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515b676 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 | """Targeted sweeps and negative controls.
ksweep - paper Fig. 2: error vs k at fixed n (predicted sqrt(k) growth)
sigma - sigma-dependence of the over-specification term (Theorem 3.2)
rho - rho-dependence of the under-specification floor (Theorem 3.2, k<0)
ktrue - |k|-dependence of the floor vs the EXACT Lemma 2.1 bound
rankctrl - negative control: same d, but the discarded direction carries no signal
sparse - sparse binary RDPG, rho_n = n^{-gamma} (Conjecture 1 / Sec. 4.2)
hubctrl - negative control for Conjecture 1: variance condition c <= E|E|^2 broken
"""
import json
import sys
import numpy as np
import rdpg
R = 5
def _ase_err(A, Xt, dims, r=R):
s, U = rdpg.full_spectrum(A)
out = {}
for d in dims:
Xh = rdpg.ase_from_spectrum(s, U, d)
e, _ = rdpg.err_2inf(Xh, Xt)
out[d] = {"err": e}
if d > r:
out[d]["trail"] = rdpg.two_inf(Xh[:, r:d])
out[d]["U_trail_2inf"] = rdpg.two_inf(U[:, r:d])
out[d]["s_hat_d"] = float(abs(s[d - 1]))
return out, s, U
def ksweep(rng, reps=8):
"""Paper Fig. 2: exponential noise, error as a function of embedding dim."""
dims = [3, 4, 5, 6, 7, 8, 10, 14, 20, 30, 40, 60, 80]
res = {}
for n in [1000, 2000, 4000]:
acc = {d: [] for d in dims}
tacc = {d: [] for d in dims if d > R}
for _ in range(reps):
A, Xt = rdpg.weighted_rdpg(n, R, rng, kind="exponential", scale=0.1)
o, _, _ = _ase_err(A, Xt, dims)
for d in dims:
acc[d].append(o[d]["err"])
if d > R:
tacc[d].append(o[d]["trail"])
res[n] = {"dims": dims,
"err_mean": [float(np.mean(acc[d])) for d in dims],
"err_se": [float(np.std(acc[d], ddof=1) / np.sqrt(reps)) for d in dims],
"trail_mean": {str(d): float(np.mean(tacc[d])) for d in dims if d > R}}
ks = [d - R for d in dims if d > R]
res[n]["k_slope_err"] = rdpg.loglog_slope(
ks, [np.mean(acc[d]) for d in dims if d > R])
res[n]["k_slope_trail"] = rdpg.loglog_slope(
ks, [np.mean(tacc[d]) for d in dims if d > R])
res[n]["argmin_dim"] = int(dims[int(np.argmin([np.mean(acc[d]) for d in dims]))])
return res
def sigma(rng, reps=8):
"""Theorem 3.2 writes the over-specification term as sqrt(sigma^2 k)(...)/n^{1/4},
i.e. sigma^1. Lemma 2.1 builds that term as ||Uhat_{r+1:r+k}||_{2,inf}*||E||^{1/2},
and ||E|| ~ 2 sigma sqrt(n), which predicts sigma^{1/2}. We measure the exponent."""
n, d = 2000, 10
sigmas = [0.025, 0.05, 0.1, 0.2, 0.4, 0.8]
T, EV, SV, EX = [], [], [], []
for sg in sigmas:
t, ev, sv, ex = [], [], [], []
for _ in range(reps):
A, Xt = rdpg.weighted_rdpg(n, R, rng, kind="normal", scale=sg)
o, s, U = _ase_err(A, Xt, [R, d])
t.append(o[d]["trail"])
ev.append(o[d]["U_trail_2inf"])
sv.append(np.sqrt(o[d]["s_hat_d"]))
ex.append(o[d]["err"] - o[R]["err"])
T.append(float(np.mean(t))); EV.append(float(np.mean(ev)))
SV.append(float(np.mean(sv))); EX.append(float(np.mean(ex)))
return {"n": n, "d": d, "k": d - R, "sigmas": sigmas,
"trail_mean": T, "U_trail_2inf_mean": EV, "sqrt_s_hat_mean": SV,
"excess_err_mean": EX,
"exp_trail": rdpg.loglog_slope(sigmas, T),
"exp_U": rdpg.loglog_slope(sigmas, EV),
"exp_sqrt_s": rdpg.loglog_slope(sigmas, SV),
"exp_excess": rdpg.loglog_slope(sigmas, EX),
"bbp_ratio": [float(n / 30 / (sg * np.sqrt(n))) for sg in sigmas]}
def rho(rng, reps=8):
"""Theorem 3.2 (k<0): floor >~ sqrt(|k| rho_n). Sweep rho at fixed n, |k|=2."""
n, d = 4000, 3
rhos = [1.0, 0.5, 0.25, 0.125, 0.0625, 0.03125]
E, B = [], []
for rh in rhos:
e, b = [], []
for _ in range(reps):
X = rdpg.dirichlet_latent(n, R, rng)
A, Xt = rdpg.weighted_rdpg(n, R, rng, rho=rh, kind="normal",
scale=0.02, X=X)
o, _, _ = _ase_err(A, Xt, [d])
s_nz = np.sort(np.linalg.eigvalsh(rh * (X.T @ X)))[::-1]
e.append(o[d]["err"])
b.append(float(np.sqrt(s_nz[d:].sum() / n)))
E.append(float(np.mean(e))); B.append(float(np.mean(b)))
return {"n": n, "d": d, "k": d - R, "rhos": rhos, "floor_mean": E,
"lemma21_bound_mean": B,
"ratio": [float(a / b) for a, b in zip(E, B)],
"sqrt_k_rho": [float(np.sqrt(2 * rh)) for rh in rhos],
"exp_floor": rdpg.loglog_slope(rhos, E),
"exp_bound": rdpg.loglog_slope(rhos, B)}
def ktrue(rng, reps=8):
"""|k|-dependence of the under-specification floor vs the exact Lemma 2.1 bound."""
n = 4000
dims = [1, 2, 3, 4]
E = {d: [] for d in dims}
B = {d: [] for d in dims}
for _ in range(reps):
X = rdpg.dirichlet_latent(n, R, rng)
A, Xt = rdpg.weighted_rdpg(n, R, rng, kind="normal", scale=0.1, X=X)
o, _, _ = _ase_err(A, Xt, dims)
s_nz = np.sort(np.linalg.eigvalsh(X.T @ X))[::-1]
for d in dims:
E[d].append(o[d]["err"])
B[d].append(float(np.sqrt(s_nz[d:].sum() / n)))
return {"n": n, "dims": dims,
"floor_mean": [float(np.mean(E[d])) for d in dims],
"lemma21_bound_mean": [float(np.mean(B[d])) for d in dims],
"ratio": [float(np.mean(E[d]) / np.mean(B[d])) for d in dims],
"sqrt_k_rho": [float(np.sqrt(R - d)) for d in dims]}
def rankctrl(rng, reps=8):
"""NEGATIVE CONTROL for the k<0 lower bound. At the SAME embedding
dimension d=4 we compare (i) a true rank-5 P (one signal direction is
discarded -> floor) with (ii) a true rank-4 P (nothing is discarded ->
no floor). If a floor appeared in (ii) as well, the effect would be an
artefact of the dimension count rather than of discarded signal."""
dims = [4]
out = {"n": [], "rank5_d4": [], "rank4_d4": [], "rank5_d5": []}
for n in [500, 1000, 2000, 4000]:
a, b, c = [], [], []
for _ in range(reps):
X5 = rdpg.dirichlet_latent(n, 5, rng)
A5, Xt5 = rdpg.weighted_rdpg(n, 5, rng, kind="normal", scale=0.1, X=X5)
o5, _, _ = _ase_err(A5, Xt5, [4, 5])
X4 = rdpg.dirichlet_latent(n, 4, rng)
A4, Xt4 = rdpg.weighted_rdpg(n, 4, rng, kind="normal", scale=0.1, X=X4)
o4, _, _ = _ase_err(A4, Xt4, [4], r=4)
a.append(o5[4]["err"]); b.append(o4[4]["err"]); c.append(o5[5]["err"])
out["n"].append(n)
out["rank5_d4"].append(float(np.mean(a)))
out["rank4_d4"].append(float(np.mean(b)))
out["rank5_d5"].append(float(np.mean(c)))
for key in ["rank5_d4", "rank4_d4", "rank5_d5"]:
out["slope_" + key] = rdpg.loglog_slope(out["n"], out[key])
return out
def sparse(rng, reps=6):
"""Sparse binary RDPG, rho_n = n^{-gamma} (paper Eq. 18). The k<0 floor is
predicted to be sqrt(|k| rho_n) ~ n^{-gamma/2}: a *decaying* floor whose
slope is set by gamma. This is a sharp, falsifiable prediction."""
dims = [3, 4, 5, 6, 7, 10, 20]
ngrid = [500, 1000, 2000, 4000, 8000]
res = {}
for gam in [0.0, 0.25, 0.5]:
per = {d: [] for d in dims}
bnd, dl = [], []
for n in ngrid:
acc = {d: [] for d in dims}
bb, dd = [], []
for _ in range(reps):
X = rdpg.dirichlet_latent(n, R, rng)
rh = n ** (-gam)
A, Xt = rdpg.binary_rdpg(n, R, rng, rho=rh, X=X)
o, s, U = _ase_err(A, Xt, dims)
for d in dims:
acc[d].append(o[d]["err"])
s_nz = np.sort(np.linalg.eigvalsh(rh * (X.T @ X)))[::-1]
bb.append(float(np.sqrt(s_nz[3:].sum() / n)))
dd.append(float(np.abs(U[:, R]).max()))
for d in dims:
per[d].append(float(np.mean(acc[d])))
bnd.append(float(np.mean(bb)))
dl.append(float(np.mean(dd)))
res[str(gam)] = {"ngrid": ngrid,
"err": {str(d): per[d] for d in dims},
"lemma21_bound_d3": bnd,
"deloc_rp1": dl,
"slopes": {str(d): rdpg.loglog_slope(ngrid, per[d]) for d in dims},
"slope_bound_d3": rdpg.loglog_slope(ngrid, bnd),
"slope_deloc": rdpg.loglog_slope(ngrid, dl),
"predicted_floor_slope": -gam / 2}
return res
def hubctrl(rng, reps=6):
"""NEGATIVE CONTROL for Conjecture 1. The conjecture relaxes Assumption A7
to c <= E|E_ij|^2 <= C, i.e. entry variances bounded AWAY FROM ZERO. A
bounded-degree binary graph (rho_n = c/n) violates the lower bound c, and
there eigenvector localisation is expected. We compare the delocalisation
statistic in the dense (conjecture-compliant) and bounded-degree regimes."""
ngrid = [1000, 2000, 4000]
out = {"ngrid": ngrid, "dense": [], "bounded_degree": [],
"dense_ipr": [], "bounded_degree_ipr": []}
for n in ngrid:
a, b, ai, bi = [], [], [], []
for _ in range(reps):
A, _ = rdpg.binary_rdpg(n, R, rng, rho=1.0)
s, U = rdpg.full_spectrum(A)
a.append(float(np.abs(U[:, R]).max())); ai.append(float((U[:, R] ** 4).sum()))
A, _ = rdpg.binary_rdpg(n, R, rng, rho=12.0 / n)
s, U = rdpg.full_spectrum(A)
b.append(float(np.abs(U[:, R]).max())); bi.append(float((U[:, R] ** 4).sum()))
out["dense"].append(float(np.mean(a)))
out["bounded_degree"].append(float(np.mean(b)))
out["dense_ipr"].append(float(np.mean(ai)))
out["bounded_degree_ipr"].append(float(np.mean(bi)))
out["slope_dense"] = rdpg.loglog_slope(ngrid, out["dense"])
out["slope_bounded_degree"] = rdpg.loglog_slope(ngrid, out["bounded_degree"])
out["slope_dense_ipr"] = rdpg.loglog_slope(ngrid, out["dense_ipr"])
out["slope_bd_ipr"] = rdpg.loglog_slope(ngrid, out["bounded_degree_ipr"])
return out
def decompctrl(rng, reps=8):
"""NEGATIVE CONTROL for the two-term decomposition of Lemma 2.1. We rebuild
the d-dimensional embedding with its trailing columns SET TO ZERO,
[Xhat_{1:r} | 0], and re-solve the same O_d Procrustes problem. If the
n^{-1/4} degradation really comes from the trailing block, this ablated
embedding must fall back exactly onto the n^{-1/2} base curve."""
dims = [10, 20]
out = {"ngrid": [], "base": [], "full": {str(d): [] for d in dims},
"ablated": {str(d): [] for d in dims}}
for n in [500, 1000, 2000, 4000]:
b, f, a = [], {d: [] for d in dims}, {d: [] for d in dims}
for _ in range(reps):
A, Xt = rdpg.weighted_rdpg(n, R, rng, kind="normal", scale=0.1)
s, U = rdpg.full_spectrum(A)
Xr = rdpg.ase_from_spectrum(s, U, R)
b.append(rdpg.err_2inf(Xr, Xt)[0])
for d in dims:
Xh = rdpg.ase_from_spectrum(s, U, d)
f[d].append(rdpg.err_2inf(Xh, Xt)[0])
Xz = Xh.copy()
Xz[:, R:] = 0.0
a[d].append(rdpg.err_2inf(Xz, Xt)[0])
out["ngrid"].append(n)
out["base"].append(float(np.mean(b)))
for d in dims:
out["full"][str(d)].append(float(np.mean(f[d])))
out["ablated"][str(d)].append(float(np.mean(a[d])))
out["slope_base"] = rdpg.loglog_slope(out["ngrid"], out["base"])
out["slope_full"] = {k: rdpg.loglog_slope(out["ngrid"], v) for k, v in out["full"].items()}
out["slope_ablated"] = {k: rdpg.loglog_slope(out["ngrid"], v) for k, v in out["ablated"].items()}
return out
if __name__ == "__main__":
which = sys.argv[1]
rng = np.random.default_rng(hash(which) % (2 ** 31))
fn = {"ksweep": ksweep, "sigma": sigma, "rho": rho, "ktrue": ktrue,
"rankctrl": rankctrl, "sparse": sparse, "hubctrl": hubctrl,
"decompctrl": decompctrl}[which]
res = fn(rng)
with open(f"outputs/extra_{which}.json", "w") as f:
json.dump(res, f, indent=1)
print(which, "done")
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