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Beyond Local Linearity: Scale-Resolved Geometry of Learned Image Encoders

Jakub Szymkowiak, Wojtek Pałubicki, Kamil Adamczewski

arXiv:2609.39115v1cs.CV

TL;DR

Existing geometric analyses describe learned representations locally, leaving unmeasured how far they predict feature displacement under finite input changes. This paper compares measured displacement with a local linear prediction across scales and finds a characteristic bump in trained encoders. The profile is associated with meaningful learning conditions, while its causal mechanism remains unresolved.

  • Problem

    Existing geometric analyses of learned representations are predominantly local, and how far they predict feature displacement beyond an input’s immediate neighborhood has not been measured.

    Source: §1 · p. 1

  • Method

    The paper introduces a scale-resolved statistic comparing an encoder’s finite feature displacement with its local linear prediction as perturbation magnitude increases.

    Source: §1 · p. 1, §2 · p. 2

  • Results

    Across trained encoders, the measured response follows a plateau–rise–peak–decay bump, which is absent under randomized-label or unstructured-random-input training controls.

    Source: §1 · pp. 1–2

  • Takeaways & Limitations

    The bump is empirically associated with learning shared input–target structure and varies with the training distribution and robustness objective.

    Source: Contributions · p. 2, §4 · p. 4

    The measure captures only response magnitude averaged over isotropic directions, larger-scale perturbations might leave the natural-image manifold, and the empirical connection does not establish a causal mechanism.

    Source: §4 · pp. 4–5

Abstract

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Understanding how learned representations respond to finite input changes is important for characterizing their sensitivity, invariances, and robustness. Yet existing geometric analyses are predominantly local and describe only infinitesimal perturbations. We introduce a scale-resolved statistic that compares an encoder's measured feature displacement with its local linear prediction as the perturbation magnitude increases. Across diverse image encoders, we discover a characteristic plateau-rise-peak-decay profile, which we call the bump. The bump is absent at initialization, emerges early during standard training, and does not form under randomized labels or random-noise inputs. Its shape also varies with the training distribution and robustness objective. These results establish departures from local geometry as a signature of how encoder representations are shaped by learning.

1. Introduction

The paper addresses how far local geometric descriptions predict feature displacement under finite input changes, introducing a scale-resolved comparison. It identifies a characteristic bump in trained encoders and reports that the profile depends on meaningful training conditions.

1. Introduction

  • The study targets an unmeasured gap: how far differential descriptions of learned representations predict feature displacement beyond an input’s immediate neighborhood.The introduction links learned representation geometry to selective sensitivity to input variations relevant to prediction.Source: §1 · p. 1

Contributions

  • A scale-resolved statistic compares an encoder’s measured feature displacement with its local linear prediction as perturbation magnitude increases.The approach averages over random perturbation directions and examines behavior across architectures, training regimes, and evaluation data.Source: §1 · p. 1, Contributions · p. 2

1. Introduction

  • The bump begins near agreement with the local prediction, rises to an intermediate-scale maximum discrepancy, then falls relative to that prediction.The paper presents this profile across trained image encoders as a characteristic response pattern.Source: §1 · pp. 1–2

Contributions

  • The bump is absent under randomized labels and unstructured random inputs, despite optimization, and is associated with learning shared input–target structure.The paper also reports that the bump emerges early during meaningful training and varies with the training distribution and robustness objective.Source: §1 · p. 2, Contributions · p. 2

2. Method

The method measures how an encoder’s finite feature response departs from the local geometry at an input. It compares actual displacement with a Jacobian-based prediction across perturbation scales and directions.

2. Method

  • The method compares an encoder’s finite-perturbation feature displacement with the change predicted by its local pullback metric.The encoder is frozen during measurement, and the metric is formed from its Jacobian at the input.Source: §2 · p. 2
  • Averaging over isotropically sampled unit directions yields a direction-free scale-dependent ratio of measured response to locally predicted response.A ratio near 1 indicates agreement in magnitude; values above or below 1 indicate larger or smaller measured responses, respectively.Source: §2 · p. 2
  • For differentiable inputs with nonzero local sensitivity, the ratio approaches 1 as perturbation magnitude tends to zero.The paper states this small-scale limit holds by construction.Source: §2 · p. 3

3. Experiments

Across trained encoders, profiles show a plateau near the local prediction, then rise to an interior peak and decay; the bump is absent at initialization. Its formation and shape vary with training progress, training conditions, and evaluation data.

3. Experiments

  • Profiles were measured at 17 logarithmically spaced perturbation scales, with the median across images reported unless otherwise stated.The experiments span convolutional and transformer encoders, standard and adversarial ImageNet supervision, synthetic-data training, and random initialization.Source: §3 · p. 3
  • Across trained architectures, median profiles show a plateau near rη = 1, a rise to an interior peak, and eventual decay; randomly initialized encoders lack a comparable peak.The bump’s height, location, and width vary across models and evaluation data.Source: §3 · p. 3
  • The bump is absent at initialization, emerges early, and remains largely preserved during training across three CIFAR-10 architectures; it appears on validation as well as training images.Its presence on validation images indicates that it is not specific to memorized examples.Source: §3 · pp. 3–4
  • Adversarial training moves the peak 10–30× outward, primarily through an approximately 2000× reduction in the local prediction; blur changes the bump with training and evaluation blur.The bump mostly grows with evaluation blur and shrinks with training blur; randomized-label and random-data controls show no peak.Source: Figure 2 · p. 4, §3 · p. 4

4. Conclusions and Limitations

The paper introduces a scale-resolved measure of finite-response departures from local linear geometry and identifies the bump across diverse image encoders. The bump emerges during meaningful training and is empirically associated with learning shared structure that transfers to unseen examples.

4. Conclusions and Limitations

  • The scale-resolved measure identifies a bump across diverse image encoders that emerges during meaningful training and is empirically associated with learning shared structure that transfers to unseen examples.The paper does not establish the bump’s causal mechanism.Source: §4 · pp. 4–5
  • The measure captures only response magnitude averaged over isotropic directions, and larger-scale perturbations might leave the natural-image manifold.This constrains what the statistic describes at larger scales; the paper also leaves extension beyond image classification as future work.Source: §4 · pp. 4–5
  • The observed connection between the bump, learning, and generalization is empirical and does not establish a causal mechanism.The analysis’s extension beyond image classification remains future work.Source: §4 · pp. 4–5

A.1. Theoretical details

The theoretical details define the pullback metric as local feature geometry induced on image space, then compare finite feature displacement with its local prediction. Direction averaging, Taylor expansion, and antipodal sampling characterize the scale-dependent ratio and its estimation.

A.1. Theoretical details

  • The pullback metric is the image-space bilinear form induced by feature-space geometry, represented in coordinates by G = J⊺J; it is degenerate when the feature dimension is smaller than the image dimension.Its null directions include directions in the Jacobian’s kernel.Source: §A.1 · p. 7
  • At scale η, the metric predicts squared feature displacement as the squared length of the first-order displacement ηJu; the analysis compares this with the actual finite displacement.The ratio averages actual response and local prediction over uniformly sampled directions to remove directionality.Source: §A.1 · pp. 7–8
  • For smooth encoders, the odd cubic term vanishes under uniform direction averaging, so rη(x) departs from unity at order η^2 rather than linearly.This avoids a linear-term bias toward detecting departures at smaller perturbation scales.Source: §A.1 · p. 8
  • With finitely sampled directions, the odd term can leave a linear-in-η estimation bias; pairing each direction with its antipode cancels it exactly for every draw.The resulting estimate departs from its small-scale level at order η^2.Source: §A.1 · p. 8
  • For piecewise-affine encoders such as ReLU networks, rη(x) stays exactly 1 within the activation region until the perturbation crosses its boundary.No Taylor expansion is needed within that region.Source: §A.1 · p. 8

A.2. Experimental details

The appendix specifies the evaluated encoders and image sets, the perturbation and aggregation protocol, and the CIFAR-10 training variants.

A.2. Experimental details

  • Nine ImageNet encoders span convolutional and transformer architectures, varied training sources, and random initialization; additional CIFAR-10 classifiers are evaluated under the same protocol.The ImageNet models include standard, adversarially trained, and synthetic-data-trained encoders; the CIFAR-10 models include ResNet-18, VGG-16, and WRN-28-10.Source: §A.2 · p. 9
  • ImageNet evaluation sets include seeded samples from validation, ImageNet-A, and ImageNet-O, plus generated and blurred images; CIFAR-10 sets include test/train samples, blurred images, and noise.The ImageNet sets use 64 images and the CIFAR-10 sets use 32; main-text figures report val64 unless stated otherwise.Source: §A.2 · p. 9
  • Profiles use 17 logarithmically spaced scales, average responses over 2,048 directions per image, and report medians across each evaluation set.The local prediction uses 256 exact Jacobian–vector products; perturbations are applied in raw pixel space and are not projected back to the valid range.Source: §A.2 · p. 9
  • CIFAR-10 controls permute labels or replace images with uniform noise, while blur-trained variants blur training images and formation experiments measure saved checkpoints.Randomized-label and random-data variants disable augmentation; blurred variants use σ ∈ {1, 2}.Source: §A.2 · pp. 9–10

A.3. Related works

The related work covers pullback metrics, Jacobian-based sensitivity, finite-scale linearity, linear-region analyses, and parameter-space linearization. The paper distinguishes its input-space, finite-scale measurement from these approaches.

Applications of the pullback metric

Jacobian-based sensitivity statistics

  • Novak et al. (2018) use a first-order expansion for expected output change under small isotropic Gaussian perturbations; this work evaluates the approximation across finite input scales.The paper uses the left and right sides of that approximation as the numerator and denominator of rη, respectively.Source: §A.3 · p. 10

Finite-scale linearity and linear regions

  • Qin et al. (2019) regularize scalar-loss Taylor violations, while this work measures representation-map response; their maximum-over-ball profile cannot peak by construction.Humayun et al. (2024) instead track ReLU linear-region boundaries in a fixed-radius neighborhood over training, observing descent, ascent, and a second descent.Source: §A.3 · p. 10

Random networks and the NTK

  • The neural tangent kernel literature studies parameter-space linearization during training, whereas this work measures input-space linearization with network parameters frozen.The monotone rise at random initialization is described as consistent with the expansivity of random deep networks (Poole et al., 2016; Raghu et al., 2017).Source: §A.3 · p. 11

Appendix B. Full response profiles

The appendix presents per-image response profiles for trained ImageNet encoders, randomly initialized controls, and adversarially trained ResNet-50 models. The profiles show a shared trained-model shape, distinct random-control trajectories, and changes with adversarial training budget.

Appendix B. Full response profiles

  • The plateau–rise–peak–decay profile persists across trained encoders and all five evaluation sets, although its height and location vary.Figure 3 displays individual images as lines, with encoders arranged by rows and evaluation sets by columns.Source: Figure 3 · p. 12, Appendix B · p. 12
  • Randomly initialized RN-50 and ViT-S/16 rise monotonically to the ladder’s end, while ConvNeXt-Tiny stays near rη = 1 before decaying; none forms an interior peak.These patterns occur across the evaluation sets.Source: Figure 4 · p. 13
  • With increasing adversarial training budget ε, the peak moves to larger η and grows, while the ε = 0 baseline retains a small-scale peak.Source: Figure 5 · p. 14
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