An Entropy Sumset Inequality and Polynomially Fast Convergence to Shannon Capacity Over All Alphabets

Abstract

We prove a lower estimate on the increase in entropy when two copies of a conditional random variable X|Y, with X supported on Zq = {0,1,..., q − 1} for prime q, are summed modulo q. Specifically, given two i.i.d. copies (X1, Y1) and (X2, Y2) of a pair of random variables (X, Y), with X taking values in Zq, we show H(X1 + X2 | Y1, Y2) - H(X|Y ) ≥ α(q) · H(X|Y)(1 - H(X|Y)) for some α(q) > 0, where H (·) is the normalized (by factor log2q) entropy. In particular, if X|Y is not close to being fully random or fully deterministic and H(X|Y) ∈ (γ, 1-γ), then the entropy of the sum increases by Ωq (γ). Our motivation is an effective analysis of the finite-length behavior of polar codes, for which the linear dependence on γ is quantitatively important. The assumption of q being prime is necessary: for X supported uniformly on a proper subgroup of Zq we have H(X + X) = H(X). For X supported on infinite groups without a finite subgroup (the torsion-free case) and no conditioning, a sumset inequality for the absolute increase in (unnormalized) entropy was shown by Tao in [20]. We use our sumset inequality to analyze Arikan's construction of polar codes and prove that for any q-ary source X, where q is any fixed prime, and any e > 0, polar codes allow efficient data compression of N i.i.d. copies of X into (H(X) + e)N q-ary symbols, as soon as N is polynomially large in 1/e. We can get capacity-achieving source codes with similar guarantees for composite alphabets, by factoring q into primes and combining different polar codes for each prime in factorization. A consequence of our result for noisy channel coding is that for all discrete memoryless channels, there are explicit codes enabling reliable communication within e > 0 of the symmetric Shannon capacity for a block length and decoding complexity bounded by a polynomial in 1/e. The result was previously shown for the special case of binary-input channels [7, 9], and this work extends the result to channels over any alphabet.