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1f48ccf | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 | """Figures for the numerical audits of the spectral statements (Claims 1 and 2)."""
from __future__ import annotations
import json
import math
import os
import numpy as np
import pandas as pd
import plotly.graph_objects as go
from analyze import PALETTE, _c, write_fig, linfit
RES = "results"
def main():
out = {}
a = pd.read_csv(f"{RES}/audit_spectrum.csv")
dims = sorted(a["d"].unique())
# --- lambda1(A*): diverges for quadratic, converges to 6 for truncated -----
fig = go.Figure()
for i, d in enumerate(dims):
q = a[(a["act"] == "quad") & (a["d"] == d)].groupby("delta")["lam1"].mean().reset_index()
t = a[(a["act"] == "trunc") & (a["M"] == 8.0) & (a["d"] == d)] \
.groupby("delta")["lam1"].mean().reset_index()
fig.add_trace(go.Scatter(x=q["delta"], y=q["lam1"], mode="lines+markers",
name=f"quad d={d}", line=dict(color=_c(i, len(dims)), dash="dash")))
fig.add_trace(go.Scatter(x=t["delta"], y=t["lam1"], mode="lines+markers",
name=f"trunc d={d}", line=dict(color=_c(i, len(dims)))))
fig.add_hline(y=6, line_dash="dot", line_color="#111",
annotation_text="population λ₁ = 6")
fig.update_layout(title="λ₁(A*) vs δ = n/d — quadratic (dashed) vs truncated M=8 (solid)",
xaxis_title="δ = n/d", yaxis_title="λ₁(A*)", xaxis_type="log",
yaxis_type="log", template="plotly_white", height=470)
write_fig(fig, "audit_lam1")
# --- the eq. (3.13) error: |lam1-6| + |lam2-2| vs the claimed rate ---------
t = a[(a["act"] == "trunc")].copy()
t["err"] = (t["lam1"] - 6).abs() + (t["lam2"] - 2).abs()
t["rate"] = np.exp(-t["M"] / 3) + t["M"] * np.sqrt(t["d"] / t["n"])
g = t.groupby(["M", "delta", "d"])[["err", "rate"]].mean().reset_index()
g["C"] = g["err"] / g["rate"]
g.to_csv(f"{RES}/audit_eq313.csv", index=False)
fig = go.Figure()
Ms = sorted(g["M"].unique())
for i, M in enumerate(Ms):
s = g[g["M"] == M]
fig.add_trace(go.Scatter(x=s["rate"], y=s["err"], mode="markers",
marker=dict(size=9, color=_c(i, len(Ms))), name=f"M={M:g}"))
lim = [float(g["rate"].min()) * 0.8, float(g["rate"].max()) * 1.2]
for C, dash in ((1.0, "dot"), (0.5, "dash")):
fig.add_trace(go.Scatter(x=lim, y=[C * lim[0], C * lim[1]], mode="lines",
line=dict(color="#444", dash=dash), name=f"C = {C}"))
fig.update_layout(
title="Eq. (3.13) audit: |λ₁−6| + |λ₂−2| vs C(e^(−M/3) + M√(d/n))",
xaxis_title="e^(−M/3) + M√(d/n)", yaxis_title="|λ₁−6| + |λ₂−2|",
xaxis_type="log", yaxis_type="log", template="plotly_white", height=470)
write_fig(fig, "audit_eq313")
out["eq313_max_C"] = float(g["C"].max())
out["eq313_max_C_largedelta"] = float(g[g["delta"] >= 16]["C"].max())
# --- uniform-in-theta BBP --------------------------------------------------
b = pd.read_csv(f"{RES}/audit_uniform_bbp.csv")
bt = b[(b["act"] == "trunc") & (b["M"] == 8.0)].copy()
bt["kind"] = np.where(bt["theta"].str.startswith("random"), "random θ", bt["theta"])
fig = go.Figure()
kinds = ["random θ", "theta_star", "adversarial"]
cols = {"random θ": PALETTE[1], "theta_star": PALETTE[4], "adversarial": PALETTE[6]}
for k in kinds:
s = bt[bt["kind"] == k]
for j, col in enumerate(("lam1", "lam2")):
fig.add_trace(go.Scatter(
x=s["delta"], y=s[col], mode="markers",
marker=dict(size=11, color=cols[k], symbol="circle" if j == 0 else "x"),
name=f"{k} — λ{j+1}", legendgroup=k, showlegend=True))
fig.add_hline(y=6, line_dash="dot", line_color="#111")
fig.add_hline(y=2, line_dash="dot", line_color="#111")
fig.update_layout(
title="Uniform-in-θ BBP transition of A(θ): λ₁ (circles) and λ₂ (crosses), truncated M=8",
xaxis_title="δ = n/d", yaxis_title="eigenvalue of A(θ)", xaxis_type="log",
template="plotly_white", height=470)
write_fig(fig, "audit_uniform_bbp")
s64 = bt[bt["delta"] == 64.0]
out["bbp_delta64"] = dict(lam1_min=float(s64["lam1"].min()), lam1_max=float(s64["lam1"].max()),
lam2_min=float(s64["lam2"].min()), lam2_max=float(s64["lam2"].max()),
ov_min=float(s64["sq_overlap_v1"].min()),
ov_max=float(s64["sq_overlap_v1"].max()))
# --- indicator mass -------------------------------------------------------
c = pd.read_csv(f"{RES}/audit_indicator.csv")
ct = c[c["act"] == "trunc"].copy()
ct["kind"] = np.where(ct["theta"].str.startswith("random"), "random θ", ct["theta"])
fig = go.Figure()
for i, k in enumerate(["random θ", "theta_star", "adversarial"]):
s = ct[ct["kind"] == k]
fig.add_trace(go.Box(x=s["M"], y=s["ratio"], name=k,
marker_color=[PALETTE[1], PALETTE[4], PALETTE[6]][i]))
fig.add_hline(y=1.0, line_dash="dot", line_color="#111",
annotation_text="bound with C = 1")
fig.update_layout(
title="Uniform indicator-mass bound: measured mass ÷ (e^(−M/2) + √(d/n)·log(n/d))",
xaxis_title="M", yaxis_title="ratio", template="plotly_white", height=440,
boxmode="group")
write_fig(fig, "audit_indicator")
out["indicator_max_ratio"] = float(ct["ratio"].max())
out["indicator_max_ratio_adv"] = float(ct[ct["kind"] == "adversarial"]["ratio"].max())
# --- Theorem 3.2 deficit bound --------------------------------------------
th = pd.read_csv(f"{RES}/thm32_bound.csv")
gg = th.groupby(["M", "delta"])[["deficit", "rate", "C_implied"]].mean().reset_index()
fig = go.Figure()
Ms = sorted(gg["M"].unique())
for i, M in enumerate(Ms):
s = gg[gg["M"] == M]
fig.add_trace(go.Scatter(x=s["rate"], y=s["deficit"], mode="markers+lines",
marker=dict(size=10, color=_c(i, len(Ms))),
line=dict(color=_c(i, len(Ms))), name=f"M={M:g}"))
lim = [float(gg["rate"].min()) * 0.9, float(gg["rate"].max()) * 1.1]
fig.add_trace(go.Scatter(x=lim, y=lim, mode="lines", line=dict(color="#444", dash="dot"),
name="C = 1"))
fig.update_layout(
title="Theorem 3.2 audit: realised deficit 1 − |⟨θ_∞,θ*⟩| vs e^(−M/2) + (d/n)^(1/5), d=512",
xaxis_title="e^(−M/2) + (d/n)^(1/5)", yaxis_title="1 − |⟨θ_∞, θ*⟩|",
xaxis_type="log", yaxis_type="log", template="plotly_white", height=470)
write_fig(fig, "audit_thm32")
out["thm32_max_C_Mge4"] = float(th[th["M"] >= 4]["C_implied"].max())
out["thm32_max_C_all"] = float(th["C_implied"].max())
# --- smooth vs hard truncation robustness ---------------------------------
p = f"{RES}/sweep_smooth.csv"
if os.path.exists(p):
sm = pd.read_csv(p).groupby(["d", "delta"])["sq_overlap"].mean().reset_index()
hd = pd.read_csv(f"{RES}/sweep_trunc.csv").groupby(["d", "delta"])["sq_overlap"] \
.mean().reset_index()
dims2 = sorted(sm["d"].unique())
fig = go.Figure()
for i, d in enumerate(dims2):
s = sm[sm["d"] == d].sort_values("delta")
h = hd[hd["d"] == d].sort_values("delta")
fig.add_trace(go.Scatter(x=s["delta"], y=s["sq_overlap"], mode="lines+markers",
name=f"smooth d={d}", line=dict(color=_c(i, len(dims2)))))
fig.add_trace(go.Scatter(x=h["delta"], y=h["sq_overlap"], mode="lines",
name=f"hard d={d}",
line=dict(color=_c(i, len(dims2)), dash="dot")))
fig.update_layout(
title="Robustness: C^∞ truncation (eq. 3.10, solid) vs hard truncation (eq. 4.3, dotted)",
xaxis_title="δ = n/d", yaxis_title="squared overlap",
template="plotly_white", height=470)
write_fig(fig, "audit_smooth_vs_hard")
mrg = sm.merge(hd, on=["d", "delta"], suffixes=("_smooth", "_hard"))
mrg["absdiff"] = (mrg["sq_overlap_smooth"] - mrg["sq_overlap_hard"]).abs()
mrg.to_csv(f"{RES}/smooth_vs_hard.csv", index=False)
out["smooth_vs_hard_maxdiff_delta_ge_4"] = float(mrg[mrg["delta"] >= 4]["absdiff"].max())
out["smooth_vs_hard_meandiff_delta_ge_4"] = float(mrg[mrg["delta"] >= 4]["absdiff"].mean())
with open(f"{RES}/audit_summary.json", "w") as f:
json.dump(out, f, indent=2, default=float)
print(json.dumps(out, indent=2, default=float))
if __name__ == "__main__":
main()
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