High-Dimensional Simulation-Based Inference in Latent Spaces

Lars Kühmichel; Stefan T. Radev; Bhanu Prasanna Koppolu; Masoumeh Davoudi; Jerry M. Huang; Paul-Christian Bürkner

cs.LGSep 29, 2026

High-Dimensional Simulation-Based Inference in Latent Spaces

Authors: Lars Kühmichel, Stefan T. Radev, Bhanu Prasanna Koppolu, Masoumeh Davoudi, Jerry M. Huang, Paul-Christian Bürkner

Organizations: Department of Statistics, TU Dortmund University, Germany · Department of Cognitive Science, Rensselaer Polytechnic Institute

Abstract

Neural simulation-based inference (SBI) has been widely successful in inferring a relatively small number of interpretable parameters from potentially high-dimensional observations, such as images or time series. Accordingly, representation learning in SBI has focused almost exclusively on compressing the observations used to condition the posterior. More recently, however, SBI has begun to target increasingly high-dimensional parameter spaces, raising the complementary question of whether the inference target itself should be compressed. Our answer is a practical merger of SBI and latent generative modeling, which learns a low-dimensional representation of the simulator parameters, performs posterior inference directly in this latent space, and maps posterior samples back to the original parameter space. We characterize the conditions under which latent-space inference recovers the desired target posterior and systematically study its empirical trade-offs. Across four case studies and three generative families, we compare latent and standard estimators while controlling for network capacity, regularization, optimization, and training compute. At matched training compute, latent-space inference achieves accuracy and marginal calibration comparable to direct target-space inference while sampling up to more than an order of magnitude faster.

Figures & tables

Figure 1: Side-by-side comparison between standard ( left ) and latent ( right ) simulation-based inference (SBI) approaches. In latent SBI, high-dimensional inference targets are compressed into a low-dimensional representation using an encoder. This representation is later used for target reconstruction. The encoder maps parameters

\boldsymbol{\theta}

to a code

\boldsymbol{\vartheta}=\mathcal{E}(\boldsymbol{\theta})

; given the summary

s(\mathbf{x})

, the inference network transports base noise

\mathbf{z}\sim\mathcal{N}(\mathbf{0},I)

\boldsymbol{\vartheta}

, and the decoder returns

\boldsymbol{\theta}=\mathcal{D}(\boldsymbol{\vartheta})

Panel of Figure 2 — Figure 2: Recovery and calibration on the correlated Gaussian. Parameter recovery ( top ) and calibration ECDFs ( Säilynoja et al., 2022 ) ( bottom ) of standard SBI ( blue , first row of each) and latent SBI ( purple , second row of each) for the correlated Gaussian with flow matching, one column per block variance $\sigma_{1}^{2},\dots,\sigma_{8}^{2}$ . Latent SBI achieves notably better calibration than standard SBI, as no ECDF leaves the simultaneous confidence bands. A comparison against the exact posterior is in Appendix G .

Panel of Figure 3 — Figure 3: Posterior draws with latent and standard flow matching for two conditions of the Bayesian denoising and GRF estimation case studies. For the random fields, the posterior is the whole Gaussian process, so draws should match the amplitude and smoothness of the ground truth, not its pixel values. Numerical results are in Table 1 and random conditions are in Appendix I .

Figure 4: Posterior draws for map to satellite with diffusion at $256\times 256$ under a matched training budget. Latent draws follow the street layout of the map, while target-space draws mostly show generic vegetation. Latent sampling is also 19 $\times$ faster. Numerical results are in Table 1 and random conditions are in Appendix I .

		Accuracy		Calibration					Cost
		NRMSE $\ddagger$ $\downarrow$	CRPS $\ddagger$ $\downarrow$	Cal. err. $\ddagger$ $\downarrow$	Cov 50 $\ast$	Cov 90 $\ast$	C2ST $\dagger$	SSIM $\ddagger$ $\uparrow$	Speedup $\uparrow$
Correlated Gaussian
FM	Standard	4.1 $\pm$ 0.0	23.58 $\pm$ 0.01	39.6 $\pm$ 0.0	0.99 $\pm$ 0.00	1.00 $\pm$ 0.00	0.58 $\pm$ 0.00	–	–
	Latent	2.9 $\pm$ 0.0	2.27 $\pm$ 0.01	6.8 $\pm$ 3.6	0.45 $\pm$ 0.05	0.83 $\pm$ 0.05	0.51 $\pm$ 0.00	–	$1.8\times$
DM	Standard	3.6 $\pm$ 0.0	16.31 $\pm$ 0.05	39.6 $\pm$ 0.0	0.97 $\pm$ 0.00	1.00 $\pm$ 0.00	0.61 $\pm$ 0.01	–	–
	Latent	3.0 $\pm$ 0.0	2.26 $\pm$ 0.00	7.5 $\pm$ 3.5	0.44 $\pm$ 0.05	0.82 $\pm$ 0.05	0.51 $\pm$ 0.00	–	$3.6\times$

Table 1: Posterior quality and sampling cost of standard (target-space) and latent inference at an equal training budget. We report one seed for the map to satellite case study and the mean

\pm

standard error over three seeds otherwise. Definitions of the metrics are in Section 4.3 .

Figure 5: Compression ratio study for the Correlated Gaussian example. Given latent dimension

d

, the latent compression ratios are

64/d

. For each ratio, we evaluate the continuous ranked probability score (CRPS; left ), covariance error ( middle ), and calibration error ( right ) against their standard SBI baselines for three inference networks: flow matching, normalizing flow, and diffusion model.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Case study	Parameter dimension	Condition dimension	Simulation budget
Correlated Gaussian	$2{,}080$	$64$	$10{,}000$
Fashion-MNIST deblurring	$32\times 32=1{,}024$	$32\times 32=1{,}024$	$60{,}000$
Gaussian random fields	$32\times 32=1{,}024$	$2$	$60{,}000$
Map to satellite	$256\times 256\times 3=196{,}608$	$256\times 256\times 3=196{,}608$	$40{,}555$

Appendix

Table 2: Overview of the case studies. Parameter dimension denotes the dimensionality of the inference target

\boldsymbol{\theta}

, condition dimension that of the conditioning variable, and simulation budget the number of training examples.

Study	Space	Subnet widths	Autoencoder (enc. / dec. filters $\to$ code)	Summary
Fashion-MNIST /	target	UNet [68, 140], 2 stages	—	Identity
random fields ( $32^{2}{\times}1$ )	latent	UNet [64], 1 stage	[32,64,128,128] / [128,128,64,32] $\to 4^{2}{\times}32$ (2:1, 64:1 spatial)	ResNet $\to 4^{2}{\times}64$
Correlated Gaussian	target	MLP [186, 186]	—	Identity
( $2{,}080$ , $k{=}8$ )	latent	MLP [139, 139] $\dagger$	MLP [128,128] / [128,128] $\to 16$ (130:1)	Identity
Map $\to$ satellite	target	UNet [32,64,128,256,256], 5 stages	—	Identity
( $256^{2}{\times}3$ )	latent	UNet [140,280], 2 stages	[32,64,128,256] / [256,128,64,32] $\to 16^{2}{\times}8$ (96:1, 256:1 spatial)	ResNet $\to 16^{2}{\times}64$

Appendix

Table 3: Network architecture per case study and space ( Appendix B ). Subnet widths are the stage widths of the U-Net (image studies) or the hidden widths of the MLP or normalizing-flow subnet (correlated Gaussian). Dropout

0.1

and, for the U-Net, group normalization with

4

groups are shared by every variant and every stage. The autoencoder column gives the encoder / decoder ResNet filters and the resulting code shape, with the compression ratio by value count (and, for the image studies, by spatial grid position). Latent generators condition on the summary network’s output, concatenated channel-wise on the code grid. The correlated Gaussian’s conditioning variable is already its sufficient statistic, so both variants use the identity summary. Fashion-MNIST deblurring and the random fields share one configuration. On the random fields, the two conditioning scalars enter as two constant channels, which add a few hundred weights to the first layer ( Table 4 ).

Study	total	AE (frozen)	Summ.	Gen.	Latent total	(h)
	Target-space	Latent				Budget
Fashion-MNIST	2,763,426	1,479,215	811,362	566,164	2,856,741	6
Random fields	2,764,038	1,479,215	811,684	566,164	2,857,063	6
Corr. Gaussian	944,635	840,458	0	105,566	946,024	0.25 $\dagger$
Map $\to$ sat.	15,424,368	3,242,856	959,598	11,157,865	15,360,319	36

Appendix

Table 4: Parameter counts and training budgets. Trainable and frozen parameter counts, computed from the configured architecture ( Table 3 ), and the per-run training budget of Appendix C . Latent total

=

frozen autoencoder

+

trainable summary network

+

trainable generator, matched to the target-space total within

4\%

in every study. The first stage trains for

1{,}600

autoencoder epochs on Fashion-MNIST and the random fields and

300

on the correlated Gaussian, a fixed design choice in both cases. Its share of the run’s budget depends on the measured epoch time (25% on Fashion-MNIST and the random fields), except on the map study, where the epoch count is instead solved so the first stage takes exactly one third of the budget.

	correlated Gaussian	deblurring	random fields
$d_{\boldsymbol{\vartheta}}$	$16$	$512$	$512$
$H(\boldsymbol{\vartheta}\mid\boldsymbol{\theta})$	$-46.4\pm 3.7$	$-315.0\pm 2.8$	$-891.1\pm 1.5$
$H(\boldsymbol{\vartheta})$ (bound)	$-6.6\pm 4.3$	$573.2\pm 0.9$	$646.5\pm 1.0$
$I(\boldsymbol{\theta};\boldsymbol{\vartheta})$ (bound)	$39.8\pm 0.7$	$888.2\pm 3.5$	$1537.6\pm 0.9$
per code dimension	$2.49\pm 0.04$	$1.73\pm 0.01$	$3.00$
held-out MSE, $\sigma^{2}$	–	$0.0006$	$0.0186$

Appendix

Table 5: Information terms of the error decomposition. Information terms of Proposition 1 after training, in nats, mean

\pm

standard deviation over three seeds (omitted where it rounds to zero). Code entropies are taken in the standardized code space the generator models;

I(\boldsymbol{\theta};\boldsymbol{\vartheta})

is invariant to that map.

H(\boldsymbol{\vartheta})

is the held-out cross-entropy of an unconditional normalizing flow fitted to encoded training codes, an upper bound, so

I(\boldsymbol{\theta};\boldsymbol{\vartheta})

and the decoder term are upper bounds. The decoder term is the reconstruction negative log-likelihood under

\mathcal{N}(\mathcal{D}(\boldsymbol{\vartheta}),\sigma^{2}I)

with

\sigma^{2}

the held-out mean squared error, minus

H(\boldsymbol{\theta})

, plus

I(\boldsymbol{\theta};\boldsymbol{\vartheta})

; on the deblurring study

H(\boldsymbol{\theta})

is unknown and the reconstruction likelihood stands alone. The linear reference is the reconstruction-optimal linear code of the same dimension, whose code term is closed form. Generator likelihoods are on encoded test targets (normalizing flows exact, flow matching by the instantaneous change of variables); diffusion models expose none. The latent-space two-sample test separates

(\mathbf{x},\mathcal{E}(\boldsymbol{\theta}))

from

(\mathbf{x},\boldsymbol{\vartheta}\sim q_{\phi}(\boldsymbol{\vartheta}\mid\mathbf{x}))

pairs, one draw per condition, against a label-permuted null.

Figure 6: Comparison against the exact posterior on the correlated Gaussian. With flow matching, on the

k=8

block variances (

1{,}000

test conditions,

100

draws each). Left: posterior standard deviation relative to the exact one. Latent widths scatter evenly around the exact width, while standard widths are too narrow for most conditions and far too wide for some. Right: Wasserstein-1 distance (

W_{1}

) to exact draws, relative to the exact standard deviation; the dashed outline shows the sampling noise of the exact posterior itself.

Targets			NRMSE $\downarrow$	CRPS $\downarrow$	Cal. err. $\downarrow$	Cov 50	Cov 90
$1{,}000$	FM	Standard	0.052 $\pm$ 0.000	0.0186 $\pm$ 0.0001	0.061 $\pm$ 0.020	0.58 $\pm$ 0.05	0.92 $\pm$ 0.01
		Latent	0.061 $\pm$ 0.000	0.0243 $\pm$ 0.0002	0.062 $\pm$ 0.003	0.57 $\pm$ 0.01	0.86 $\pm$ 0.01
	DM	Standard	0.055 $\pm$ 0.000	0.0196 $\pm$ 0.0002	0.056 $\pm$ 0.005	0.54 $\pm$ 0.04	0.89 $\pm$ 0.01
		Latent	0.060 $\pm$ 0.000	0.0240 $\pm$ 0.0001	0.063 $\pm$ 0.001	0.54 $\pm$ 0.01	0.85 $\pm$ 0.01
$10{,}000$	FM	Standard	0.044 $\pm$ 0.000	0.0151 $\pm$ 0.0001	0.090 $\pm$ 0.012	0.43 $\pm$ 0.06	0.84 $\pm$ 0.02
		Latent	0.048 $\pm$ 0.000	0.0178 $\pm$ 0.0001	0.041 $\pm$ 0.002	0.50 $\pm$ 0.01	0.85 $\pm$ 0.01

Appendix

Table 6: Training-set size ablation. Fashion-MNIST deblurring at three training-set sizes, at the epoch counts of the main comparison ( Appendix B ). Mean

\pm

standard error over three seeds.

Figure 7: Random conditions for Fashion-MNIST deblurring. Posterior samples for four Fashion-MNIST test conditions drawn at random, not selected, with flow matching.

Figure 8: Random conditions for the Gaussian random fields. Posterior samples for four random-field test conditions drawn at random, not selected, as in 3(b) . Colors share one symmetric scale per condition.

Figure 9: Smoothness sweep for the Gaussian random fields. Posterior samples for the random fields at $\log\sigma=0$ and four equally spaced values of $\alpha$ . Smoothness increases with $\alpha$ in both variants, as in the ground truth.

Figure 10: Random conditions for map to satellite. Posterior samples for six map-to-satellite test conditions drawn at random, not selected, as in Figure 4 .

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