Latent Laplace Diffusion (LLapDiff)

• LLapDiff is a generative framework that models low-dimensional latent trajectories to forecast and impute irregular multivariate time series without step-by-step integration.
• It features an end-to-end architecture combining a latent encoder/decoder, gap-aware history summarizer, and a latent diffusion model with stable Laplace-domain modal parameterization.
• Empirical results show that LLapDiff offers highly stable, efficient long-horizon predictions under irregular and sparse observation regimes, outperforming many baselines.

Searching arXiv for the LLapDiff paper and closely related latent/Laplace diffusion work to ground the article in current papers. Latent Laplace Diffusion (LLapDiff) is a generative framework for irregular multivariate time series that models the target as a low-dimensional latent trajectory and performs horizon-wide generation without step-by-step integration over physical time. It combines a latent target encoder/decoder, a gap-aware history summarizer, and a latent diffusion model whose reverse process is guided by a stable modal parameterization motivated by stochastic port-Hamiltonian dynamics. Its mean evolution is parameterized in the Laplace domain via learnable complex-conjugate poles, so the latent trajectory can be evaluated directly at arbitrary irregular timestamps in closed form (You et al., 19 May 2026). In the broader literature, LLapDiff sits at the intersection of nonlocal Laplacian diffusion, spectral Laplacian restoration, and latent diffusion modeling: the self-similar Laplacian provides a physically derived nonlocal diffusion generator (Michelitsch et al., 2012), Laplacian-eigenbasis restoration operationalizes diffusion in a Laplace-structured spectral space (Mukherjee et al., 2024), and latent diffusion in other domains supplies the compression-first rationale for moving generative modeling away from raw observation space (Estad et al., 27 May 2026, Siraudin et al., 20 Jan 2026).

1. Problem setting and conceptual motivation

LLapDiff addresses long-horizon forecasting for irregular multivariate time series, where discrete methods and continuous-time methods induce different failure modes. Discrete methods usually re-grid or interpolate irregular data onto a regular timeline, which can distort temporal structure via re-gridding. Continuous-time models preserve timestamps, but usually require step-by-step numerical integration with ODE/CDE/SDE solvers, which is expensive and can accumulate drift over long horizons. Standard diffusion time-series models denoise directly in observation space and treat irregularity mostly through masks and embeddings, so they lack explicit stability and dynamical structure (You et al., 19 May 2026).

The observed history is written as

$$H_{t_i}=\{(t_j,X_{t_j},M_{t_j})\}_{j=i-\ell+1}^{i},$$

and the target or query window as

$V_{t_i}=\{(t_q,Y_{t_q},M_{y,t_q})\}_{q=1}^{h}.$

Queries may be future forecast points or historical missing-value locations. LLapDiff models

$p(z \mid E_{t_i}),$

where $z$ is a latent target trajectory over the query window and $E_{t_i}$ is a learned summary of the irregular history (You et al., 19 May 2026).

The model’s central design choice is to generate a low-dimensional latent trajectory rather than raw observations. This places LLapDiff in a broader latent diffusion lineage. In missing-data imputation, latent diffusion was studied through a two-stage framework in which a robust VAE-based imputer first learns compact semantic features from incomplete observations and a diffusion model is then trained in the resulting latent space; on MNIST, that model remained stable up to $50\%$ missingness and achieved consistently better performance than pixel-space diffusion (Estad et al., 27 May 2026). In graph generation, latent diffusion was paired with Laplacian autoencoders to obtain a latent representation whose dimensionality scales linearly with the number of nodes, eliminating the quadratic bottleneck of adjacency-space diffusion (Siraudin et al., 20 Jan 2026). These results do not define LLapDiff, but they support the general premise that latent-space modeling can stabilize generative learning when raw-space diffusion is brittle.

2. End-to-end architecture

LLapDiff has three main components. First, a latent target encoder/decoder maps the ground-truth query window into a low-dimensional latent sequence,

$$z_0 = \mathrm{VAEenc}(V_{t_i}) \in \mathbb{R}^{h\times d_z}.$$

Second, a history summarizer maps the irregular history into a token sequence,

$E_{t_i} = S_\theta(H_{t_i}).$

Third, a latent diffusion model corrupts the latent target and reverses that corruption while predicting stable modal parameters

$\{(p_k,\omega_k,C_k,b_k)\}_{k=1}^K,$

which define a closed-form latent trajectory over all query timestamps (You et al., 19 May 2026).

The target is represented as

$$z = \{z(t_q)\}_{q=1}^h \in \mathbb{R}^{h\times d_z}.$$

During training, the model encodes the target window with a pretrained VAE, diffuses the latent trajectory, and learns to reconstruct the clean latent $V_{t_i}=\{(t_q,Y_{t_q},M_{y,t_q})\}_{q=1}^{h}.$0. At inference, it samples $V_{t_i}=\{(t_q,Y_{t_q},M_{y,t_q})\}_{q=1}^{h}.$1, reverse diffuses to obtain $V_{t_i}=\{(t_q,Y_{t_q},M_{y,t_q})\}_{q=1}^{h}.$2, and decodes $V_{t_i}=\{(t_q,Y_{t_q},M_{y,t_q})\}_{q=1}^{h}.$3 to produce forecasts or imputations (You et al., 19 May 2026).

A defining property is that the model is solver-free over physical time: it does not integrate an ODE step-by-step across the forecast horizon. Once modal parameters are predicted, the whole trajectory is synthesized in one parallel pass. This differs sharply from standard continuous-time models, whose evolution typically takes the form

$V_{t_i}=\{(t_q,Y_{t_q},M_{y,t_q})\}_{q=1}^{h}.$4

A plausible implication is that LLapDiff is designed to decouple horizon length from numerical rollout cost in physical time, replacing solver dependence with latent denoising plus closed-form modal evaluation (You et al., 19 May 2026).

3. Laplace-domain dynamics and stability bias

The model’s stability prior is motivated by stochastic port-Hamiltonian dynamics. LLapDiff introduces an auxiliary state $V_{t_i}=\{(t_q,Y_{t_q},M_{y,t_q})\}_{q=1}^{h}.$5 evolving according to

$V_{t_i}=\{(t_q,Y_{t_q},M_{y,t_q})\}_{q=1}^{h}.$6

where $V_{t_i}=\{(t_q,Y_{t_q},M_{y,t_q})\}_{q=1}^{h}.$7 is skew-symmetric, $V_{t_i}=\{(t_q,Y_{t_q},M_{y,t_q})\}_{q=1}^{h}.$8 is dissipative, $V_{t_i}=\{(t_q,Y_{t_q},M_{y,t_q})\}_{q=1}^{h}.$9 is the Hamiltonian, $p(z \mid E_{t_i}),$0 is context or input, and $p(z \mid E_{t_i}),$1 is stochastic noise. The port-collocated output is

$p(z \mid E_{t_i}),$2

Applying Itô’s lemma yields the expected energy balance

$p(z \mid E_{t_i}),$3

Between context updates, the input term vanishes, so the expected energy is non-increasing in the noise-free unforced case (You et al., 19 May 2026).

LLapDiff uses this as a prior for the clean latent trajectory,

$p(z \mid E_{t_i}),$4

biasing the learned mean dynamics of the diffusion denoiser toward dissipative, stable behavior. The paper then linearizes the dynamics around a local operating point $p(z \mid E_{t_i}),$5, defining

$p(z \mid E_{t_i}),$6

with linearized SDE

$p(z \mid E_{t_i}),$7

where

$p(z \mid E_{t_i}),$8

Taking expectations removes the martingale term:

$p(z \mid E_{t_i}),$9

For a linear readout $z$0, the mean output is

$z$1

with

$z$2

The unilateral Laplace transform gives the transfer function

$z$3

This is the basis for LLapDiff’s Laplace-domain parameterization of mean dynamics (You et al., 19 May 2026).

Rather than explicitly learning $z$4, LLapDiff learns a stable rational modal form:

$z$5

where $z$6 is the decay rate, $z$7 is the oscillation frequency, and $z$8 are low-rank residue vectors. The corresponding complex-conjugate poles are

$z$9

The positivity constraint $E_{t_i}$0 ensures stability. Each mode can be realized by a real $E_{t_i}$1 block

$E_{t_i}$2

Stacking these yields a full stable state-space system with

$E_{t_i}$3

Since each block has eigenvalues $E_{t_i}$4, the system is Hurwitz whenever $E_{t_i}$5 (You et al., 19 May 2026).

4. Irregular timestamps, renewal averaging, and the history summarizer

Once poles and residues are predicted, the latent trajectory can be evaluated at any relative query time $E_{t_i}$6 using

$E_{t_i}$7

optionally plus a small residual correction MLP. This permits parallel closed-form synthesis over all query timestamps without recursive rollout from one query point to the next (You et al., 19 May 2026).

A major theoretical contribution is the renewal-averaging analysis. Let irregular gaps be

$E_{t_i}$8

For a single continuous-time mode

$E_{t_i}$9

the event-time recursion becomes

$50\%$0

Taking expectations yields

$50\%$1

where

$50\%$2

is the effective discrete or event-domain multiplier, and the effective event-domain log-pole is

$50\%$3

This shows that irregular sampling maps continuous-time poles into effective discrete dynamics (You et al., 19 May 2026).

Because $50\%$4,

$50\%$5

hence

$50\%$6

which implies

$50\%$7

The paper therefore concludes that random sampling preserves mean stability and often increases attenuation. It also gives the second-order approximation

$50\%$8

showing that the effective observed dynamics depend not only on intrinsic pole parameters but also on the gap distribution (You et al., 19 May 2026).

This directly motivates the gap-aware history summarizer $50\%$9. LLapDiff uses three token types: port tokens based on observed values, dynamics tokens based on local differences or finite-difference proxies, and temporal tokens based on timestamps, per-step gaps, and masks. These tokens are fused into a history summary token sequence $$z_0 = \mathrm{VAEenc}(V_{t_i}) \in \mathbb{R}^{h\times d_z}.$$0, which conditions the reverse diffusion and modal pole prediction (You et al., 19 May 2026). The token ablations reported in the paper show that the temporal token is the most important, with dynamics tokens next and port tokens less severely but still adversely affecting performance when removed. This supports the renewal-gap theory (You et al., 19 May 2026).

5. Diffusion formulation and inference

LLapDiff uses a standard DDPM forward process on the latent trajectory:

$$z_0 = \mathrm{VAEenc}(V_{t_i}) \in \mathbb{R}^{h\times d_z}.$$1

equivalently,

$$z_0 = \mathrm{VAEenc}(V_{t_i}) \in \mathbb{R}^{h\times d_z}.$$2

The denoiser predicts the clean latent trajectory from the noisy latent, diffusion step, and history summary:

$$z_0 = \mathrm{VAEenc}(V_{t_i}) \in \mathbb{R}^{h\times d_z}.$$3

where $$z_0 = \mathrm{VAEenc}(V_{t_i}) \in \mathbb{R}^{h\times d_z}.$$4 is implemented as a modal predictor plus modal synthesizer. The training objective is the standard $$z_0 = \mathrm{VAEenc}(V_{t_i}) \in \mathbb{R}^{h\times d_z}.$$5-parameterization with MSE,

$$z_0 = \mathrm{VAEenc}(V_{t_i}) \in \mathbb{R}^{h\times d_z}.$$6

During inference, DDIM-style sampling is used with conditional and unconditional predictions combined through classifier-free guidance:

$$z_0 = \mathrm{VAEenc}(V_{t_i}) \in \mathbb{R}^{h\times d_z}.$$7

The modal predictor constrains poles by construction,

$$z_0 = \mathrm{VAEenc}(V_{t_i}) \in \mathbb{R}^{h\times d_z}.$$8

so $$z_0 = \mathrm{VAEenc}(V_{t_i}) \in \mathbb{R}^{h\times d_z}.$$9 (You et al., 19 May 2026).

The training and inference pipelines are explicit. During training, a history-target pair $E_{t_i} = S_\theta(H_{t_i}).$0 is sampled, the target is encoded as $E_{t_i} = S_\theta(H_{t_i}).$1, the history summary $E_{t_i} = S_\theta(H_{t_i}).$2 is built, conditioning may be dropped with probability $E_{t_i} = S_\theta(H_{t_i}).$3, a diffusion step and noise are sampled, the latent is corrupted, and modal parameters are predicted to synthesize $E_{t_i} = S_\theta(H_{t_i}).$4, minimizing $E_{t_i} = S_\theta(H_{t_i}).$5. During inference, the model builds $E_{t_i} = S_\theta(H_{t_i}).$6, samples $E_{t_i} = S_\theta(H_{t_i}).$7, reverse diffuses with DDIM-style steps, predicts conditional and unconditional modal parameters at each step, applies guidance, and decodes $E_{t_i} = S_\theta(H_{t_i}).$8 to obtain forecasts or imputations (You et al., 19 May 2026).

The paper emphasizes that LLapDiff’s generation does not scale with horizon length via sequential physical-time integration. Modal synthesis is parallel over query points, so inference time is nearly flat across forecast horizons, with runtime dominated by the diffusion loop rather than trajectory rollout (You et al., 19 May 2026).

6. Empirical behavior, ablations, and operational scope

LLapDiff is evaluated on seven datasets: BMS Air, UCI Air, PhysioNet, NOAA US, NOAA UK, Crypto, and US Equity. The baselines include DLinear, PatchTST, mr-Diff, TimeGrad, mTAN, T-PATCHGNN, ContiFormer, NeuralCDE, and CSDI for imputation comparisons (You et al., 19 May 2026).

The paper reports that LLapDiff achieves the best average rank overall and is especially strong at longer horizons, more irregular datasets, and sparser observation regimes. A representative example is NOAA UK at horizon $E_{t_i} = S_\theta(H_{t_i}).$9, where LLapDiff reduces MSE from $\{(p_k,\omega_k,C_k,b_k)\}_{k=1}^K,$0 for the strongest baseline in the table to $\{(p_k,\omega_k,C_k,b_k)\}_{k=1}^K,$1 (You et al., 19 May 2026). Qualitatively, forecasts remain coherent through regions with multiple missing values, where many baselines drift or oversmooth. For imputation, the same model reconstructs masked historical values by querying historical timestamps instead of future ones (You et al., 19 May 2026).

The imputation mechanism follows directly from the continuous-time latent trajectory formulation. To impute missing entries inside the observed window, the model includes the missing historical timestamps in the query set, keeps the same conditioning history $\{(p_k,\omega_k,C_k,b_k)\}_{k=1}^K,$2, synthesizes the latent at those historical times, and decodes to recover imputed values. The paper characterizes this as causal or filtering-style imputation rather than bidirectional smoothing (You et al., 19 May 2026).

The ablations isolate the role of each major component. Removing learned poles significantly hurts performance. Removing the latent trajectory formulation degrades results, indicating that compact latent generative modeling is important for long-horizon stability. Removing history conditioning causes the largest degradation, confirming that the summary sequence is essential. The learned poles all satisfy $\{(p_k,\omega_k,C_k,b_k)\}_{k=1}^K,$3, staying safely away from instability, and conditioning shifts the pole distribution, indicating that the summarizer adapts damping and frequency to the observed gap pattern (You et al., 19 May 2026). Under artificially increased missingness, performance degrades gracefully, and the model remains relatively stable unless informative channels are severely lost. Controlled regime shift experiments show that the learned modal dynamics generalize under frequency and decay shifts, though extreme shifts degrade performance somewhat (You et al., 19 May 2026).

The stated limitations are twofold: dependence on pretrained latent representations and a locally linear modal approximation that may not fully capture richer nonlinear behavior (You et al., 19 May 2026).

7. Relation to Laplacian diffusion and neighboring research directions

The name “Latent Laplace Diffusion” invites comparison with several distinct uses of Laplace or Laplacian structure in generative modeling.

The oldest of these in the present set is the construction of a self-similar Laplacian in $\{(p_k,\omega_k,C_k,b_k)\}_{k=1}^K,$4 dimensions from a quasi-continuous linear chain with self-similar harmonic interparticle springs. There, self-similarity at a point is imposed by

$\{(p_k,\omega_k,C_k,b_k)\}_{k=1}^K,$5

and Hamilton’s variational principle yields a non-local integral operator with power-law kernel. In $\{(p_k,\omega_k,C_k,b_k)\}_{k=1}^K,$6 dimensions the operator is

$\{(p_k,\omega_k,C_k,b_k)\}_{k=1}^K,$7

for $\{(p_k,\omega_k,C_k,b_k)\}_{k=1}^K,$8, and it coincides, up to a strictly positive prefactor, with the fractional Laplacian,

$\{(p_k,\omega_k,C_k,b_k)\}_{k=1}^K,$9

The corresponding Fokker–Planck equation generates spatially isotropic Lévy stable distributions and Lévy flights rather than Brownian motion (Michelitsch et al., 2012). This is not LLapDiff’s mechanism, but it supplies a rigorous example of a Laplace-type diffusion generator whose nonlocality and stability properties are physically derived rather than inserted ad hoc. This suggests a conceptual analogy: LLapDiff also replaces naive local dynamics with a structured operator-level parameterization designed to stabilize long-range generation.

A second line is diffusion restoration for PDEs governed by the Laplace operator. In the Poisson setting

$$z = \{z(t_q)\}_{q=1}^h \in \mathbb{R}^{h\times d_z}.$$0

on $$z = \{z(t_q)\}_{q=1}^h \in \mathbb{R}^{h\times d_z}.$$1 with homogeneous Dirichlet boundary conditions, Laplacian eigenpairs

$$z = \{z(t_q)\}_{q=1}^h \in \mathbb{R}^{h\times d_z}.$$2

are used as the analogue of the singular-vector basis in DDRM. The method performs restoration in Laplacian spectral coordinates and significantly improves both the forward task $$z = \{z(t_q)\}_{q=1}^h \in \mathbb{R}^{h\times d_z}.$$3 and the inverse task $$z = \{z(t_q)\}_{q=1}^h \in \mathbb{R}^{h\times d_z}.$$4; for example, in the dry forward process the MAE improves from $$z = \{z(t_q)\}_{q=1}^h \in \mathbb{R}^{h\times d_z}.$$5 without restoration to $$z = \{z(t_q)\}_{q=1}^h \in \mathbb{R}^{h\times d_z}.$$6 with DDRM, versus a finite difference baseline of $$z = \{z(t_q)\}_{q=1}^h \in \mathbb{R}^{h\times d_z}.$$7, while in the dry inverse process the MAE improves from $$z = \{z(t_q)\}_{q=1}^h \in \mathbb{R}^{h\times d_z}.$$8 to $$z = \{z(t_q)\}_{q=1}^h \in \mathbb{R}^{h\times d_z}.$$9 (Mukherjee et al., 2024). This paper does not define LLapDiff, but it strongly motivates a latent or spectral Laplacian diffusion framework by showing that diffusion restoration can be generalized from matrix inverse problems to PDE operators using the Laplacian eigenbasis (Mukherjee et al., 2024).

A third line is latent diffusion as a general robustness strategy. In missing-data imputation, LDMiss uses a $V_{t_i}=\{(t_q,Y_{t_q},M_{y,t_q})\}_{q=1}^{h}.$00-VAE with $V_{t_i}=\{(t_q,Y_{t_q},M_{y,t_q})\}_{q=1}^{h}.$01 to learn semantic latent features from incomplete observations and then trains a variance-preserving SDE diffusion model in the resulting latent space. On MNIST, the VAE compresses $V_{t_i}=\{(t_q,Y_{t_q},M_{y,t_q})\}_{q=1}^{h}.$02, about an $V_{t_i}=\{(t_q,Y_{t_q},M_{y,t_q})\}_{q=1}^{h}.$03 compression, and the latent diffusion model remains robust up to $V_{t_i}=\{(t_q,Y_{t_q},M_{y,t_q})\}_{q=1}^{h}.$04 missingness while achieving the best overall imputation quality at $V_{t_i}=\{(t_q,Y_{t_q},M_{y,t_q})\}_{q=1}^{h}.$05 test missingness across training missing rates (Estad et al., 27 May 2026). In graph generation, LG-Flow uses Laplacian positional encodings and magnetic Laplacian positional encodings to build a permutation-equivariant autoencoder whose latent representation is sufficiently adjacency-identifying for near-lossless reconstruction, with sample reconstruction accuracies such as $V_{t_i}=\{(t_q,Y_{t_q},M_{y,t_q})\}_{q=1}^{h}.$06 on Extended Planar and $V_{t_i}=\{(t_q,Y_{t_q},M_{y,t_q})\}_{q=1}^{h}.$07 on GuacaMol, while achieving up to $V_{t_i}=\{(t_q,Y_{t_q},M_{y,t_q})\}_{q=1}^{h}.$08 speed-up over graph-space diffusion in some settings (Siraudin et al., 20 Jan 2026).

Within this landscape, LLapDiff is specific to irregular multivariate time series. It does not use the self-similar fractional Laplacian of anomalous diffusion (Michelitsch et al., 2012), nor does it formulate denoising directly in a Laplacian eigenbasis as in Poisson restoration (Mukherjee et al., 2024). Its “Laplace” component refers to Laplace-domain modal parameterization of the latent mean dynamics, not to the graph Laplacian or spatial Laplacian operator. A common misconception is therefore to equate LLapDiff with a fractional-Laplacian diffusion process or a generic Laplacian spectral model. The available evidence instead indicates a different construction: a latent diffusion model whose reverse denoiser is structurally constrained by stable Laplace-domain poles and whose timestamp handling is derived from renewal-averaged continuous dynamics (You et al., 19 May 2026).

Taken together, these neighboring works clarify LLapDiff’s place in the literature. It combines the latent compression rationale of latent diffusion (Estad et al., 27 May 2026, Siraudin et al., 20 Jan 2026), the operator-aware structuring impulse seen in Laplacian restoration (Mukherjee et al., 2024), and a stability-oriented continuous-time formalism grounded in Laplace-domain dynamics (You et al., 19 May 2026). Its distinctive contribution is to make that combination work for irregular multivariate time series without physical-time solver rollout.