Train generative models from noisy measurements of a single ill-posed operator

Ascertain whether generative models such as variational autoencoders, generative adversarial networks, or diffusion models can be trained purely self-supervised using noisy measurements produced by a single ill-posed forward operator, for example via constraints analogous to the Equivariant Imaging formulation.

Background

The manuscript reviews progress on training generative models (VAEs, GANs, diffusion) from noisy or incomplete data, often requiring multiple forward operators or low-noise settings. It highlights that current methods typically assume access to multiple operators or particular measurement regimes.

The authors pose the question of whether similar training is feasible using only noisy measurements from a single ill-posed operator, potentially leveraging the equivariant imaging constraints introduced earlier.

References

However, these methods generally rely on low-noise measurements from multiple operators, and it remains an open question as to whether generative models could be trained with noisy measurements taken from a single ill-posed measurement operator in a similar manner to~\Cref{eq: ei}.

— Self-Supervised Learning from Noisy and Incomplete Data  (2601.03244 - Tachella et al., 6 Jan 2026) in Chapter 5 (Extensions and open problems), Section "Uncertainty quantification and generative modelling" (Generative models)

The model is therefore not intrinsically tied to either the number of acquired voltage points or the spectral grid used to evaluate the reconstructed output. Although the present model is trained and evaluated exclusively on single-peak spectra ($K=1$), the decoder is formulated for an arbitrary number of Gaussian components $K$. Increasing the capacity from $K=1$ to $K=6$, for example, raises the parameter count only from 14,799 to approximately 15,279, while the training objective suppresses superfluous components when the available model capacity exceeds the spectral complexity of the input. Multi-peak reconstruction is therefore an intrinsic capability of the architecture, although its predictive performance for complex spectra remains to be experimentally established.

— Compact Variational Neural Networks for Spectral Inference from a Single Nonlinear 2D Perovskite Photodetector  (2608.27977 - Riisnaes et al., 28 Aug 2026) in Section 2, subsection “Model”