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Register a new userA note on the expected minimum error probability in equientropic channels
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While the channel capacity reflects a theoretical upper bound on the achievable information transmission rate in the limit of infinitely many bits, it does not characterise the information transfer of a given encoding routine with finitely many bits. In this note, we characterise the quality of a code (i. e. a given encoding routine) by an upper bound on the expected minimum error probability that can be achieved when using this code. We show that for equientropic channels this upper bound is minimal for codes with maximal marginal entropy. As an instructive example we show for the additive white Gaussian noise (AWGN) channel that random coding---also a capacity achieving code---indeed maximises the marginal entropy in the limit of infinite messages.
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The ability to integrate information in the brain is considered to be an essential property for cognition and consciousness. Integrated Information Theory (IIT) hypothesizes that the amount of integrated information ($Phi$) in the brain is related to the level of consciousness. IIT proposes that to quantify information integration in a system as a whole, integrated information should be measured across the partition of the system at which information loss caused by partitioning is minimized, called the Minimum Information Partition (MIP). The computational cost for exhaustively searching for the MIP grows exponentially with system size, making it difficult to apply IIT to real neural data. It has been previously shown that if a measure of $Phi$ satisfies a mathematical property, submodularity, the MIP can be found in a polynomial order by an optimization algorithm. However, although the first version of $Phi$ is submodular, the lat
Place cells in the hippocampus are active when an animal visits a certain location (referred to as a place field) within an environment. Grid cells in the medial entorhinal cortex (MEC) respond at multiple locations, with firing fields that form a periodic and hexagonal tiling of the environment. The joint activity of grid and place cell populations, as a function of location, forms a neural code for space. An ensemble of codes is generated by varying grid and place cell population parameters. For each code in this ensemble, codewords are generated by stimulating a network with a discrete set of locations. In this manuscript, we develop an understanding of the relationships between coding theoretic properties of these combined populations and code construction parameters. These relationships are revisited by measuring the performances of biologically realizable algorithms implemented by networks of place and grid cell populations, as well as constraint neurons, which perform de-noising operations. Objectives of this work include the investigation of coding theoretic limitations of the mammalian neural code for location and how communication between grid and place cell networks may improve the accuracy of each populations representation. Simulations demonstrate that de-noising mechanisms analyzed here can significantly improve fidelity of this neural representation of space. Further, patterns observed in connectivity of each population of simulated cells suggest that inter-hippocampal-medial-entorhinal-cortical connectivity decreases downward along the dorsoventral axis.
The problem of estimating an arbitrary random vector from its observation corrupted by additive white Gaussian noise, where the cost function is taken to be the Minimum Mean $p$-th Error (MMPE), is considered. The classical Minimum Mean Square Error (MMSE) is a special case of the MMPE. Several bounds, properties and applications of the MMPE are derived and discussed. The optimal MMPE estimator is found for Gaussian and binary input distributions. Properties of the MMPE as a function of the input distribution, SNR and order $p$ are derived. In particular, it is shown that the MMPE is a continuous function of $p$ and SNR. These results are possible in view of interpolation and change of measure bounds on the MMPE. The `Single-Crossing-Point Property (SCPP) that bounds the MMSE for all SNR values {it above} a certain value, at which the MMSE is known, together with the I-MMSE relationship is a powerful tool in deriving converse proofs in information theory. By studying the notion of conditional MMPE, a unifying proof (i.e., for any $p$) of the SCPP is shown. A complementary bound to the SCPP is then shown, which bounds the MMPE for all SNR values {it below} a certain value, at which the MMPE is known. As a first application of the MMPE, a bound on the conditional differential entropy in terms of the MMPE is provided, which then yields a generalization of the Ozarow-Wyner lower bound on the mutual information achieved by a discrete input on a Gaussian noise channel. As a second application, the MMPE is shown to improve on previous characterizations of the phase transition phenomenon that manifests, in the limit as the length of the capacity achieving code goes to infinity, as a discontinuity of the MMSE as a function of SNR. As a final application, the MMPE is used to show bounds on the second derivative of mutual information, that tighten previously known bounds.
Coprime arrays enable Direction-of-Arrival (DoA) estimation of an increased number of sources. To that end, the receiver estimates the autocorrelation matrix of a larger virtual uniform linear array (coarray), by applying selection or averaging to the physical arrays autocorrelation estimates, followed by spatial-smoothing. Both selection and averaging have been designed under no optimality criterion and attain arbitrary (suboptimal) Mean-Squared-Error (MSE) estimation performance. In this work, we design a novel coprime array receiver that estimates the coarray autocorrelations with Minimum-MSE (MMSE), for any probability distribution of the source DoAs. Our extensive numerical evaluation illustrates that the proposed MMSE approach returns superior autocorrelation estimates which, in turn, enable higher DoA estimation performance compared to standard counterparts.
We consider geometrical optimization problems related to optimizing the error probability in the presence of a Gaussian noise. One famous questions in the field is the weak simplex conjecture. We discuss possible approaches to it, and state related conjectures about the Gaussian measure, in particular, the conjecture about minimizing of the Gaussian measure of a simplex. We also consider antipodal codes, apply the v{S}idak inequality and establish some theoretical and some numerical results about their optimality.
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