Lecturer:
Nicolas Gillis, University of Mons, https://sites.google.com/site/nicolasgillis/
Schedule and place:
When: This 15-hour course will take place in 3 days, with morning/afternoon sessions of 2.5 hours, June 3 – 5 – 10, 2024.
Where: UCLouvain - Euler building (room A.002) Avenue Georges Lemaître,4 - 1348 Louvain la Neuve.
Travel instructions are available here
Planned Schedule:
9:20 - 9:45: Welcome coffee
9:45 - 12:35: session of 2.5h: lecture - coffee break 20min – exercise sessions (or more lecture)
13:50 - 16:40: session of 2.5h: lecture - coffee break 20min - exercise sessions (or more lecture)
Abstract: Nonnegative matrix factorization (NMF) in its modern form has become a standard tool in the analysis of high-dimensional nonnegative data sets, such as images, documents, and spectra. Like PCA, NMF is a dimensionality reduction technique but it is tailored for nonnegative data, with easily interpretable factors that may have physical of probabilistic interpretations. This course will cover:
- Applications in imaging, topic modeling, audio source separation, analytical chemistry, and recommender systems.
- Models: choice of the objective function and regularizations, link with well-known techniques such as k-means, and use of additional constraints such as orthogonality or symmetry.
- Theory: geometric interpretation, nonnegative rank, complexity, and uniqueness/identifiability.
- Algorithms: heuristic algorithms using standard nonlinear optimization schemes such as (inertial) block coordinate descent methods, and provably correct algorithms under appropriate assumptions.
The course will explain why understanding NMF better is key to using this computational tool effectively and meaningfully in practice.
Description:
Lecture 1 - Intro: Context, introduction, applications, and basic algorithms (1.5 hours)
Exercise 1: Implementation of basic algorithms and use on applications (1 hour)
Lecture 2 – Theory 1: Exact NMF, computational complexity, geometric interpretation, and nonnegative rank. (2.5 hours)
Lecture 3 – Theory 2: Uniqueness - Identifiability of NMF: under which conditions are the factors of NMF unique/identifiable? (1.25 hour)
Exercise 2/3: Exercises on the geometric interpretation of NMF and identifiability. (1.25 hours)
Lecture 4 - Models: NMF models and more applications (1.25 hours)
Exercise 4: implementation of basic algorithms for various models, and use on applications (1.25 hour)
Lecture 5 - Separable NMF: a class of NMF models that can be solved in polynomial time. (1.25 hours)
Exercise 5: implementation of separable NMF algorithms and use on applications (1.25 hours)
Lecture 6 - Advanced algorithms: majorization-minimization algorithms, extrapolated and inertial block coordinate descent methods (1.25 hours)
Exercise 6: implementation of such algorithms, and comparison with standard variants (1.25 hours).
Course material :
- Slides, exercises, and references will be available.
- Main reference: the book
N. Gillis, "Nonnegative Matrix Factorization", SIAM, Philadelphia, 2020,
available from https://bit.ly/NMFbookReprint
Evaluation:
Choose 2 exercise sessions and write your solution using, if possible, a nonnegative data set not covered in the course. Even better: a data set that you use in your research but have not analyzed yet with a meaningful NMF model or analyze it with a more advanced/another NMF model. In particular, try to propose (a) meaningful NMF model(s) for your data set with a justification/motivation, design (an) algorithm(s), and analyze the results.