3,322 research outputs found
Barycentric Subspace Analysis on Manifolds
This paper investigates the generalization of Principal Component Analysis
(PCA) to Riemannian manifolds. We first propose a new and general type of
family of subspaces in manifolds that we call barycentric subspaces. They are
implicitly defined as the locus of points which are weighted means of
reference points. As this definition relies on points and not on tangent
vectors, it can also be extended to geodesic spaces which are not Riemannian.
For instance, in stratified spaces, it naturally allows principal subspaces
that span several strata, which is impossible in previous generalizations of
PCA. We show that barycentric subspaces locally define a submanifold of
dimension k which generalizes geodesic subspaces.Second, we rephrase PCA in
Euclidean spaces as an optimization on flags of linear subspaces (a hierarchy
of properly embedded linear subspaces of increasing dimension). We show that
the Euclidean PCA minimizes the Accumulated Unexplained Variances by all the
subspaces of the flag (AUV). Barycentric subspaces are naturally nested,
allowing the construction of hierarchically nested subspaces. Optimizing the
AUV criterion to optimally approximate data points with flags of affine spans
in Riemannian manifolds lead to a particularly appealing generalization of PCA
on manifolds called Barycentric Subspaces Analysis (BSA).Comment: Annals of Statistics, Institute of Mathematical Statistics, A
Para\^itr
Conditional Density Estimation by Penalized Likelihood Model Selection and Applications
In this technical report, we consider conditional density estimation with a
maximum likelihood approach. Under weak assumptions, we obtain a theoretical
bound for a Kullback-Leibler type loss for a single model maximum likelihood
estimate. We use a penalized model selection technique to select a best model
within a collection. We give a general condition on penalty choice that leads
to oracle type inequality for the resulting estimate. This construction is
applied to two examples of partition-based conditional density models, models
in which the conditional density depends only in a piecewise manner from the
covariate. The first example relies on classical piecewise polynomial densities
while the second uses Gaussian mixtures with varying mixing proportion but same
mixture components. We show how this last case is related to an unsupervised
segmentation application that has been the source of our motivation to this
study.Comment: No. RR-7596 (2011
Self-organized synchronization of mechanically coupled resonators based on optomechanics gain-loss balance
We investigate collective nonlinear dynamics in a blue-detuned optomechanical
cavity that is mechanically coupled to an undriven mechanical resonator. By
controlling the strength of the driving field, we engineer a mechanical gain
that balances the losses of the undriven resonator. This gain-loss balance
corresponds to the threshold where both coupled mechanical resonators enter
simultaneously into self-sustained limit cycle oscillations regime. Rich sets
of collective dynamics such as in-phase and out-of-phase synchronizations
therefore emerge, depending on the mechanical coupling rate, the optically
induced mechanical gain and spring effect, and the frequency mismatch between
the resonators. Moreover, we introduce the quadratic coupling that induces
enhancement of the in-phase synchronization. This work shows how phonon
transport can remotely induce synchronization in coupled mechanical resonator
array and opens up new avenues for metrology, communication, phonon-processing,
and novel memories concepts.Comment: Comments are welcome
Thresholding methods to estimate the copula density
This paper deals with the problem of the multivariate copula density
estimation. Using wavelet methods we provide two shrinkage procedures based on
thresholding rules for which the knowledge of the regularity of the copula
density to be estimated is not necessary. These methods, said to be adaptive,
are proved to perform very well when adopting the minimax and the maxiset
approaches. Moreover we show that these procedures can be discriminated in the
maxiset sense. We produce an estimation algorithm whose qualities are evaluated
thanks some simulation. Last, we propose a real life application for financial
data
Thresholding methods to estimate the copula density
This paper deals with the problem of the multivariate copula density
estimation. Using wavelet methods we provide two shrinkage procedures based on
thresholding rules for which the knowledge of the regularity of the copula
density to be estimated is not necessary. These methods, said to be adaptive,
are proved to perform very well when adopting the minimax and the maxiset
approaches. Moreover we show that these procedures can be discriminated in the
maxiset sense. We produce an estimation algorithm whose qualities are evaluated
thanks some simulation. Last, we propose a real life application for financial
data
Gaussian Mixture Regression model with logistic weights, a penalized maximum likelihood approach
We wish to estimate conditional density using Gaussian Mixture Regression
model with logistic weights and means depending on the covariate. We aim at
selecting the number of components of this model as well as the other
parameters by a penalized maximum likelihood approach. We provide a lower bound
on penalty, proportional up to a logarithmic term to the dimension of each
model, that ensures an oracle inequality for our estimator. Our theoretical
analysis is supported by some numerical experiments
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