Manuscript Home

[Figure moi]

Welcome to Jeremy Cohen’s “Habilitation à Diriger des Recherches” (HDR) manuscript. This document is a synthesis of all the work I have done since my PhD thesis, and took almost two years to complete. The main topic under study in this manuscript is Low-Rank Approximations, and alongside my personal contributions, I summarize the current state-of-the-art to the best of my knowledge.

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, 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 [TODO].

The scientific content of the book is summarized in the ./intro/introduction.md. Here is the table of content for quick access.

TODOlist:

• ajout refs dans les .ipynb, il reste aussi des TODOs pas vu au 1er tour car ipynb

• Figures design

  • Tikz si maths, main si principe, code python si simple

  • Geometrie NMF book Gillis, 20min max at figure time

  • Figure inline integration with glue and environments

• Relecture

  • Relecture deja faite du rapport CNRS

  • Cohérence style (we, I, forme passive…)

  • Coherence between parts, intro, perspectives.

  • Acronyms, vocabulary and math notations list (NNKL or NN-KL)

  • Broken links

• Binder build

• Precise Version and package dependence

• Latex build fork ?

• How to use (0.25d)

Introduction

• Summary of HDR contents
• Acknowledgements

About the manuscript

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

Introduction to rLRA

• Low-rank models: survival kit
• Nonnegative Regressions: NNLS and NNKL
• Alternating Optimization

Theory of rLRA

• Summary
• Implicit regularization in regularized low-rank approximations
• Identifiability of complete Dictionary Learning
• Dictionary-based Low-Rank Approximations

Fast algorithms for rLRA

• 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

• Summary
• Automatic Music Transcription
• Single pixel hyperspectral image reconstruction and unmixing

Perspectives and open problems

• Numerical Optimization for KL-based Regularized Inverse Problems
• Other topics

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

Document	Modified	Method	Run Time (s)	Status
introduction/summary	2026-03-16 16:44	force	0.98	✅
part1/AlternatingOptimization	2026-03-16 16:44	force	7.1	✅
part1/lra	2026-03-16 16:44	force	3.5	✅
part1/lra_code	2026-03-16 16:44	force	3.73	✅
part1/nnls	2026-03-16 16:44	force	2.2	✅
part1/nnls_code	2026-03-16 16:45	force	33.98	✅
part2/Applications_of_rLRA/AMT	2026-03-16 16:45	force	38.79	✅
part2/Applications_of_rLRA/Single_pixel_spectral_imaging	2026-03-16 16:47	force	93.55	✅
part2/Applications_of_rLRA/intro	2026-03-16 16:47	force	3.71	✅
part2/Applications_of_rLRA/single_pixel	2026-03-16 16:48	force	92.41	✅
part2/Fast_algorithms_for_rLRA/CMTF	2026-03-16 16:48	force	0.88	✅
part2/Fast_algorithms_for_rLRA/NNParafac2	2026-03-16 16:49	force	15.89	✅
part2/Fast_algorithms_for_rLRA/UnrolledNMF	2026-03-16 16:49	force	7.36	✅
part2/Fast_algorithms_for_rLRA/inertial_BCD	2026-03-16 16:49	force	3.78	✅
part2/Fast_algorithms_for_rLRA/mSOM	2026-03-16 16:49	force	4.08	✅
part2/Fast_algorithms_for_rLRA/msom_plot	2026-03-16 16:49	force	4.47	✅
part2/Fast_algorithms_for_rLRA/proco-als	2026-03-16 16:49	force	6.26	✅
part2/Fast_algorithms_for_rLRA/sparse_nnls	2026-03-16 16:50	force	66.13	✅
part2/Theory_of_rLRA/DLRA	2026-03-16 16:50	force	4.02	✅
part2/Theory_of_rLRA/DL_identifiability	2026-03-16 16:50	force	2.37	✅
part2/Theory_of_rLRA/HRSI_algorithm	2026-03-16 16:50	force	7.72	✅
part2/Theory_of_rLRA/HRSI_theory	2026-03-16 16:51	force	9.16	✅
part2/Theory_of_rLRA/MixedSparseCoding	2026-03-16 16:51	force	9.87	✅
part2/Theory_of_rLRA/multiple_dictionaries	2026-03-16 16:51	force	4.89	✅
part2/Theory_of_rLRA/onesparseDLRA	2026-03-16 16:51	force	2.59	✅
part2/intro2	2026-03-16 16:51	force	0.76	✅
part3/KarpCoi	2026-03-16 16:51	force	1.73	✅
part3/bregman	2026-03-16 16:51	force	2.69	✅