8. Subset Selection in Regression (Monographs on Statistics and Applied Probability, no. 40). By A. J. Miller. ISBN 0 412 35380 6. Chapman and Hall, London, 1990. 240 pp. £25.00.

Subset Selection in Regression

2017 25th European Signal Processing Conference (EUSIPCO)

The Jacobi inversion theorem leads to functions of several variables with many periods. So, we are led to the problem of developing a theory of them which is analogous to the theory of elliptic functions. First of all, we need to give an introduction to the theory of functions of several complex variables. In this chapter, we give an elementary introduction which essentially follows Weierstrass. One of the main topics will be the proof of the theorem that any meromorphic function on ℂn can be written as a quotient of two entire functions. Weierstrass called this a very difficult problem. The first proof was given by Poincare. In the case n = 1, it is not difficult to show this by means of the theory of Weierstrass products, even for arbitrary domains D ⊂ ℂ instead of ℂ. The case n > 1 is more involved, since the zero sets and pole sets of analytic functions of several complex variables are not discrete.

Analytic Functions of Several Complex Variables

Sparse linear arrays, such as coprime arrays and nested arrays, have the attractive capability of providing enhanced degrees of freedom. By exploiting the coarray structure, an augmented sample covariance matrix can be constructed and MUtiple SIgnal Classification (MUSIC) can be applied to identify more sources than the number of sensors. While such a MUSIC algorithm works quite well, its performance has not been theoretically analyzed. In this paper, we derive a simplified asymptotic mean square error (MSE) expression for the MUSIC algorithm applied to the coarray model, which is applicable even if the source number exceeds the sensor number. We show that the directly augmented sample covariance matrix and the spatial smoothed sample covariance matrix yield the same asymptotic MSE for MUSIC. We also show that when there are more sources than the number of sensors, the MSE converges to a positive value instead of zero when the signal-to-noise ratio (SNR) goes to infinity. This finding explains the “saturation” behavior of the coarray-based MUSIC algorithms in the high-SNR region observed in previous studies. Finally, we derive the Cramer–Rao bound for sparse linear arrays, and conduct a numerical study of the statistical efficiency of the coarray-based estimator. Experimental results verify theoretical derivations and reveal the complex efficiency pattern of coarray-based MUSIC algorithms.

Coarrays, MUSIC, and the Cramér–Rao Bound

Cramér-Rao bounds for coprime and other sparse arrays, which find more sources than sensors

Although Cramer–Rao Bounds (CRB) for direction-of-arrival (DOA) estimation have been extensively studied for decades, existing results are mainly applicable when there are fewer sources than sensors. In contrast, this letter considers an underdetermined signal model (more sources than sensors) and investigates conditions under which CRB exist. Necessary and sufficient conditions are derived for the associated Fisher information matrix to be nonsingular, which in turn, leads to closed-form expressions for the CRBs for underdetermined DOA estimation. These conditions highlight crucial roles played by the array geometry, as well as the correlation between source signals. The CRB for different array geometries are numerically compared both in the overdetermined and underdetermined settings.

Cramér–Rao Bounds for Underdetermined Source Localization

Sparse linear arrays such as co-prime and nested arrays can resolve more sources than the number of sensors. In contrast, uniform linear arrays (ULA) cannot resolve more sources than the number of sensors. This paper demonstrates this using Sparse Bayesian learning (SBL) and co-array MUSIC for single frequency beamforming. For approximately the same number of sensors, co-prime and nested arrays are shown to outperform ULA in root mean squared error. This paper shows that multi-frequency SBL can significantly reduce spatial aliasing. The effects of different sparse sub-arrays on SBL performance are compared qualitatively using the Noise Correlation 2009 experimental data set.

Sparse Bayesian learning for beamforming using sparse linear arrays

In this paper, based on a successively accuracy-increasing approximation of the $\ell _0$ norm, we propose a new algorithm for recovery of sparse vectors from underdetermined measurements. The approximations are realized with a certain class of concave functions that aggressively induce sparsity and their closeness to the $\ell _0$ norm can be controlled. We prove that the series of the approximations asymptotically coincides with the $\ell _1$ and $\ell _0$ norms when the approximation accuracy changes from the worst fitting to the best fitting. When measurements are noise-free, an optimization scheme is proposed that leads to a number of weighted $\ell _1$ minimization programs, whereas, in the presence of noise, we propose two iterative thresholding methods that are computationally appealing. A convergence guarantee for the iterative thresholding method is provided, and, for a particular function in the class of the approximating functions, we derive the closed-form thresholding operator. We further present some theoretical analyses via the restricted isometry, null space, and spherical section properties. Our extensive numerical simulations indicate that the proposed algorithm closely follows the performance of the oracle estimator for a range of sparsity levels wider than those of the state-of-the-art algorithms.

Successive Concave Sparsity Approximation for Compressed Sensing

This paper provides new probabilistic guarantees for recovering the common support of jointly sparse vectors in multiple measurement vector (MMV) models In recent times, Bayesian approaches for sparse signal recovery (such as sparse Bayesian learning and correlation-aware LASSO) have shown preliminary evidence that under appropriate conditions (such as access to ideal covariance matrix of the measurements or certain restrictive orthogonality condition on the signals), it is possible to recover supports of size ( $K$ ) larger than the dimension ( $M$ ) of each measurement vector However, no results exist that characterize the probability with which this can be achieved for a finite number of measurement vectors ( $L$ ) This paper bridges this gap by formulating the support recovery problem in terms of a multiple hypothesis testing framework Chernoff-type upper bounds on the probability of error are established, and new sufficient conditions are derived that guarantee its exponential decay with respect to $L$ even when $K=O(M^2)$  Our sufficient conditions are based on the properties of the so-called Khatri–Rao product of the measurement matrix and reveal the importance of a sampler design Negative results are also established indicating that when $K$ exceeds a certain threshold (in terms of $M$ ), there will exist a class of measurement matrices for which any support recovery algorithm will fail Using results from the geometric probability, we characterize the probability with which a randomly generated measurement matrix will belong to this class and show that this probability tends to 1 asymptotically in the size ( $N$ ) of the sparse vectors

On Fundamental Limits of Joint Sparse Support Recovery Using Certain Correlation Priors

Realtime simulation of RoboCup small size soccer robots is a challenging task due to the high frame rate of input vision data and complex dynamic model of robots. In this paper we describe a new multirobot 3D simulator for small size robot soccer domain named 'grSim'. In order to decrease the model complexity and increase simulation speed, a simplified dynamic model for omni wheels is implemented. grSim has a distributed architecture, feature-rich user interface and supports all aspects of a small size robot soccer game, thus it can completely replace all hardware used by teams during software development. grSim can help software/AI developers design smarter SSL robot teams.

/pdf/grsim-robocup-small-size-robot-soccer-simulator-ckj8lvlczo.pdf

Ali Koochakzadeh

Papers

Cramér–Rao Bounds for Underdetermined Source Localization

Sparse Bayesian learning for beamforming using sparse linear arrays

Successive Concave Sparsity Approximation for Compressed Sensing

On Fundamental Limits of Joint Sparse Support Recovery Using Certain Correlation Priors

grSim: RoboCup small size robot soccer simulator