Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data

We estimate the quantum state of a light beam from results of quantum homodyne measurements performed on identically prepared quantum systems. The state is represented through the Wigner function, a generalized probability density on $\mathbb{R}^2$ which may take negative values and must respect intrinsic positivity constraints imposed by quantum physics. The effect of the losses due to detection inefficiencies, which are always present in a real experiment, is the addition to the tomographic data of independent Gaussian noise. We construct a kernel estimator for the Wigner function, prove that it is minimax efficient for the pointwise risk over a class of infinitely differentiable functions, and implement it for numerical results. We construct adaptive estimators, that is, which do not depend on the smoothness parameters, and prove that in some setups they attain the minimax rates for the corresponding smoothness class.Comment: Published at http://dx.doi.org/10.1214/009053606000001488 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Recovering convex boundaries from blurred and noisy observations

We consider the problem of estimating convex boundaries from blurred and noisy observations. In our model, the convolution of an intensity function $f$ is observed with additive Gaussian white noise. The function $f$ is assumed to have convex support $G$ whose boundary is to be recovered. Rather than directly estimating the intensity function, we develop a procedure which is based on estimating the support function of the set $G$. This approach is closely related to the method of geometric hyperplane probing, a well-known technique in computer vision applications. We establish bounds that reveal how the estimation accuracy depends on the ill-posedness of the convolution operator and the behavior of the intensity function near the boundary.Comment: Published at http://dx.doi.org/10.1214/009053606000000326 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Testing the isotropy of high energy cosmic rays using spherical needlets

For many decades, ultrahigh energy charged particles of unknown origin that can be observed from the ground have been a puzzle for particle physicists and astrophysicists. As an attempt to discriminate among several possible production scenarios, astrophysicists try to test the statistical isotropy of the directions of arrival of these cosmic rays. At the highest energies, they are supposed to point toward their sources with good accuracy. However, the observations are so rare that testing the distribution of such samples of directional data on the sphere is nontrivial. In this paper, we choose a nonparametric framework that makes weak hypotheses on the alternative distributions and allows in turn to detect various and possibly unexpected forms of anisotropy. We explore two particular procedures. Both are derived from fitting the empirical distribution with wavelet expansions of densities. We use the wavelet frame introduced by [SIAM J. Math. Anal. 38 (2006b) 574-594 (electronic)], the so-called needlets. The expansions are truncated at scale indices no larger than some ${J^{\star}}$, and the $L^p$ distances between those estimates and the null density are computed. One family of tests (called Multiple) is based on the idea of testing the distance from the null for each choice of $J=1,\ldots,{J^{\star}}$, whereas the so-called PlugIn approach is based on the single full ${J^{\star}}$ expansion, but with thresholded wavelet coefficients. We describe the practical implementation of these two procedures and compare them to other methods in the literature. As alternatives to isotropy, we consider both very simple toy models and more realistic nonisotropic models based on Physics-inspired simulations. The Monte Carlo study shows good performance of the Multiple test, even at moderate sample size, for a wide sample of alternative hypotheses and for different choices of the parameter ${J^{\star}}$. On the 69 most energetic events published by the Pierre Auger Collaboration, the needlet-based procedures suggest statistical evidence for anisotropy. Using several values for the parameters of the methods, our procedures yield $p$-values below 1%, but with uncontrolled multiplicity issues. The flexibility of this method and the possibility to modify it to take into account a large variety of extensions of the problem make it an interesting option for future investigation of the origin of ultrahigh energy cosmic rays.Comment: Published in at http://dx.doi.org/10.1214/12-AOAS619 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Minimax risks for sparse regressions: Ultra-high-dimensional phenomenons

Consider the standard Gaussian linear regression model $Y=X\theta+\epsilon$, where $Y\in R^n$ is a response vector and $X\in R^{n*p}$ is a design matrix. Numerous work have been devoted to building efficient estimators of $\theta$ when $p$ is much larger than $n$. In such a situation, a classical approach amounts to assume that $\theta_0$ is approximately sparse. This paper studies the minimax risks of estimation and testing over classes of $k$-sparse vectors $\theta$. These bounds shed light on the limitations due to high-dimensionality. The results encompass the problem of prediction (estimation of $X\theta$), the inverse problem (estimation of $\theta_0$) and linear testing (testing $X\theta=0$). Interestingly, an elbow effect occurs when the number of variables $k\log(p/k)$ becomes large compared to $n$. Indeed, the minimax risks and hypothesis separation distances blow up in this ultra-high dimensional setting. We also prove that even dimension reduction techniques cannot provide satisfying results in an ultra-high dimensional setting. Moreover, we compute the minimax risks when the variance of the noise is unknown. The knowledge of this variance is shown to play a significant role in the optimal rates of estimation and testing. All these minimax bounds provide a characterization of statistical problems that are so difficult so that no procedure can provide satisfying results

Statistical Inference for Structured High-dimensional Models

High-dimensional statistical inference is a newly emerged direction of statistical science in the 21 century. Its importance is due to the increasing dimensionality and complexity of models needed to process and understand the modern real world data. The main idea making possible meaningful inference about such models is to assume suitable lower dimensional underlying structure or low-dimensional approximations, for which the error can be reasonably controlled. Several types of such structures have been recently introduced including sparse high-dimensional regression, sparse and/or low rank matrix models, matrix completion models, dictionary learning, network models (stochastic block model, mixed membership models) and more. The workshop focused on recent developments in structured sequence and regression models, matrix and tensor estimation, robustness, statistical learning in complex settings, network data, and topic models

Sequential learning without feedback

In many security and healthcare systems a sequence of features/sensors/tests are used for detection and diagnosis. Each test outputs a prediction of the latent state, and carries with it inherent costs. Our objective is to {\it learn} strategies for selecting tests to optimize accuracy \& costs. Unfortunately it is often impossible to acquire in-situ ground truth annotations and we are left with the problem of unsupervised sensor selection (USS). We pose USS as a version of stochastic partial monitoring problem with an {\it unusual} reward structure (even noisy annotations are unavailable). Unsurprisingly no learner can achieve sublinear regret without further assumptions. To this end we propose the notion of weak-dominance. This is a condition on the joint probability distribution of test outputs and latent state and says that whenever a test is accurate on an example, a later test in the sequence is likely to be accurate as well. We empirically verify that weak dominance holds on real datasets and prove that it is a maximal condition for achieving sublinear regret. We reduce USS to a special case of multi-armed bandit problem with side information and develop polynomial time algorithms that achieve sublinear regret