Titre : « Denoising image geometry, from sparse estimators to neural network denoisers »

Résumé : 

Denoising (white noise) has historically been a central task in signal processing, and is today at the heart of generative diffusion models. This talk focuses on denoising the geometry of images, understood in the sense of anisotropic regularity along lower-dimensional substructures, e.g. a contour. For such regularity, sparse estimators in frames have been shown to be asymptotically optimal, but deep neural networks have recently achieved substantially stronger numerical performance.
In a first part, we introduce a denoiser for cartoon images based on a wavelet scattering transform, a simplified convolutional neural network model, and study its properties. We show how joint minimization and maximization of scattering L1 norms adaptively capture different types of geometric regularity. In a second part, we compare two of the most prominent architectures used for neural network denoisers, UNet and Transformer. Despite strong empirical performance, it remains unclear how they achieve it, at what parameter cost, and what role attention plays. For both architectures, we derive parameter scaling to optimally denoise a Gaussian Mixture Model of geometry and show that attention offers an asymptotic advantage to the Transformer over the UNet. We further show how full quadratic attention can be sparsified, its complexity reduced to quasi-linear, thanks to the multiscale structure of geometric regularity. Numerical experiments on real-world datasets support that this result carries over beyond the idealized setting in which it was derived.

Titre : « Filtering through a topological lens: homology for point processes on the time-frequency plane »

Résumé : 

TBA