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1ca0407 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 | """Young's convolution inequality quotient ||f*g||_r / (||f||_p ||g||_q) for non-negative step functions.
f = sum_i a_i 1_[i, i+1), g = sum_j b_j 1_[j, j+1); then f*g is piecewise linear with knot values
(f*g)(k) = sum_{i+j=k-1} a_i b_j, so all three norms are integrated exactly (up to float64 rounding).
Upstream (classical_inequalities.ipynb) used np.convolve(mode="same") on samples of user callables, which
truncates the convolution's support; this closed form replaces it.
"""
import numpy as np
from run import Invalid, float_list
def _seg_power_integral(c0, c1, r):
"""int_0^1 ((1-s) c0 + s c1)^r ds for non-negative c0, c1 (vectorised, cancellation-free)."""
hi, lo = np.maximum(c0, c1), np.minimum(c0, c1)
out = np.zeros_like(hi)
pos = hi > 0
t = lo[pos] / hi[pos]
with np.errstate(divide="ignore"):
u = np.log(t)
num, den = np.expm1((r + 1) * u), np.expm1(u)
ratio = np.where(u == 0, r + 1, num / np.where(u == 0, 1.0, den)) # (1 - t^{r+1}) / (1 - t)
out[pos] = hi[pos] ** r * ratio / (r + 1)
return out
def _vec(v, name, max_len):
a = np.array(float_list(v, None, name))
if not 1 <= len(a) <= max_len:
raise Invalid(f"need 1 <= len({name}) <= {max_len}")
if a.min() < 0 or a.max() > 1e6 or a.max() <= 0:
raise Invalid(f"{name}: entries must be in [0, 1e6] and not all zero")
return a / a.max()
def check(inst, ans):
p, q = inst["p"], inst["q"]
r = 1.0 / (1.0 / p + 1.0 / q - 1.0)
a = _vec(ans.get("f"), "f", inst["max_len"])
b = _vec(ans.get("g"), "g", inst["max_len"])
c = np.concatenate([[0.0], np.convolve(a, b), [0.0]])
conv_r = float(np.sum(_seg_power_integral(c[:-1], c[1:], r))) ** (1.0 / r)
fp = float(np.sum(a ** p)) ** (1.0 / p)
gq = float(np.sum(b ** q)) ** (1.0 / q)
return conv_r / (fp * gq), {"r": r}
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