universal approximators (neural networks). or for a variational inference algorithm the family of the variational posterior contains the true posterior. For the drone example, most likely the drone dynamics is nonlinear, and hence modeling the transition function f by a linear function would violate this assumption. In Assumption (AT), the injectivity of the augmented emission function implies that we have one-to-one mapping between the latent states and the observation manifold. In the drone example, this implies that the 3D position should map to a unique drone position in the pixel space. This could be violated in natural videos as they exhibit complex dependencies such as occlusions. The bijectivity of the augmented latent transition function captures the functional dependence between a latent pair (z. z,) and the process noise 8,, in contrast to Morioka et al. (202T), which use an augmented transition model on observations (x--1, x.). For the drone example, for given 3D position 2 the transition function f must map two distinct wind intensities S, to distinct 3D positions 2,. This assumption would not hold, c.g.. if the physical system that applies the torque set by the controller has some intrinsic noise and the transition function is stochastic. Assumptions (A2, A3, A4) generalize the nonstationary noise and the sufficient variability assumptions in Yao et al. (202T) to general auxiliary variable For the drone example, they imply that the underlying system can be decomposed into (conditionally) independent ID variables, e.g., the 3D position of the drone to ID variables x, y, z, which are affected independently by the æ, y, values of the wind intensity (noise). Assumptions (A4, A5) imply that the latent states and the process noise show enough non-i.i.d. structure in the form of autocorrelation or non-stationarity, i.e., they change sufficiently differently with respect to 2K values of u: uI..... 42K (Hyvärinen et af. 2019; 2023). For (A5) to hold, we need to observe the drone in sufficiently many environments with sufficiently different wind (noise) conditions. For (A4) to hold, the changes in the wind distribution should sufficiently change the distribution of 3D positions (latent states). These assumptions are violated, e.g., if the wind is constant across the observed environments.
Yao al. 2021) used nonstationarity of the process noise for identifiability of the conditionally independent latent states (Theorem 1). If the variable u is an observed categorical variable (e.g., domain indicator), the assumptions (A4, A5) can be written in an alternative form without partial derivatives with respect to u (Hyvärinen et al. 2019 (see App. C.3 for the alternative version).
Remark 1
the presence of control inputs, we can write the augmented dynamics function given the control input faug R2K - 22K The identifiability proof requires that the augmented dynamics function to be bijective. Then, the identifiability proof continues to hold as long as the augmented dynamics function conditioned for given control input a, f-ug(+la) RR + 22 is also bijective for any given a.
Remark 2
C. PROOF OF THEOREM 1: IDENTIBABILITY OF THE LATENT STATES Ze
This result is already shown in Yao al. 2021 Appendix A.3.2). Here, we follow Klindt et al. 20200) Yao et al. (202T 2022) and repeat their results in our notation as we also make use of this result in App. C.2
The injective functions g,g : RK + R are bijective between the latent space RRK and the
observation space X' C TR We denote the inverse functions from the restricted observation space
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the latent space by g g
This is also implicitly assumed in (Klindt et al. 2020 Yao et al. 2021
2022 Song et al. 2023).
Using g and g, relate the ground-truth and estimated latents Z, and 2, to each other:
X g(2))
(g G 0 (24) g = h 2 h h((2),
(19)
where the function h 2, +> 2z maps the learned latents to the ground-truth latents. To show it bi- jective, we need to show it is both injective and surjective. Following Klindt et (2020), it is injective since it a composition of injective functions. Assume it is not surjective, then there exists a neigh- borhood Uz for which g( 22) $ (( This implies that the neighborhood of images generated by g(U,)) has zero density under the learned observation density P fuogg,
(g(U2)) = 0, while hav-
ing non-zero density under the ground-truth observation density fg. (x): Puo (g(U+)) 0. This contradicts the assumption that the observation densities match everywhere. Then, h is surjective.
Using 2, = h(2)). ). we can analyze how the conditional densities of the ground-trumt and estimated latents Z, and 2, are related. For this, perform change of variables on the conditional latent density log p( 2, 2 1, u) as follows:
log p(p 21-1, u) = log 2 Z, u) + log H
(20)
K
K log pl ( Z1-1, u + log Hg
log (2ke 11
(21)
(3)).
K
K
Σ
M (Ssts
Σ mi (ZKe, u) + log H k=1
(22)
where H, (z.) is the Jacobian matrix of evaluated at 2,. When we take derivatives both sides with respect to 2x in Eq. (22). the left-hand side reduces to the single term since the conditional density of the estimated latents factorize:
Oij (Est, u)
K Ome u) Özst
i log H.
(23)
Ozkt
k=1
Next, we take derivatives with respect to 25e in Eq. 23 and the left-hand side becomes 0 since term f) (2,,, u) does not depend on Z....
K
(Zke, u) 0zke Ozu:
3m (EMe, u)
O=
3 log HH|]
+
(24)
dz
dā Oāzt
Ozkt
O2n
k=1
Lastly, take derivatives with respect ii in Eq.
Amm u) Ozkt Ozkt
o dm u)
0z
0=
Σ
(25)
02 dr Dās 023t
02.02
K
2d ne u) 0zt,Ori
pm u) 0z2
Σ
He He]as +
(26)
k=1
since the Jacobian H, does not depend on u. Using the sufficient variability assumption (A3) for the latent states z,, we can plug 2K values of uj...., U2K for which the partial derivatives of the log conditional density (ZRe, u) form linearly independent vectors v(zr, u). We see that the coefficients of these linearly independent vectors have to be zero: [He]a: [H]js 0. This implies that the Jacobian matrix H, of the transformation
= h(2,) has at most nonzero element in its rows. Therefore, the learned latents 2, are equivalent to the ground hound-truth latents z; up to permutations and invertible, element- wise nonlinear transformations.
C.2 PROOF OF THEOREM 2: IDENTIFIABILITY OF THE LATENT TRANSITION f
In this section, we prove our main theoretical contribution, Theorem 2. We first define the function that relates the tround-truth and estimated latent pairs (st, 24-1) and (Sz, 2.) to each other. Similar
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to the proof of Theorem 1, do this by using the relationship between (faugs Beug) and (faugs Baug):
(gang (ang) ((2 6t (bx-.))
(g (2)- [(6
X X
(27)
g 3(21-1
(gaug faug) (gaug faug) 1 (Baug favs)
(
St
(28)
(/-11
kan
Then, the function Kaug I2KK + (St, Z(-1):
2 maps the learned pair ($e, 21-1) the ground-truth pair
St Zi
Kaug
( 1)
St
(St. 2c-1) (2, 21-1)
(29)
Similar to the function being bijective in App. C.I follows that kaug is also bijective.
Now, we want to show that the augmented function kang decomposes into invertible block-wise functions k1, k2 Rx + Rº such that (i) kj only depends on $,, i.c., = (1r) and (ii) k2 only depends on 2;-1. i... z,-1 k2)2(-1).
easy to show (ii) the second case, since k2- = idz g- og o idN- = h and we have already shown in App. C. that the function h :2; + Ze equal g g and bijective. Hence, we can write h(2,- [2,-1]
in the place of kę (5,, 21-1):
8t Kaug (1 St (1
k (6,, 2.-1)
h(2,11)
(30)
To show (i), start by performing change of variables the log density of the pair (St, 22-1) for the transformation kaug" (8,, 11) + (8), 22-1):
log p(s,, 2 (u-1/u) log p(kaug (t-1)/u) + log Jk... ($c. 21-1)1; log ((k] (ße, 2, 1). h h(2
(31)
()]] + log (uk.. (8e, 21-1
(32)
where J Jk... (Se, 1 is the Jacobian matrix for the augmented function kaug evaluated at ($,, 21-1). As the process noise is temporally independent given 8, IL 2 u and s IL zi-1/u, lu, we factorize the densities in Eq. (32):
log p(s; u) + log p( 2 u) (st, + log p ((h(2)-1))) log J Jknn (Sc, 2c-1)] 1)].
(33)
Us
The Jacobian Jk.. is upper block-diagonal since z does not depend on Sy: J Junn
O
H,
Hence, its log determinant factorizes as log J... ($,, 2, 1)] log H. log factorization into Eq.
des I.
We can add this
28,
Os:
log ((s; u) + log p 2 u) = log p(k1 (St, 2 + log ((h(()-1))1a + log H + log 0s: (34)
In addition, the noise is conditionally independent over its dimensions given Therefore, we can further factorize the noise densities p(š; u)) = II p(ška u) and p(kı (St, 2, 2 2 = p(s; u) =
He (kkkll u) with s ($1, 21. We incorporate these factorizations into Eq. 34 follows:
Ōst
Σ log logp($kclu) + p(g (u)
Σ log logr(skllu) + logp(h(2;-1)lu) + log H, + log
0s,
qa (=(Sk..u)
(35)
When we take the derivative both sides of Eq. 35 with respect to &s,, the term log p( ((2 1 [u) at the left-hand side, and the terms log p(h(2,11))))) uu) and log H. vanish as they do not depend on su:
(Bst, u.
Ogu (skr, u) Oskr
log
8s
+ ass
0%.
(36)
Iskt 08s
&
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Next, take the derivative with respect to i, with / being arbitrary dimension, and the term
dhs log m9 vanishes since
DeL Us,
does not depend on u.
Uöet
0 (.2,, u)
(ske; u) Oskr
Σ
(37)
0s:0m
Osuiuu O8s
K
Lastly, take the derivative both sides with respect to 21
8 qu (ske, u) aske Oskt
O2 qakks u) 0 Sh
0 Σ
+ Oun 08,
(38)
05 0bs 02j,e-1
Inspecting the Eq- (38 to ensure the sufficient variability assumption (A4) for the process noise St,
the term U8a 0851 O. Following a similar reasoning with Morioka et al. 2021)) this implies that
Db:s 0E,,a-1 any dimension 8, does not depend on both & and 2 at the same time. Since St Zt-1 u and 2:-1 h has to depend solely on šy: = k1 (2.-1, 8x) k($.). This concludes that the augmented function kaug decomposes into invertible bisek-wise functions k and kaug = [k. h]:
St
k Kang
((=1))=
81 (
k($;) h h(2
(39)
7
We still need to show that the function that relates the ground-truth and estimated noise variables St and $ 8; k($,), ), is an element-wise function. For this, get back Eq. (37). We denote the Jacobian matrix of function by JR and its evaluation at & by (s,) = K, We take the derivative of both sides in Eq. (37) with respect to Smz for some index me:
@n u)
(ske, u) Skt
0 Σ
[K K.]km
(40)
Ös Oru
OsH 08
Now, inspecting the Eq. (40). we see that to ensure the sufficient variability assumption (A4), the product [KJAs[,, O. This implies that each dimension 8kt of the true process noise depends only on a single dimension of the learned process noise &t. Hence, the function k is equal to a
composition of permutation and element-wise, invertible nonlinear transformation: k = π T.
As the final step, we follow Eq. (28 and write the relationship between the augmented functions as:
gaug faug o Kaug = gaug faug:
(41)
Sugg
Baug
ofaug Kaug fang:
(42)
h
faug Kaug faug:
(43)
where hany denotes simply the concatenation [h, h), and hence have h [h-1,hh h- We have already shown that both and k are compositions permutations and element-wise invertible transformations. Inspecting Eq. 43), we see that the augmented transition function f and the true augmented transition function f are related each other only through functions h and which are compositions of permutations and element-wise invertible transformations. Hence, we conclude that the augmented transition function faug is equivalent to the true augmented transition function faug up to compositions of permutations and element-wise transformations.
C.3 ALTERNATIVE VERSIONS OF SUFFICIENT VARIABILITY ASSUMPTION
If the variable u is an observed categorical variable (e.g., domain indicator), the assumptions (A4, A5) can be written in an alternative form without partial derivatives with respect to ut, similar to Hyvärinen et al. 2019): Yao et al. (202T). For example, for the latent states 2,, the alternative version of the (A4) takes the form:
Sufficient variability of latent states for a categorical u (Yao et al. 2021). For any Z;, there exist some 2K + 1 values for u: uj.... U2K. such that the 2K vectors v(z,, u,+1) v(z,, u,) with j = 0, 1..,, 2K, are linearly independent where
Om (21e:
Ōnk (2K1, 0zgi
8n pnk((KK (2KI.
v(z,, u) =
E 2R
O2i0
0z
0=2
Kt
(44)
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A similar categorical version is provided in (Hyvärinen et al., 2019
Assumption 3), while the
continuous version is provided in the same work (Hyvärinen et al. 2019 Appendix D).
D VARIATIONAL INFERENCE
Similar to previous works (Yao et al. 2021 2022), we want to maximize the marginal log-likelihood log p(1:T) u) that is obtained by marginalizing over the latent states 21.7 and process noise $1:7:
log pe (x1:T () = log /
po(XI.T, 2o.7, S1:7 (u) dzo:T dS1:T,
(45)
2.8
where decompose the joint distribution follows:
(x(xT Zo:r, (1:r,,1rrlu) = pe (zo) II pa (e (u) po 1 s:) pe (e(xellz) Iz).
(46)
6(x; f(s..
Note that the state transitions Po (z St) are assumed to deterministic. The above integral is intractable due to non-linear dynamics f and observation functions. As typically done with the deep latent variable models, we approximate the log marginal likelihood by a variational lower bound, i.e., we introduce amortized approximate posterior distribution yo (Zo:7, 81.7 (xIT, u) that decomposes as follows:
T (o XLT, u) = 4o (zo|XITT, u) II | II XI:T, u) go 20. S1.2, u)
(o(6:lz(-1xx1))
90(3, (e(2((2
(47)
We simplify the variational posterior g(s.l.) (s(s 1, S1: u) (s( corresponding to a filtering distribution. As in the generative model, we choose go (2 22-1 St) po(z, 14-1, s:) = 8 (z. ((st,2(-)))
S1 u) = 90 (z) II 12.
(48)
where the functional forms of the densities Zo (zol-) and go (s:l.) are chosen as diagonal Gaussian distributions whose parameters are computed by recurrent neural networks. The variational lower bound takes the following form:
Po Po(Zo:r: $1:7
C(0,0) () Eg (Borr stor) log (21.T + log
(49)
go (2 81.7 X1.T, U)
po (Zu:r $1.7 (1)
B logpo(
Egg (xo.0.
log
(50)
Ro(2:17 S1:7 X1:7, u)
Reconstruction term, Cx
KL teem, ExL
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The reconstruction term can easily be computed in variational auto-encoder framework. Below, provide the derivation of of the KL term:
po (zo)
po (1:TT S1:T u)
LKL ( (..))
log
log
(51)
(0(Zu) X1: u)
9o (21:T $1:7
XI.T, u)
po(21: S1:r(u)
log
(52)
4á (ZI:T, 81:7 X1:T, 1)
T
po u)po Ze Z4 1, St)
=
+ (20
( 11-1, XI:T. u go (z log
(1,
(53)
T
log (s( uU) 4ad (8t bs 1, XI.T, u)
= - -Dxxlgol2o/xxi7 u)lpo
Eg (a.r (..)))
(54)
pe (55 [1)
-- www.golzo/xxxxio (xIT, w)llpo +
E Eig (s1 (8L 21 - las 2:91 1.81T
log (5 24- 1, X1.7
B 1/81-222 2-92 1D (g0 (sa/ 12-1 ...r. (s. |u)
(55)
E ADDITIONAL EXPERIMENTAL DISCUSSION AND RESULTS
E. IS NOISE HARDER TO ESTIMATE?
In Sec. 4 we observe that Mppl > MCC for all models. In the generative model, observations x(t) directly depend on zf)): x(t) = g(z(t)), while the dependency to s(t) indirectly through (t): x(t) = gf ((() To reconstruct observations x(t). the model has to recover 2tt) without having to fully identify the noise s(t). Hence, the noise is harder to estimate, to the best of our understanding.
E.2 ABLATION: HOW DOES EACH MODEL COMPONENT AFFECT THE PERFORMANCE?
To understand how each model component affects the perfor- mance, we set up an ablation study. We chose 3-VAE (Higgins)
et al. 2017) as the base model. On top of it, we add three model components one by one: (i) 1-MLP-DYN: a single MLP
MsD X Rutord
modeling K outputs of the transition function {fx 1k (i)
K-MLP-DYN: K independent MLPS modeling outputs of the transition function {fr 1 and (iii) OURS: additional conditional normalizing flow to model the nonstationarity p(s(u). We present MSE results for prediction steps T pred E (2, 4, 8) in Fig. 7 together with the MCC [appu values in the legend. 8- VAE cannot make future predictions, hence we only report its McC(Zumpul result. We see that adding K-MLP transitions results in larger performance improvements both in terms of disentanglement (McC Zimpu)) and future predictions (MSE (fsure)) compared to adding 1-MLP transitions or conditional flows. As expected, the improvement in future prediction accuracy is larger compared to the improvement in disentanglement performance, since conditional flows only indirectly affect the future predictions through the noise samples while directly affecting the noise distribution. More importantly, the results suggest that as we predict for longer horizons (e.g. 8 steps), a higher Mcc on the latent states corresponds to larger improvement on the future prediction performance.
0.0
Prediction Steps
1-MLL Towetion, Miristoad 0.0
R-Mar Transtion, Meciau 0.54
Conditional Flow Mec/Boarl
Figure 7: McC vs. MSE results for ablations for different time steps.
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%
de
d
3c fo(2)-1188) Bt)
Eo (2
Zi ()
2, 10(2)-1188)
yo (aalaal/es: (
Pig R
82
89
84
qa (St Me 12 ) Se qo Me-1; X1te)
Sy pa (stelt))
pul (sc
AA of
Flow
Flow
GRUE
GRU!
GRU1
%
X
Xo
X2
Training on X::T
Future prediction of
Figure 6: Diagram of the model architecture: In training, the observation x is passed through the the initial latent
encoder es to get the intermediate representation x. We learn the distribution state Zn conditional on the representations of the first Tie observations (in the diagram, we show The 1; our experiments, we use Ti 2). The latent state decoded by the decoder de to produce the predicted observation & (which is trained to match the corresponding actual observation). The next value of the latent state computed by transition function fz, which depends both on the previous state and the process noise In training, the process noise is sampled from the variational posterior that depends on the previous state as well as on the representation created by recurrent neural network (GRU) that has received up to the current observation. In future prediction, the process noise s sampled from the prior, which is a learned normalizing flow.
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Labent Treperary
Observal (aentAegee)
Lavil mestiry Denoee
Convantion Spain plaiee))
Figure 8: Test trajectory prediction samples, their corresponding skeletons and latent trajectories for MOCAP-SINGIFF At test time, the model takes first 3 time points as input, and predicts the full horizon (T = 300).
Osstrttin Spece
Osservtton Space
Obeervr on Space
Ohsevvnion Spece
Figure 9: Test predictions for MOCAP-SINGLE. (Top row) From left to right, each column shows latent trajectory samples until times [50, 100, 200, 300]. (Bottom row) From left right, each column shows samples at times (50, 100, 200, 300] transformed into skeletons, together with test data.
with Denn
Figure 10: Extension of Fig. 2 (Lef' columns) The same columns as Fig. 2
(Right columns)
The same plots produced for the 8-step future predictions.
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Dewrske
Brerice?
Figure 11: For the synthetic experiment, model predictions in the data space with the estimated uncertainties. Our model uses the first To 2 data points to encode the initial latent state 2o and Tayn = 4 data points to encode the noise sequence Swain- We unroll our model for To + Tayn +Tuuure = 14 steps ahead, where the future noise variables S7.14 follow the learned prior flow. We draw 32 trajectory samples X9:14 by sampling from the initial state and process noise. Above, black and blue dots show the training and test data points. The red curves are the mean trajectories and the red region corresponds to +2 standard deviation computed empirically. We observe near cear-perfect predictions and low uncertainty for the input data (the first Tiain To Tayn 6 time points) while the uncertainty grows as we unroll over time. Further, the uncertainty grows even more when the model predictions are off. Therefore, almost all test points lie the #2 std region, reflecting the high calibration level our probabilistic model attains.
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Narke
Meat
1
Der hage Serel
Test trajectory predictions for all 50 dimensions for LEAP-LIN and OURS in MOCAP- MULTI. At test time, first 75 time points are given as input, and the full prediction horizon is T 150 time points.
Figure 12:
F EXPERIMENTAL DETAILS
F. DATASETS
We chose three datasets with different characteristics to keep our experimental setup (i) general as possible, and (ii) comparable with the related works that are closest to ours (LEAP (aao al.
2021) and TDRL. Yao et al.) 2022)). Our datasets include: (i) synthetic dataset satisfying our modeling assumptions as common nonlinear ICA literature (Hyvärinen & Morioka 2016 2017 Hyvärinen et al. 2019; Yao et al.
2021: (022), (ii) cartpole dataset with high- high-dimensional video observations whose low-dimensional ground-truth dynamics are governed by set nonlinear differential equations (Yao et al.
2022), and (iii) the high-dimensional (50-dim) real-world Mocap
dataset with longer training and
prediction horizons (ranging between 75-300) (Yao et al. 2021
2022 Yildiz al. 2019
Li et al.
2020 Auzina et al. 2024).
Our motivation for the synthetic experiment is to validate our theory, and for the cartpole experiment to show model's ability recover low-dimensional latent dynamics from different kind of
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high-dimensional input modality (videos). To test model's 's capability to predict longer sequences of complex real-world dynamics, we also included the MOCAP experiment.
Note on nonstationarity and observation regimes. In the experiments, we observe (x,, Ur) pairs where each observation X; has an associated regime label uN, For the synthetic experiment, we have R 20 regimes. For the cartpole experiment, we have R 6 regimes (5 for training, 1 for testing). For the MOCAP experiment, the different regimes correspond to different persons walking. In MOCAP-SINGLE we have just one person, hence R = 1, and u, is constant, demonstrating case violating our assumptions. In MOCAP-MULTI and MOCAP-SHIFT we have 6 persons, hence R = 6. For each observation X,, the model (or the prior p(s, u)) takes Uz i.e., the one-hot encoded regime label, as input.
Synthetic dataset. We first set up synthetic experiment evaluate latent dynamics identification. Same as Yao et al. (2021] 2022), we generate multivariate time-series following our data generating process Eqs.
the dimension of s, z and x is K 8, the number of regimes is R 20, the distribution of noise changes between distinct regimes to provide non-stationarity to satisfy assumptions, Skz au(u)
Et (0 0.1), the functions and g are 2-linear layer random MLPs, and we choose Z1:7 to be a second-order Markov process, i.e, Zz = f(z 11 We
summarize the generative process as follows:
Zo: 21 ~ NN 1),
(56)
1
8kt 0g (1) Etz
E ((0,00),))
1 2, ... T
(57)
2. f(2: 2, Ze- 8:),
1 = 2, .... T
(58)
X = g(ze),
t = 0, ...y T,
(59)
where as (u) U(, 1) is uniform random variable that modulates the noise variance between distinct regimes. Hence, the noise represents random or unmodeled external variations in the process, not captured by deterministic dynamics. Here, the distribution of simply changes between distinct regimes to provide non-stationarity to satisfy the identifiability assumptions (see App. F.2)
For each environment, we generate the synthetic training sequences as To + Tayn = 2 + 4 = as in LEAP Yao et al. 202T) and TDRL (Yao et al. 2022), to keep our setup comparable to the two closest related works. Including the future time points, each sequence has the total length = To + Tayn + T future
sequences as train/validation/test data. We chose
+ 4 + 8 = 14. As we have second-order Markov process, first To = 2 states (Zo, Z1) are spared as initial states. The next 4 observations X2.5 are used for training the dynamical model. The last 8 observations X6.13 are used for evaluating future predictions. We choose the future prediction horizon Tiumue 8 the double of the training sequence length. If the dynamics are truly identified, the model should predict future states well, even for this longer horizon.
Cartpole dataset. Next, we use a more challenging video dataset of a cartpole system (Brockman al. 2016). The environment simulates a latent physical system with a pole attached cart from pivot point. The system has 4 states: position and velocity of the cart, and the angle and the angular velocity of the pole ([rean, Opole Teart, p. Observations consist of video sequences of the system and we use the true states only to compute MCC.
As in Huang et al.] (202T): Yao et al. (2022), we use a modified setup with: (i) 5 source domains used for training with different levels of gravity g = (5. 10, 20, 30, 40) and mass m 1.0. and (ii) a target domain with = 90 to check the methods' ability to extrapolate. The regime labels are non-informative and observed, while the gravity values are unobserved. However, Yao et al. (2022) assign a single constant action (only left or only right) to each environment, which is very unrealistic. In contrast to this, we observe the system under random binary actions, which move the cart to left or right at each step. We generate sequences of length T' = To Tayn TTuure = + 7 8 16: the first frame for the initial state, the next 7 frames for training the dynamical model, and the last 8 frames for future prediction. For each source domain, we have 900/100/100 sequences as train/validation/tes data. For the target domain, we have data sets with different numbers of samples Nunget (20, 50, 100, 1000).
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Mocap datasets. To evaluate future predictions long horizons with complex real-world dynamics, we use three datasets from the CMU motion capture (Mocap) library containing human action
trajectories (Wang& et al., 2007:
Yao et al. 2021 2022): : MOCAP-SINGLE (Yildiz et al. 2019).
MOCAP-MULTII (Auzina et al. 2024) and MOCAP-SHIFT (Auzina et al., 2024). For all Mocap
datasets, each observation X, R:00 has 50 dimensions, after the preprocessing used in (Yildiz et al. 2019
Li et al. 2020).
MOCAP-SINGLE contains 23 walking trials of single subject 35, split into 16/3/4 train/val/test sequences. Each sequence has 300 time points. At test time, the models take first 3 data points as input and predict the whole trajectory. MOCAP-MULE walking trials for 6 subjects. Each sequence is down-sample by factor of 2 to 150 time points. The models take first data points as input and predict the whole trajectory. The detailed subject and trial ids can be found in the Appendix of Auzina et al. (2024).
and MOCAP-SHIFT contain a total of 56
F.2 ASSUMPTIONS (A3, A4) ON SYNTHETIC AND CARTPOLE DATASETS
In the synthetic experiment, we use noise with (randomly) modulated variance between environments. Under this modulated noise, sufficient variability assumptions (A3, A4) hold.
Assumption (A4) in the synthetic experiment. We start with the analysis for the noise (assump- tion (A4)). Our ground-trute functions for this experiment are:
Skt
(60)
a4 (u)
Zkt (Zt- 22-2 8kt-
(61)
Without loss of generality, let us assume is a standard normal random variable: NN to simplify the derivations. The log probability of noise is:
0 2
log p( u) = Ikk = - log 2n + log ax
2 kkr-
Then, the required partial derivatives take the following form:
8qqt.
dak (u)skts
Osg 20
(62)
Zm
jrque qkt
Dak
-=x
(u).
(63)
as
Ou
These equations tell that if the 2K functions -2a (u) Dur
Bog (u)sgæ and -ag (u) Bos
dus (u) for
k 1...., K in are linearly independent, then the assumption (A4) holds.
For our synthetic setup, we choose the modulator variables (u) uniformly random between [0, 1]. To simplify the analysis, let assume that have dę /2wrw with w₁. UO, 1). Then, the 2K functions become - -2 and -w,, which are random functions. This results in 2K x 2K matrix with random entries the form of ww and -2w with w (O, 1) and Skr A ag 3). We conjecture that the probability of these random vectors to be linearly independent is 1. To validate this empirically, we sampled 10000 2K x 2K matrices with entries -2 and Wkk where (0, 1) and Skt ~ N(0, w(). They all had rank 2K.
Assumption (A3) in the synthetic experiment. The analysis for the latent (assumption (A3)) is similar. We write the log probability as follows:
ax (ZKe - fA (zi-1)))
log p(ke 1 u) = Te - log 2T + log ag 2
Then, the required partial derivatives take the following form:
d -2nx (u)
dag
(u)(zut fa (z(-1)))
(64)
Ozu
Ou
Ths -2a (u)
Da
(u).
(65)
2z dw:
Ou:
Since 2kz - (2,-1) =
Sk.. the 2K equations here have the same form as in Eqs. 62 and (63 in
the analysis of assumption (A4). Then, using the same reasoning, the assumption (A3) holds.
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Table 4: Architecture components for the synthetic experiment (Tinpot 2 44 6, 4,
8, L = 2).
Encuder
ICEncoder
NoiseEn
TransitioNMLP MLP)
Decoder
Input: R Topu PC 64, Leaky-ReLU
Ioput: R X64 FC 64, Leaky-ReLU FC 64, Leaky ReLU FC XCLX2XX D
Input: R. Pupu x44
Dx Input: B Rxx0+1
Inpot: R R
GRU 64, layer
Dx FC 64, Lea
PC 64, Leaky-RoLU
PC 64, Leaky- ReLU
Concall x,- Lit-1l
Dx FC PC64, Leaky ReLU
PC 64, Leaky ReLU FC
PC 64, Leaky-Rel.11
FC 64, Lenky-Ret
Dx FC1
FC
Leaky-ReLU
FC2 x D
Table 5: Other hyperparameters for our method for the synthetic experiment.
Parameter
Value
Ir
2e-3
8
2.5e-03
Bn
1e-03
weight decay
le-6
batch size
32
latent dim. 8
Cartpole dataset. While we simulate cartpole data under different gravity values to create nonsta- tionarity, Assumptions (A3, A4) do not exactly hold in the cartpole experiment since the nonlincar differential equations underlying the cartpole system are deterministic.
F.3 ARCHITECTURE AND OPTIMIZATION DETAILS
In this section, we report the architecture details. We optimize our model with Adam optimizer with default parameters, except the learning rate which is chosen by validation. We chose all hyperparameters for our method, KALMANVAE, two versions LEAP, CRU and TDRL with cross- validation. 3- 3-VAE uses the hyperparameters chosen for our method. In particular, performed random search as well as Bayesian optimization (in some cases) over learning rate, loss weights (e.g.. (9, weight regularization, the number of layers in all MLPs, and latent dimensionality.
Conditional normalizing flows. In all experiments, we use ID conditional normalizing flows to model the nonstationary prior for the noise variables. We employ the implementation from [Stimper al. (2023). These flows are I-layer neural spline flows conditioned on the auxiliary variable u. The auxiliary variable u first one-hot encoded and then embedded using I linear layer before taken as input.
F.3.1 SYNTHETIC DATA EXPERIMENTS
Our method. For the synthetic experiment, the architectural components of our model are detailed in Tab. 4 In addition the architecture layers, our method has several other hyperparameters selected by validation. They are listed in Tab.
Baseline methods. For the baselines KALMAN VAE, CRU, TDRL, LEAP-LIN and LEAP-NP, we the architectures provided for multivariate time-series in their public code base. The identifiable representation learning methods TDRL. LEAP-LIN and LEAP-NP have similar architectures to ours, since they also perform synthetic experiments validate their theoretical findings. The rest of the hyperparameters are specified in Tab. 6 For the hyperparameters with multiple values, we perform a sweep over them determine the final value by validation. For the others with single value, keep the recommended values in the publicly available code base.
F.3.2 CARTPOLE EXPERIMENTS
For the cartpole experiment, the architectural components of our model used in the cartpole experiment detailed in Tab. 7 We present the rest of the hyperparameters for method in Tab.
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Table 6: Hyperparameters of the baseline methods for the synthetic experiment.
Model
Parameter
Value
learning rate
[2e-3, le-3]
[1e-2, 3e-3, 2e-3, le-3, le-4] le-2
TDRL
?
latent dim.
8
learning rate
[2e-3, le-3]
3
[le-2, 3e-3, 2e-3, le-3, le-4]
LEAP-LIN and LEAP-NP
7
2e-2 (LEAP-LIN), 9e-3 (LEAP-NP) le-6
latent dim.
8
learning rate
[5e-3, 3e-3, le-3]
CRU
latent dim.
[16,
learning rate
1e-3
3
[10, 1, 0.1, le-2, le-3]
KALMANVAE
init. cov
[10, 1, 0.1]
latent dim.
[2, 4, 8]
Table 7: Architecture components for the cartpole experiment (Tinput = = + 7 8, = 4,
K 8).
Encoder CNN
Decoder Decony
Input: Tipput x x 64x ne
Input: T ipput R.K
x 3 Conv2d, 32 GeLU, stride 2
FC 64, Unflatten
x 3 Conv2d, 32 GeLU, stride
3 x Upconv2d, 64 GeLU, stride
x 3 Conv2d, 64 GeLU, stride 2
3 x 3 Upconv2d, 32 GeLU, stride 2
x 3 Conv2d, 64 GeLU, stride
3 x 3 Upconv2d, 32 GeLU, stride
Flatten
3 x 3 Upconv2d, Sigmoid, stride
ICEncoder
NoiseEncoder
TransitionMLP (Kx MLPs)
Input: x
Input: R x
Dx Input: RLxD+1
FC 64, Leaky-ReLU FC 64, Leaky-ReLU FC L x 2 D
GRU 64, 1 layer
Dx FC 64, Leaky-ReLU
Concatl 2t -at--1
Dx FC 64, Leaky-ReLU
FC 64, Leaky-ReLU FC 64, Leaky-ReLU
Dx FC
FC 2 x D
Table Other hyperparameters for our method for the cartpole experiment.
Name
Value
Ir
2e-4
B
le-1
Bep
le-1
weight decay
0.0
batch size
latent dim.
Baseline methods. Among the baselines, TDRL Yao et al. 2022) also has an experiment on the modified cartpole setup. We keep their architecture, including the backbone CNN and its deconvolution decoder. For CRU, LEAP-LIN (Yao et al., 2021 and LEAP-NP (Yao et al. 202T). we the same backbone CNN and the deconvolutional decoder from Yao et al. (2022). since it is
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larger network compared to ours with additional conv2d layer (filters: 32,32,64,64,644 i.e., it has more parameters. For KALMANVAE, we tried the CNN encoder/decoder components from TDRL Yao al. (2022) and our architecture, but the algorithm kept throwing numerical errors among seeds (it finished only 1/5 seed without errors). Therefore, we used the simple CNN encoder used in their implementation. The rest of the hyperparameters specified in Tab.
Table 9: Hyperparameters of the baseline methods for the cartpole experiment.
Model
Parameter
Value
learning rate
[2e-3, le-3]
3
[1.0, le-1, le-2, 1e-3]
TDRL
[100, 10, 1]
[le-2, le-3]
latent dim.
learning rate
[2e-3, 1e-3, 5e-4]
8
[1.0, le-1, le-2]
LEAP-LIN and LEAP-NP
7
[20. 10, 1]
[1e-6, 0.0]
A
[1c-1, 1e-2, 0.0]
latent dim.
learning rate
[5e-3, 3e-3, le-3] [16, 32]
CRU
latent dim.
learning rate
[7e-3, 3e-3, 1e-3] 0.1]
8
KALMANVAE
init. cov
[20, 0.11 4]
latent dim.
F.3.3 MOCAP EXPERIMENTS
For Mocap experiments, we adapt the architecture used in Yildiz et al. (2019) to our model, e.g., we the number of layers and hidden dimensions for the base encoder and the decoder. We also use its observational density which is diagonal Gaussian where a single diagonal variance parameter estimated. The details presented in Tab. TO
Table 10: Architecture components for the Mocap experiments (Ti = 3, D 50, K = L. 1).
Base Encoder
ICEncoder
Noeseer
TransitionMLP
Decoder
Input: R2 X FC Leaky-ReLU
Input: IR Fi, x30
Input: R Tawn X30
Dx Inpat: B D41
Input: Tispu D.D FC Leaky-ReLU FC 30, Leaky-ReLU FC +C++1
30, Leaky-ReLU L x x 2 K
GRU layer
30, Leaky-ReLU
FC:30
Concatl Z:- Lit-1l FC 30, Leaky-ReLU TC x
PC 30, Leaky-ReLU
PC
The rest of the hyperparameters for our method are shown in Tab. 11 Additionally, we have observed that gradually increasing the training sequence length during training improves training stability for our method. For example, start training with shorter sequences, e.g., with 25-step long and we train the model with these for epoch. Afterwards, the training continues with 75-step long sequences until convergence.
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Table 11: Other hyperparameters for our method for the Mocap experiment.
Name
Value
Ir
2e-4
B
le-1
8zo
le-1
weight decay
0.0
batch size
latent dim.
its recommended architecture for Mocap datasets, since it also has an experiment on another Mocap dataset. For CRU, we adapt the encoder and decoder layers in Yildiz et (2019). The rest of the hyperparameters are specified in Tab. 12 Particularly, we tried to increase the input sequence length, but we observed that the model cannot be trained with sequences longer than 10-steps.
Baseline methods. For LEAP-LIN Yao et 2021).
Table 12: Hyperparameters of the baseline methods for the Mocap experiment.
Model
Parameter
Value
learning rate
[5e-3, 3e-3, 1e-3] [16, 32]
CRU
latent dim.
learning rate
le-4,, lc-4]
[1.0, le-1]
LEAP-LIN
[20, 10,
[le-6, 0.0]
A
[le-3, 0.0]
latent dim.
8 [5.7,10]
sequence length
30


Reviews:
[{"review_id": "NdtTZq4lnZ", "reviewer": "Reviewer_i2e2", "summary": "This paper proposes a structure with a theoretical foundation for identifying latent dynamics under certain assumptions. The implementation is standard. The experiments' results show various advantages in different scenarios, even when the assumptions are violated.", "questions": "1. Does the identifiability still exist when I have any external input, like the control input in the state-space model?\n2. What's the exact definition of $s_t$, and what's the motivation for incorporating $s_t$ into the model?\n3. (minor) For Table 1, do you use any stats test to confirm your approach is the best?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "In Table 3, the author(s) claim, \"Our approach is the first to predict future states of a dynamical system by utilizing a transition function that identifies the underlying, true dynamics.\" As far as I know, many approaches can do similar things without a theoretical foundation for the identification.", "weaknesses": "Perhaps a theoretical or practical time complexity analysis would be more appropriate.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Bj9HlluqQx", "reviewer": "Reviewer_XAHn", "summary": "In this paper, the authors introduce a theory under which one can identify both the latent states as well as the latent dynamics from observational data (up to permutations and invertible transformations). Next, the authors introduce a practical algorithm, based on sequential variational autoencoders, for learning identifiable state-space models. Experiments provide empirical evidence that proposed approach can correctly recover the latent states and the latent dynamics.", "questions": "1. Why was such a small prediction horizon used for testing? I found this odd for the synthetic data.\n2. I see that training sequences used to train the model are relatively short. Is there a reason why?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "My favorite thing about this paper is the simplicity of the proposed approach. By just simply changing the form of the latent dynamics model, i.e., as opposed to additive noise, which is the standard, the noise is an input to the transition function, and assuming a dimension-dependent dynamics function, one can recover the states and dynamics (under some assumptions). I think this is a massive break through for people who learn state-space models.", "weaknesses": "While the paper is strong, there a couple of minor weaknesses. Firstly, is that assumption 2 doesn't seem to align with one of the motivating examples in the introduction, which is model-based RL. Specifically, assumption 2 does not allow for process noise/control values that are temporally correlated. But in model-based RL, the process noise *is* temporally correlated because the control value chosen at time $t$ depends on the previous state of the system. I found this contrast jarring.\n\nSecond, the experiments section is great but I think it would be great to have a seqVAE baseline that is more similar to proposed generative model. In the experiments section, the baseline seqVAE is the KalmanVAE, which is a hierarchical SSM. While the proposed approach outperforms KalmanVAE, its not clear whether it outperforms it is truly better or if there is a model mismatch between the true generative model and the one used by KalmanVAE. I think training a seqVAE using the generative model as described in section 2.1 would strengthen the results. \n\nAlso, section 4.4 doesn't make sense at a first glance. If I had the true latents then training a model in a supervised fashion should be better than learning both in an unsupervised fashion. When doing the comparison, were the number of parameters matched? I ask because if your latent state has $K$ dimensions then one would need $K$ neural networks, so it might be the case that the proposed approach is working just because the model has more expressive power.\n\nLastly, I think the proofs in the appendix could benefit with a little more hand-holding to follow along!", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "psV14ztpyT", "reviewer": "Reviewer_iU39", "summary": "This work deals with the general problem of identifying (i.e. inferring) the nonlinear transition function underlying the dynamics of complex dynamical systems. The authors extend recent theoretical work on identifiability of latent states in dynamical systems to the identifiability of their transition function. Furthermore, the authors provide a neural-based inference model that implements the key aspects of the theory and relies on neural variational inference. The authors then test their methodology on three settings: (i) a synthetic dataset that mirrors their generative model; (ii) a cartpole simulation which is represented as a video signal; and (iii) three sets of experimental motion capture data. The results demonstrate the proposed models outperform the baselines, not only on the identification of the latent variables, but also on forecasting tasks.", "questions": "1. Is there a reason why the noise posterior $q(\\mathbf{s}\\_{t} | \\mathbf{\\tilde x}, \\mathbf{z}\\_{t-1} )$ explicitly depends on $\\mathbf{z}\\_{t-1}$? Does this follow from your theory? If so, how?\n\n2. In line 140 the authors write “setting $\\mathbf{u}$ to an *observed regime index*...” What do you mean by observed regime index? Do you mean the time index $t$ of the observation $\\mathbf{x}(t)$? Or do you mean the observation value $\\mathbf{x}(t)$ itself?\n\n3. How is $\\mathbf{u}$ used in your practical implementation? Put another way, do you set $\\mathbf{u}$ to be an *observed regime index* or $\\mathbf{s}\\_{t-1}$? Or is it only implicitly define via the prior $p(\\mathbf{s}\\_t| \\mathbf{u})$? In Figure 6, $\\mathbf{u}$ is shown as an input to the prior noise model, which is used for forecasting. Can you please explain how you choose $\\mathbf{u}$ in each experiment?\n\n4. What does $\\mathbf{s}$ represent in the synthetic experiments? Similarly, what does it represent in the Cartpole dataset?\n\n5. In Table 1, why is $\\text{MCC}[\\mathbf{z}] > \\text{MCC}[\\mathbf{s}]$ for all models? Is the noise variable harder to infer in practice? \n\n6. How do you interpret the inferred $\\mathbf{s}$ in the MOCAP datasets? \n\n7. In the synthetic dataset description of the appendix (lines 1235, 1236) the authors write: the number of environments is R = 20. What do these environments represent?\n\n8. Are the sufficient variability assumptions (A3, A4) satisfied in the synthetic and Cartpole experiments? If so, can you please explain why?\n\n9. Why are the simulated trajectories (of the synthetic dataset) so short? Meaning: what is the motivation behind such a setting?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- This work tackles a key problem of (nonlinear) dynamical system modelling, namely the identifiability of the transition function underlying the dynamics;\n- The authors naturally extend the identifiability theory from previous works to the identification of the nonlinear transition function of the hidden process. Their proofs are well written and easy to follow;\n- The authors demonstrate that their model implementation of their theory outperforms different baselines, in different settings. The results are very convincing (see however the questions below).", "weaknesses": "In my view, the main weaknesses of the paper have to do with its presentation. The three most pressing points are:\n\n1. There is somewhat of a disconnect between the theory section and the practical implementations, which I think could easily be improved. Especially when it comes to the experimental section. Questions 2 – 8 below point to concepts or ideas which are not clearly explained, that could be explained better or that are simply absent. In particular, it is not clear to me how the auxiliary ($\\mathbf{u}$) and noise ($\\mathbf{s}$) variables are used (or represented) in the experiments. To improve the readability of the paper, I'd suggest the authors explicitly describe the role of $\\mathbf{u}$ and $\\mathbf{s}$ in each experiment.\n\n2. The target datasets could be explained and motivated better. It is not clear from a first reading why these datasets are relevant in the context of the model (see e.g. questions 3 and 4 below). This point is clearly related to point 1 above. I believe that if the authors explain what $\\mathbf{u}$ and $\\mathbf{s}$ represent in each dataset, it will become clearer why those target datasets are interesting.\n \n3. The paper builds on a large collection of previous works which deal with identifiability and go all the way back to independent component analysis (ICA). The authors assumed the reader is versed in the development of ICA, from its linear to its nonlinear versions, and simply present e.g. the augmented dynamics (line 126) and the auxiliary variable (line 130), or the identifiability assumption, without any further explanations and context. There is however some nice physical interpretation underlying e.g. the conditional independence assumption of the noise process given the auxiliary variable, which can be traced back to Packard et al. [1980], and that is also nicely explained (albeit in a different sense) by e.g. Hyvarinen et al. [2019]. Even in the introduction, lines 41-49, the authors could already mention that their work builds on classical ideas of ICA. In short, I think the authors could better exploit the history of nonlinear ICA and its application to dynamical systems to better motivate their proposal. \n\nNote:  *I will happily increase my score once the authors answer the questions below* and explain how they will try to incorporate this information back into the paper.\n\n**Other comments and/or typos** \n- in line 88 the authors write *the generative function*. Do the authors mean the function $\\mathbf{g}$? I’d suggest, for the sake of clarity, to either stick to the “instantaneous mapping” label used later in line 112, or better use the more standard “emission model” label.  \n- in your KL loss term (lines 290 inside Algorithm 1) you write that the posterior over the noise $\\mathbf{s}$ is conditioned on $\\mathbf{u}$. This is a typo, isn’t it?\n\n**References**\n- Packard et al. [1980]: Geometry from a time series\n- Hyvarinen et al. [2019]: Nonlinear ICA Using Auxiliary Variables and Generalized Contrastive Learning", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "w0rwFTIXbq", "reviewer": "Reviewer_oxFU", "summary": "This paper presents a framework for identifying the unknown transition function in latent dynamical systems. The authors emphasize that previous research focused on identifying latent states, and lack the ability to accurately predict future states, particularly in complex nonlinear systems.\n\nThe authors establish the theoretical identifiability of both the latent states and the transition function. They leverage the assumption of \"sufficient variability,\" meaning that the latent noise and states exhibit enough variation across different environments to disentangle the underlying dynamics. The framework also relies on the assumption that the augmented transition function is bijective. \n\nThe framework was tested across a range of scenarios, including synthetic datasets, a simulated cartpole environment, and real-world Mocap data. The results demonstrate that the model effectively recovers the true underlying dynamics, and is accurate in predicting future states.\n\nAfter the discussion, the authors have addressed my concerns. I have decided to raise the score.", "questions": "Does the proposed framework requires the dimensionality of latent space and noise to be known?\nIf not, what would be the procedure of estimation?\n\nDoes the proposed framework allow for sparse noise?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The identifiability was theoretically defined and justified with given assumptions.\n\nThe proposed framework was empirically evaluated and compared with existing methods across synthetic and real-world data.", "weaknesses": "The claimed contribution \"the first framework that allows for the identification of the unknown transition function alongside latent states and the generative function\" seems overstated. There are existing methods to identify latent states, transition and generative functions together, e.g. [Sussillo16, Schimel21, Hess23, Zhao23]. \n\nThe proposed framework is not demonstrated as general as claimed like \"... improves generalization and interpretability of dynamical systems...\". The authors focused on typical RL and robotics scenario. There exist dynamical systems in other fields do not satisfy those assumptions or are in completely different regime. This could disadvantage the compared methods with weaker assumptions (e.g. NODE).\nThough the authors used Mocap dataset to showcase then it violates the identifiability assumptions, the system itself is still of RL and robotics scenario. It is totally valid to narrow down problems to solve them better. However, the statements and claims should be more specific.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)