Latent Equivariant Operators for Robust Object Recognition: Promise and Challenges

TL;DR

Here, using simple datasets of rotated and translated noisy MNIST, it is illustrated how such architectures can successfully be harnessed for out-of-distribution classification, thus overcoming the limitations of both traditional and equivariant networks.

Abstract

Despite the successes of deep learning in computer vision, difficulties persist in recognizing objects that have undergone group-symmetric transformations rarely seen during training$\unicode{x2013}$for example objects seen in unusual poses, scales, positions, or combinations thereof. Equivariant neural networks are a solution to the problem of generalizing across symmetric transformations, but require knowledge of transformations a priori. An alternative family of architectures proposes to learn equivariant operators in a latent space, from examples of symmetric transformations. Here, using simple datasets of rotated and translated noisy MNIST, we illustrate how such architectures can successfully be harnessed for out-of-distribution classification, thus overcoming the limitations of both traditional and equivariant networks. While conceptually enticing, we discuss challenges ahead on the path of scaling these architectures to more complex datasets.

Figures (5)

• Figure 1: Pipeline for handling single transformation. Two transformed views of the same input are encoded by a shared encoder $f_E$ and mapped to a canonical representation using inverse shift operators $\varphi^{-k_1}$ and $\varphi^{-k_2}$, yielding embeddings $Z_1$ and $Z_2$. The embedding $Z_1$ is used for classification via a MLP $f_{\text{CLF}}$ optimized with the cross-entropy loss $\mathcal{L}_{\mathrm{CE}}$, while a representation consistency loss $\mathcal{L}_{\mathrm{reg}}$ encourages alignment between $Z_1$ and $Z_2$ that both correspond to the canonical pose. When the operator is learned, we add an extra term $\mathcal{L}_{\mathrm{op}}$ to the loss.
• Figure 2: Classification accuracy as a function of transformations on MNIST. The shaded region denotes the range of translations observed during training.
• Figure 3: Test accuracy heatmaps under joint horizontal (rows) and vertical (columns) translations. The bordered cross indicates transformations observed during training.
• Figure 4: Pipeline of operator-based classification under compound transformations. Top (Training): A canonical input is transformed along individual axes to generate augmented views. Shared encoders map each view into a representation space, where inverse operators $\varphi^{-k}$ align embeddings back to a canonical pose. Bottom (Inference): Given an input undergoing a compound transformation $T_{x,y}(k_1,k_2)$, the model encodes the input and applies the corresponding inverse operators to recover a canonical representation for classification.
• Figure 5: Extrapolation behavior of operator-based models on MNIST. Solid lines use ground-truth transformation degrees; dashed lines "(auto)" use k-NN pose inference. Shaded regions denote training degrees.