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Register a new userFundamental limits of detection in the spiked Wigner model
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We study the fundamental limits of detecting the presence of an additive rank-one perturbation, or spike, to a Wigner matrix. When the spike comes from a prior that is i.i.d. across coordinates, we prove that the log-likelihood ratio of the spiked model against the non-spiked one is asymptotically normal below a certain reconstruction threshold which is not necessarily of a spectral nature, and that it is degenerate above. This establishes the maximal region of contiguity between the planted and null models. It is known that this threshold also marks a phase transition for estimating the spike: the latter task is possible above the threshold and impossible below. Therefore, both estimation and detection undergo the same transition in this random matrix model. We also provide further information about the performance of the optimal test. Our proofs are based on Gaussian interpolation methods and a rigorous incarnation of the cavity method, as devised by Guerra and Talagrand in their study of the Sherrington--Kirkpatrick spin-glass model.
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We consider the problem of estimating a large rank-one tensor ${boldsymbol u}^{otimes k}in({mathbb R}^{n})^{otimes k}$, $kge 3$ in Gaussian noise. Earlier work characterized a critical signal-to-noise ratio $lambda_{Bayes}= O(1)$ above which an ideal estimator achieves strictly positive correlation with the unknown vector of interest. Remarkably no polynomial-time algorithm is known that achieved this goal unless $lambdage C n^{(k-2)/4}$ and even powerful semidefinite programming relaxations appear to fail for $1ll lambdall n^{(k-2)/4}$. In order to elucidate this behavior, we consider the maximum likelihood estimator, which requires maximizing a degree-$k$ homogeneous polynomial over the unit sphere in $n$ dimensions. We compute the expected number of critical points and local maxima of this objective function and show that it is exponential in the dimensions $n$, and give exact formulas for the exponential growth rate. We show that (for $lambda$ larger than a constant) critical points are either very close to the unknown vector ${boldsymbol u}$, or are confined in a band of width $Theta(lambda^{-1/(k-1)})$ around the maximum circle that is orthogonal to ${boldsymbol u}$. For local maxima, this band shrinks to be of size $Theta(lambda^{-1/(k-2)})$. These `uninformative local maxima are likely to cause the failure of optimization algorithms.
In this paper, we draw attention to a problem that is often overlooked or ignored by companies practicing hypothesis testing (A/B testing) in online environments. We show that conducting experiments on limited inventory that is shared between variants in the experiment can lead to high false positive rates since the core assumption of independence between the groups is violated. We provide a detailed analysis of the problem in a simplified setting whose parameters are informed by realistic scenarios. The setting we consider is a $2$-dimensional random walk in a semi-infinite strip. It is rich enough to take a finite inventory into account, but is at the same time simple enough to allow for a closed form of the false-positive probability. We prove that high false-positive rates can occur, and develop tools that are suitable to help design adequate tests in follow-up work. Our results also show that high false-negative rates may occur. The proofs rely on a functional limit theorem for the $2$-dimensional random walk in a semi-infinite strip.
Let $G$ be a non--linear function of a Gaussian process ${X_t}_{tinmathbb{Z}}$ with long--range dependence. The resulting process ${G(X_t)}_{tinmathbb{Z}}$ is not Gaussian when $G$ is not linear. We consider random wavelet coefficients associated with ${G(X_t)}_{tinmathbb{Z}}$ and the corresponding wavelet scalogram which is the average of squares of wavelet coefficients over locations. We obtain the asymptotic behavior of the scalogram as the number of observations and scales tend to infinity. It is known that when $G$ is a Hermite polynomial of any order, then the limit is either the Gaussian or the Rosenblatt distribution, that is, the limit can be represented by a multiple Wiener-It^o integral of order one or two. We show, however, that there are large classes of functions $G$ which yield a higher order Hermite distribution, that is, the limit can be represented by a a multiple Wiener-It^o integral of order greater than two.
We propose a Bayesian methodology for estimating spiked covariance matrices with jointly sparse structure in high dimensions. The spiked covariance matrix is reparametrized in terms of the latent factor model, where the loading matrix is equipped with a novel matrix spike-and-slab LASSO prior, which is a continuous shrinkage prior for modeling jointly sparse matrices. We establish the rate-optimal posterior contraction for the covariance matrix with respect to the operator norm as well as that for the principal subspace with respect to the projection operator norm loss. We also study the posterior contraction rate of the principal subspace with respect to the two-to-infinity norm loss, a novel loss function measuring the distance between subspaces that is able to capture element-wise eigenvector perturbations. We show that the posterior contraction rate with respect to the two-to-infinity norm loss is tighter than that with respect to the routinely used projection operator norm loss under certain low-rank and bounded coherence conditions. In addition, a point estimator for the principal subspace is proposed with the rate-optimal risk bound with respect to the projection operator norm loss. These results are based on a collection of concentration and large deviation inequalities for the matrix spike-and-slab LASSO prior. The numerical performance of the proposed methodology is assessed through synthetic examples and the analysis of a real-world face data example.
We mainly study the M-estimation method for the high-dimensional linear regression model, and discuss the properties of M-estimator when the penalty term is the local linear approximation. In fact, M-estimation method is a framework, which covers the methods of the least absolute deviation, the quantile regression, least squares regression and Huber regression. We show that the proposed estimator possesses the good properties by applying certain assumptions. In the part of numerical simulation, we select the appropriate algorithm to show the good robustness of this method
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