Inverse Problems Conditioned on Observation Ensembles: Applications and Methods

The paper proposes ensemble inverse generative models, a non-iterative posterior-sampling framework that conditions on an individual observation and permutation-invariant information extracted from its observation set, to infer posteriors for unseen priors without explicitly using the forward model at inference time.

Source: [5], [28], [32]

The comparison methods include single-observation conditional DDPM/FM, GDDPM using observation moments, OmniFold in best-initialization and combined-initialization variants, SBUnfold, and Sourcerer; the compared neural-network structures are kept the same where specified for a fair comparison.

Source: [9], [10], [12], [16], [22], [25]

The empirical evaluation covers synthetic inverse problems, MNIST image inversion, particle-physics data unfolding, and seismic full waveform inversion (FWI).

Source: [21], [26], [27]

The evaluation uses sliced Wasserstein distance (SWD) for distribution similarity in the synthetic Gaussian experiment, pixel-wise MSE and SSIM for MNIST, Wasserstein-1 distance for HEP jet kinematics, and MSE, MAE, and SSIM for FWI. The reported comparison includes 40,000 samples for the Gaussian recovery evaluation, a t sweep from 0 to 1 in increments of 0.01 for MNIST, and 1-D Wasserstein distances on selected HEP components for four unseen processes.

Source: [1], [4], [11], [17], [19], [31]

The proposed EI-DDPM and EI-FM models use a permutation-invariant encoder ϕ_w(Y) to summarize the observation set and condition a conditional generative sampler on that summary together with the individual observation y; their training jointly learns the encoder and sampler from truth–observation pairs.

Source: [2], [3], [13], [23]

Training datasets contain truth–observation pairs from multiple priors, with the same observation conditional distribution and hence the same forward model across datasets; the paper assumes the model is not directly known. The method also assumes that the observation set is sufficiently informative to aid prior or posterior inference, with the required set size depending on how distinguishable the induced observation distributions are.

Source: [14], [20]

The authors identify dependence on observation-set informativeness and on the quality, capacity, and stability of the permutation-invariant encoder as limitations; optimal encoder design is not addressed. They also identify validation on directly measured real-world data and theoretical recovery-discrepancy guarantees as future work.

Source: [8]

In the synthetic Gaussian experiment, training uses priors with γ in [−0.75, −0.25] ∪ [0.25, 0.75] and evaluates recovery over γ in [−1, 1], with observation-set size N = 4000 and ensemble dimension k = 3. In the MNIST experiment, training uses interpolation parameters t in [0.1, 0.4] ∪ [0.6, 0.9] and evaluates recovery at unknown t, with N = 128. In HEP, the training pairs span 18 physics processes and evaluation includes four unseen processes; in FWI, eight families are used for training and two entirely unseen families are evaluated.

Source: [6], [18], [24], [26], [27]