Adaptive estimation of the density matrix in quantum homodyne tomography with noisy data

In the framework of noisy quantum homodyne tomography with efficiency parameter $1/2 < \eta \leq 1$, we propose a novel estimator of a quantum state whose density matrix elements $\rho_{m,n}$ decrease like $Ce^{-B(m+n)^{r/ 2}}$, for fixed $C\geq 1$, $B>0$ and $0<r\leq 2$. On the contrary to previous works, we focus on the case where $r$, $C$ and $B$ 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 $\mathbb{L}_2$ -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 $(r,B,C)$. 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 &lt; 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 $\pi$ by simulating a Markov chain with transition kernel $P$ such that $\pi$ is invariant under $P$. However, there are many situations for which it is impractical or impossible to draw from the transition kernel $P$. 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 $P$ by an approximation $\hat{P}$. 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 $\hat{P}$ is to the chain given by $P$. 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 $\chi^{2}$ 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