arXiv subjects

Cristina Butucea

Publications and source records attributed to Cristina Butucea.

At least 37 records · Page 2Linked to original sources

arXiv2020

Locally private non-asymptotic testing of discrete distributions is faster using interactive mechanisms

We find separation rates for testing multinomial or more general discrete distributions under the constraint of local differential privacy. We construct efficient randomized algorithms and test procedures, in both the case where only non-interactive privacy mechanisms are allowed and also in the case where all sequentially interactive privacy mechanisms are allowed. The separation rates are faster in the latter case. We prove general information theoretical bounds that allow us to establish the optimality of our algorithms among all pairs of privacy mechanisms and test procedures, in most usual cases. Considered examples include testing uniform, polynomially and exponentially decreasing distributions.

math.ST↗

arXiv2019

Classification under local differential privacy

We consider the binary classification problem in a setup that preserves the privacy of the original sample. We provide a privacy mechanism that is locally differentially private and then construct a classifier based on the private sample that is universally consistent in Euclidean spaces. Under stronger assumptions, we establish the minimax rates of convergence of the excess risk and see that they are slower than in the case when the original sample is available.

math.ST↗

arXiv2019

Local differential privacy: Elbow effect in optimal density estimation and adaptation over Besov ellipsoids

We address the problem of non-parametric density estimation under the additional constraint that only privatised data are allowed to be published and available for inference. For this purpose, we adopt a recent generalisation of classical minimax theory to the framework of local $α$-differential privacy and provide a lower bound on the rate of convergence over Besov spaces $B^s_{pq}$ under mean integrated $\mathbb L^r$-risk. This lower bound is deteriorated compared to the standard setup without privacy, and reveals a twofold elbow effect. In order to fulfil the privacy requirement, we suggest adding suitably scaled Laplace noise to empirical wavelet coefficients. Upper bounds within (at most) a logarithmic factor are derived under the assumption that $α$ stays bounded as $n$ increases: A linear but non-adaptive wavelet estimator is shown to attain the lower bound whenever $p \geq r$ but provides a slower rate of convergence otherwise. An adaptive non-linear wavelet estimator with appropriately chosen smoothing parameters and thresholding is shown to attain the lower bound within a logarithmic factor for all cases.

math.ST↗

arXiv2018

Variable selection with Hamming loss

We derive non-asymptotic bounds for the minimax risk of variable selection under expected Hamming loss in the Gaussian mean model in $\mathbb{R}^d$ for classes of $s$-sparse vectors separated from 0 by a constant $a > 0$. In some cases, we get exact expressions for the nonasymptotic minimax risk as a function of $d, s, a$ and find explicitly the minimax selectors. These results are extended to dependent or non-Gaussian observations and to the problem of crowdsourcing. Analogous conclusions are obtained for the probability of wrong recovery of the sparsity pattern. As corollaries, we derive necessary and sufficient conditions for such asymptotic properties as almost full recovery and exact recovery. Moreover, we propose data-driven selectors that provide almost full and exact recovery adaptively to the parameters of the classes.

math.ST↗

arXiv2016

Fast adaptive estimation of log-additive exponential models in Kullback-Leibler divergence

We study the problem of nonparametric estimation of density functions with a product form on the domain $\triangle=\{( x_1, \ldots, x_d)\in \mathbb{R}^d, 0\leq x_1\leq \dots \leq x_d \leq 1\}$. Such densities appear in the random truncation model as the joint density function of observations. They are also obtained as maximum entropy distributions of order statistics with given marginals. We propose an estimation method based on the approximation of the logarithm of the density by a carefully chosen family of basis functions. We show that the method achieves a fast convergence rate in probability with respect to the Kullback-Leibler divergence for densities whose logarithm belongs to a Sobolev function class with known regularity. In the case when the regularity is unknown, we propose an estimation procedure using convex aggregation of the log-densities to obtain adaptability. The performance of this method is illustrated in a simulation study.

math.ST↗

arXiv2016

Adaptive test for large covariance matrices with missing observations

We observe $n$ independent $p-$dimensional Gaussian vectors with missing coordinates, that is each value (which is assumed standardized) is observed with probability $a>0$. We investigate the problem of minimax nonparametric testing that the high-dimensional covariance matrix $Σ$ of the underlying Gaussian distribution is the identity matrix, using these partially observed vectors. Here, $n$ and $p$ tend to infinity and $a>0$ tends to 0, asymptotically. We assume that $Σ$ belongs to a Sobolev-type ellipsoid with parameter $α>0$. When $α$ is known, we give asymptotically minimax consistent test procedure and find the minimax separation rates $\tilde φ_{n,p}= (a^2n \sqrt{p})^{- \frac{2 α}{4 α+1}}$, under some additional constraints on $n,\, p$ and $a$. We show that, in the particular case of Toeplitz covariance matrices,the minimax separation rates are faster, $\tilde ϕ_{n,p}= (a^2n p)^{- \frac{2 α}{4 α+1}}$. We note how the "missingness" parameter $a$ deteriorates the rates with respect to the case of fully observed vectors ($a=1$). We also propose adaptive test procedures, that is free of the parameter $α$ in some interval, and show that the loss of rate is $(\ln \ln (a^2 n\sqrt{p}))^{α/(4 α+1)}$ and $(\ln \ln (a^2 n p))^{α/(4 α+1)}$ for Toeplitz covariance matrices, respectively.

math.ST↗

arXiv2016

Sharp minimax tests for large covariance matrices and adaptation

We consider the detection problem of correlations in a $p$-dimensional Gaussian vector, when we observe $n$ independent, identically distributed random vectors, for $n$ and $p$ large. We assume that the covariance matrix varies in some ellipsoid with parameter $α>1/2$ and total energy bounded by $L>0$. We propose a test procedure based on a U-statistic of order 2 which is weighted in an optimal way. The weights are the solution of an optimization problem, they are constant on each diagonal and non-null only for the $T$ first diagonals, where $T=o(p)$. We show that this test statistic is asymptotically Gaussian distributed under the null hypothesis and also under the alternative hypothesis for matrices close to the detection boundary. We prove upper bounds for the total error probability of our test procedure, for $α>1/2$ and under the assumption $T=o(p)$ which implies that $n=o(p^{ 2 α})$. We illustrate via a numerical study the behavior of our test procedure. Moreover, we prove lower bounds for the maximal type II error and the total error probabilities. Thus we obtain the asymptotic and the sharp asymptotically minimax separation rate $\tildeφ = (C(α, L) n^2 p )^{- α/(4 α+ 1)}$, for $α>3/2$ and for $α>1$ together with the additional assumption $p= o(n^{4 α-1})$, respectively. We deduce rate asymptotic minimax results for testing the inverse of the covariance matrix. We construct an adaptive test procedure with respect to the parameter $α$ and show that it attains the rate $\tildeψ= ( n^2 p / \ln\ln(n \displaystyle\sqrt{p}) )^{- α/(4 α+ 1)}$.

math.ST↗

arXiv2016

Optimal exponential bounds for aggregation of estimators for the Kullback-Leibler loss

We study the problem of model selection type aggregation with respect to the Kullback-Leibler divergence for various probabilistic models. Rather than considering a convex combination of the initial estimators $f_1, \ldots, f_N$, our aggregation procedures rely on the convex combination of the logarithms of these functions. The first method is designed for probability density estimation as it gives an aggregate estimator that is also a proper density function, whereas the second method concerns spectral density estimation and has no such mass-conserving feature. We select the aggregation weights based on a penalized maximum likelihood criterion. We give sharp oracle inequalities that hold with high probability, with a remainder term that is decomposed into a bias and a variance part. We also show the optimality of the remainder terms by providing the corresponding lower bound results.

math.ST↗

arXiv2015

Maximum entropy distribution of order statistics with given marginals

We consider distributions of ordered random vectors with given one-dimensional marginal distributions. We give an elementary necessary and sufficient condition for the existence of such a distribution with finite entropy. In this case, we give explicitly the density of the unique distribution which achieves the maximal entropy and compute the value of its entropy. This density is the unique one which has a product form on its support and the given one-dimensional marginals. The proof relies on the study of copulas with given one-dimensional marginal distributions for its order statistics.

math.ST↗

arXiv2015

Adaptive variable selection in nonparametric sparse additive models

We consider the problem of recovery of an unknown multivariate signal $f$ observed in a $d$-dimensional Gaussian white noise model of intensity $\varepsilon$. We assume that $f$ belongs to a class of smooth functions ${\cal F}^d\subset L_2([0,1]^d)$ and has an additive sparse structure determined by the parameter $s$, the number of non-zero univariate components contributing to $f$. We are interested in the case when $d=d_\varepsilon \to \infty$ as $\varepsilon \to 0$ and the parameter $s$ stays "small" relative to $d$. With these assumptions, the recovery problem in hand becomes that of determining which sparse additive components are non-zero. Attempting to reconstruct most non-zero components of $f$, but not all of them, we arrive at the problem of almost full variable selection in high-dimensional regression. For two different choices of ${\cal F}^d$, we establish conditions under which almost full variable selection is possible, and provide a procedure that gives almost full variable selection. The procedure does the best (in the asymptotically minimax sense) in selecting most non-zero components of $f$. Moreover, it is adaptive in the parameter $s$.

math.ST↗

arXiv2015

Sharp minimax tests for large Toeplitz covariance matrices with repeated observations

We observe a sample of $n$ independent $p$-dimensional Gaussian vectors with Toeplitz covariance matrix $ Σ= [σ_{|i-j|}]_{1 \leq i,j \leq p}$ and $σ_0=1$. We consider the problem of testing the hypothesis that $Σ$ is the identity matrix asymptotically when $n \to \infty$ and $p \to \infty$. We suppose that the covariances $σ_k$ decrease either polynomially ($\sum_{k \geq 1} k^{2α} σ^2_{k} \leq L$ for $ α>1/4$ and $L>0$) or exponentially ($\sum_{k \geq 1} e^{2Ak} σ^2_{k} \leq L$ for $ A,L>0$). We consider a test procedure based on a weighted U-statistic of order 2, with optimal weights chosen as solution of an extremal problem. We give the asymptotic normality of the test statistic under the null hypothesis for fixed $n$ and $p \to + \infty$ and the asymptotic behavior of the type I error probability of our test procedure. We also show that the maximal type II error probability, either tend to $0$, or is bounded from above. In the latter case, the upper bound is given using the asymptotic normality of our test statistic under alternatives close to the separation boundary. Our assumptions imply mild conditions: $n=o(p^{2α- 1/2})$ (in the polynomial case), $n=o(e^p)$ (in the exponential case). We prove both rate optimality and sharp optimality of our results, for $α>1$ in the polynomial case and for any $A>0$ in the exponential case. A simulation study illustrates the good behavior of our procedure, in particular for small $n$, large $p$.

math.ST↗

arXiv2015

Spectral thresholding quantum tomography for low rank states

The estimation of high dimensional quantum states is an important statistical problem arising in current quantum technology applications. A key example is the tomography of multiple ions states, employed in the validation of state preparation in ion trap experiments \cite{Haffner2005}. Since full tomography becomes unfeasible even for a small number of ions, there is a need to investigate lower dimensional statistical models which capture prior information about the state, and to devise estimation methods tailored to such models. In this paper we propose several new methods aimed at the efficient estimation of low rank states in multiple ions tomography. All methods consist in first computing the least squares estimator, followed by its truncation to an appropriately chosen smaller rank. The latter is done by setting eigenvalues below a certain "noise level" to zero, while keeping the rest unchanged, or normalising them appropriately. We show that (up to logarithmic factors in the space dimension) the mean square error of the resulting estimators scales as $r\cdot d/N$ where $r$ is the rank, $d=2^k$ is the dimension of the Hilbert space, and $N$ is the number of quantum samples. Furthermore we establish a lower bound for the asymptotic minimax risk which shows that the above scaling is optimal. The performance of the estimators is analysed in an extensive simulations study, with emphasis on the dependence on the state rank, and the number of measurement repetitions. We find that all estimators perform significantly better that the least squares, with the "physical estimator" (which is a bona fide density matrix) slightly outperforming the other estimators.

quant-ph↗

arXiv2014

Semiparametric topographical mixture models with symmetric errors

Motivated by the analysis of a Positron Emission Tomography (PET) imaging data considered in Bowen et al. (2012), we introduce a semiparametric topographical mixture model able to capture the characteristics of dichotomous shifted response-type experiments. We propose a local estimation procedure, based on the symmetry of the local noise, for the proportion and locations functions involved in the proposed model. We establish under mild conditions the minimax properties and asymptotic normality of our estimators when Monte Carlo simulations are conducted to examine their finite sample performance. Finally a statistical analysis of the PET imaging data in Bowen et al. (2012) is illustrated for the proposed method.

math.ST↗

arXiv2013

Detection of a sparse submatrix of a high-dimensional noisy matrix

We observe a $N\times M$ matrix $Y_{ij}=s_{ij}+ξ_{ij}$ with $ξ_{ij}\sim {\mathcal {N}}(0,1)$ i.i.d. in $i,j$, and $s_{ij}\in \mathbb {R}$. We test the null hypothesis $s_{ij}=0$ for all $i,j$ against the alternative that there exists some submatrix of size $n\times m$ with significant elements in the sense that $s_{ij}\ge a>0$. We propose a test procedure and compute the asymptotical detection boundary $a$ so that the maximal testing risk tends to 0 as $M\to\infty$, $N\to\infty$, $p=n/N\to0$, $q=m/M\to0$. We prove that this boundary is asymptotically sharp minimax under some additional constraints. Relations with other testing problems are discussed. We propose a testing procedure which adapts to unknown $(n,m)$ within some given set and compute the adaptive sharp rates. The implementation of our test procedure on synthetic data shows excellent behavior for sparse, not necessarily squared matrices. We extend our sharp minimax results in different directions: first, to Gaussian matrices with unknown variance, next, to matrices of random variables having a distribution from an exponential family (non-Gaussian) and, finally, to a two-sided alternative for matrices with Gaussian elements.

math.ST↗

arXiv2013

Maximum entropy copula with given diagonal section

We consider copulas with a given diagonal section and compute the explicit density of the unique optimal copula which maximizes the entropy. In this sense, this copula is the least informative among the copulas with a given diagonal section. We give an explicit criterion on the diagonal section for the existence of the optimal copula and give a closed formula for its entropy. We also provide examples for some diagonal sections of usual bivariate copulas and illustrate the differences between them and the maximum entropy copula with the same diagonal section.

math.ST↗

arXiv2013

Rank penalized estimation of a quantum system

We introduce a new method to reconstruct the density matrix $ρ$ of a system of $n$-qubits and estimate its rank $d$ from data obtained by quantum state tomography measurements repeated $m$ times. The procedure consists in minimizing the risk of a linear estimator $\hatρ$ of $ρ$ penalized by given rank (from 1 to $2^n$), where $\hatρ$ is previously obtained by the moment method. We obtain simultaneously an estimator of the rank and the resulting density matrix associated to this rank. We establish an upper bound for the error of penalized estimator, evaluated with the Frobenius norm, which is of order $dn(4/3)^n /m$ and consistency for the estimator of the rank. The proposed methodology is computationaly efficient and is illustrated with some example states and real experimental data sets.

math.ST↗

arXiv2013

Sharp Variable Selection of a Sparse Submatrix in a High-Dimensional Noisy Matrix

We observe a $N\times M$ matrix of independent, identically distributed Gaussian random variables which are centered except for elements of some submatrix of size $n\times m$ where the mean is larger than some $a>0$. The submatrix is sparse in the sense that $n/N$ and $m/M$ tend to 0, whereas $n,\, m, \, N$ and $M$ tend to infinity. We consider the problem of selecting the random variables with significantly large mean values. We give sufficient conditions on $a$ as a function of $n,\, m,\,N$ and $M$ and construct a uniformly consistent procedure in order to do sharp variable selection. We also prove the minimax lower bounds under necessary conditions which are complementary to the previous conditions. The critical values $a^*$ separating the necessary and sufficient conditions are sharp (we show exact constants). We note a gap between the critical values $a^*$ for selection of variables and that of detecting that such a submatrix exists given by Butucea and Ingster (2012). When $a^*$ is in this gap, consistent detection is possible but no consistent selector of the corresponding variables can be found.

math.ST↗

arXiv2013

Sharp detection of smooth signals in a high-dimensional sparse matrix with indirect observations

We consider a matrix-valued Gaussian sequence model, that is, we observe a sequence of high-dimensional $M \times N$ matrices of heterogeneous Gaussian random variables $x_{ij,k}$ for $i \in\{1,...,M\}$, $j \in \{1,...,N\}$ and $k \in \mathbb{Z}$. The standard deviation of our observations is $\ep k^s$ for some $\ep >0$ and $s \geq 0$. We give sharp rates for the detection of a sparse submatrix of size $m \times n$ with active components. A component $(i,j)$ is said active if the sequence $\{x_{ij,k}\}_k$ have mean $\{θ_{ij,k}\}_k$ within a Sobolev ellipsoid of smoothness $τ>0$ and total energy $\sum_k θ^2_{ij,k} $ larger than some $r^2_\ep$. Our rates involve relationships between $m,\, n, \, M$ and $N$ tending to infinity such that $m/M$, $n/N$ and $\ep$ tend to 0, such that a test procedure that we construct has asymptotic minimax risk tending to 0. We prove corresponding lower bounds under additional assumptions on the relative size of the submatrix in the large matrix of observations. Except for these additional conditions our rates are asymptotically sharp. Lower bounds for hypothesis testing problems mean that no test procedure can distinguish between the null hypothesis (no signal) and the alternative, i.e. the minimax risk for testing tends to 1.

math.ST↗