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In 1959, Renyi proposed the information dimension and the $d$-dimensional entropy to measure the information content of general random variables. This paper proposes a generalization of information dimension to stochastic processes by defining the information dimension rate as the entropy rate of the uniformly-quantized stochastic process divided by minus the logarithm of the quantizer step size $1/m$ in the limit as $mtoinfty$. It is demonstrated that the information dimension rate coincides with the rate-distortion dimension, defined as twice the rate-distortion function $R(D)$ of the stochastic process divided by $-log(D)$ in the limit as $Ddownarrow 0$. It is further shown that, among all multivariate stationary processes with a given (matrix-valued) spectral distribution function (SDF), the Gaussian process has the largest information dimension rate, and that the information dimension rate of multivariate stationary Gaussian processes is given by the average rank of the derivative of the SDF. The presented results reveal that the fundamental limits of almost zero-distortion recovery via compressible signal pursuit and almost lossless analog compression are different in general.
The authors have recently defined the Renyi information dimension rate $d({X_t})$ of a stationary stochastic process ${X_t,,tinmathbb{Z}}$ as the entropy rate of the uniformly-quantized process divided by minus the logarithm of the quantizer step siz
The rate-distortion dimension (RDD) of an analog stationary process is studied as a measure of complexity that captures the amount of information contained in the process. It is shown that the RDD of a process, defined as two times the asymptotic rat
In an effort to develop the foundations for a non-stochastic theory of information, the notion of $delta$-mutual information between uncertain variables is introduced as a generalization of Nairs non-stochastic information functional. Several propert
Given a probability measure $mu$ over ${mathbb R}^n$, it is often useful to approximate it by the convex combination of a small number of probability measures, such that each component is close to a product measure. Recently, Ronen Eldan used a stoch
We investigate the asymptotic rates of length-$n$ binary codes with VC-dimension at most $dn$ and minimum distance at least $delta n$. Two upper bounds are obtained, one as a simple corollary of a result by Haussler and the other via a shortening app