Universal Local Error and Realized Amplification for the First-Order EDM Predictor

Nicolas Brosse; Arnak S. Dalalyan

Figure 5: The split index

K_{\mathrm{hi}}

of eq. 25 is the last level at or above the threshold

\sigma_{\mathrm{hi}}

. The step

\sigma_{K_{\mathrm{hi}}}\to\sigma_{K_{\mathrm{hi}}+1}

that crosses the threshold (shaded) is the first step of the low-noise block, so every high-noise step ends at or above

\sigma_{\mathrm{hi}}

.

Figure 6: Schematic, not to scale. (a) Along one direction, the local expansion rate of the learned velocity (solid) peaks between the two modes of the law. The shaded density, with one lobe per mode, is the squared-displacement occupation of the optimal-coupling segments between

\mu_{j}

and

\widehat{\mu}_{j}

, which is the weight in eq. 81 : the transport moves mass within each mode, not across the gap where the expansion peaks. The global one-sided rate

\mathcal{L}^{\mathrm{os}}_{j}

is the supremum of the expansion rate (dotted); the directional rate

\mathcal{E}^{\mathrm{OT}}_{j}

is its average under this weight (dashed), and stays small. In higher dimension the supremum also runs over all directions. (b) Majorants of

\Lambda_{K}

; an arrow from

A

to

B

means

A\leq B

, and each arrow discards the information named beside it. The directional and map-level majorants are not ordered, and Proposition 10 shows that the chain cannot be reversed.

quantity of the analysis	one-dimensional mixture	CIFAR–10
final error $e_{K}$ , factors $\gamma_{j,K}$ , $\Lambda_{K}$ ( Definition 8 )	measured by quadrature	not accessible
one-step recursion ( 10 )	every term measured	not accessible
directional majorant $\Lambda_{K}^{\mathrm{dir}}$ eq. 30	measured	displacement rate $\lambda^{\parallel}$ on coarse–fine segments eq. 41
field-level majorant ( 33 )	finite-domain proxy	worst-direction rate $\lambda^{\max}$ eq. 41 ; empirical EDM margin ( 40 )
stability hypotheses	tested quantitatively	not certified

Appendix

Table 1: Quantities of the analysis and what the two experiments measure of them.

$K$	$e_{K}$	order	$\sum_{j}\widetilde{\delta}_{j,K}$	order	$\Lambda_{K}$	ampl. order	$c_{K}$	measured	universal
	final error		summed biases		amplification			bound / error
17	0.113	–	0.166	–	0.325	–	1.388	1.35	134
34	0.0747	0.597	0.0934	0.826	0.548	0.679	1.469	1.40	127
68	0.0497	0.590	0.0496	0.913	0.795	0.643	1.524	1.40	122
136	0.0332	0.583	0.0255	0.959	1.065	0.612	1.555	1.39	119
272	0.0222	0.581	0.0129	0.981	1.344	0.596	1.572	1.38	118

Appendix

Table 2: Oracle denoiser with exact initialization, under refinement. Orders are the local orders eq. 38 on the pair

(K/2,K)

; the amplification-based order is

1-(\Lambda_{K}-\Lambda_{K/2})/\log 2

, and

c_{K}=Ke_{K}e^{-\Lambda_{K}}

. Bound / error divides the recursion bound ( 39 ) by

e_{K}

, with the measured local biases

\widetilde{\delta}_{j,K}

or with their universal bounds

\Delta_{j}

(

B_{1,1}=16

), propagated by the same measured factors.

Figure 7: Oracle denoiser on the four-component mixture, exact initialization,

M=65{,}536

quantiles. (a) Final error

e_{K}

, the recursion bound ( 39 ) built from the measured local biases and factors, and the unweighted sum of the measured biases; triangles mark slopes

1

and

0.58

. (b) Measured

\Lambda_{K}

, with a logarithmic fit on

K\geq 136

. (c) Local orders eq. 38 on each pair

(K/2,K)

, against the amplification-based order

1-(\Lambda_{K}-\Lambda_{K/2})/\log 2

computed from (b). The summed biases converge at first order; the final error does not, and the growth of

\Lambda_{K}

accounts for the gap.

	$K=17$			$K=1088$
denoiser	$\Lambda_{K}$	directional	field	$\Lambda_{K}$	directional	field
oracle	0.325	1.976	143	1.921	1.022	11.1
trained, seed 0	0.324	1.970	27.5	1.328	1.020	5.1
trained, seed 1	0.334	1.934	28.0	1.875	1.017	3.8
trained, seed 2	0.320	1.990	29.6	1.879	1.018	3.8
misspecified mean	0.205	3.938	944	0.673	1.012	75.6

Appendix

Table 3: Measured

\Lambda_{K}

and its two majorants, as ratios to

\Lambda_{K}

, on the coarsest and finest grids. Directional:

\Lambda_{K}^{\mathrm{dir}}

eq. 30 ; field: the finite-domain proxy for eq. 33 . Canonical Gaussian initialization, common noise threshold.

Figure 8: Where expansion sits, and what each stability control costs. Canonical Gaussian initialization. (a) Oracle on

G_{272}

at one low noise level: local expansion rate

\lambda_{j}(x)

, the error weight

w_{j}

(shaded, arbitrary scale), and the two rates each bound pays at this step, the supremum of

\lambda_{j}

(field) and its

w_{j}

-average

\mathcal{E}^{\mathrm{OT}}_{j}

(directional). The error sits where the predictor contracts. (b) Measured

\Lambda_{K}

by denoiser; trained networks as median and range over three seeds. (c) Directional bound (dashed) and finite-domain field proxy (dotted), divided by

\Lambda_{K}

, with the colours of (b).

Figure 9: Stability diagnostics for the pretrained CIFAR–10 EDM model. Shading marks the noise levels at which the 90th-percentile empirical margin of (b) is positive. (a) Normalized denoising loss

r_{\mathrm{reg}}

per coordinate. (b) Empirical margin ( 40 ) from the median and the 90th percentile of the local stretch; dotted lines mark where the latter crosses

0

and the supercritical threshold

6/7

. (c) Energy-weighted rates on the coarse–fine segments of

G_{68}

, at the same nodes: worst direction, displacement, secant, and random direction eq. 41 . Absolute values; open markers are negative rates. (d) Accumulated synchronous rate over

\sigma\leq 2

, energy-weighted over all 1280trajectories with its leave-one-trajectory-out range, and the same accumulation of the level-wise median of the per-trajectory rates.

Figure 10: Terms of the measured recursion ( 99 ) from the canonical Gaussian initialization: final error, recursion bound, and its propagated discretization, learning and initialization contributions. (a) Oracle denoiser. (b) Trained networks, median and range over three seeds. The contributions add up to an upper bound, not to the observed error.

$\rho_{\sf EDM}$	final error	measured biases	universal bound	ratio
2	0.0229	0.0164	4.64	282
3	0.0173	0.0095	5.13	539
7	0.0139	0.0065	7.48	1150

Appendix

Table 4: Clock control at

K=544

, oracle with exact initialization; every grid satisfies

\theta_{K}<1/2

. Ratio: universal bound over measured biases.

$\overline{\sigma}$	$K$	final error	change	$\sum_{j}\Delta_{j}$	$\sum_{j}\Pi_{j+1,K_{\mathrm{hi}}}\Delta_{j}$
19.8	209	0.02176	$-1.6\times 10^{-6}$	3.36	1.54
80	272	0.02176	–	13.77	2.81
318	348	0.02176	$+0.1\times 10^{-6}$	47.60	4.00

Appendix

Table 5: Starting-noise control,

\rho_{\sf EDM}=7

with a fixed clock step; oracle started from the exact top law. The change in final error is relative to

\overline{\sigma}=80

. The last two columns sum the universal bias bounds

\Delta_{j}

over the high-noise steps (

\sigma_{j+1}\geq\sigma_{\mathrm{hi}}=4

), without and with the weights

\Pi_{j+1,K_{\mathrm{hi}}}

of the high-noise kernel.

denoiser	$K$	synchronous	optimal coupling	ratio	$\Lambda$ on window
oracle	68	0.909	0.704	1.29	0.793
	272	1.605	1.296	1.24	1.326
misspecified	68	1.112	0.762	1.46	0.804
	272	1.870	0.687	2.72	0.699

Appendix

Table 6: Calibration of the synchronous diagnostic: accumulated linear rates

\sum_{j}\ell_{j}(\mathcal{E}^{\mathrm{sync}}_{j,K})_{+}

and

\sum_{j}\ell_{j}(\mathcal{E}^{\mathrm{OT}}_{j,K})_{+}

over the levels

\sigma_{j}\leq 2

, canonical Gaussian initialization. Ratio: synchronous over optimal coupling. Last column:

\Lambda_{K}

restricted to the same levels,

\sum_{j}\log\max\{1,\gamma_{j,K}\}

.

pair $(K/2,K)$	quantiles	error order	ampl.-based order	bias-sum order
$(272,544)$	65536	0.5811	0.5886	0.9908
$(272,544)$	262144	0.5779	0.5854	0.9908
$(544,1088)$	65536	0.5850	0.5882	0.9955
$(544,1088)$	262144	0.5782	0.5814	0.9955

Appendix

Table 7: Paired quadrature check with exact initialization. Local orders eq. 38 on the pair

(K/2,K)

; the amplification-based order is

1-(\Lambda_{K}-\Lambda_{K/2})/\log 2

, and the bias-sum order refers to

\sum_{j}\widetilde{\delta}_{j,K}

. Digits show numerical drift rather than certified precision.

quantity	check	criterion	outcome
One-dimensional mixture
endpoint error $e_{K,K}$	$M$ vs $4M$ quantiles on $G_{1088}$ (analytic arms, trained seed 0)	discrepancy $<e_{K,K}/10$	pass; oracle error changes by $<1\%$ . Bound/error ratio not reported for arms without a $4M$ rerun
cumulative log-amplification $\Lambda_{K}$	$M$ vs $4M$ on $G_{1088}$	small relative change	Gaussian start: about $0.1\%$ ; exact start: about $0.45\%$ . Paired orders checked separately in Table 7
one-step recursion ( 98 )	measured slack	nonnegative up to quadrature	pass on every arm and step
field proxy eq. 97	domain $(R,N_{x})=(12,4097)\to(16,8193)$	small relative change	pass; $\leqslant 0.8\%$ . The maximizer hits the domain boundary at 225 of 507 oracle levels on $G_{1088}$ (never for trained arms) but sits on a flat plateau, not clipped
noise threshold	re-summation at the four candidate thresholds of Section G.1.2	range reported	$\Lambda_{K}$ bit-identical; $\Lambda_{K}^{\rm dir}$ varies by $0.13\%$ at $K=1088$

Appendix

Table 8: Numerical reliability checks. Failures lead to omission, not to a qualified report.

Universal Local Error and Realized Amplification for the First-Order EDM Predictor

Abstract

Figures & tables

Appendix

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