
Dimension
reduction (extraits
du rapport d'évaluation du LJK)
In most highdimensional
problems, samples are sparse and classical
estimators are thus unreliable. This is the
socalled curse of dimensionality. Under the
assumption that highdimensional phenomena lie near
subspace of lower dimension, these problems can be
tackled using dimension reduction strategies.
SAM and Mistis teams both worked on the estimation
of effective dimension reduction subspaces. Main
applications are hyperspectral imaging and pattern
recognition. Compressed sensing techniques
have also been developed through various extensions
of the Dantzig selector and of the Lasso method.
–
High dimensionality, Non parametric
statistics. High dimensional supervised
and unsupervised clustering.
Stéphane
Girard has proposed Gaussian
models for high dimensional data based on a
parsimonious parametrisation of the covariance
matrix. The model is applied to supervised, unsupervised and
semisupervised classification contexts.
[halinria00071243v1, hal00022183v1, hal00325263v1]. The associated R software (HDDA/HDDC package)
is described in [hal00541203v3].
The approach has been adapted to the
classification of spectroscopy data [hal00459947v2
]
where the observations were curves (joint work
with J. Jacques from INRIA team MODAL in Lille). A
dimension selection technique has also been
proposed [hal00440372v3] in collaboration with
G. Celeux from INRIA team SELECT in Orsay. Recently, the approach
was extended to non Gaussian
distributions through the use of kernel techniques [hal00687304v1].
– High dimension
regression.
While regression has
been extensively studied, situations where the
input variable is of high dimension, and where the
response variable may not be fully observed, still
challenge the current state of the art. Mistis
team addressed this point following two
lines.
 On one hand, Stéphane
Girard extended the standard Slice
Inverse Regression method to make it
tractable in realdata
problems, namely with
very high dimensional inputs or
[inria00180458v3]
multivariate outputs [hal00714981v3] and for
datastreams [hal00688609v3]. Both
contributions were joint work with J. Saracco
from INRIA team CQFD in Bordeaux. An
application to the estimation of dominant
parameters for leakage
variability in
microelectronics [hal00846806v1] was
carried out in a collaboration
with STmicroelectronics in the context of the
PhD thesis of S. Joshi. In the
context of the PhD
of A. Chiancone
(started Oct. 2013)
application to hyperspectral images
analysis is investigated.
 On the other hand,
starting from standard
mixture of linear regressions, Florence Forbes
proposed a novel mixture of
locallylinear regression model that unifies
regression and dimensionality reduction into a
common framework [hal00863468v3]. The approach
compares favourably to a number of existing
regression techniques and was apply to the
retrieval of Mars surface physical properties
from hyperspectral images [hal00863468v3] and sound source
separation [hal00960796v1]. The model
has been implemented in a Matlab toolbox
(GLLiM) available at https://team.inria.fr/perception/gllim_toolbox/.
–
Dimensionality
reduction for functional
models.
A.
Antoniadis, P. Fryzlewicz
(LondonSchool of Economics) and F.
Letué adapted the CandèsTao “Dantzig
Selector” algorithm to the framework of the
nonparametric Cox model and applied it to
microarray gene expression problems [hal00568233v1, hal00853895v1, hal00379716v1].
M.
Giacofci worked on dimensionality reduction
in waveletbased mixed effects models for genomics
data [tel00987441v1].
S. LambertLacroix (TIMCGrenoble) and L.
Zwald combined Huber’s criterion with Lasso
to create a Least Absolute Deviation regression
technique that is resistant to heavytailed errors
and outliers [hal00661864v1].
With Z.
Harchaoui (LEAR team, LJK),
F. Enikeeva studied the problem of detecting
abrupt multidimensional changes of mean in
sequences of Gaussian vectors. Their decision
procedure provides optimal performance for both
highly and moderately sparse changes, and it is
rateoptimal in the minimax sense [hal00933185v1].
In the context of dimensionality reduction for
nonparametric estimation, A.
Iouditski with V.Spokoiny and coworkers [hal00978264v1,hal00981927v1], aimed to estimate the
“Effective Dimension Reduction” (EDR) subspace –
the subspace of the observation universe that is
“charged” by the function/features of interest.
The current state of this research, the most
recent algorithms for dimensionality reduction by
direct estimation of the EDRsubspace using
semidefinite programming, and an application of
these techniques to genomic data are described in
[hal00381120v1].
– Certified
modelling for compressed
sensing.
With A.
Nemirovski and F. Kilinc
̧Karzan, A.
Iouditski studied the computational
tractability of sensor synthesis for compressed
sensing [hal00978268v1, hal00978266v1, hal00976864v1, hal00372141v2, hal00450983v2,hal00981868v1,hal00981904v1, hal00981896v1, hal00981921v1, hal00981926v1, hal00981929v1, hal00981928v1,hal00981940v1]. One statistical
outcome of this research is new optimal methods
for sparse recovery based on L1
minimization. These are
closely related to
classical Lasso and the Dantzig
Selector, but they outperform these “conventional”
techniques by exploiting a computational analysis
of the sensing matrix.
– Adaptive nonparametric
estimation.
C.
Dion developed an adaptive nonparametric
deconvolution technique based on the recent
GoldenshlugerLepski selection method [hal01023300v3].
With A. Nemirovski (Georgia Institute of
Technology), A.
Iouditski worked on minimax and adaptive
affine estimators, showing that under rather
general assumptions and up to an absolute constant
factor, the minimaxoptimal affine estimator is
minimaxoptimal among all estimators [hal00976658v1, hal00981924v1]. The resulting
estimators have been successfully applied to
dynamical system identification [hal00853893v1], testing and detection
applications [hal00978374v1, hal00978362v1, hal00981883v1], and distribution
recovery from noisy observations [hal00976668v1]. A.
Iouditski and A. Nemirovski also studied
the problem of denoising signals of unknown
structure [hal00318084v1, hal00365531v1]. This relies on the
adaptive reconstruction of an optimal linear
filter from the observations (which thus depends
on the unknown signal itself).
