Dissipation of information in channels with input constraints

One of the basic tenets in information theory, the data processing inequality states that output divergence does not exceed the input divergence for any channel. For channels without input constraints, various estimates on the amount of such contraction are known, Dobrushin's coefficient for the total variation being perhaps the most well-known. This work investigates channels with average input cost constraint. It is found that while the contraction coefficient typically equals one (no contraction), the information nevertheless dissipates. A certain non-linear function, the \emph{Dobrushin curve} of the channel, is proposed to quantify the amount of dissipation. Tools for evaluating the Dobrushin curve of additive-noise channels are developed based on coupling arguments. Some basic applications in stochastic control, uniqueness of Gibbs measures and fundamental limits of noisy circuits are discussed. As an application, it shown that in the chain of $n$ power-constrained relays and Gaussian channels the end-to-end mutual information and maximal squared correlation decay as $\Theta(\frac{\log\log n}{\log n})$, which is in stark contrast with the exponential decay in chains of discrete channels. Similarly, the behavior of noisy circuits (composed of gates with bounded fan-in) and broadcasting of information on trees (of bounded degree) does not experience threshold behavior in the signal-to-noise ratio (SNR). Namely, unlike the case of discrete channels, the probability of bit error stays bounded away from $1\over 2$ regardless of the SNR.Comment: revised; include appendix B on contraction coefficient for mutual information on general alphabet

Mutual Information in Frequency and its Application to Measure Cross-Frequency Coupling in Epilepsy

We define a metric, mutual information in frequency (MI-in-frequency), to detect and quantify the statistical dependence between different frequency components in the data, referred to as cross-frequency coupling and apply it to electrophysiological recordings from the brain to infer cross-frequency coupling. The current metrics used to quantify the cross-frequency coupling in neuroscience cannot detect if two frequency components in non-Gaussian brain recordings are statistically independent or not. Our MI-in-frequency metric, based on Shannon's mutual information between the Cramer's representation of stochastic processes, overcomes this shortcoming and can detect statistical dependence in frequency between non-Gaussian signals. We then describe two data-driven estimators of MI-in-frequency: one based on kernel density estimation and the other based on the nearest neighbor algorithm and validate their performance on simulated data. We then use MI-in-frequency to estimate mutual information between two data streams that are dependent across time, without making any parametric model assumptions. Finally, we use the MI-in- frequency metric to investigate the cross-frequency coupling in seizure onset zone from electrocorticographic recordings during seizures. The inferred cross-frequency coupling characteristics are essential to optimize the spatial and spectral parameters of electrical stimulation based treatments of epilepsy.Comment: This paper is accepted for publication in IEEE Transactions on Signal Processing and contains 15 pages, 9 figures and 1 tabl