Nicolas Boumal
Assistant professor in mathematics at EPFL


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I joined the institute of mathematics at EPFL on July 1st, 2020.

I am an applied mathematician working on optimization, statistical estimation and numerical analysis. Much of what I do is related to nonconvex optimization and optimization on manifolds. For the latter, I develop a toolbox called Manopt and I wrote a book.

Research topics:

  • Nonconvex optimization (computational complexity, global optimality)
  • Optimization on Riemannian manifolds
  • Semidefinite programs and relaxations in low-rank form
  • Low-rank optimization
  • Statistical estimation, bounds, notably under group actions
  • Synchronization (estimation from pairwise information)
  • Single particle reconstruction in cryo-electron microscopy
  • curve fitting on manifolds


Positions prior to EPFL:


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Conference papers

 

Talks

 

Posters

 
Nicolas Boumal
EPFL SB MATH
MA C2 627 (Bâtiment MA)
Station 8
CH-1015 Lausanne
Switzerland

Office: MA C2 627 (2nd floor)
E-mail: nicolas.boumal@epfl.ch

Miscellaneous facts

At UCLouvain, my office mate was Romain Hollanders.
In Paris, my office mates were Amit Bermanis, Damien Scieur and Vianney Perchet.

My Erdös number is 3, courtesy of my co-author and PhD advisor Vincent Blondel.

Research will get you places! It got me in: Palo Alto, Boston, Princeton, London, Prague, Cannes, Lisbon, Milan, Dagstuhl, Granada, Sierra Nevada, Valencia, Berlin, Les Houches, Costa da Caparica, Paris, Florence, San Diego, Bordeaux, Montréal, Bonn, Pittsburgh, Oxford, Geneva, New York City, Barcelona, Vancouver, Ames, Minneapolis, Washington DC, Stockholm, Lausanne, Oaxaca, Villars-sur-Ollon, Pasadena, Interlaken, Chexbres... and various places in Belgium (Louvain-la-Neuve, Leuven, Liège, La Roche, Mons, Knokke, Daverdisse, Spa, Namur, Bruxelles...).

Teaching at EPFL

Teaching at Princeton University

  • Linear algebra with applications (MAT202), Spring 2016, 2017
  • Numerical methods (MAT321), Fall 2016, 2017, 2018, 2019, lecture notes
  • Junior seminar (optimization on manifolds, MAT982), Spring 2018
  • Junior seminar (math of data science through cryo-electron microscopy, MAT982), Fall 2018
  • Junior seminar (math of data science, MAT982), Fall 2019
  • Optimization on smooth manifolds (graduate course, MAT588), Spring 2019, 2020, book

Teaching at UCLouvain

  • Mathématiques 1 (FSAB1101), TA, autumn 2008, autumn 2009
  • Projet 1 (FSAB1501), TA, autumn 2010
  • Théorie des Matrices (INMA2380), TA, spring 2011, autumn 2013
  • Signaux et Systèmes (LFSAB1106), TA, autumn 2011
  • Analyse numérique : approximation, interpolation, intégration (LINMA2171), TA, autumn 2011 and 2012
  • Mathématiques 2 (LFSAB1102), TA, spring 2012
  • Modélisation et analyse des systèmes dynamiques (LINMA2370), TA, autumn 2012
  • Projet en ingénierie mathématique (LINMA2360), TA, spring 2012 and 2013
  • Projet en mathématiques appliquées (LINMA1375), TA, spring 2013
  • Systèmes dynamiques non linéaires (LINMA2361), TA, autumn 2013
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An introduction to optimization on smooth manifolds

This is a book about optimization on smooth manifolds for readers who are comfortable with linear algebra and multivariable calculus. There are no prerequisites in geometry or optimization. Chapters 3 and 5 in particular can serve as a standalone introduction to differential and Riemannian geometry, focused on embedded submanifolds of linear spaces, with full proofs. A distinguishing feature is that these early chapters do not involve charts.

Click here to go to the book webpage.

You may also be interested in the Manopt toolboxes (Matlab, Python, Julia) and in the book Optimization Algorithms on Matrix Manifolds by Absil, Mahony and Sepulchre (Princeton University Press, 2008), all freely available online.

Here are a one-hour video and a two-hour video introducing the basics of differential geometry and Riemannian geometry for optimization on smooth manifolds.

Table of contents (expand/collapse)
  • Preface
  • 1. Introduction
  • 2. Simple examples
    • 2.1 Logistic regression
    • 2.2 Sensor network localization from directions
    • 2.3 Single extreme eigenvalue or singular value
    • 2.4 Dictionary learning
    • 2.5 Principal component analysis
    • 2.6 Synchronization of rotations
    • 2.7 Low-rank matrix completion
    • 2.8 Gaussian mixture models
    • 2.9 Smooth semidefinite programs
  • 3. Embedded geometry: first order
    • 3.1 Euclidean space
    • 3.2 Embedded submanifolds of Euclidean space
    • 3.3 Smooth maps on embedded submanifolds
    • 3.4 The differential of a smooth map
    • 3.5 Vector fields and the tangent bundle
    • 3.6 Moving on a manifold: retractions
    • 3.7 Riemannian manifolds and submanifolds
    • 3.8 Riemannian gradients
    • 3.9 Local frames*
    • 3.10 Notes and references
  • 4. First-order optimization algorithms
    • 4.1 A first-order Taylor expansion on curves
    • 4.2 First-order optimality conditions
    • 4.3 Riemannian gradient descent
    • 4.4 Regularity conditions and iteration complexity
    • 4.5 Backtracking line-search
    • 4.6 Local convergence*
    • 4.7 Computing gradients
    • 4.8 Numerically checking a gradient
    • 4.9 Notes and references
  • 5. Embedded geometry: second order
    • 5.1 The case for another derivative of vector fields
    • 5.2 Another look at differentials of vector fields in linear spaces
    • 5.3 Differentiating vector fields on manifolds: connections
    • 5.4 Riemannian connections
    • 5.5 Riemannian Hessians
    • 5.6 Connections as pointwise derivatives*
    • 5.7 Differentiating vector fields on curves
    • 5.8 Acceleration and geodesics
    • 5.9 A second-order Taylor expansion on curves
    • 5.10 Second-order retractions
    • 5.11 Notes and references
  • 6. Second-order optimization algorithms
    • 6.1 Second-order optimality conditions
    • 6.2 Riemannian Newton's method
    • 6.3 Computing Newton steps: conjugate gradients
    • 6.4 Riemannian trust regions
    • 6.5 The trust-region subproblem: truncated CG
    • 6.6 Local convergence
    • 6.7 Numerically checking a Hessian
    • 6.8 Notes and references
  • 7. Embedded submanifolds: examples
    • 7.1 Euclidean spaces as manifolds
    • 7.2 The unit sphere in a Euclidean space
    • 7.3 The Stiefel manifold: orthonormal matrices
    • 7.4 The orthogonal group and rotation matrices
    • 7.5 Fixed-rank matrices
    • 7.6 The hyperboloid model
    • 7.7 Manifolds defined by $h(x) = 0$
    • 7.8 Notes and references
  • 8. General manifolds
    • 8.1 A permissive definition
    • 8.2 The atlas topology, and a final definition
    • 8.3 Embedded submanifolds are manifolds
    • 8.4 Tangent vectors and tangent spaces
    • 8.5 Differentials of smooth maps
    • 8.6 Tangent bundles and vector fields
    • 8.7 Retractions
    • 8.8 Coordinate vector fields as local frames
    • 8.9 Riemannian metrics and gradients
    • 8.10 Lie brackets as vector fields
    • 8.11 Riemannian connections and Hessians
    • 8.12 Covariant derivatives, velocity and geodesics
    • 8.13 Taylor expansions and second-order retractions
    • 8.14 Submanifolds embedded in manifolds
    • 8.15 Notes and references
  • 9. Quotient manifolds
    • 9.1 A definition and a few facts
    • 9.2 Quotient manifolds through group actions
    • 9.3 Smooth maps to and from quotient manifolds
    • 9.4 Tangent, vertical and horizontal spaces
    • 9.5 Vector fields
    • 9.6 Retractions
    • 9.7 Riemannian quotient manifolds
    • 9.8 Gradients
    • 9.9 A word about Riemannian gradient descent
    • 9.10 Connections
    • 9.11 Hessians
    • 9.12 A word about Riemannian Newton's method
    • 9.13 Total space embedded in a linear space
    • 9.14 Horizontal curves and covariant derivatives
    • 9.15 Acceleration, geodesics and second-order retractions
    • 9.16 Notes and references
  • 10. Additional tools
    • 10.1 Distance, geodesics and completeness
    • 10.2 Exponential and logarithmic maps
    • 10.3 Parallel transport
    • 10.4 Lipschitz conditions and Taylor expansions
    • 10.5 Transporters
    • 10.6 Finite difference approximation of the Hessian
    • 10.7 Tensor fields and their covariant differentiation
    • 10.8 Notes and references
  • 11. Geodesic convexity
    • 11.1 Convex sets and functions
    • 11.2 Geodesically convex sets and functions
    • 11.3 Differentiable geodesically convex functions
    • 11.4 Positive reals and geometric programming
    • 11.5 Positive definite matrices
    • 11.6 Notes and references
  • Bibliography


Last update: Sep. 12, 2021.