MDA 2026 Winter course

This course is given in the Master Mathématiques de l’aléatoire of Université Paris-Saclay.

  • Course Time: Mondays 9.30 - 12.00, LMO Salle 1A14.
  • Dates: January 26th - March 23rd 2026. No lecture on March 2nd (8 sessions in total).
  • Evaluation: Paper presentations on March 30th (see details below).
  • Lecture Notes: Available here (last update February 12th 2026).
  • Requirements: Some knowledge in high-dimensional probability is highly recommended.

Course description and temptative schedule

Title: Statistical and computational phase transitions in high-dimensional statistics

Many problems in modern statistics and machine learning involve detecting or estimating low-dimensional structures hidden in high-dimensional noise. A central theme is to understand when such problems are feasible: first, at the information-theoretic level, and second, via efficient algorithms. These two notions of feasibility sometimes diverge, associated to phenomena called computational phase transitions. In this course we will survey some mathematical techniques that allow one to locate these thresholds with precision. We will leverage concepts and results from high-dimensional probability (random matrix theory, concentration inequalities, moment method …), information theory, computer science, and statistical physics (cavity method, approximate message-passing algorithms, …).

The following is a (very) tentative list of topics, and approximate schedule:

  1. Gaussian additive models and basics of statistical inference (January 26th 2026)
    • Posterior measure, free entropy, and mutual information
    • Scalar denoising
    • 1-sparse denoising / Gaussian mean location: a first phase transition
    • The spiked matrix and spiked tensor models: definition
  2. Spectral algorithms in the spiked matrix model (February 2nd 2026)
    • Asymptotic spectrum of Wigner matrices
    • The Baik-Ben Arous-Péché transition and the emergence of outliers in the spectra
  3. Optimal estimation: approaches from statistical physics (February 9th - February 23rd 2026)
    • The replica-symmetric formula for the free entropy
    • Heuristic derivation: the cavity method
    • Proof of the replica-symmetric formula: interpolation methods
    • Algorithms: approximate message-passing
    • Conclusion: computational phase diagrams in the spiked matrix model
  4. Contiguity and the low-degree method (March 9th 2026)
    • The second moment method for contiguity
    • The low-degree likelihood ratio method
  5. Optimization: local minima in high-dimensional landscapes (March 16th - March 23rd 2026)
    • The Kac-Rice formula
    • Topological transitions in the optimization landscape of the spiked tensor model
    • Beyond Gaussian loss landscapes

Presentation & evaluation

If you would like to validate the class, you will give a presentation on a research paper.

  • When: March 30, 9:30 AM — in our usual classroom.
  • Length: 20 minutes per student + ~5 minutes for questions and discussion. You may also work in pairs (40 minutes total + Q&A).
  • Format: Slides or blackboard — whichever you prefer, in English or in French. If you use slides, please bring your laptop or send them to me in advance.
  • Choosing a paper: Please select one from this Google Sheets file, on a first come-first served basis. If you have another paper in mind that fits the course topics, you are very welcome to propose it, just email it to me for approval first.

What to focus on

There is no need to cover every technical detail. A clear and thoughtful presentation of the main ideas is much more important. In particular, try to address:

  1. The context, motivation, and general setting.
  2. The main result(s) (or a meaningful subset) and, if relevant, a concrete example or application.
  3. For at least one key result, a high-level sketch of the proof: what is the central idea? Why does the argument work? What is the core mathematical/statistical insight?

It is perfectly fine to focus on a few important results rather than trying to say everything.

I will ask a few questions during or after the presentation, mainly to deepen the discussion. And of course, feel free to reach out after class or by email if you would like to discuss your choice of paper or your presentation beforehand.