Statistical model selection is a great challenge when the number of accessible measurements is much smaller than the dimension of the parameter space. We study the problem of model selection in the context of subset selection for highdimensional linear regressions. Accordingly, we propose a new model selection criterion with the Fisher information that leads to the selection of a parsimonious model from all the combinatorial models up to some maximum level of sparsity. We analyze the performance of our criterion as the number of measurements grows to infinity, as well as when the noise variance tends to zero. In each case, we prove that our proposed criterion gives the true model with a probability approaching one. Additionally, we devise a computationally affordable algorithm to conduct model selection with the proposed criterion in practice. Interestingly, as a side product, our algorithm can provide the ideal regularization parameter for the Lasso estimator such that Lasso selects the true variables. Finally, numerical simulations are included to support our theoretical findings.
Spectrum sensing is a fundamental component in cognitive radio. A major challenge in this area is the requirement of a high sampling rate in the sensing of a wideband signal. In this paper a wideband spectrum sensing model is presented that utilizes a sub-Nyquist sampling scheme to bring substantial savings in terms of the sampling rate. The correlation matrix of a finite number of noisy samples is computed and used by a subspace estimator to detect the occupied and vacant channels of the spectrum. In contrast with common methods, the proposed method does not need the knowledge of signal properties that mitigates the uncertainty problem. We evaluate the performance of this method by computing the probability of detecting signal occupancy in terms of the number of samples and the SNR of randomly generated signals. The results show a reliable detection even in low SNR and small number of samples.
Abstract-We consider the problem of direction-of-arrival (DOA) estimation in unknown partially correlated noise environments where the noise covariance matrix is sparse. A sparse noise covariance matrix is a common model for a sparse array of sensors consisted of several widely separated subarrays. Since interelement spacing among sensors in a subarray is small, the noise in the subarray is in general spatially correlated, while, due to large distances between subarrays, the noise between them is uncorrelated. Consequently, the noise covariance matrix of such an array has a block diagonal structure which is indeed sparse. Moreover, in an ordinary nonsparse array, because of small distance between adjacent sensors, there is noise coupling between neighboring sensors, whereas one can assume that nonadjacent sensors have spatially uncorrelated noise which makes again the array noise covariance matrix sparse. Utilizing some recently available tools in low-rank/sparse matrix decomposition, matrix completion, and sparse representation, we propose a novel method which can resolve possibly correlated or even coherent sources in the aforementioned partly correlated noise. In particular, when the sources are uncorrelated, our approach involves solving a second-order cone programming (SOCP), and if they are correlated or coherent, one needs to solve a computationally harder convex program. We demonstrate the effectiveness of the proposed algorithm by numerical simulations and comparison to the Cramer-Rao bound (CRB).
We investigate the task of model selection for highdimensional data. For this purpose, we propose an extension to the Bayesian information criterion. Our information criterion is asymptotically consistent either as the number of measurements tends to infinity or as the variance of noise decreases to zero. The numerical results provided support our claim. Additionally, we highlight the link between model selection for high-dimensional data and the choice of hyper-parameter in 1-constrained estimators, specifically the LASSO.
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