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Welcome to Jeremy Cohen’s “Habilitation à Diriger des Recherches” (HDR) manuscript. This document is a synthesis of the work I have done since my PhD thesis, and took almost two years to complete. The main scientific topic of this manuscript is Low-Rank Approximations, and alongside my personal contributions, I introduce relevant existing results and tools from numerical optimization, having in mind to reuse this material for teaching.

Fig. 1 (Generated with ChatGPT)

The book contains code snippets with barebone implementations of algorithms, numerical simulations and visualisations. This was done so that most results shown in the manuscript are easily reproduced by the reader. The book is available in .pdf upon request, but it is advised to read the content from this website to get the best interface experience. The philosophy of the book design is further discussed in How to read this manuscript.

The scientific content of the book is summarized in the Summary of HDR contents. Here is the table of content for quick access.

Use of AI tools

Except for the picture above, generative AI has been used in this manuscript only for grammar and style checking, and generate javascript codes for interactive plots.

Introduction

• Summary of HDR contents
• Acknowledgements

About the manuscript

• How to read this manuscript
• Terminology and Acronyms Reference
• Notation
• Bibliography

Introduction to rLRA

• Low-rank matrix and tensor models
• Nonnegative regressions: NNLS and NNKL
• Alternating optimization

Theory of rLRA

• Theory of rLRA: summary
• Implicit regularization in rLRA
• Identifiability of complete dictionary learning
• Dictionary-based LRA

Fast algorithms for rLRA

• Fast algorithms: summary
• Proco-ALS for fast nCPD
• Exact sparse \(\ell_0\) nonnegative least squares
• Median second-order majorant for faster NNLS
• Extrapolated BCD algorithms for unconstrained matrix and tensor decompositions
• Constrained coupled matrix and tensor factorization
• Unrolled MU for data-driven NMF

Applications of rLRA

• Applications of rLRA: summary
• Automatic music transcription
• Single-pixel hyperspectral image reconstruction and unmixing

Perspectives and open problems

• Numerical optimization for KL-based regularized inverse problems
• Other perspectives

Many pages contain code executed upon compilation of the book for publication, below is their status.

Document	Modified	Method	Run Time (s)	Status
Howtouse/howtoread	2026-04-27 15:32	force	0.82	✅
introduction/summary	2026-04-27 15:32	force	4.31	✅
part1/AlternatingOptimization	2026-04-27 15:32	force	6.32	✅
part1/lra	2026-04-27 15:32	force	3.29	✅
part1/nnls	2026-04-27 15:32	force	3.74	✅
part2/Applications_of_rLRA/AMT	2026-04-27 15:33	force	40.03	✅
part2/Applications_of_rLRA/Single_pixel_spectral_imaging	2026-04-27 15:34	force	63.4	✅
part2/Applications_of_rLRA/intro	2026-04-27 15:34	force	3.35	✅
part2/Fast_algorithms_for_rLRA/CMTF	2026-04-27 15:34	force	0.73	✅
part2/Fast_algorithms_for_rLRA/NNParafac2	2026-04-27 15:34	force	15.51	✅
part2/Fast_algorithms_for_rLRA/UnrolledNMF	2026-04-27 15:34	force	6.7	✅
part2/Fast_algorithms_for_rLRA/inertial_BCD	2026-04-27 15:34	force	3.82	✅
part2/Fast_algorithms_for_rLRA/mSOM	2026-04-27 15:34	force	3.87	✅
part2/Fast_algorithms_for_rLRA/proco-als	2026-04-27 15:35	force	6.45	✅
part2/Fast_algorithms_for_rLRA/sparse_nnls	2026-04-27 15:36	force	100.85	✅
part2/Theory_of_rLRA/DLRA	2026-04-27 15:36	force	0.81	✅
part2/Theory_of_rLRA/DL_identifiability	2026-04-27 15:36	force	2.38	✅
part2/Theory_of_rLRA/HRSI_theory	2026-04-27 15:37	force	15.11	✅
part2/Theory_of_rLRA/MixedSparseCoding	2026-04-27 15:37	force	11.98	✅
part2/Theory_of_rLRA/multiple_dictionaries	2026-04-27 15:37	force	4.88	✅
part2/Theory_of_rLRA/onesparseDLRA	2026-04-27 15:37	force	2.62	✅
part3/KarpCoi	2026-04-27 15:37	force	1.92	✅