On a question of Babadi and Tarokh

In a recent remarkable paper, Babadi and Tarokh proved the "randomness" of sequences arising from binary linear block codes in the sense of spectral distribution, provided that their dual distances are sufficiently large. However, numerical experiments conducted by the authors revealed that Gold sequences which have dual distance 5 also satisfy such randomness property. Hence the interesting question was raised as to whether or not the stringent requirement of large dual distances can be relaxed in the theorem in order to explain the randomness of Gold sequences. This paper improves their result on several fronts and provides an affirmative answer to this question

Analysis of Fisher Information and the Cram\'{e}r-Rao Bound for Nonlinear Parameter Estimation after Compressed Sensing

In this paper, we analyze the impact of compressed sensing with complex random matrices on Fisher information and the Cram\'{e}r-Rao Bound (CRB) for estimating unknown parameters in the mean value function of a complex multivariate normal distribution. We consider the class of random compression matrices whose distribution is right-orthogonally invariant. The compression matrix whose elements are i.i.d. standard normal random variables is one such matrix. We show that for all such compression matrices, the Fisher information matrix has a complex matrix beta distribution. We also derive the distribution of CRB. These distributions can be used to quantify the loss in CRB as a function of the Fisher information of the non-compressed data. In our numerical examples, we consider a direction of arrival estimation problem and discuss the use of these distributions as guidelines for choosing compression ratios based on the resulting loss in CRB.Comment: 12 pages, 3figure

Sparsity enables estimation of both subcortical and cortical activity from MEG and EEG

Subcortical structures play a critical role in brain function. However, options for assessing electrophysiological activity in these structures are limited. Electromagnetic fields generated by neuronal activity in subcortical structures can be recorded noninvasively, using magnetoencephalography (MEG) and electroencephalography (EEG). However, these subcortical signals are much weaker than those generated by cortical activity. In addition, we show here that it is difficult to resolve subcortical sources because distributed cortical activity can explain the MEG and EEG patterns generated by deep sources. We then demonstrate that if the cortical activity is spatially sparse, both cortical and subcortical sources can be resolved with M/EEG. Building on this insight, we develop a hierarchical sparse inverse solution for M/EEG. We assess the performance of this algorithm on realistic simulations and auditory evoked response data, and show that thalamic and brainstem sources can be correctly estimated in the presence of cortical activity. Our work provides alternative perspectives and tools for characterizing electrophysiological activity in subcortical structures in the human brain