101 research outputs found
Adaptive estimation of the density matrix in quantum homodyne tomography with noisy data
In the framework of noisy quantum homodyne tomography with efficiency
parameter , we propose a novel estimator of a quantum state
whose density matrix elements decrease like ,
for fixed , and . On the contrary to previous works,
we focus on the case where , and are unknown. The procedure
estimates the matrix coefficients by a projection method on the pattern
functions, and then by soft-thresholding the estimated coefficients.
We prove that under the -loss our procedure is adaptive
rate-optimal, in the sense that it achieves the same rate of conversgence as
the best possible procedure relying on the knowledge of . Finite
sample behaviour of our adaptive procedure are explored through numerical
experiments
Time series prediction via aggregation : an oracle bound including numerical cost
We address the problem of forecasting a time series meeting the Causal
Bernoulli Shift model, using a parametric set of predictors. The aggregation
technique provides a predictor with well established and quite satisfying
theoretical properties expressed by an oracle inequality for the prediction
risk. The numerical computation of the aggregated predictor usually relies on a
Markov chain Monte Carlo method whose convergence should be evaluated. In
particular, it is crucial to bound the number of simulations needed to achieve
a numerical precision of the same order as the prediction risk. In this
direction we present a fairly general result which can be seen as an oracle
inequality including the numerical cost of the predictor computation. The
numerical cost appears by letting the oracle inequality depend on the number of
simulations required in the Monte Carlo approximation. Some numerical
experiments are then carried out to support our findings
Periodic hypokalemic paralysis disclosing thyrotoxicosis
BACKGROUND:
Hypokaliemic periodic paralysis is an uncommon complication of hyperthyroidism occurring sporadically almost exclusively in young Asian men. The clinical presentation is the same as in familial hypokaliemic periodic paralysis. Treatment consists of conventional management for thyrotoxicosis.
CASE REPORT:
A Laotian man aged 42 years had suffered episodes of pain and fatigue in the lower limbs lasting 2 to 7 days over the last few months. The patient was hospitalized with severe limb pain. Clinical examination found severe motor deficit involving all four limbs. Laboratory findings induced hypokaliemia (1.8 mmol/l) and hyperthyroidism (free thyroxin 36 pmol/l, TSH < 0.005 mlU/l). Thyroid echography revealed multinodular goitre with two heterogeneous nodules. Strong uptake was seen on the scintigram. The motor deficit regressed within 8 hours and the kaliemia was restored with 1.50 g KCl. The patient was discharged with carbimazole (30 mg/d). Three months later he was euthyroid and symptom free. Total thyroidectomy was performed and L-thyroxin prescribed. The patient remains symptom-free at the last follow-up, 5 months after thyroidectomy.
DISCUSSION:
The pathogenesis of hypokaliemic periodic paralysis involves the ATPase-dependent sodium-potassium pump whose activity is stimulated by thyroid hormones. The reasons for the ethnic and male predominance are poorly elucidated
PAC-Bayesian Bounds for Randomized Empirical Risk Minimizers
The aim of this paper is to generalize the PAC-Bayesian theorems proved by
Catoni in the classification setting to more general problems of statistical
inference. We show how to control the deviations of the risk of randomized
estimators. A particular attention is paid to randomized estimators drawn in a
small neighborhood of classical estimators, whose study leads to control the
risk of the latter. These results allow to bound the risk of very general
estimation procedures, as well as to perform model selection
Noisy Monte Carlo: Convergence of Markov chains with approximate transition kernels
Monte Carlo algorithms often aim to draw from a distribution by
simulating a Markov chain with transition kernel such that is
invariant under . However, there are many situations for which it is
impractical or impossible to draw from the transition kernel . For instance,
this is the case with massive datasets, where is it prohibitively expensive to
calculate the likelihood and is also the case for intractable likelihood models
arising from, for example, Gibbs random fields, such as those found in spatial
statistics and network analysis. A natural approach in these cases is to
replace by an approximation . Using theory from the stability of
Markov chains we explore a variety of situations where it is possible to
quantify how 'close' the chain given by the transition kernel is to
the chain given by . We apply these results to several examples from spatial
statistics and network analysis.Comment: This version: results extended to non-uniformly ergodic Markov chain
Rank-based model selection for multiple ions quantum tomography
The statistical analysis of measurement data has become a key component of
many quantum engineering experiments. As standard full state tomography becomes
unfeasible for large dimensional quantum systems, one needs to exploit prior
information and the "sparsity" properties of the experimental state in order to
reduce the dimensionality of the estimation problem. In this paper we propose
model selection as a general principle for finding the simplest, or most
parsimonious explanation of the data, by fitting different models and choosing
the estimator with the best trade-off between likelihood fit and model
complexity. We apply two well established model selection methods -- the Akaike
information criterion (AIC) and the Bayesian information criterion (BIC) -- to
models consising of states of fixed rank and datasets such as are currently
produced in multiple ions experiments. We test the performance of AIC and BIC
on randomly chosen low rank states of 4 ions, and study the dependence of the
selected rank with the number of measurement repetitions for one ion states. We
then apply the methods to real data from a 4 ions experiment aimed at creating
a Smolin state of rank 4. The two methods indicate that the optimal model for
describing the data lies between ranks 6 and 9, and the Pearson test
is applied to validate this conclusion. Additionally we find that the mean
square error of the maximum likelihood estimator for pure states is close to
that of the optimal over all possible measurements.Comment: 24 pages, 6 figures, 3 table
Revisiting clustering as matrix factorisation on the Stiefel manifold
International audienceThis paper studies clustering for possibly high dimensional data (e.g. images, time series, gene expression data, and many other settings), and rephrase it as low rank matrix estimation in the PAC-Bayesian framework. Our approach leverages the well known Burer-Monteiro factorisation strategy from large scale optimisation, in the context of low rank estimation. Moreover, our Burer-Monteiro factors are shown to lie on a Stiefel manifold. We propose a new generalized Bayesian estimator for this problem and prove novel prediction bounds for clustering. We also devise a componentwise Langevin sampler on the Stiefel manifold to compute this estimator
- …