A Study of Posterior Stability in Time-Series Latent Diffusion
Authors: Yangming Li
Yixin Cheng
Mihaela van der Schaar
Venue: ICLR 2026 Poster (2026)
Keywords: Latent Diffusion
Time Series
Posterior Collapse
Primary area: generative models
TL;DR: Conducted a solid analysis of posterior collapse in time-series latent diffusion, and presented a new framework that is free from the problem.

Abstract: Latent diffusion has achieved remarkable success in image generation, with high sampling efficiency. However, this framework might suffer from posterior collapse when applied to time series. In this work, we first show that latent diffusion with a collapsed posterior degenerates into a much weaker generative model: variational autoencoder (VAE). This finding highlights the significance of addressing the problem. We then introduce a principled method: dependency measures, which quantify the sensitivity of a recurrent decoder to input variables. Through this method, we confirm that posterior collapse seriously affects latent time-series diffusion on real time series. For example, the latent variable has an exponentially decreasing impact on the decoder over time. Building on our theoretical and empirical studies, we finally introduce a new framework: posterior-stable latent diffusion, which interprets the diffusion process as a type of variational inference. In this way, it eliminates the use of risky KL regularization and penalizes decoder insensitivity. Extensive experiments on multiple real time-series datasets show that our new framework is with a highly stable posterior and notably outperforms previous baselines in time series synthesis.

Published as a conference paper at ICLR 2026
A STUDY OF POSTERIOR STABILITY IN TIME-SERIES LATENT DIFFUSION
Yangming Li', Yixin Cheng 1.23, MMihaela van der Schaar' 'DMMTP, University of Cambridge LIONS, EPFL
Hithink Research
y187 7 774ccmm...m aa. uk
ABSTRACT
Latent diffusion has achieved remarkable success in image generation, with high sampling efficiency. However, this framework might suffer from posterior col- lapse when applied time series. In this work, we first show that latent diffu- sion with collapsed posterior degenerates into much weaker generative model: variational autoencoder (VAE). This finding highlights the significance of address- ing the problem. We then introduce principled method: dependency measures, which quantify the sensitivity of a recurrent decoder to input variables. Through this method, we confirm that posterior collapse seriously affects latent time-series diffusion on real time series. For example, the latent variable has an exponen- tially decreasing impact on the decoder over time. Building our theoretical and empirical studies, we finally introduce a new framework: posterior-stable latent diffusion, which interprets the diffusion process In this way, it eliminates the use of risky KL regularization and penalizes decoder insensitivity. Extensive experiments on multiple real time-series datasets show that our new framework is with a highly stable posterior and notably outperforms previous baselines in time series synthesis.
type of variational inference.
INTRODUCTION
Latent diffusion (Rombach et al. 2022) has shown strong performance in image synthesis (Podell 2024). offering substantially faster sampling than standard diffusion models (Ho et al. 2020). However, when applied to time-series data, this framework might suffer from posterior collapse (Bowman al. (016)) a common and important problem that happens some generative models (e.g., autoencoder (Baldi) [012]] Lucas et al. (2019), where the latent variable only captures limited information from the data. In that case, the decoder tends to ignore the variable during condi- tional generation. In this paper, present systematic analysis of posterior collapse in time-series latent diffusion and propose an improved framework that builds upon our analysis.
Impact analysis of posterior collapse. We first show that a strictly collapsed posterior reduces the latent diffusion to vanilla variational autoencoder (VAE) (Kingma & Welling 2013) in formu- lation, indicating that this problem renders the framework inexpressive, even weaker than a vanilla diffusion model. We then introduce a principled method: dependency measure, which quantifies the dependencies of an autoregressive decoder on the latent variable and the observed time series. Through performing empirical estimation of these measures, we confirm the latent variable has nearly exponentially vanishing impact on the recurrent decoder, indicating that time-series latent diffusion indeed suffers from posterior collapse.
From this empirical study, we also observe an interesting symptom of posterior collapse: depen- dency illusion: when time series are randomly shuffled and thus lack structural dependencies, the estimated dependency measures show that the autoregressive decoder still heavily relies on previous observations (instead of the latent variable) for predicting the next one.

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Our posterior-stable latent diffusion. We identify two main causes of posterior collapse in latent diffusion: KL regularization and strong decoder (Bowman et al. 2016). The first cause stems from the design of VAE, ensuring that the latent variable follows simple prior distribution. Such reg- ularization is fact unnecessary in latent diffusion, as its diffusion component supports sampling latent variables from complex distribution. The second cause is the absence of mechanism that enforces decoder sensitivity to the latent variable.
In light of this analysis. we first propose reinterpret the diffusion process form of variational inference, thereby eliminating the use for risky KL regularization in latent diffusion and allowing unrestricted prior distributions for latent variable sampling. To ensure that the decoder is sensitive to the latent variable, we also apply the diffusion process to simulate collapsed posterior, imposing a significant penalty on dependency illusion.
Contributions and roadmap. In summary, our contributions are as follows:
We provide rigorous analysis of posterior collapse in latent diffusion, showing that collapsed posterior will render it no more expressive than a simple VAE. We also introduce dependency measures that confirm the problem on real time-series data;
We present new type of time-series generative model: posterior-stable latent diffusion, which is free from risky KI. regularization and ensures that the autoregressive decoder is sensitive to the input latent variable;
We conduct extensive experiments on multiple real time-series datasets. The results demon- strate that our framework remains robust against posterior collapse and significantly out- performs a number of previous baselines.
The remainder of this paper is organized as follows. In Sec. we review latent diffusion. Sec. presents a systematic analysis of posterior collapse in time-series latent diffusion. Sec. 4 intro- duces our proposed framework address this problem. Finally, Sec. reports experimental results demonstrating the effectiveness of the framework.
BACKGROUND: LATENT DIFFUSION
The architecture of latent diffusion consists of two parts: 1) autoencoder (Baldi) 2012) that maps high-dimensional or structured data into low-dimensional latent variables; 2) a diffusion model (Sohl-Dickstein al. 2015) that learns the distribution of latent variables.
The autoencoder is typically implemented as VAE Kingmaa & Welling 2013). Let X denote free-form raw sample, following an underlying distribution g graw (X) The encoder fone
Autoencoder:
of VAE aims to convert the sample into a fow-dimensional vector v = fene (X). Through a repa- rameterization trick, the vector can be used to compute latent variable z as
pe diag(o) E, 1),, μ W exp.or-ex((W.avv (W.),
(I)
where W n. W, are learnable matrices, and diag(.) is an operation that casts a vector into a diagonal matrix. The above procedure, which differentially samples latent variable from the posterior
(L(z X) = =N(z; μl, diag(o²)), is termed variational inference (Blei et al. 2017). In VAE, the decoder pdec is used to parameterize conditional generation distribution p [X z), recovering the real sample X from latent variable z.
The exact negative log- likeeliodoood loss function of VAE computationally infeasible, its opti- mization relies on an upper bound of the loss:
VAE
E V In pen (X z)] + DKL X) prior (z)),
(2)
where the prior distribution prior (z) is typically set standard Gaussian A/ I). The last KL divergence term to ensure that the prior pprior (z) is compatible with decoder gree at test time, though it is one cause of posterior collapse (Bowman al. 2016).
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Diffusion model. The diffusion model can be implemented DDPM (Ho et al. 2020). The model consists of two Markov chains of L E N+ steps. One of them is the diffusion process, which incrementally applies the forward transition kernel:
form (z' zz ==~ / 311''11''... 11),
(3)
the latent variable 2 ulc. Here the distribution of latent variable glatent(z) is defined as (z X)qqui (X)dX. The
glatent (z), where 3i [L, Li is some predefined variance sched-
outcomes of this process form series of new latent variables {z x²" ... z²), with the last one z
approximately following a standard Gaussian N\ I) for >
The other is the reverse process, which iteratively applies the backward transition kernel:
1 Vaa (2 3r
back (z'
(4) ITk= a², bback(( (.) is a neural network, zt an initial sample drawn from (0, I), and of some backward variance schedule. The outcome of this process a reversed sequence of latent variables {z- z f-2 ,2"). where the last one z" is expected to follow the
p back (2" 1- 22) = -('===111 ; bucckk(k (z', i), IT),
back bubck((2) (z',
where a² = 1 10-3--,0
distribution real samples: q glatent (z").
To optimize the diffusion model, common practices adopt loss function below:
DDM = Ei.o,,lll back (Vā'z) + V - -at,, 1)]]
(5)
where 6 ~ (O,, I), ~ -
3 PROBLEM ANALYSIS
In this section, we first show the significant impact of posterior collapse on time-series latent diffu- sion. Then, define proper measures that can empirically quantify this impact. Finally, confirm that time-series diffusion indeed suffers from this problem on real datasets.
3.1 SIGNIFICANCE OF POSTERIOR COLLAPSE
Let focus time series X X2, XT). where every observation x,, [1,7] is D-dimensional vector and T denotes the number of observations. When applying latent diffusion to time series, problem that might arise is posterior collapse, which occurs to many types of autoencoders, especially VAE. The problem can be formulated as follows.
Problem formulation. The posterior q (z X) of VAE is said to collapse if it reduces to the Gaussian prior prrior (z) = [2; 0, I), irrespective of the conditional X:
q VI X) = pptior(z)) (z). vX E RTD.
In that case, the latent variable contains no information about time series X, otherwise the pos- terior distribution (z (z | X) would vary depending on different conditionals. Above is strict definition. In practice, one is mostly faced with a situation where 9 V X) =s pprior (z) and is still appropriate to say that the posterior collapses.
Implications of posterior collapse. A typical symptom of this problem that, since the latent variable carries very limited information of time series X, the decoder fdee tends to ignore this input variable z, which is undesired for conditional generation psen (X z). Besides this empirical finding from previous works, find that posterior collapse also seriously impacts the expres- siveness of latent diffusion. Let us first see the below conclusion.
Proposition 3.1 (Gaussian Latent Variables). For standard latent diffussio,, suppose its posterior y (z X) collapses, then the distribution glatent(z) (z) of latent variable z will shape as a standard Gaussian (0, I), which is trivial for the diffusion model approximate.
Simply put, if posterior collapse happens, latent variable z will be Gaussian at test time. In that case, the diffusion model that aims to approximate the complex variable distribution platent is redundant module. Therefore, posterior collapse reduces latent diffusion to a weak VAE, making inexpressive. The proof for this conclusion is provided in Appendix A
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3.2 INTRODUCTION OF DEPENDENCY MEASURES
A typical symptom [Bowman et al. 2016) of posterior collapse is that the input latent variable loses control of decoder fae during conditional generation (x z). To verify whether this is the case for time-series latent diffusion on real datasets, we introduce some proper measures that quantify the dependencies of decoder fdec on various inputs.
Autoregressive decoder. Consider that decoder pdee has an autoregressive structure, which con- ditions on latent variable z and prefix XI:- = [ X ,x,- to predict the next observation With abuse of notation, we set Xo- and formulate the decoder as
h = pdee (),, Xox-1 = [Xo, X1, X2, ,
(6)
where the representation h, 2 is linearly projected multiple parameters (e.g.. mean vector and covariance matrix) that determine the distribution p*t (xt z, x1:-1) of some family (e.g., Gaussian). Examples of such decoder include recurrent neural networks (RNN) (Hochreiter & Schhidhuber] (197)) and Transformer (Vaswani et al. 2017). We present the formulation details of these example in Appendix B
Dependency measure. As mentioned, clear symptom of posterior collapse is that the decoder pdee heavily relies on prefix X1a-1 (especially the last observation x,-1) to compute the representa- tion h, ignoring the guidance of latent variable Xo z. The variable that loses control of decoder
pdec is undesired for conditional generation peen (X 2))
Inspired by integrated gradients (Sundararajan al. 2017), present technique: dependency measure, which quantifies the impacts of latent variable X z and prefix X1a-1 on decoder fdee,
Specifically, we first set baseline input Op:-1 as [xo 0, X 0, X 0] and denote the
term fdce 10.--1 as h, Then, we parameterize a straight line 7(s) [0. 1] + RAD between the actual input X and the input baseline Oa:
(s) sXor-1 + (1 - ss) s) = [sXo, sX1, sx1-1].
Applying the chain rule in differential calculus, we have
de (r())
(-1k=D (r(s)) dr)
1-1 df (r))s)
ds ΣΣ
dr))((s)
ds
Ljik
(7)
(8)
j=
1=0
where (s) denote the k-th dimension 8 s-araa of the j-th vector sx, in point 7(s). With the above elements, we can define the below measures.
Definition 3.2 (Dependency Measures). For an autoregressive decoder fdee that conditions on both latent variable X = z and the prefix X1:t-1 to compute representation h, the dependency measure of every input variable xj. [0, t 1] to the decoder is defined as
Σ JjK G'' de (y(()) k
h. hes
ds
h,
h2ll
dry.n(s)
where k E [1, D] and operation < represents the inner product. We name my,0 as the global dependency and mr,r-j, 1 < j < as the j-th order local dependency.
We provide the derivation for dependency measure maj and detail its relations to integrated gradi- ents in Appendix C Plus, the integral term can be approximated as
L. dedec (T(s))
dedec (7(s))
ds
(10)
S
(r)
SES
where S is the set of independent samples drawn from uniform distribution (1 1}. According to the law of large numbers (Sedor 2013), this approximation is unbiased and gets more accurate for bigger sample set (S). Lastly, the defined measures have the following properties.
Proposition 3.3 (Signed and Normalization Properties). The dependency measure maj: Vj E [0, t- is signed measure and always satisfies D/=D mi.j 1.
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WARDS Dataset
1.00
100
0.75
0.50
0.25
0.25
0.00
0.00
MIMIC Dataset
1.0
Averaged 1st Order Local Dependency
1.0
Averaged Global Dependency
0.0
0.0
12
2
3
(Ordered) Time Steps
(Shuffled) Time Steps
Figure Dependency measures m4,0: M1,111 averaged over 500 multivariate time series, with standard deviations as the error bars. We can see that the latent variable z of latent diffusion has vanishing impact on the decoder fdee, typical symptom of posterior collapse. We also observe phenomenon of dependency illusion in the case of shuffled time series.
We that the measure Maj be either positive negative, with normalized sum the subscript j as 1. If ma.j 2 O, then we say that vector xj has positive impact on the decoder pdec; the bigger is m..j. the larger is such an impact; Similarly, if my. < 0, then the vector has negative impact on the decoder: the smaller is m.j. the greater the negative influence. It is also not hard to understand that there exists negative impact. For example, the latent variable z glatent (z) might be an outlier for the decoder fdec, which locates at low-density region in the prior distribution priir (z). We provide the proof for this conclusion in Appendix D
3.3 EMPIRICAT. DEPENDENCY ESTIMATIONS
We are mainly interested in two types of defined measures. One the global dependency m. which estimates the impact of latent variable X = z on the decoder pdec, The other is the first-order local dependency m1,-1, which estimates the dependency of decoder fdee the last observation X1,-1- In this part, we empirically estimate these measures, with the aims to confirm that posterior collapse occurs and show its impacts.
Experiment setup. We adopt two real time-series datasets: WARDS (Alaa et al. 201 and MIMIC (Johnson al. (0166). To study the case where time series have no structural dependencies, we also try randomly shufiling the time steps of ordered time series. We train latent diffusion models on those datasets and sample time series from the models.
Insightful results. Fig. illustrates the estimated dependency measures M,,0, M1,111 of time- series latent diffusion averaged over 500 samples. We can see that, for both ordered and shuffled time series, the global dependency m,,0 exponentially converges 0 with increasing time step t, indicating posterior collapse: latent variable loses control of the decoder pdec.
More interesting results are shown the right two subfigures. When adjacent observations x,- X; not much correlated in shuffled time series, still observe that the first-order measure 7,,-1 notably different from (e.g., around 0.1 to 0.2). This phenomenon might arise as neural networks overfit and we term dependency illusion: different observations X,, Xt, 8 < t in time series X are totally or almost independent, but the decoder pdee still highly relies on X2 to predict X₁ (i.e., high dependency measure 14,a))
METHOD: POSTERIOR- SINTERIOR-SSABLE STABLE LATENT DIFFUSION
In this section, first analyze the potential causes of posterior collapse in time-series latent dif- fusion. Based on the discussion, we then introduce a new framework, which extends from latent diffusion but is free from the problem.
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4.1 CAUSES OF POSTERIOR COLLAPSE IN LATENT DIFFUSION
Previous works (Semeniuta et al. 2017 Alemi et al. 2018) have identified two main causes of posterior collapse: KL-divergence temn and strong decoder. For time-series latent diffusion, we will explain as below that those causes indeed exist, but are in fact avoidable.
Unnecessary KL regularization. The KL-divergence term DKL (q (z X) || (prio(z)) in Eq. ( moves the posterior 9 (z | X) towards prior pprior (z), which might incur posterior col- lapse by definition (see Sec. 3.I. In fact, this term is tailored for VAE, such that it is proper to sample latent variable z from Gaussian prior pprior (z) at test time. However, the variable z in latent diffusion is sampled from the diffusion model, which can approximate a non- non-Gaussian prior distribution. In this regard, sion, which also results in a limited prior pprior (z).
see that KL regularization not very necessary in latent diffu-
The strong decoder is also cause of posterior col- lapse, which happens to sequence autoencoders (Bowman et al.] 2016 Eikema & Aziz 2019). Time series X T has clear temporal structure, and thus its decoding is autoregressive, resulting in a strong decoder f'ec, In contrast, the decoder in image-based latent diffusion, is mainly configured feedforward neural network (Svozil et al., 1997), such as U-Net (Ronneberger et 20.35. This rype of feedforward decoder is naturally sensitive to the input variables, so the original design of latent diffusion did not consider address the possible insensitivity.
Time-series decoders more vulnerable.
4.2 NEW FRAMEWORK WITH STABLE POSTERIOR
In light of the above discussion, we introduce type of posterior-stable latent diffusion that elimi- nates the use of risky KL regularization, permit free-foom prior distribution perior (z), and increase the sensitivity of decoder pdec to latent variable
Importantly, we notice a conclusion (Ho et al. 2020 for the diffusion process (i.e., Eq. 3b):
forw (x. 22) N(22 Vazz. a²) a')I),
(11)
where the coefficient a² monotonically decreases from to approximately 0 for E [0, L]. In this sense, suppose the initial variable 2 is set as fene (X), then we can infer that the random variable ~ - forw (z' zz") contains 100% information about the vector v, with (1-ā') 100% pure noise. For i - O, the diffusion process is similar to the variational inference (i.c., Eq. D of
VAE, adding slight Gaussian noise to the encoder output V. For i - T, the variable z simulates the problem of posterior collapse since fforw (z' 2²0) R (222; O.
Diffusion process as variational inference. Considering the above facts, we first treat the starting few iterations of the diffusion process as the variational inference. Specifically, with fixed small integer N < L, we sample number i from uniform distribution (7,0, N) and let the diffusion process convert the encoder output v fene (X) into the latent variable:
z' gorw (z² 120)) )2 V.
(12)
In terms of the formerly defined generation distribution pron (X) z) (parameterized by the decoder pdec), negative log-likelihood loss LV is incurred as
Es l1-a" In 8
(X z zz) z')],
(13)
where 7 N+ >N < L is hyper-paramett with the aim to reduce the impact of a very noisy latent variable z. As multiplier y increases, the weight a decreases.
Similar to VAE, the variational inference in our framework also leads the latent variable z to be smooth (Bowman et al. 2016) in its effect on decoder pdee, However, our framework is free from the KL-divergence term DKL (q VT X) || priior (z)) of VAE (i.e., one cause of the posterior collapse), since we can facilitate z ~ glatent(z) (z) at test time through applying the reverse process of the diffusion model (i.e., Eq. (4)) to sample variable z i 00, N].

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Algorithm 1 Training
Algorithm 2 Sampling
1: repeat
z7 ~ p (zL.) = (0,
2: Encoding time-series sample: = (eme(X)
3: z' ~ q forw (z' x v).j ~ WM(0,
2: Set stop time: in [[
4:
ZVI a In p (X - z²)
4: z ~ back
3: for I = I,, I 1 + do
12')
5:
i ~~ Utj E I(0,T)
5: end for
6:
CDM = ||e back (Va'z' + i)l2
Conditional generation: peen (X z2)
7: return Time series X
7:
x forw (x* 22 v), ( L}
ccs - a In pren (X| 2 )²
8:
9: Gradient descent with v(Cvi + 2DM + )
10:
until converged
Diffusion process for collapse simulation. Then, we apply the last few iterations of the diffusion process to simulate posterior collapse, with the aims of increasing the impact of latent variable z on conditional generation y (X z) and reducing dependency illusion.
Following our previous variational inference, we set x penc (X) and apply the diffusion process to cast the initial variable z into a highly noisy variable z², i -> L. Considering that the variable z contains little information about the encoder output f (unc(X) it is unlikely that the decoder fdee can recover time series X from variable z otherwise there posterior collapse or dependency illusion. In this sense, we have the following regularization:
C R. [11 al ) In p [X = =2));
(14)
which penalizes the model for having high conditional density parn (X z) for non-informative
latent variable z = z', E [M, L]. Here M E N+ is close to L, i UtM L}. [-] is the ceiling function, and 1 2 1 is set to reduce the impact of informative variable x².
For strong decoder fdec the regularization L will impose a heavy penalty if the decoder solely relies on previous observations {xx k < j} to predict an observation Xj. In that situation, even if the latent variable z contains very limited information about the raw data X, a high prediction probability will still be assigned to the observation Xj.
Training, inference, and running times. While our new framework extends from latent diffusion, its training and inference procedures are very different. We respectively depict the two procedures in Algorithm| | and Algorithm 2 For training, the key points are to compute three loss functions: EV for likelihood maximization, CDM for training diffusion models, and L for collapse regu- larization. For inference, the main difference is that the stopping time of the backward process is not 0, but random variable. From these pseudo codes, we can that framework is almost as efficient as vanilla latent diffusion. We provide an in-depth analysis and empirical experiments about the running times of our framework in Appendix F.2
RELATED WORK
Besides latent diffusion, our paper is related to previous methods that aim to mitigate the problem of posterior collapse for VAE (Kingma & Welling 2013). In the following, we will first briefly introduce those baselines, explaining their limitations, and then discuss some other types of time- series generative models.
Existing methods for mitigating posterior collapse. KL annealing (Bowman et al. 2016 Fu et al. 2019 Ichikawa & Hukushima 2024) is to assign an adaptive weight to control the effect of KL-divergence term, so that VAE is unlikely to fall into the local optimum of posterior collapse at the initial optimization stage. With similar idea, there are other types of regularization (e.g., mutual information constraints (Melis et al. 2022)) in the literature. Such methods can only mitigate the problem to some degree, but cannot fully reduce it.
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WARDS Dataset
1.00
100
0.5
0.25
0.0
0.00
MIMIC Dataset
Averaged 1st Order Local Dependency
1.5
0.75
Averaged Global Dependency
10
m
0.30
0.25
0.0
10 11
12
10
11 12
(Ordered) Time Steps
(Shuffled) Time Steps
Figure 2: The results of averaged dependency measures and error bars for our framework, which should be compared with those (e.g.. Fig. [ of latent diffusion, showing that framework has stable posterior and is without dependency illusion.
Decoder weakening (Lu et al. 2021: Kinoshita al., 2023) is also a popular class of methods to address posterior collapse. A well-known example Variable Masking (Semeniuta et al.) (017)), which randomly masks input observations to the autoregressive decoder, such that the decoder is forced to rely more the latent variable for predicting the next observation. However, this method will make the model inexpressive since the decoder weakened.
To improve the impact of latent variables the recurrent decoder, method called skip connec- tions (Dieng et al. 2019 Fu et al. 2024) directly feeds the latent variable into the decoder at every step, not only the first step. However, the latent variable in that case acts constant input signal at every time step, so the decoder will still tend to ignore this redundant information.
Other time-series generative models. In addition to latent diffusion, prior studies have proposed various alternative models for time-series generation. For example, Neural ODE Rubanova et al. (2019) Li al. (2024) that works well on irregular time series, and TimeGAN (Yoon et 2019) that is based on Generative Adversarial Networks (GAN) (Goodfellow et al. 2020). We have in- cluded these three generative baselines in Table 2 for comparison.
EXPERIMENTS
6.1 STABLE POSTERIOR OF OUR FRAMEWORK
To show that our framework has a stable posterior (z [z] X). we follow the same experiment setup (e.g., datasets) as Sec. 3.3 and average the dependency measures mr M over 500 sampled time series. The results are illustrated in Fig. 2 For ordered time series in the left two subfigures, we can see that, while the global dependency m2,0 still decreases with increasing time step f, converges into value around 0.5, which is also bit higher than the converged first-order local dependency m1,-11 These results indicate that latent variable z our framework maintains its control of decoder pdee during the entire conditional generation process pgen [X zx)
For shuffled time series in the right two subfigures, we can see that the global dependency m,,0 is always around or above 1, and the local dependency m 1 is negative most of the time. These results indicate that the decoder ddec only relies on latent variable z and the context x even has negative impact on conditional generation peen z), suggesting our framework is without dependency illusion. Based on all our findings, we conclude that: compared with latent diffusion (Fig. our framework is free from posterior collapse.
6.2 PERFORMANCES IN TIME SERIES GENERATION
In this part, we aim to verify that our framework improves latent diffusion in terms of time series generation, which promising since it is free from posterior collapse. We also include other
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Model
Backbone
MIMIC
WARDS
Farthquakes
Latent Diffusion
LSTM
5.19
7.52
5.87
Latent Diffusion w/ KI. Annealing
LSTM
4.28
5.74
3.88
Latent Diffusion w/ Variable Masking
L.TMM
4.73
6.01
4.26
Latent Diffusion w/ Skip Connections
LSTM
3.91
4.95
3.74
Our Framework
[STM
2.29
3.16
2.67
Latent Diffusion
Transformer
5.02
7.46
5.91
Latent Diffusion w/ KI. Annealing
Transformer
4.31
5.54
3.51
Latent Diffusion w/ Variable Masking
Transformer
4.42
5.97
4.45
Latent Diffusion w/ Skip Connections
Transformer
3.75
4.67
3.69
Our Framework
Transfommr
2.13
3.01
2.49
Table I: Wasserstein distances of different models real time-series datasets. The lower the dis-
tance metric, the better the generation quality. More results from other datasets, with another eval-
uation metric, are placed in Table 3 Appendix F3
Model
MIMIC
Earthquakes
Latent Diffusson (Rombach et al.
2022)
5.02
5.91
Latent Diffusion Mutual Information Constraints [Melis
2022
3.59
3.85
Latent Diffusion w/ Inverse Lipschitz Constraint Kiiosshita et al.
2023
3.01
3.42
Neural STPP (Chen al.
2021)
5.13
5.82
Neural Latent Dynamic (LI al. 2024
4.31
5.12
Frequency Diffusion (Crabbe et al.
2024
4.56
5.07
Our Framework
2.13
2.49
Table 2: Comparison with more baselines, including recent new methods to mitigate posterior col-
lapse and other types of time-series generative models. Both latent diffusion baselines and our model are with Transformer as the backbone.
previous methods that can mitigate posterior collapse, including KI. annealing (Fu et al., 2019), variable masking (Bowman et al. 2016), and skip connections (Dieng et al. 2019). We adopt the Wasserstein distances (Bischoff et al. 2024) the metric.
The experiment results on three commonly used time-series datasets are shown in Table [ From that, regardless of the used dataset and the backbone of autoencoder,
the results, framework significantly outperforms latent diffusion and the baselines, which strongly confirms our framework achieves 2.53 points lower
intuition. For example, with the backbone of Transformer, than latent diffusion w/ KI. Annealing on the WARDS dataset.
We have also compared erative models and new methods to mitigate posterior collapse. The results are shown Table [] which further confirm the effectiveness of our framework.
model with recent baselines, including non-latent diffusion gen-
6.3 MORE RESULTS THE APPENDIX
Due to space limitations in the main text, additional experimental results provided in the ap- pendix: more time-series datasets and an alternative evaluation metric (Appendix [F3), results on text and image modalities (Appendix F.4), ablation studies (Appendix F. and running-time analy- sis (Appendix F2. Details the experimental setup are also included in Appendix
CONCLUSION
In this paper, we first provide a solid analysis of posterior collapse in latent diffusion, showing that the problem will reduce a weak generative model (i.e., VAE). Then, we introduce a quantitative method: dependency measures, confirming that time-series latent diffusion indeed has an unstable posterior on real data. Lastly, we introduce a new framework that eliminates the use of risky KL regularization, permits unlimited prior distribution, and ensures that the decoder is sensitive to the input latent variable. Extensive experiments on multiple real time-series datasets show that our framework has stable posterior and outperforms previous baselines.

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A THE IMPACT OF POSTERIOR COLLAPSE
Under the assumption of posterior collapse, the below equality:
( X) ppriot (z) = -,(2;0,1)
(15)
holds for any latent variable z E IR and any conditional X E RTDD, Then, note that
Jasent (z) = / (z X)qamx(X)ea /
W(220,, (X)
(16)
= 0,(; / ((an(() (am =Xxxx==((x00.)) (22
which is exactly our claim.
B RECURRENT ENCODERS
We mainly implement the backbone of decoder fdee as LSTM (Hochreiter Schmidhuber, 1997) or Transformer /aaswani al. 2017). In the former case, apply the latent variable to initialize LSTM and condition it on prefix X x to compute the representation h, Formally, the LSTM- based decoder fdec is as
8t L. x,,1, 21
(17)
h, = W3 (anh(W)ss)
where 8, is the state vector of LSTM and W7. W) are learnable matrices. In particular, for corner case = we fix So, Xy as zero vectors.
In the later case, we just treat latent variable as Xp. Therefore, we have
S sol Transformer( x0)
(18)
h, W3 tanh (W)
where the subscript alignment results from self-attention mechanism.
C DERIVATION OF DEPENDENCY MEASURES
Integrated gradient (Sundararajan et al. 2017) is a very effective method of feature attributions. Our dependency measures can be regarded as its extension to the case of sequence data and vector-valued neural networks. In the following, we provide the derivation of dependency measures.
For the computation h, pdec pdle(XXot-11 ), suppose the output of decoder fdec at origin Op:-1 is h then we apply the fundamental theorem of calculus as
h h, / d' (r(s)) ds
(19)
ds
where 7ss) is a straight line connecting the origin Oot-1 and the input X as >ss) = s 2xar-1 (1 0ot-- Based on the chain rule, the above equality can be expanded
ap (r(s)) d'i.a(s) ds +
h h,
=
dys,u(s)
ds
j
0
(20)
k=
dedec (T(())
Σ
ZjSh d7),((s)
ds
where (s) denote the k-th dimension Zj.k of the j-th vector sx, in point 7(s). Intuitively speaking, every term inside the outer sum operation Σ/ represents the additive contribution of variable x; (to the output difference h, hτ) along the integral line (s)).
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Model
Backbone
N for 1 for Les Diffusion Iterations L
MIMIC
WARDS
Latent Diffusion
Transformer
1000
5.02
7.46
LD w/ Skip Connections
Transformer
1000
3.75
4.67
Our Framework
Transformer
1000
2.13
3.01
Our Framework
Transformer
50
50
1000
2.59
3.32
Our Framework
Transformer
50
150
1000
2.71
3.46
Our Framework
Transformer
50
200
1000
2.83
3.75
Our Framework
Transformer
100
1000
2.31
3.16
Our Framework
Transformer
100
100
1000
2.38
3.24
Our Framework Transformer
150
100
1000
2.75
3.41
Table 3: Ablation studies of the hyper-parameters N. M, which are respectively used in the estima-
tions of likelihood loss CVI and collapse penalty C Here LD is short for latent diffusion and the symbol means "Not Applicable".
To simplify the notation, we denote the mentioned term as
k= D
di
(y())) dys,((s)
maj
Σ
Jisk
ds.
(21)
k=1
Since m a vector, we map the new term scalar and re-scale
mr.j h h, > ma.
(22)
h h, h, h which is exactly our definition of the dependency measure.
D PROPERTIES OF OF DEPENDENCY MEASURES
Firstly, terms of Eq. (22). obvious that the dependency measure my signed: the measure can be either positive or negative. Then, based Eq. (00 we have
h h;
m.j-
(23)
1=0
By taking an inner product with the vector h, he at both sides, we get
h h h. h, m.j. h, h, >= Σ
m h.
h >
(24)
j=0
1=0
By rearranging the term, we finally arrive at
j. h h,
Majj
(25)
==U h h, h, h >
j=0
which is exactly our claim.
E EXPERIMENT DETAILS
We have adopted three widely used real-world time-series datasets for both analysis and model eval- uation, including MIMIC (Johnson et al.) (2066), WARDS (Alaa et al. (0177), and Earthquakes (U.S.) Geological Survey 2020). For the first two datasets, we extract the observations of the first 12 hours, with the top features that have the highest variances forming multivariate time series. For MIMIC, we specially simplify it into a version of univariate time series for the illustration purpose, which only used in the experiments shown in Fig. 3. All other experiments are about multivariate time series. For the Earthquakes dataset, it is about the location and time of all earthquakes in Japan from 1990 to 2020 with magnitude of at least 2.5 from U.S. Geological Survey (2020). We follow the same preprocessing procedure for this dataset
(2023)
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Method
Training Time 2hr 10min
Inference Time
Latent Diffusion
5min 12s
Our Framework
2hr 50min
5min 17s
Table 4: Comparison of Training and Inference Times on the MIMIC dataset.
Method
Backhone
Retail
Energy
Latent Diffusion
Transformer
0.037
0.052
Latent Diffusion w/ Skip Connections
Transformer
0.033
0.043
Our Framework
Transformer
0.025
0.031
Latent Diffusion
LSTM
0.041
0.057
Latent Diffusion w/ Skip Connections
LSTM
0.035
0.047
Our Framework
LSTM
0.027
0.033
Table 5: Comparison two time-series datasets, with another metric: MMD.
We use almost the same model configurations for all experiments. The diffusion models are pa- rameterized by standard U-net (Ronneberger et al. (015), with L = 1000 diffusion iterations and hidden dimensions {128, 64, 32). The hidden dimensions of autoencoders and latent variables are fixed as 128. The conditional distribution peen z) is parameterized as Gaussian, with learnable mean vector and diagonal covariance matrix functions. For our framework, N, M are respectively selected 50, 100, with y and 1 1. We also apply dropout with ratio of 0.1 to most layers of neural networks. We adopt Adam algorithm (Kingma & Ba, 2015) with the default hyper- parameter setting optimize
model. For Table [ Table every number is averaged over 10 different random seeds, with a standard deviation less than 0.05. For the computing resources, all models can be trained I GPU (40G memory) within 10 hours. Finally, the performance gains achieved by our models are all verified by the Student's s t-test (Mishra et al. 20199.
F ADDITIONAL EXPERIMENTS
Due to the limited space of main text, we put the results of some minor experiments here in the appendix. Notably, we will adopt more datasets and another evaluation metric.
F.1 ABLATION STUDIES
We have conducted ablation studies to verify that our hyper-parameter selections = 50, 100 are optimal. The experiment results are shown in Table 3 For both N and M, either increasing or decreasing their values results in worse performance on the two datasets.
F.2 STUDY ON RUNNING TIMES
Our framework only incurs minor increase in training time and enjoys the same inference speed as the latent diffusion. For training, while our framework will run the decoder a second time for collapse simulation LCS can be made parallel with the first run of decoder fder for likelihood computation LVI Therefore, the training is still efficient on GPU devices. Our framework also has a different way of variational inference to infer latent variable z from data X. However, it admits closed-fom ffoed solution and is thus efficient the reparameterization trick of latent diffusion. For inference, our framework has no difference from the latent diffusion: sampling the latent variable with the reverse diffusion process and running the decoder pdec in one shot.
To show the running times in practice, we perform an experiment on the MIMIC dataset as shown in Table 4 We can see that that our framework is free from posterior collapse and delivers better generation performances, this slight time investment is well worth it.
framework indeed only has minor increase for training. Given
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Method
Backbone
ATIS
SNIPS
Latent Diffusion
Transformer
37.12
59.36
Latent Diffusion w/ Skip Connections
Transformer
40.56
65.41
Our Framework
Transformer
51.73
78.12
Latent Diffusion
LSTM
35.38
55.72
Latent Diffusion w/ Skip Connections
LSTM
39.27
60.31
Our Framework
LSTM
48.46
71.45
Table 6: Performance comparison on two text datasets, with BLEU as the metric.
Method
Backbone CIFAR-10
Latent Diffusion
U-Net
3,91
Latent Diffusion w/ KL Annealing
U-Net 3.87
Our Framework
U-Net 3.85
Table 7: Performance comparison on an image dataset, with FID as the metric.
F.3 MORE DATASETS AND ANOTHER EVALUATION METRIC
We conduct additional experiments on more public UCI time-series datasets (Bay et al. 2000): Retail and Energy. with another widely used evaluation metric: maximum mean discrepancy (MMD) (Dziugaite et al. 2013). Lower MMD scores indicate better generative models. From the results shown in Table 5 We can see that our framework still significantly outperforms the base- lines terms of the new benchmarks. For example, with LSTM as the backbone, our framework achieves a score that is 29.79% lower than Skip Connections on the Energy dataset.
F.4 MORE DATA MODALITIES
While our paper primarily focused on time series data, our framework is generally applicable to other types of data, including your mentioned text and images.
Experiment on text data. For text data, considering that natural language sentences exhibits sequential structure similar time series, is intuitive that the posterior of text latent diffusion might also collapse. This intuition is supported by many evidences from previous works (Bowman et al.) 2016 Fu al.] 2019). To verify that our framework is capable of improving text latent diffusion, we have conducted an experiment using two publicly available text datasets: ATIS (Hemphill al., 1990) and SNIPS (Coucke al. 2018).
The numbers in this table represent BLEU scores (Papineni et al.) 2002), a widely used metric for evaluating text generation models. Higher scores indicate better performance. As the results shown in Table/o and notably outperformed a strong baseline- Connections Connections-acros across all datasets and backbones. Therefore, our framework also applies to text data.
that our framework has significantly improved the text latent diffusion
Experiment on image data. For image data, we provide a detailed discussion Sec. 4.1 of our pa- per: Image latent diffusion rarely affected by posterior collapse because of its non-autoreerressive decoder. To confirm this claim in practice, we have conducted an experiment comparing latent diffusion with our framework on the widely used CIFAR-10 dataset (Krizhevsky et 2009)
The results are shown in Table 7 The numbers in this table represent FID scores (Naeem et al. 2020), common metric for evaluating image generation models. Lower scores indicate better per- formance. Our results show that both the baseline model (i.e., KI. Annealing) and our framework improve the image latent diffusion to some extent. However, the improvements are minor, suggest- ing that image models are almost free from posterior collapse.
F.5 CASE STUDIES OF DEPENDENCY MEASURES
Fig. Bishows several examples of applying the dependency measures, where each subfigure contains sample of time series (i.e., blue curve) generated by some model and two types of dependency
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Latent Diffusion iosiion
1st Order Dependency
0.00
6015
0,00
Global Dependency
5ampled Time Series
0.00
0.4
02-
0.120
0.000
0.0
64
Our Tramework
0.
7.-
00000
10-
-0,000
69
-0,075
04
-0.
(Orderec) Time Steps
(Shuffled) Time Steps
The global and local dependency measures 7,, 1,,,1111 (as defined in Sec. 3.2) respec- tively quantify the impacts of latent variable z and observation X;-1 on predicting the next one X We can see that the latent variable z of latent diffusion loses control over the condition generation pren (X 2), with dependency illusion (as introduced Sec. S.3. in the case shuffled time series. In contrast, our framework has such symptoms of posterior collapse.
Figure 3:
measures (i.e., red and green bar charts) estimated by Eq. ()). Specifically, every point X; in time series corresponds to green bar that indicates the global dependency ma,0 and red bar the upper left subfigure, we can see
that represents the first-order local dependency M4,1-1- that the positive impact of latent variable on the decoder (e.g., m2,0) decreases over time and vanishes eventually. From the lower right subfigure, we can even see that some bars (i.e., local dependency m4,1-1) negative, indicating that the variable x provides the decoder with rather false information in predicting the next observation X
An example (i.e., the green bar chart) is shown in the upper left subfigure of Fig. 3 More interest- ingly, the upper right subfigure illustrates a phenomenon we call dependency illusion: Even when the time series is randomly shuffled and thus lacks structural dependencies, the decoder of latent diffusion still heavily relies on input observations (instead of the latent variable) for prediction (i.e., the global dependency measures m,.0
- Exer My.s converges to 0 as time step increases). As demonstrated in the lower two subfigures of Fig. 3 our framework exhibits no signs of posterior collapse, such as the vanishing impact of latent variables over time. Note that the dataset that is used to train the baseline and our model is constructed by extracting the observations of the first 12 hours, with the top feature that has the highest variance to form univariate time series.
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Reviews:
[{"review_id": "FQJZQKwFoS", "reviewer": "Reviewer_xYR6", "summary": "The paper studies posterior collapse in latent diffusion models (LDMs) for time-series generation. It argues that, when paired with recurrent decoders, the latent variable’s influence decays (nearly exponentially) across time, leading the decoder to ignore latents. The paper identifies two main causes: unnecessary KL regularization in the latent path and vulnerability of time-series decoders. To handle this, this paper proposes new framework that treating the diffusion process as variational inference with regularization losses. Experiments show clear gains across several benchmarks.", "questions": "Please refer to the Weaknesses section.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Well-articulated background on posterior collapse and its time-series specifics.\n- Empirical improvements appear consistent across datasets.", "weaknesses": "- The paper reports no experimental results on WARDS in Table 2, despite its inclusion in Table 1. Please clarify this.\n- The baselines shown in Figure 2 are insufficiently explained in the text, and the related-work section lacks a discussion of how these baselines address posterior collapse. Given that posterior collapse is already a well-recognized problem in time-series generative models, the paper should more clearly articulate what differentiates its approach.", "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": "K0hgifVqhu", "reviewer": "Reviewer_uFie", "summary": "The authors study the effect of posterior collapse in latent diffusion models for time series data. The authors show that under posterior collapse a latent diffusion model collapses to a simple VAE. The authors introduce dependency measures which quantify the impact of the latent variables on the recurrent decoder for time-series data. The authors then leverage these findings to propose the so-called posterior-stable latent diffusion where the VAE decoder is free from the impact of posterior collapse.", "questions": "See Weaknesses", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. I found the formulation of the dependency measure quite intuitive. However this intuition could be better clarified in the main text. See my comments on the presentation in the weaknesses section. I also liked Figure 1 which clearly justifies the intuition presented by the authors.\n\n2. I like the idea of penalizing the decoder for large noise levels in the diffusion model.", "weaknesses": "I dont have major comments about the method itself but some comments on the presentation itself and some minor issues.\n\n**Presentation issues**\n\n1. The impact of the VAE posterior collapse on latent diffusion is trivial to understand. Its totally fine if the authors dont stress on this aspect too much in their introduction and instead just keep the current discussion in Section 3.1. I dont think this impacts the significance of the contributions.\n\n2. The dependence on autoregressive time t in the decoder $f^{dec}$ is missing and could be made more explicit starting Section 3.2.\n\n3. The notation in Eqs 8 and 9 can be vectorized for better readability.\n\n4. The definition of the dependency measure in Eq. 9 at a first glance looks quite messy and unintuitive. The authors should provide more intuition around this definition in the main text for better clarity for the readers. For instance something like: “the first component in the dot product denotes the overall change in the output of the time-series decoder while the second component denotes the overall impact of the jth context vector as s is modulated from 0 to 1”. Moreover, the equation itself can be cleaned up a bit as highlighted in the point (c) also.\n\n5. The idea of dependency illusion is nice but not surprising as some work in image classification also shows that deep neural nets can learn arbitrary mappings between images and class labels with high accuracy. Maybe the authors can clarify this in a bit more detail in the main text around the results in Fig. 1\n\n6. I think the authors can clarify the key differences between their approach and the classic latent diffusion approach in terms of training and sampling in more details in Section 4.2 or Section 5.\n\n**Unconvincing arguments**: The second point in Section 4.1 regarding the strong decoder in time-series data is not convincing. More specifically, it is not clear to me why the decoder used in time-series data inherently strong? Can the authors elaborate more on this?\n\n**Minor**: Is there a typo in Line 6 of the Training algorithm? I think the superscript over the variable z should be i and not j. Otherwise there is a discrepancy between the time embedding passed in the diffusion model and the actual noise level added.", "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": "Zb2ofQxFCc", "reviewer": "Reviewer_pXBK", "summary": "This paper investigates the problem of posterior collapse in latent diffusion models when applied to time-series data, claiming that the autoregressive nature of time-series decoders makes them uniquely susceptible to this issue. The authors' key contributions include a theoretical argument that a fully collapsed posterior degenerates the powerful latent diffusion model into a much weaker Variational Autoencoder, and a novel diagnostic tool called \"dependency measures\" to empirically quantify this effect. Using this tool, the paper provides empirical evidence that standard latent diffusion does suffer from a vanishing latent variable impact on time-series data. To solve this, the authors propose a \"posterior-stable latent diffusion\" framework which removes the standard KL regularization.", "questions": "See Weaknesses.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 1, "strengths": "- The introduction of \"dependency measures\"  is a principled and interesting contribution. Basing this metric on integrated gradients  provides a solid theoretical grounding for quantifying the type of decoder insensitivity (global vs. local) that leads to posterior collapse in sequence models.\n\n- The proposed solution is technically sound.", "weaknesses": "- Insufficient Motivation from Time-Series Literature: The most significant concern is the lack of motivation for this problem within the time-series domain. The introduction (Section 1, Paragraph 1) frames the success of latent diffusion by citing image synthesis papers (Rombach et al., 2022; Podell et al., 2024) and the problem of posterior collapse by citing general VAE and NLP papers (e.g., Bowman et al., 2016). The paper fails to cite any existing work that first applies latent diffusion to time series and then identifies posterior collapse as a critical failure mode.\n\n- The paper fails to benchmark against a sufficiently broad set of recent, state-of-the-art (SOTA) time-series generation models. Table 2 does include some more recent and relevant baselines, such as \"Neural Latent Dynamic\" (Li et al., 2024) and \"Frequency Diffusion\" (Crabbé et al., 2024), which is commendable. However, this list is far from comprehensive. To make a convincing claim of superiority for a top-tier conference like ICLR, the method should be compared against a wider array of SOTA models such as [1, 2, 3, 4, 5]. \n\n\n- One of the paper's baselines is \"Latent Diffusion (Rombach et al., 2022)\". This model was designed for and is famous for image synthesis. The paper (e.g., in Table 2) directly cites this image paper when applying it to time series data. It is unclear if the authors adapted this model for sequential data or are applying an image-generation architecture directly. This choice is confusing.\n\n\n[1] Galib, Asadullah Hill, Pang-Ning Tan, and Lifeng Luo. \"FIDE: Frequency-Inflated Conditional Diffusion Model for Extreme-Aware Time Series Generation.\" NeurIPS 2024.\n\n[2] Naiman, Ilan, et al. \"Utilizing image transforms and diffusion models for generative modeling of short and long time series.\" NeurIPS 2024\n\n[3] Jeon, Jinsung, et al. \"GT-GAN: General purpose time series synthesis with generative adversarial networks.\" NeurIPS 2022\n\n[4] Naiman, Ilan, et al. \"Generative modeling of regular and irregular time series data via Koopman VAEs.\" ICLR 2024.\n\n[5] Coletta, Andrea, et al. \"On the constrained time-series generation problem.\" NeurIPS 2023.", "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": "xwL4tHcZen", "reviewer": "Reviewer_iFvJ", "summary": "This paper investigates posterior collapse in time-series latent diffusion models. It first argues that, under a collapsed posterior, latent diffusion effectively degenerates into a weaker VAE-like formulation, motivating a careful re-examination of the learning objective. The authors propose dependency-based diagnostics to quantify decoder sensitivity to the latent variable versus past observations and report evidence of both collapse and a ``dependency illusion'' on real datasets. Building on this analysis, the paper introduces a posterior-stable latent diffusion framework that removes KL regularization by reinterpreting early diffusion steps as variational inference and adds a collapse-simulation loss to encourage decoder sensitivity. Experiments on multiple time-series datasets show consistent improvements over latent-diffusion baselines.", "questions": "1. **Status of “diffusion as variational inference.”** Is the connection theoretically exact under stated assumptions (and if so, which), or primarily a heuristic that performs well in practice? Can you please specify the conditions under which objective functions align and where gaps remain ?\n\n2. **Sensitivity of dependency measures.** How robust are the proposed dependence metrics to decoder architecture and training choices? Could particular design decisions (e.g., capacity, regularization, normalization) spuriously inflate measured reliance on the latent variable? Ablations or controls would help.\n\n3. **Beyond time series: image latents.** Do the diagnostics detect collapse in image latent diffusion as well, or is the phenomenon predominantly temporal? A brief analysis, or a clear negative result, would contextualize the time-series specificity.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. **Problem framing and clarity.** The paper clearly articulates an important and under-explored failure mode of latent diffusion for time series, identifies likely causes, and presents them in a well-structured manner.\n\n2. **Diagnostic contribution.** While I cannot fully assess the novelty of the proposed dependency measure, it appears useful and practically informative for characterizing posterior collapse and decoder reliance. The empirical analyses derived from this diagnostic are convincing.\n\n3. **Methodological direction.** The proposed loss regime, interpreting early diffusion steps as variational inference and adding a collapse-simulation term, is intriguing. Beyond the present setting, the idea may also be applicable to addressing the well-known over-smoothing tendencies in VAEs.", "weaknesses": "1. **Related work coverage and baselines.** The coverage of prior work on time-series generative modeling is too narrow. Several state-of-the-art diffusion-based and alternative generative approaches for time series [1, 2, 3, 4] are missing.\n\n2. **Experimental setting.** The experimental section does not compare against these stronger baselines on the standard generation task, making it difficult to gauge relative progress. A comparison is expected to be in a broader context and in a standard manner in unconditional generative modeling.\n\n3. **Positioning w.r.t. KL-related remedies in images.** Prior work in images has also linked KL terms to posterior collapse and proposed mitigations such as vector-quantized autoencoders or very small KL weights [5, 6]. Empirically, latent spaces in these models exhibit non-Gaussian structure despite the nominal prior. The paper should better delineate what is fundamentally new here (beyond the sequence-specific pathologies) and why the proposed approach is preferable to these alternatives in the time-series setting.\n\n[1] On the constrained time-series generation problem\n\n[2] Utilizing Image Transforms and Diffusion Models for Generative Modeling of Short and Long Time Series.\n\n[3] A Non-Isotropic Time Series Diffusion Model with Moving Average Transitions.\n\n[4] Generative Modeling of Regular and Irregular Time Series Data via Koopman VAEs.\n\n[5] Taming transformers for high-resolution image synthesis\n\n[6] High-resolution image synthesis with latent diffusion models", "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)