Training-Free Bottleneck Width Planning for Convolutional Autoencoders

Guannan Guo

cs.CVSep 25, 2026

Training-Free Bottleneck Width Planning for Convolutional Autoencoders

Authors: Guannan Guo

Organizations: Beihang University

Abstract

Multiscale Spectral Rate-Distortion (MS-SRD) estimates the bottleneck channels required at user-supplied spatial cuts from training images and a normalized mean-squared error (NMSE) bound, without fitting a neural network. Its covariance-tail rule is exact for shared linear block-convolutional autoencoders under squared error. A nested-scale dominance result motivates reporting the activation-parameter Pareto frontier alongside the minimum-latent candidate. At NMSE <= 0.01 on thirteen grayscale datasets, its latent-size prediction has 0.84% mean absolute percentage error against nonlinear patch-autoencoder boundaries; ten predictions are exact and the remaining three differ by one channel. In a four-dataset deployable comparison, MS-SRD matches all retrospective external widths and all four selected models pass, without training a selector; a 46-fit validation grid and four Least-Volume fits each pass on two datasets. In a skip-closed U-shaped autoencoder at the same bound, five predictions are exact, nine are within one channel, and every failing prediction is one channel short. Experiments at looser bounds show progressively larger nonlinear savings.

Figures & tables

Figure 1: The MS-SRD pipeline. The mean image is estimated from the training split. “Pooling patches” means collecting patch vectors from every training image and spatial block into one covariance sample; it is not a neural pooling layer. Network training is used only for subsequent evaluation.

Dataset	Shape	Development	External	Dataset	Shape	Development	External
MNIST	$28^{2}$	20,000	5,000	CIFAR-100	$32^{2}$	20,000	5,000
KMNIST	$28^{2}$	20,000	5,000	ChestMNIST	$28^{2}$	20,000	5,000
Fashion-MNIST	$28^{2}$	20,000	5,000	PneumoniaMNIST	$28^{2}$	5,232	624
CIFAR-10	$32^{2}$	20,000	5,000	BreastMNIST	$28^{2}$	624	156
Oxford-IIIT Pet	$64^{2}$	3,680	3,669	OrganAMNIST	$28^{2}$	20,000	5,000
EuroSAT	$64^{2}$	20,000	5,000	RetinaMNIST	$28^{2}$	1,200	400

Table 1: Dataset pools after resizing and grayscale conversion. “Development” supplies the mean image, covariance estimate, and optimization images. “External” supplies the retrospective patch-model comparison boundary. For the U-shaped experiment, an internal validation subset is removed from Development and External is evaluated only after checkpoint selection. Counts are the images actually used after deterministic caps, not necessarily the full released split sizes.

Dataset	$d_{\rm PCA}$	MS-SRD prediction		Empirical oracle			Error
		Tensor	$M_{\rm pred}$	Tensor	$M_{\rm emp}$	NMSE
MNIST	330	$4\times 4\times 38$	608	$4\times 4\times 38$	608	.0089	0.0%
KMNIST	496	$4\times 4\times 37$	592	$4\times 4\times 38$	608	.0087	2.6%
Fashion-MNIST	452	$4\times 4\times 37$	592	$4\times 4\times 37$	592	.0092	0.0%
CIFAR-10	431	$4\times 4\times 32$	512	$4\times 4\times 32$	512	.0092	0.0%
CIFAR-100	408	$4\times 4\times 30$	480	$4\times 4\times 30$	480	.0097	0.0%

Table 2: Training-free predictions and empirical nonlinear boundaries at

\delta=0.01

d_{\rm PCA}

is the global PCA dimension at the same NMSE bound;

M_{\rm pred}

and

M_{\rm emp}

are the predicted and empirical latent scalar counts. Error is their absolute percentage difference.

Figure 2: Training-free predicted latent scalar counts versus the empirical minimum tested counts. The dashed line is equality. Numbers identify datasets in the legend.

Figure 3: Global PCA, MS-SRD, and empirical nonlinear bottleneck sizes. Global PCA uses unrestricted whole-image coefficients; MS-SRD and the empirical model count coefficients repeated over a spatial bottleneck grid.

Figure 4: Activation–parameter trade-off for the same Oxford-IIIT Pet training images rendered at three geometries. Blue points are Pareto-optimal; gray points are dominated. Larger blocks reduce repeated latent activations but require larger shared linear transforms.

Dataset	Predicted tensor	$M_{\rm pred}$	Empirical tensor	$M_{\rm emp}$	NMSE	Bootstrap
Oxford-IIIT Pet	$8\times 8\times 33$	2,112	$8\times 8\times 33$	2,112	.0096	$c=33$ : 12; $c=32$ : 8
EuroSAT	$8\times 8\times 26$	1,664	$8\times 8\times 26$	1,664	.0096	$c=26$ : 14; $c=25$ : 6

Table 3: Training-free predictions and nonlinear patch-autoencoder boundaries on the

64\times 64

extension at

\delta=0.01

. The bootstrap column reports selection counts over 20 image-level resamples.

Figure 5: External-pool NMSE for the two

64\times 64

datasets. The dashed horizontal line is

\delta=0.01

and the dotted vertical line is the training-free predicted scalar count. Both curves cross at the predicted width.

Selector	Selected channels	Fits	MAPE	External passes	Mean external NMSE
MS-SRD	38, 32, 33, 26	0	0.00%	4/4	.00933
Block PCA	38, 32, 33, 26	0	0.00%	3/4	.00959
Validation grid	38, 31, 32, 26	46	1.54%	2/4	.00971
Least Volume	39, 31, 32, 26	4	2.20%	2/4	.00970

Table 4: Deployable fixed-cut width selection on MNIST, CIFAR-10, Oxford-IIIT Pet, and EuroSAT. The channel vector follows that order. “Fits” counts neural models used to select width; the one nonlinear fit used only to evaluate the MS-SRD-selected width is not charged as selection. MAPE compares channels with the retrospective external oracle.

Figure 6: Patch covariance eigenvalues, normalized by total patch variance, at each candidate scale for four representative datasets.

Figure 7: External-pool NMSE against latent scalar count for representative datasets and scales. The horizontal line is

\delta=0.01

; the vertical dotted line is the MS-SRD prediction.

Figure 8: Mean and range across two additional seeds at

\delta=0.01

. Every empirical-boundary run passes; one of 26 one-channel-smaller runs also falls just below the bound.

$\delta$	Median $c_{\rm test}/c_{\rm MS}$	MS MAE	Exact	Within $\pm 1$	Pass	Val. MAE
0.01	1.000	1.36	5/11	9/11	7/11	0.64
0.02	0.967	1.91	5/11	7/11	11/11	0.45
0.05	0.692	4.55	1/11	2/11	10/11	0.55
0.10	0.429	4.36	2/11	2/11	11/11	0.45

Table 5: Relation between MS-SRD channel counts and protocol-specific test boundaries for the skip-closed U-shaped autoencoder. “Pass” counts datasets whose unmodified MS-SRD width meets the test bound. “Val. MAE” compares the validation-selected and retrospective test widths.

Figure 9: Test NMSE versus terminal channels for the skip-closed U-shaped autoencoder. Filled circles are retrospective adjacent-channel crossings; open squares are MS-SRD widths. The test boundary is an analysis oracle, not a deployable selector.

Figure 10: Left: MS-SRD width versus the protocol-specific test boundary. Right: their ratio by NMSE bound; diamonds connect per-bound medians. The primary

\delta=0.01

points are emphasized; looser bounds show the change in nonlinear savings.

Dataset	$M$	$S$	$S/M$	Zero NMSE	Dataset	$M$	$S$	$S/M$	Zero NMSE
MNIST	608	21,952	36.1	.0246%	CIFAR-100	480	28,672	59.7	.0710%
KMNIST	592	21,952	37.1	.0406%	ChestMNIST	272	21,952	80.7	.0798%
Fashion-MNIST	592	21,952	37.1	.0457%	PneumoniaMNIST	384	21,952	57.2	.1089%
CIFAR-10	512	28,672	56.0	.0654%	BreastMNIST	352	21,952	62.4	.0866%
OrganAMNIST	720	21,952	30.5	.0465%	RetinaMNIST	288	21,952	76.2	.0888%
BloodMNIST	448	21,952	49.0	.0808%

Table 6: Full-skip U-Net audit.

M

is the MS-SRD-predicted terminal scalar count at

\delta=0.01

S

is raw skip-path activation count, and zero NMSE is measured with the terminal activation identically zero.

Figure 11: External-split NMSE for full-skip U-Nets with a zeroed terminal path or one terminal channel. Every run lies below

\delta=0.01

Convolutional Autoencoder

Convolutional autoencoders learn compact spatial or spectral representations by encoding inputs into latent codes and reconstructing them with convolutional decoders.

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Autoencoders (AEs) learn low-dimensional representations by mapping data into a latent space while minimizing reconstruction error. Despite their empirical success, theoretical understanding remains limited and largely restricted to linear models or settings without a bottleneck. In this work, we study nonlinear AEs with a fixed finite-dimensional bottleneck in the mean-field (MF) regime. We derive explicit MF learning dynamics for both encoder and decoder, providing a tractable characterization of training in the nonlinear setting. We show that, over finite time horizons, the empirical risk of finite-width networks trained with stochastic gradient descent closely tracks the MF risk trajectory with high probability. At optimality, we further establish that the finite-width risk converges to the MF optimum, demonstrating that finite networks are sufficiently expressive to approximate the infinite-width solution.

Abstract

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