Nonnegative Matrix Factorization
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.
Praktische info:
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Leertraject
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.
Schedule
This 15-hour course will take place in 3 days, with morning/afternoon sessions of 2.5 hours on 3, 5 and 10 June 2024.
- 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)
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.
Lecturer
Nicolas Gillis
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