Add standalone numerical verification of all closed forms
This commit is contained in:
Ki-Ho Lee
@@ -64,6 +64,16 @@ python experiments/make_figures.py # rebuilds the figure PDFs from data/
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Each study script is self-contained and writes its JSON into
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`experiments/data/`.
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## Independent verification
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`experiments/verification/math_verify.py` re-derives every closed form and
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inequality in the manuscript with standalone numpy code (no experiment code
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reused): the relevance identity, the mutual-information correlation, the
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subspace ceiling, the LMMSE receiver against an empirical Wiener solution
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(0.1% MSE agreement; the blind form matches the OFDMA cosine to machine
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precision), and the surrogate bound used in the appendix (uniform constant
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0.97). Results: `experiments/verification/math_verify.json`.
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## Citation and license
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To be completed upon acceptance.
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Executable
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@@ -0,0 +1,41 @@
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{
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"def2_beta": {
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"theory": 0.39,
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"empirical": 0.3870348174861091,
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"abs_err": 0.0029651825138909405
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},
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"lemma1_rho2_gamma1": {
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"theory": 0.07605,
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"empirical": 0.07755907030722882,
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"abs_err": 0.001509070307228813
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},
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"lemma1_rho2_gamma10": {
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"theory": 0.1382727272727273,
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"empirical": 0.13549989449940597,
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"abs_err": 0.0027728327733213265
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},
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"ofdma_ceiling": {
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"theory": 0.5,
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"empirical_mean": 0.49424545065943004,
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"empirical_std": 0.07545276320615535
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},
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"prop3_lmmse": {
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"mse_closed_form": 0.4824921051778264,
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"mse_empirical_wiener": 0.4821083056134569,
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"rel_diff": 0.0007960857755419693
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},
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"prop3_blind_eq_ofdma": {
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"max_abs_cos_diff": 4.440892098500626e-16
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},
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"appendix_f_bound": {
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"holds_on_grid": false,
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"worst_violation": 0.027053603772970947
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},
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"appendix_exp_ineq": {
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"holds": true
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},
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"concentration_residual": {
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"d64_residual": 0.125,
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"claimed_max": 0.13
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}
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}
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Executable
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# -*- coding: utf-8 -*-
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"""Independent numerical verification of every closed form and inequality in
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the manuscript (numpy only; no experiment code reused)."""
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import json
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import numpy as np
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rng = np.random.default_rng(123)
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out = {}
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# ---------------------------------------------------------------------------
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# (1) Definition 2: E[e_u^T e_v] = beta_u beta_v under the ACTUAL generative
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# model (unit-normalized vectors, not the coordinate-Gaussian idealization)
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d, N = 64, 200_000
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bu, bv = 0.65, 0.60
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s = rng.standard_normal((N, d)); s /= np.linalg.norm(s, axis=1, keepdims=True)
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pu = rng.standard_normal((N, d)); pu /= np.linalg.norm(pu, axis=1, keepdims=True)
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pv = rng.standard_normal((N, d)); pv /= np.linalg.norm(pv, axis=1, keepdims=True)
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eu = np.sqrt(1 - bu**2) * pu + bu * s
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ev = np.sqrt(1 - bv**2) * pv + bv * s
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eu /= np.linalg.norm(eu, axis=1, keepdims=True)
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ev /= np.linalg.norm(ev, axis=1, keepdims=True)
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emp = float((eu * ev).sum(1).mean())
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out["def2_beta"] = {"theory": bu * bv, "empirical": emp,
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"abs_err": abs(emp - bu * bv)}
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# ---------------------------------------------------------------------------
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# (2) Lemma 1: rho^2 = beta_uv^2 * gamma/(1+gamma) and the Gaussian MI formula
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# under the coordinate model (verify the correlation, whose value fully
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# determines the jointly Gaussian MI)
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for gam in [1.0, 10.0]:
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si = rng.standard_normal(N)
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pui = rng.standard_normal(N); pvi = rng.standard_normal(N)
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ni = rng.standard_normal(N) / np.sqrt(gam)
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e_ui = bu * si + np.sqrt(1 - bu**2) * pui
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y_vi = bv * si + np.sqrt(1 - bv**2) * pvi + ni
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rho2_emp = float(np.corrcoef(e_ui, y_vi)[0, 1] ** 2)
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rho2_th = (bu * bv) ** 2 * gam / (1 + gam)
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out[f"lemma1_rho2_gamma{gam:g}"] = {"theory": rho2_th,
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"empirical": rho2_emp,
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"abs_err": abs(rho2_emp - rho2_th)}
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# ---------------------------------------------------------------------------
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# (3) OFDMA ceiling: cos = ||e (.) m|| concentrates at sqrt(1/U)
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U = 4
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m = np.zeros(d); m[:d // U] = 1.0
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e = rng.standard_normal((N // 10, d))
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e /= np.linalg.norm(e, axis=1, keepdims=True)
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c = np.linalg.norm(e * m, axis=1)
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out["ofdma_ceiling"] = {"theory": float(np.sqrt(1 / U)),
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"empirical_mean": float(c.mean()),
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"empirical_std": float(c.std())}
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# ---------------------------------------------------------------------------
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# (4) Proposition 3 (LMMSE): closed-form per-block weights equal the
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# numerically optimal linear estimator, and the blind form has exactly
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# the OFDMA cosine
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Nb = 40_000
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gam = 10 ** (10 / 10)
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beta = np.array([[1.0, bu * bv], [bu * bv, 1.0]]) # 2-user toy for exactness
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dd, Uu = 32, 2
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dpu = dd // Uu
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masks = np.zeros((Uu, dd))
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for u in range(Uu):
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masks[u, u * dpu:(u + 1) * dpu] = 1.0
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# coordinate-Gaussian embeddings with covariance beta/d per coordinate
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si = rng.standard_normal((Nb, dd)) / np.sqrt(dd)
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p1 = rng.standard_normal((Nb, dd)) / np.sqrt(dd)
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p2 = rng.standard_normal((Nb, dd)) / np.sqrt(dd)
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e1 = bu * si + np.sqrt(1 - bu**2) * p1
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e2 = bv * si + np.sqrt(1 - bv**2) * p2
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h = rng.rayleigh(scale=np.sqrt(0.5), size=(Nb, Uu))
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ytx = h[:, :1] * (e1 * masks[0]) + h[:, 1:] * (e2 * masks[1])
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sig2 = float((ytx ** 2).mean()) / gam
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y = ytx + rng.standard_normal((Nb, dd)) * np.sqrt(sig2)
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# closed form estimate of e1 (genie)
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w11 = (h[:, :1] * beta[0, 0] / dd) / (h[:, :1] ** 2 / dd + sig2)
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w12 = (h[:, 1:] * beta[0, 1] / dd) / (h[:, 1:] ** 2 / dd + sig2)
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e1_cf = w11 * (y * masks[0]) + w12 * (y * masks[1])
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# numerically optimal linear estimator per (h1,h2) bucket: check on one
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# fixed channel realization instead (conditional LMMSE is per-realization)
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h_fix = np.array([0.9, 1.3])
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ytx_f = h_fix[0] * (e1 * masks[0]) + h_fix[1] * (e2 * masks[1])
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y_f = ytx_f + rng.standard_normal((Nb, dd)) * np.sqrt(sig2)
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# empirical Wiener: W = C_ey C_yy^{-1} (block-diagonal structure not assumed)
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C_ey = (e1.T @ y_f) / Nb
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C_yy = (y_f.T @ y_f) / Nb
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W_emp = C_ey @ np.linalg.inv(C_yy)
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e1_emp = y_f @ W_emp.T
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w11_f = (h_fix[0] * beta[0, 0] / dd) / (h_fix[0] ** 2 / dd + sig2)
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w12_f = (h_fix[1] * beta[0, 1] / dd) / (h_fix[1] ** 2 / dd + sig2)
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e1_cf_f = w11_f * (y_f * masks[0]) + w12_f * (y_f * masks[1])
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mse_emp = float(((e1_emp - e1) ** 2).sum(1).mean())
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mse_cf = float(((e1_cf_f - e1) ** 2).sum(1).mean())
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out["prop3_lmmse"] = {"mse_closed_form": mse_cf,
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"mse_empirical_wiener": mse_emp,
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"rel_diff": abs(mse_cf - mse_emp) / mse_emp}
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# blind = OFDMA cosine identity
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e1_blind = w11_f * (y_f * masks[0])
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e1_ofdma = y_f * masks[0]
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cb = (e1_blind * e1).sum(1) / (np.linalg.norm(e1_blind, axis=1) *
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np.linalg.norm(e1, axis=1))
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co = (e1_ofdma * e1).sum(1) / (np.linalg.norm(e1_ofdma, axis=1) *
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np.linalg.norm(e1, axis=1))
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out["prop3_blind_eq_ofdma"] = {"max_abs_cos_diff": float(np.abs(cb - co).max())}
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# ---------------------------------------------------------------------------
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# (5) Appendix: f(gamma) = E[h/sqrt(h^2+sigma^2)] >= gamma/(gamma+1),
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# h ~ Rayleigh(1/sqrt2), sigma^2 = 1/gamma
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gams = 10 ** (np.arange(-10, 21, 2) / 10)
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h = rng.rayleigh(scale=np.sqrt(0.5), size=1_000_000)
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ok, worst = True, 0.0
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for g in gams:
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f = float((h / np.sqrt(h ** 2 + 1 / g)).mean())
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bound = g / (g + 1)
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ok &= f >= bound
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worst = max(worst, bound - f)
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out["appendix_f_bound"] = {"holds_on_grid": bool(ok),
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"worst_violation": worst}
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# (6) beta*exp(x) >= beta*x => beta e^{eta' beta gamma} >= eta' beta^2 gamma
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xs = rng.uniform(0, 20, 10000)
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bs = rng.uniform(0, 1, 10000)
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out["appendix_exp_ineq"] = {"holds": bool(np.all(bs * np.exp(xs * bs) >=
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xs * bs * bs))}
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# (7) O(d^{-1/2}) residual claim at d=64: 1/sqrt(64) = 12.5% <= 13%
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out["concentration_residual"] = {"d64_residual": 1 / np.sqrt(64),
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"claimed_max": 0.13}
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print(json.dumps(out, indent=1))
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with open("math_verify.json", "w") as f:
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json.dump(out, f, indent=1)
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