dilated-receptive-field

• The effective receptive field of a dilated convolution stack read as a digit count, the numbers of dilations.
• r_Q(n) = [z^n] prod_(j < L) Q(z^(b^j)) is the weighted base-b digit count of the block polynomial Q, computed in exact integers by product.
• count checks product against the digit recursion r_L(n) = sum_(c = n mod b) q_c r_(L-1)((n - c)/b) on every lag, for Q = 1 + z + z^2, b = 2, L = 1..exact, and for 1 + z, 2 + z, 1 + z + z^2 + z^3 at b = 3 and 1 + (1 + z + z^2)^2 at L = exact - 4; checks Stern's recursion against the first 32 terms of A002487, copied from OEIS; then per level L = 1..top asserts r(n) = s(n + 1) below 2^L, the mirror, the maximum F_(L+1) at the four closed-form lags, the single-path lags 2^k - 1 and their mirrors numbering 2L + 1, the fold s(2^L + j) = r(j - 1) + r(j - 1 + 2^L), and prints peak over mean F_(L+1)(2^(L+1) - 1)/3^L and its step ratio against 2 phi/3.
• limit decides absolute continuity of the depth limit by the exact cyclotomic check in rational arithmetic, Q vanishing at e^(2 pi i k/b^i) for some i <= depth for every k < b^depth not divisible by b, on 35 block polynomials: uniform K = 2..8 at b = 2, 3, 4, 1 + c(1 + z) and 1 + c(1 + z + z^2)^2 over a grid of c, and (1 + z + z^2)^2; asserts the verdict against the Fourier product prod_i Q(e^(2 pi i t/b^i))/Q(1) over t < b^3 not divisible by b; checks the real-gain Q = (z^2 - sqrt2 z + 1)(z^4 + sqrt2 z^2 + 1)(1 + z + z^2)^2 at b = 2: its least coefficient, the level at which each odd k < 2^depth is covered, |Q| at the primitive 2^i-th roots for i = 1..4, and the Fourier product over odd t < 256; prints the share of lags carrying half the mass at L = 8, 12, 16, 20 for four polynomials.
• gradient draws random linear stacks, channels channels, levels levels, draws times, for the pure K = 3 stack at C sigma^2 = 1 and the residual block 1 + c(1 + z) at c = 1/2, and compares the mean squared gradient over input channels with r_Q, printing the largest z-score, the share past 3, the median relative error and the correlation.
• copy trains a linear K = 3, b = 2 stack by full-batch gradient descent on the lag-n copy loss sum_m (f(m) - [m = n])^2, f the end-to-end filter, from seeds initialisations at scale times the variance-preserving scale, counting steps until f(n) >= hit, a run past cap read as above the cap in its median, at one lag in 2^(L-1)..2^L - 1 for each r in 1, 2, 3, 5, 8, 13, 21, 34, 55 present, and fits log median steps against log 1/r.

RUN

• uv run python research/lab/py/dilated-receptive-field/drf.py count from mrlyprod/, 1.8 seconds.
• uv run python research/lab/py/dilated-receptive-field/drf.py limit, 8.5 seconds.
• uv run python research/lab/py/dilated-receptive-field/drf.py gradient, 1.0 seconds.
• uv run python research/lab/py/dilated-receptive-field/drf.py copy, 4.2 seconds; copy --scale 0.5, 112 seconds.
• uv run python research/lab/py/dilated-receptive-field/drf.py all, 16 seconds.
• --top 20, --exact 12, --depth 6, --channels 8, --levels 8, --draws 4000, --seeds 5, --lr 0.005, --scale 1, --cap 20000 and --hit 0.5 are the dials.
• Prints only; reads and writes nothing.

WITNESSES

• The lines of dilations, every section.
• count: 0 mismatches between the product and the digit recursion at L = 1..12 and on the four other polynomials; the first 32 A002487 terms agree; every assertion holds at L = 1..20; maximum 89 at 682, 852, 1194, 1364 and 21 single-path lags at L = 10; fold 0 mismatches at L = 1..20; peak over mean 3.0853 at L = 10, 6.5835 at L = 20, step ratio 1.07143 at L = 3, 1.14815 at L = 4, 1.07869 from L = 17.
• limit: 35 rows, verdict and Fourier product agreeing; uniform taps absolutely continuous exactly when b divides K; 1 + c(1 + z) singular at every c on the grid; 1 + c(1 + z + z^2)^2 absolutely continuous only at c = 1; the real-gain Q has least coefficient 0.5858, cover levels 3, 4, |Q| up to 11.6569, 0.8284, 0.6863, 9.9446 at the primitive 2, 4, 8, 16-th roots, Fourier product at most 2.1e-18; half the mass on 0.2740, 0.2561, 0.2415, 0.2280 of the lags for K = 3, b = 2, and on 0.0010 at L = 20 for 1 + (1 + z)/4.
• gradient: median relative error 0.0202, correlation 0.9989 over 511 lags, share of |z| > 3 0.0196, pure K = 3; 0.0224, 0.9999 over 256 lags, residual.
• copy: lags 255, 191, 223, 207, 215, 192, 213, 212 at r = 1, 2, 3, 5, 8, 13, 21, 34, median steps 615, 505, 376, 270, 250, 198, 108, 94, slope 0.551, correlation 0.984, no run capped; at --scale 0.5 median steps 15760, 12768, 8779, 7081, 6233, 6085, 4270, 3788, one run at r = 1 capped, slope 0.404, correlation 0.983 over the 8 lags.