A Lower Bound on the Differential Entropy of Log-Concave Random Vectors with Applications

We derive a lower bound on the differential entropy of a log-concave random variable X in terms of the p-th absolute moment of X. The new bound leads to a reverse entropy power inequality with an explicit constant, and to new bounds on the rate-distortion function and the channel capacity. Specifically, we study the rate-distortion function for log-concave sources and distortion measure |x− x̂|, and we establish that the difference between the rate distortion function and the Shannon lower bound is at most log(2 √ πe) ≈ 2.5 bits, independently of r and the target distortion d. For mean-square error distortion, the difference is at most log( √ 2πe) ≈ 2 bits, regardless of d. The bounds can be further strengthened if the source, in addition to being log-concave, is symmetric. In particular, we establish that for mean-square error distortion, the difference is at most log( √ πe) ≈ 1.5 bits, regardless of d. We also provide bounds on the capacity of memoryless additive noise channels when the noise is log-concave. We show that the difference between the capacity of such channels and the capacity of the Gaussian channel with the same noise power is at most log( √ 2πe) ≈ 2 bits, and at most log( √ πe) ≈ 1.5 bits if the noise is symmetric log-concave. Our results generalize to the case of vector X with possibly dependent coordinates, and to γ-concave random variables. Our proof technique leverages tools from convex geometry.