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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 2 - LENA (non-LinEar sigNal processing for solving data challenges in Astrophysics)

Teaser

Summary

The goal of the LENA project is to investigate new data analysis methods based on non-linear signal processing, with applications to challenges in Astrophysics. In this particular, we will mainly focus on solving inverse problems for multivalued (e.g. multispectral/hyperspectral) signal and imaging processing, which are central in a large number of astrophysical applications. These models and methods will take their roots in recent advances in applied mathematics: sparse signal modelling, proximal algorithms and machine learning. It will allow extending sparse models and methods to the non-linear world. These developments will further provide a bridge between signal processingÂ and machine learning providing new approaches to model and restore signal and image beyond the standard linear methods.

These algorithms will be deployed to the following applications in Astrophysics:

- A new look at the Planck data: the ability to use sparse non-linear physical models in addition to effective numerical algorithms will allow for a precise decomposition of the sky seen by Planck into its elementary constituents: CMB, SZ, galactic emissions, etc.
-With the current LoFAR project and the advent of the next generation large radio-telescopes such as SKA, fundamental signals such as the cosmological signal at the epoch or reionization (EoR signal) will be accessible. However, this requires designing highly effective component separation methods, which share strongly similarities with the ones we are developing for Planck.

-Euclid is the next European space telescope that will be able to investigating the distribution and nature of the so-called Dark Matter. This type of matter is not observed directly but via the weak gravitational lensing effect, which is measured by evaluating the shape of observed galaxies. However, these measures are highly tricky measure: they are tiny and highly sensitive to all sorts of instrumental effects and noise. In this context, the numerical tools developed in the LENA project are expected to significantly improve the estimation of the lensing effect.

Work performed

Since the beginning of the project, the investigations have focused on the following items:

I. Novel multispectral data analysis methods beyond standard models: we have investigated novel multivalued data analysis methods based on sparse matrix factorisation algorithms to tackle unsupervised component separation problems in more general settings. This includes the development of novel numerical methods to tackle such problems when the data are corrupted with outliers or Poisson noise. The former has been applied to X-ray astrophysical images.

II. New numerical methods for large-scale matrix factorisation: we have investigated the design of a new optimisation framework for sparse matrix factorisation so as to obtain algorithms that are theoretically well-grounded and efficient in practical applications. As well, these investigations have also focused on extended such algorithms to tackle large-scale sparse matrix factorisation problems.

III. Component separation with non-linear models: these investigations have been dedicated to the development new component separations methods that can combine sparse signal modelling and non-linear physical models. The resulting numerical methods have been applied to the Planck data to produce a high resolution snapshot of the thermal dust emission of our galaxy.

IV. High precision shear measurement from galaxy surveys : this work has focused on studying the relationship between shear bias and galaxy morphology, which is key to design efficient shear calibration methods for weak lensing surveys such as Euclid.

Final results

The investigations conducted since the beginning of the LENA project have led to the following advances beyond the state-of-the-art:

I. The development of a novel robust sparse matrix factorisation to tackle unsupervised component separation methods in the presence of outliers. The novelty lies in the use sparse modelling of both the components to be retrieved and the outliers, which allows to dramatically improve the discrimination between these two types of signals. We showed that the proposed approach performs very well in regimes where state-of-the-art methods fail, specifically when the number of observed data is low.

II. We proposed a novel optimisation framework for sparse matrix factorisation, with application to unsupervised component separation. The proposed framework combines theoretically well-grounded algorithms and more robust numerical heuristics. The resulting algorithms has been showed to provide enhanced factorisation results, with algorithmically robust minimisation schemes. Furthermore, the underlying parameters of these algorithms can be set automatically, which makes them good candidates to tackle real-work component separation problems. As well, we have designed extensions of these methods, based on block optimisation, which have been showed to solve accurately sparse matrix factorisation when the number of components or factors is large, which was known to be a highly challenging case.

III. We have introduced the first component separation method that combines sparse modelling and non-linear models, which allows to account for physical models in practical applications. The resulting algorithm has been applied to the Planck data to produce a high-resolution image of the galactic dust emission in the microwave wavelength.