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Quantifying Translation-Invariance in Convolutional Neural Networks
Eric Kauderer-Abrams
TL;DR
CNNs are widely considered capable of translation-invariant representations, but the mechanism remains unclear, with receptive-field growth and pooling proposed as explanations. This paper uses translation-sensitivity tools to compare architectures and finds that training-data augmentation matters more than architectural choices.
Problem
Although CNNs are considered capable of translation-invariant representations, the mechanism is unclear and receptive-field growth and pooling have been proposed as explanations.
Source: §1 · p. 1Method
The authors use translation-sensitivity maps and radial functions to visualize and quantify translation invariance across CNN architectures.
Source: §1 · p. 1, §3.1 · p. 2, §5 · p. 5Results
Training-data variation due to translation is the most important factor; augmented networks are more translation-invariant than non-augmented ones, while architecture plays a secondary role.
Source: §4.2 · pp. 4–5, §5 · p. 5Takeaways & Limitations
CNN architecture alone does not provide translation invariance, but CNNs can learn it when trained on data with substantial translation variation.
Source: §5 · p. 5
Abstract
from arXiv · showhide
A fundamental problem in object recognition is the development of image representations that are invariant to common transformations such as translation, rotation, and small deformations. There are multiple hypotheses regarding the source of translation invariance in CNNs. One idea is that translation invariance is due to the increasing receptive field size of neurons in successive convolution layers. Another possibility is that invariance is due to the pooling operation. We develop a simple a tool, the translation-sensitivity map, which we use to visualize and quantify the translation-invariance of various architectures. We obtain the surprising result that architectural choices such as the number of pooling layers and the convolution filter size have only a secondary effect on the translation-invariance of a network. Our analysis identifies training data augmentation as the most important factor in obtaining translation-invariant representations of images using convolutional neural networks.
1. Introduction
Translation-invariant representations matter for object recognition, but the mechanism by which CNNs obtain them is not established. This study investigates the competing architectural explanations and identifies training-data augmentation as the most important factor.
1. Introduction
- CNN translation invariance lacks a definitive explanation, despite widespread agreement that CNNs can learn such representations.The paper asks how invariant CNN representations are and which features are responsible.Source: §1 · p. 1
- The study tests receptive-field growth in successive convolution layers against the hypothesis that pooling operations produce translation invariance.The authors note that the nature of CNN translation invariance has received little direct exploration.Source: §1 · p. 1
- Translation-sensitivity maps are introduced to visualize and quantify invariance across CNN architectures, and training-data augmentation emerges as the most important factor.The study applies the tools to architectures trained on an expanded MNIST digit dataset.Source: §1 · p. 1
2. Related Work
Prior work has quantified transformation invariance, modified CNN architectures to increase it, and developed theoretically grounded alternatives. The paper's tools are intended to quantify translation invariance across these approaches.
2. Related Work
- Prior work quantified invariance to affine transformations, while CNN modifications include spatial transformer networks that learn example-specific affine transforms.Spatial transformer networks aim to standardize the location and orientation of relevant image information.Source: §2 · p. 1
- The Scattering Transform uses fixed wavelet transforms and nonlinearities to provide provable affine invariance and stability to small deformations without learning.The paper notes that its performance matches CNNs on many datasets.Source: §2 · p. 2
- The proposed methods can also quantify translation invariance in the related approaches discussed in this section.Source: §2 · p. 2
3. Methods
The methods measure how network outputs change under input translations, using maps for visualization and radial functions for cross-architecture comparison. Experiments compare CNNs trained on centered MNIST with and without translation augmentation.
3.1. Convolutional Neural Networks
- The paper treats “translation-invariant” as shorthand for outputs that vary little under input translation, and compares architectures by their output sensitivity.The study constructs and trains multiple CNN architectures; Figure 1 shows a sensitivity map for a network with two convolution and two pooling layers.Source: §3.1 · p. 2, Figure 1 · p. 2
3.2. Translation-Sensitivity Maps
- A translation-sensitivity map displays normalized score-space distances from an untranslated base image to its translated copies as a heat map.The map averages results across multiple test images of a class; Figure 2 shows untranslated and translated MNIST images.Source: §3.1 · pp. 2–3, §3.2 · p. 3, Figure 2 · p. 3
3.3. Radial Translation-Sensitivity Fuctions
- Radial translation-sensitivity functions summarize average map values by translation distance, enabling rough comparisons across architectures despite map anisotropy.Source: §3.3 · p. 3
3.4. Translated MNIST Dataset
- Experiments compare CNNs trained on centered MNIST with and without translation augmentation.The original and augmented training sets each contain 50,000 images; augmentation uses four translated copies of 12,500 images, with shifts of 1–10 pixels in each direction. The data also includes 10,000 validation and 10,000 test images.Source: §3.4 · p. 3, §3.3 · p. 3
4. Results
Across the experiments, training augmentation produced the clearest difference in translation-invariance, while architecture and filter size had secondary effects. The section compares five CNN configurations and tests filter sizes in a deeper network.
4.1. Experiment One: Five Architectures With and Without Training Data Augmentation
- Five CNN configurations varying convolution and pooling arrangements were tested with and without augmentation; each reached 98–99% test accuracy.Translation-sensitivity maps and radial functions were used to compare the networks.Source: §4.1 · pp. 3–4
4.2. The Importance of Data Augmentation
- Augmented and non-augmented training produced the largest reported difference in translation-invariance across the five architectures.Figure 3 shows brighter maps for augmented training, and Figure 4 distinguishes augmented (dashed) from non-augmented (solid) networks.Source: §4.2 · p. 4, Figure 3 · p. 4, Figure 4 · p. 4
4.3. Architectural Features Play A Secondary Role
- Architectural choices had a secondary role: without augmentation, layer counts did not correlate with invariance, while augmented deeper models were more invariant than shallower ones.Among the three deeper augmented configurations, the passage reports no distinction, suggesting depth rather than layer type contributes to invariance.Source: §4.3 · p. 5
4.4. Experiment Two: The Effect of Filter-Size in a Deeper Network
- In the deeper network, larger filters increased translation-invariance, but filter size remained secondary to whether training data were augmented.The experiment compared 3- and 5-pixel filters under both augmented and non-augmented training.Source: §4.4 · p. 5
5. Conclusion
The paper introduces translation-sensitivity maps and radial translation-sensitivity functions to measure translation invariance, and finds that training-data variation matters more than architectural choices. It also points to future comparisons with spatial transformer networks and hybrid models combining CNNs with The Scattering Transform.
5. Conclusion
- The paper introduces translation-sensitivity maps and radial translation-sensitivity functions to visualize and quantify classification algorithms’ translation invariance.The conclusion says these tools were applied to measure translation invariance across CNN architectures and isolate influential network and training features.Source: §5 · p. 5
- Figure 5 compares three- and five-pixel filters and reports that augmentation has a larger effect on translation invariance than filter size.The paper also describes larger filters as having a secondary role.Source: Figure 5 · p. 5, §5 · p. 5
- Training on data with significant translation variation is the most important factor; depth, pooling-layer count, and filter size play secondary roles.Larger filters and deeper networks have the capacity for greater translation invariance, but that capacity is realized only with augmented training data.Source: §5 · p. 5
- The authors propose using translation-sensitivity maps to quantify the translation invariance achieved by spatial transformer networks.This is presented as planned follow-up work.Source: §5 · p. 5
- The paper suggests exploring hybrid algorithms that combine CNNs’ pattern-learning ability with invariance structure such as that of The Scattering Transform.The Scattering Transform is described as having built-in translation invariance regardless of training, unlike CNNs, whose capacity depends on appropriate data.Source: §5 · pp. 5–6