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Register a new userNon-Gaussian Observations in Nonlinear Compressed Sensing via Stein Discrepancies
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Performance guarantees for compression in nonlinear models under non-Gaussian observations can be achieved through the use of distributional characteristics that are sensitive to the distance to normality, and which in particular return the value of zero under Gaussian or linear sensing. The use of these characteristics, or discrepancies, improves some previous results in this area by relaxing conditions and tightening performance bounds. In addition, these characteristics are tractable to compute when Gaussian sensing is corrupted by either additive errors or mixing.
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We propose the novel augmented Gaussian random field (AGRF), which is a universal framework incorporating the data of observable and derivatives of any order. Rigorous theory is established. We prove that under certain conditions, the observable and its derivatives of any order are governed by a single Gaussian random field, which is the aforementioned AGRF. As a corollary, the statement ``the derivative of a Gaussian process remains a Gaussian process is validated, since the derivative is represented by a part of the AGRF. Moreover, a computational method corresponding to the universal AGRF framework is constructed. Both noiseless and noisy scenarios are considered. Formulas of the posterior distributions are deduced in a nice closed form. A significant advantage of our computational method is that the universal AGRF framework provides a natural way to incorporate arbitrary order derivatives and deal with missing data. We use four numerical examples to demonstrate the effectiveness of the computational method. The numerical examples are composite function, damped harmonic oscillator, Korteweg-De Vries equation, and Burgers equation.
In this paper we consider the nonparametric functional estimation of the drift of Gaussian processes using Paley-Wiener and Karhunen-Lo`eve expansions. We construct efficient estimators for the drift of such processes, and prove their minimaxity using Bayes estimators. We also construct superefficient estimators of Stein type for such drifts using the Malliavin integration by parts formula and stochastic analysis on Gaussian space, in which superharmonic functionals of the process paths play a particular role. Our results are illustrated by numerical simulations and extend the construction of James-Stein type estimators for Gaussian processes by Berger and Wolper.
We establish exponential inequalities for a class of V-statistics under strong mixing conditions. Our theory is developed via a novel kernel expansion based on random Fourier features and the use of a probabilistic method. This type of expansion is new and useful for handling many notorious classes of kernels.
This paper analyzes the impact of non-Gaussian multipath component (MPC) amplitude distributions on the performance of Compressed Sensing (CS) channel estimators for OFDM systems. The number of dominant MPCs that any CS algorithm needs to estimate in order to accurately represent the channel is characterized. This number relates to a Compressibility Index (CI) of the channel that depends on the fourth moment of the MPC amplitude distribution. A connection between the Mean Squared Error (MSE) of any CS estimation algorithm and the MPC amplitude distribution fourth moment is revealed that shows a smaller number of MPCs is needed to well-estimate channels when these components have large fourth moment amplitude gains. The analytical results are validated via simulations for channels with lognormal MPCs such as the NYU mmWave channel model. These simulations show that when the MPC amplitude distribution has a high fourth moment, the well known CS algorithm of Orthogonal Matching Pursuit performs almost identically to the Basis Pursuit De-Noising algorithm with a much lower computational cost.
Random divisions of an interval arise in various context, including statistics, physics, and geometric analysis. For testing the uniformity of a random partition of the unit interval $[0,1]$ into $k$ disjoint subintervals of size $(S_k[1],ldots,S_k[k])$, Greenwood (1946) suggested using the squared $ell_2$-norm of this size vector as a test statistic, prompting a number of subsequent studies. Despite much progress on understanding its power and asymptotic properties, attempts to find its exact distribution have succeeded so far for only small values of $k$. Here, we develop an efficient method to compute the distribution of the Greenwood statistic and more general spacing-statistics for an arbitrary value of $k$. Specifically, we consider random divisions of ${1,2,dots,n}$ into $k$ subsets of consecutive integers and study $|S_{n,k}|^p_{p,w}$, the $p$th power of the weighted $ell_p$-norm of the subset size vector $S_{n,k}=(S_{n,k}[1],ldots,S_{n,k}[k])$ for arbitrary weights $w=(w_1,ldots,w_k)$. We present an exact and quickly computable formula for its moments, as well as a simple algorithm to accurately reconstruct a probability distribution using the moment sequence. We also study various scaling limits, one of which corresponds to the Greenwood statistic in the case of $p=2$ and $w=(1,ldots,1)$, and this connection allows us to obtain information about regularity, monotonicity and local behavior of its distribution. Lastly, we devise a new family of non-parametric tests using $|S_{n,k}|^p_{p,w}$ and demonstrate that they exhibit substantially improved power for a large class of alternatives, compared to existing popular methods such as the Kolmogorov-Smirnov, Cramer-von Mises, and Mann-Whitney/Wilcoxon rank-sum tests.
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