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mathematics
information-theory
experimental-design
memory-compression
linear-algebra
quantum-information
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9552b49 | 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 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | """Constructive checks of jointly optimized archive and repair geometry.
Analytic curves and finite algebraic/numerical checks only; no measured
hardware or dataset performance is represented by these values.
"""
import json
import sys
from pathlib import Path
import numpy as np
from scipy.linalg import block_diag
ROOT = Path(__file__).resolve().parents[1]
sys.path.insert(0, str(ROOT / "src"))
from oqaivaei.continuation import balanced_real_projector
TOL = 2e-10
def omitted_basis(q, r):
projection = balanced_real_projector(q, r)
eigenvalues, vectors = np.linalg.eigh(projection)
return vectors[:, eigenvalues > 0.5]
def main():
rng = np.random.default_rng(20261020)
records = []
assertions = 0
residuals = []
for q in range(2, 25):
for r in range(1, q + 1):
B = 3.0
U = omitted_basis(q, r)
assert U.shape == (q, r)
assert np.allclose(U.T @ U, np.eye(r), atol=TOL)
assertions += 2
norms = np.einsum("ij,ij->i", U, U)
shared_primal = float(r * norms.max() / B)
shared_formula = r * r / (q * B)
announced_primal = float(norms.max() / B)
announced_formula = r / (q * B)
for actual, expected in [(shared_primal, shared_formula),
(announced_primal, announced_formula)]:
residuals.append(abs(actual - expected))
assert abs(actual - expected) < TOL
assertions += 1
# Arbitrary dual mixtures cannot exceed the exhibited certificate.
max_random_dual = 0.0
for _ in range(8):
pi = rng.dirichlet(np.ones(q))
G = U.T @ (pi[:, None] * U)
score = float(np.sqrt(np.maximum(0, np.linalg.eigvalsh(G))).sum() ** 2 / B)
max_random_dual = max(max_random_dual, score)
assert score <= shared_formula + TOL
assertions += 1
# Check the actual unbiased signal reconstruction for a random state.
h = rng.normal(size=q)
residual = U.T @ h
retained = h - U @ residual
S = np.sqrt(B / r) * np.eye(r)
y = S @ residual
decoded = retained + np.sqrt(r / B) * U @ y
assert np.allclose(decoded, h, atol=TOL)
assertions += 1
records.append({"q": q, "p": q - r, "r": r, "B": B,
"shared_optimum": shared_formula,
"announced_optimum": announced_formula,
"coordinate_storage_shared": r / B,
"shared_constructed": shared_primal,
"max_random_dual": max_random_dual})
blocks = []
for q in [4, 8, 12, 16, 24]:
for r in range(1, q + 1):
for k in range(1, r + 1):
if q % k or r % k:
continue
s, d = q // k, r // k
U = block_diag(*[omitted_basis(s, d) for _ in range(k)])
assert U.shape == (q, r)
assert np.allclose(U.T @ U, np.eye(r), atol=TOL)
# Each active cell uses all gain B=1 in d dimensions.
costs = []
for g in range(k):
rows = U[g * s:(g + 1) * s]
costs.append(float(d * np.sum(rows * rows, axis=1).max()))
formula = r * r / (k * q)
assert abs(max(costs) - formula) < TOL
residuals.append(abs(max(costs) - formula))
assertions += 3
blocks.append({"q": q, "r": r, "k": k, "B": 1,
"optimum": formula, "constructed": max(costs)})
# Curve used in a worked chapter example: q64, p32, B8.
announcements = []
for bits in range(7):
k = 2 ** bits
r, q, B = 32, 64, 8
v = max(r / q, r * r / (min(k, r) * q)) / B
announcements.append({"q": q, "p": q - r, "B": B,
"advance_bits": bits, "cells": k,
"optimal_variance": v,
"rho": 0.8, "stationary_mse_bound": v / 0.36})
result = {"status": "PASS", "seed": 20261020,
"scope": "analytic constructions; floating-point finite checks",
"checks": assertions, "subspace_cases": len(records),
"block_cases": len(blocks), "tolerance": TOL,
"max_abs_equality_residual": max(residuals),
"memory_frontier": records, "divisible_blocks": blocks,
"announcement_example": announcements}
destination = ROOT / "results" / "codesign_results.json"
destination.parent.mkdir(exist_ok=True)
destination.write_text(json.dumps(result, indent=2) + "\n", encoding="utf-8")
print(json.dumps({key: result[key] for key in
["status", "checks", "subspace_cases", "block_cases",
"max_abs_equality_residual", "tolerance"]}, indent=2))
if __name__ == "__main__":
main()
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