# Séminaire de Mécanique d'Orsay

## Le 12 décembre 2024 à 14h00 - Salle de conférences du LIMSI Bât. 507

### Unifying framework for data-driven linear modeling with rank
constraints

## Jean-Christophe Loiseau

ENSAM Paris

Many practical models in engineering sciences belong to the class of generalized linear
models y = Kx + ε,
with y ∈ Rm, x ∈ Rn, and K ∈ Rm×n a linear operator mapping x to y. In system
identification, these include OKID where y is the response of an unknown linear system, K a
Toeplitz matrix constructed from an input sequence and x is the vector of Markov parameters
of the unknown system. Likewise, linear stochastic estimation or the linear deconvolution
problem can be cast as generalized linear models.
Of interest to us are situations where the operator K itself is unknown. Given training pairs
(xi, yi), it is possible to formulate an ordinary least-squares regression problem to identify
it. Yet, for typical engineering problems, x and y are in general high-dimensional vectors.
Hence, we are unlikely to have sufficient data to obtain a good statistical estimate of K. One
can however regularize the problem by assuming K to be a low-rank linear operator. A good
estimate can then be obtained by solving
minimize the norm ∥Y − KX∥ subject to rank(K) = r,
where X and Y are data matrices and r is the rank of the desired approximation. Although
non-convex, this rank-constrained problem admits a closed-form solution. We will in particu-
lar illustrate how many of the classical modal decompositions used in the community fall into
this framework, e.g. POD, DMD or CCA to name just a few. One key feature of this work is
that it actually provides a tractable optimal formulation of Dynamic Mode Decomposition as
well as clearly establishing the connection between POD and DMD modes, a long standing
question in the community. Under certain conditions, it also relates the problem of finding
the best rank-r DMD model to that of maximizing the mutual information between x and y,
providing a better understanding of the statistical interpretation of DMD analysis.

Accès Salle de conférences du LIMSI Bât. 507