14,417 research outputs found
Comparison of the main forms of half-quadratic regularization
We consider the reconstruction of images by minimizing regularized cost-functions. To accelerate the computation of the estimate, two forms of half-quadratic regularization, multiplicative and additive, are often used. The goal of this paper is to compare both theoretically and experimentally the efficiency of these two forms. We provide a theoretical and experimental analysis of the speed of convergence that they allow. We show that the multiplicative form gives rise to a better rate of convergence.published_or_final_versio
Regularized Optimal Transport and the Rot Mover's Distance
This paper presents a unified framework for smooth convex regularization of
discrete optimal transport problems. In this context, the regularized optimal
transport turns out to be equivalent to a matrix nearness problem with respect
to Bregman divergences. Our framework thus naturally generalizes a previously
proposed regularization based on the Boltzmann-Shannon entropy related to the
Kullback-Leibler divergence, and solved with the Sinkhorn-Knopp algorithm. We
call the regularized optimal transport distance the rot mover's distance in
reference to the classical earth mover's distance. We develop two generic
schemes that we respectively call the alternate scaling algorithm and the
non-negative alternate scaling algorithm, to compute efficiently the
regularized optimal plans depending on whether the domain of the regularizer
lies within the non-negative orthant or not. These schemes are based on
Dykstra's algorithm with alternate Bregman projections, and further exploit the
Newton-Raphson method when applied to separable divergences. We enhance the
separable case with a sparse extension to deal with high data dimensions. We
also instantiate our proposed framework and discuss the inherent specificities
for well-known regularizers and statistical divergences in the machine learning
and information geometry communities. Finally, we demonstrate the merits of our
methods with experiments using synthetic data to illustrate the effect of
different regularizers and penalties on the solutions, as well as real-world
data for a pattern recognition application to audio scene classification
Bayesian Estimation for Continuous-Time Sparse Stochastic Processes
We consider continuous-time sparse stochastic processes from which we have
only a finite number of noisy/noiseless samples. Our goal is to estimate the
noiseless samples (denoising) and the signal in-between (interpolation
problem).
By relying on tools from the theory of splines, we derive the joint a priori
distribution of the samples and show how this probability density function can
be factorized. The factorization enables us to tractably implement the maximum
a posteriori and minimum mean-square error (MMSE) criteria as two statistical
approaches for estimating the unknowns. We compare the derived statistical
methods with well-known techniques for the recovery of sparse signals, such as
the norm and Log (- relaxation) regularization
methods. The simulation results show that, under certain conditions, the
performance of the regularization techniques can be very close to that of the
MMSE estimator.Comment: To appear in IEEE TS
A Stochastic Majorize-Minimize Subspace Algorithm for Online Penalized Least Squares Estimation
Stochastic approximation techniques play an important role in solving many
problems encountered in machine learning or adaptive signal processing. In
these contexts, the statistics of the data are often unknown a priori or their
direct computation is too intensive, and they have thus to be estimated online
from the observed signals. For batch optimization of an objective function
being the sum of a data fidelity term and a penalization (e.g. a sparsity
promoting function), Majorize-Minimize (MM) methods have recently attracted
much interest since they are fast, highly flexible, and effective in ensuring
convergence. The goal of this paper is to show how these methods can be
successfully extended to the case when the data fidelity term corresponds to a
least squares criterion and the cost function is replaced by a sequence of
stochastic approximations of it. In this context, we propose an online version
of an MM subspace algorithm and we study its convergence by using suitable
probabilistic tools. Simulation results illustrate the good practical
performance of the proposed algorithm associated with a memory gradient
subspace, when applied to both non-adaptive and adaptive filter identification
problems
A Causal, Data-Driven Approach to Modeling the Kepler Data
Astronomical observations are affected by several kinds of noise, each with
its own causal source; there is photon noise, stochastic source variability,
and residuals coming from imperfect calibration of the detector or telescope.
The precision of NASA Kepler photometry for exoplanet science---the most
precise photometric measurements of stars ever made---appears to be limited by
unknown or untracked variations in spacecraft pointing and temperature, and
unmodeled stellar variability. Here we present the Causal Pixel Model (CPM) for
Kepler data, a data-driven model intended to capture variability but preserve
transit signals. The CPM works at the pixel level so that it can capture very
fine-grained information about the variation of the spacecraft. The CPM
predicts each target pixel value from a large number of pixels of other stars
sharing the instrument variabilities while not containing any information on
possible transits in the target star. In addition, we use the target star's
future and past (auto-regression). By appropriately separating, for each data
point, the data into training and test sets, we ensure that information about
any transit will be perfectly isolated from the model. The method has four
hyper-parameters (the number of predictor stars, the auto-regressive window
size, and two L2-regularization amplitudes for model components), which we set
by cross-validation. We determine a generic set of hyper-parameters that works
well for most of the stars and apply the method to a corresponding set of
target stars. We find that we can consistently outperform (for the purposes of
exoplanet detection) the Kepler Pre-search Data Conditioning (PDC) method for
exoplanet discovery.Comment: Accepted for publication in the PAS
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