A Data-Driven Prism: Multi-View Source Separation with Diffusion Model Priors

A common challenge in the natural sciences is to disentangle distinct, unknown sources from observations. Examples of this source separation task include deblending galaxies in a crowded field, distinguishing the activity of individual neurons from overlapping signals, and separating seismic events from an ambient background. Traditional analyses often rely on simplified source models that fail to accurately reproduce the data. Recent advances have shown that diffusion models can directly learn complex prior distributions from noisy, incomplete data. In this work, we show that diffusion models can solve the source separation problem without explicit assumptions about the source. Our method relies only on multiple views, or the property that different sets of observations contain different linear transformations of the unknown sources. We show that our method succeeds even when no source is individually observed and the observations are noisy, incomplete, and vary in resolution. The learned diffusion models enable us to sample from the source priors, evaluate the probability of candidate sources, and draw from the joint posterior of the source distribution given an observation. We demonstrate the effectiveness of our method on a range of synthetic problems as well as real-world galaxy observations.

about the sources. Similarly, most deep-learning-based methods require access to samples from the source priors to generate training sets [20][21][22][23][24][25]. When the source distributions are not wellunderstood, this poses a degeneracy: isolating and measuring the source signals requires a source prior, but constraining the source prior requires isolated measurements of the sources.

Alternatively, some source separation methods assume a known mixing process and thereby relax the need for a source prior [26][27][28][29]. To break the degeneracies between the sources, these methods rely on distinct collections of observations, or views, with each view offering a different linear mixture of the underlying sources. These works focus on contrastive datasets, where the goal is to separate a signal that is enriched in a target view compared to a background view. While relevant for a number of scientific datasets, these source separation methods are either limited in their expressivity [28,29] or are not designed for incomplete data [26]. Additionally, the contrastive assumption fails in domains where no source is ever individually measured.

Recent work has shown that score-based diffusion models [30] can serve as expressive Bayesian priors. Notably, once a diffusion model prior is trained, it enables effective posterior sampling for Bayesian inverse problems [31][32][33][34][35][36][37][38][39]. In the setting of noisy, incomplete observations, embedding diffusion models within an expectation-maximization framework can be used to learn an empirical prior [40]. In this work, we extend the use of diffusion model priors to MVSS. By leveraging the ability to sample joint diffusion posteriors over independent sources, our method directly learns a prior for each source. The main contributions of our method are:

Generalist method for multi-view source separation: Our method is designed for any MVSS problem that is identifiable and linear. We show experimentally that our method works even when the data is incomplete, noisy, and varies in dimensionality. Additionally, our method does not require contrastive examples and succeeds even if every source is present in every observation.

Source priors and posteriors: Our method results in independent diffusion models for each source. This affords all of the sampling and probability density evaluation benefits of diffusion models.

Our method outperforms existing methods on the contrastive MVSS problem despite having a more generalist framework.