All Years Seminars
[INMA] 2011-10-25 (14.00) : Stability of Linear Switched Systems
At Euler building
Speaker :
Moussa BALDE (UCAD Dakar)
Abstract :
The talk is about asymptotic stability properties of linear switched
systems.
For the planar linear switched system d/dt x(t) = u(t) A_1
x(t)+(1-u(t)) A_2 x(t), where the real matrices A_1, A_2 in R^{2×2} are
Hurwitz and u(·) : [0, infty[ --> {0, 1} is a measurable function, we
give a coordinate-invariant necessary and sufficient condition on
A_1 and A_2 under which the system is asymptotically stable for
arbitrary switching functions u(.). This condition is mainly given
in terms of the polar form of the determinant map Gamma(A_1,A_2) = 1/2
(tr(A_1)tr(A_2) - tr(A_1 A_2)).
Then under the hypothesis that p subsystems in R^d share a non strict
quadratic Lyapunov function, we provide a large class of switching
signals for which a large class of switched systems are asymptotically
stable. For this purpose we define what we call non chaotic inputs,
which generalize the different notions of inputs with dwell time.
Keywords: Switched systems; Asymptotic stability; Quadratic Lyapunov
functions; Chaotic signals; Omega-limit sets.
[INMA] 2011-10-18 (14.00) : Robust 1-Bit Compressive Sensing: How the Sign of Random Projections Distinguishes Sparse Vectors
At Euler building
Speaker :
Jacques, Laurent
Abstract : The Compressive Sensing (CS) framework aims to ease the
burden on analog-to-digital converters (ADCs) by reducing the sampling
rate required to acquire and stably recover sparse signals. Practical
ADCs not only sample but also quantize each measurement to a finite
number of bits; moreover, there is an inverse relationship between the
achievable sampling rate and the bit depth.
In this talk, an alternative CS approach that shifts the emphasis from
the sampling rate to the number of bits per measurement is presented. In
particular, we explore the extreme case of 1-bit CS measurements,
which capture just their sign.
Our results come in two flavors. First, we consider ideal reconstruction from noiseless 1-bit measurements and
provide a lower bound on the best achievable reconstruction error. We
also demonstrate that a large class of measurement mappings achieve
this optimal bound.
Second, we consider reconstruction robustness to
measurement errors and noise and introduce the Binary
{\epsilon}-Stable Embedding (B{\epsilon}SE) property, which
characterizes the robustness measurement process to sign changes.
We show the same class of matrices that provide optimal noiseless
performance also enable such a robust mapping.
On the practical side, we introduce the Binary Iterative Hard Thresholding (BIHT) algorithm
for signal reconstruction from 1-bit measurements that offers state-of-the-art performance.
Joint work with J. Laska (Rice Univ., USA), P. Boufounos (MERL, USA)
and R. Baraniuk (Rice Univ., USA)
[INMA] 2011-10-11 (14.00) : Nearest Stable System
At Euler building
Speaker :
François-Xavier ORBAN DE XIVRY (UCL)
Abstract : Stability is a crucial property in the study of dynamical
systems. We focus on the problem of enforcing the stability of a
system a posteriori. The system can be a matrix or a polynomial either
in continuous-time or in discrete-time. We present an algorithm that
constructs a sequence of successive stable iterates that tend to a
nearest stable approximation of a given unstable system. This kind of
algorithm can be useful in identification where the measured system is
stable but the identified model of the system appears to be unstable
due to some factors.
[INMA] 2011-10-04 (14.00) : Optimal placement of Marine Protected Areas
At Euler building
Speaker :
Patrick DE LEENHEER (University
of Florida)
Abstract : Overfishing can lead to the reduction or elimination of fish populations and the degradation or even destruction of their habitats.
This can be prevented by introducing Marine Protected Areas (MPA's), regions in the ocean or along coastlines where fishing is prohibited. MPA's can also lead to larger fish densities outside the protected area through spill-over,
which in turn may increase the fishing yield. A natural question in this context, is where exactly to establish
an MPA, in order to maximize these benefits. We address this problem along a one-dimensional stretch of coast-line, by first proposing a model for the fish dynamics. Fish are assumed to move diffusively, and are subject to recruitment, natural death and harvesting through fishing. Our problem is then cast as an optimal control problem
for the steady state equation corresponding to the PDE which models the fish dynamics. The cost index being maximized is a weighted sum of the average fish density and the average fishing yield. We show that optimal controls exist, and will see that the form of an optimal control -and hence the location of the MPA- is determined by two key model parameters, namely the length of the coastline, and the weight of the average fish density appearing in the cost index. If these parameters are large enough -and precisely how large, can be calculated exactly- our results indicate when and where an MPA should be established.
The main mathematical tool used to prove our results is Pontryagin's maximum principle.
[INMA] 2011-09-20 (14.00) : Linear regression on fixed-rank matrices: a Riemannian approach
At Euler building
Speaker :
Meyer, Gilles
Abstract : In this work, we address the problem of learning a linear regression
model parameterized by a fixed-rank matrix. We deal with the case of
fixed-rank symmetric positive semidefinite matrices and the case of
fixed-rank non-symmetric matrices. The focus is on the nonlinear
nature of the search space and on scalability to high-dimensional
problems. The mathematical developments rely on the theory of batch
and online gradient descent algorithms adapted to the Riemannian
geometry that underlies the set of fixed-rank matrices. The resulting
algorithms scale to high-dimensional problems, enjoy local convergence
properties, favorably compete with the state-of-the-art and confer a
geometric basis to recent contributions on learning fixed-rank
matrices. The considered linear regression framework encompasses many
modern machine learning applications including the learning of a
low-rank distance, low-rank matrix completion, learning on data pairs,
ranking or multi-task learning.
Seminars
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