Applied mathematics at Inria & ENS Paris

Since October 2014, I am a post-doctoral researcher at the Inria offices in Paris, affiliated with the computer science department of the Ecole Normale Supérieure. There, I work with Alexandre d'Aspremont in the SIERRA team on topics at the intersection of optimization and statistics.

I obtained my Ph.D. working with Pierre-Antoine Absil and Vincent Blondel at the Université catholique de Louvain, in the department of mathematical engineering. My dissertation is entitled optimization and estimation on manifolds.

I look into the theory and applications of **optimization**,
especially, **optimization on manifolds** (for
which I develop a
toolbox called Manopt). A reference in this field is the
book Optimization
Algorithms on Matrix Manifolds. This led me to study curve fitting on manifolds
(master
thesis), **low-rank matrix completion**
and **synchronization of rotations**. I
collaborate with Amit
Singer and his group at PACM
(the program in applied and computational mathematics at
Princeton University) on synchronization problems and **semidefinite
relaxations**. Lately, I work at the **intersection
of Riemannian and convex optimization**.

Manopt, available at manopt.org,
is a user-friendly, open source and **documented **Matlab
toolbox which can be used to leverage the power of modern
Riemannian optimization algorithms with ease. Manopt won the ORBEL
Wolsey Award 2014 for best open source operational
research implementation.

Synchronization is the problem of estimating elements $g_1, \ldots, g_N$ in a group $G$, given measurements of relative quantities: $h_{ij} \approx g_i^{}g_j^{-1}$. These elements are best visualized on a graph (undirected), where each element $g_i$ is a node and there exists an edge between two nodes $g_i$ and $g_j$ if a measurement $h_{ij}$ is available. I focus on $G = \mathrm{SO}(n)$, the group of rotations:

$$\mathrm{SO}(n) = \{ R \in \mathbb{R}^{n\times n} \colon R^TR = I_n \ \mathrm{ and } \ \operatorname{det}(R) = +1 \}.$$

SynchronizeMLE is the distribution of Matlab codes for this project, available under BSD license. It contains code both to perform the estimation and to compute Cramér-Rao bounds.

This is an algorithm to compute KKT points for problems of the form $$\min_X f(X)$$ with $X$ a symmetric matrix of size $n\times n$ such that $$X\succeq 0 \textrm{ and } X_{ii} = I_d \forall i,$$ meaning that the $d\times d$ diagonal blocks of $X$ are identity matrices. The cost function $f$ must be twice continuously differentiable. If $f$ is convex, KKT points are global optimizers.

The main idea is to attain a solution by tracking intermediate solutions of low-rank, increasing the rank as needed. This is in contrast with interior point methods, which work with full-rank matrices to ultimately converge to (often) low-rank solutions.

Here is our Matlab code for what we call the Riemannian staircase method. It is readily usable to solve such problems with $f(X) = \operatorname{Trace}(CX)$ and a pseudo-Huber-loss smoothed version of $f(X) = \sum_{(i,j)\in E} \|C_{ij}Y_j - Y_i\|_F$ (notice the absence of square). These two functions are concave (the linear cost is also convex), which promotes solutions at extreme points. The latter have low rank. See also my slides.

Let $M \in \mathbb{R}^{m\times n}$ be a matrix with low rank $r \ll \min(m, n)$. Low-rank matrix completion is the task of estimating (or recovering) $M$ from measurements $\hat M_{ij} \approx M_{ij}$ of a few entries $(i, j) \in \Omega$.

At NIPS 2011, we proposed RTRMC, a Riemannian trust-region method for low-rank matrix completion:

In this project, which was the topic of my master's
thesis, we are given time-labeled points on a Riemannian
manifold $\mathcal{M}$ (for example, on a sphere, on the group
of rotations, on the set of positive-definite matrices, etc.):
$p_1, \ldots, p_n$, associated to timestamps $t_1 \leq \ldots
\leq t_n$. The goal is to propose a curve (a model) on the
manifold, $\gamma \colon [t_1, t_n] \to \mathcal{M}$, **such
that the curve fits the data** (exactly for
interpolation, reasonably for regression): $\gamma(t_i)
\approx p_i$ **and **such that $\gamma$ **is
smooth** in some suitable sense. Interpolation and
regression are fundamental operations in signals processing.
They serve the goals of** denoising and resampling
acquired data**. These tasks are well understood when
the data belongs to a Euclidean space such as $\mathbb{R}^n$,
but much less so **when the data belon****g****s
to a nonlinear manifold**.

23 Avenue d'Italie

75013 Paris, France

Office: 61 (5

E-mail: nicolasboumal@gmail.com

At UCL, my office mate was Romain
Hollanders.

In Paris, my office mates are Amit Bermanis 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... and various places in Belgium (Louvain-la-Neuve, Leuven, Liège, La Roche, Mons, Knokke, Daverdisse, Spa, Namur...).

In Paris, my office mates are Amit Bermanis 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... and various places in Belgium (Louvain-la-Neuve, Leuven, Liège, La Roche, Mons, Knokke, Daverdisse, Spa, Namur...).

- 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