RotaTouille: Equivariant Contour Learning

• RotaTouille is a deep learning framework for contour data that encodes rotation and cyclic shift symmetries using complex-valued circular convolution.
• It integrates equivariant non-linearities, pooling layers, and invariant global pooling to enable robust shape classification, reconstruction, and contour regression.
• The framework leverages the intrinsic symmetry group G_n to ensure exact equivariance, yielding consistent performance improvements over traditional CNN and graph-based methods.

RotaTouille is a deep learning framework for learning from contour data, where a contour is treated as a closed planar curve represented as a complex-valued signal $x:\mathbb{Z}_n\to\mathbb{C}^k$. Its central objective is to build models that are exactly equivariant to planar rotations and to cyclic shifts of the contour starting point, and exactly invariant at output when required. The framework realizes these symmetries through complex-valued circular convolution, and complements that linear layer with equivariant non-linearities, equivariant coarsening layers, and invariant global pooling for downstream tasks such as shape classification, reconstruction, and contour regression (Gardaa et al., 22 Aug 2025).

1. Problem setting and symmetry structure

RotaTouille addresses learning problems in which the input is a contour or closed planar curve appearing, for example, as an object boundary in computer vision, an isoline in meteorology, or the orbit of rotating machinery (Gardaa et al., 22 Aug 2025). In this setting, two symmetries are intrinsic. First, planar rotations of the contour should induce correspondingly rotated outputs. Second, contours are typically represented as an ordered sequence of edge points, and the choice of starting point is arbitrary; this makes equivariance under cyclic shifts of the sequence equally desirable.

The framework formalizes these symmetries through the group

$G_n=\mathbb{Z}_n\times S^1,$

acting on contour signals by

$(l,w)\cdot x(q)=w\cdot x(q-l)$

for $l\in\mathbb{Z}_n$, $w\in S^1$, and $q\in\mathbb{Z}_n$ (Gardaa et al., 22 Aug 2025). Here, $\mathbb{Z}_n$ captures cyclic re-indexing of the contour points, while $S^1$ captures planar rotation through complex phase multiplication. The resulting formulation makes rotation and starting-point ambiguity part of the model class rather than a nuisance to be handled through augmentation alone.

This symmetry structure clarifies the design target of the framework: hidden layers should commute with the $G_n$-action, and the terminal representation should be invariant when the task requires a symmetry-insensitive output. A plausible implication is that contour learning can be expressed in a form where geometric prior information is built directly into the network architecture rather than left to optimization.

2. Complex-valued circular convolution

The core linear operator in RotaTouille is circular convolution on complex-valued contour signals. For single-channel contours $x,y\in X_n^1$, the convolution is defined as

$G_n=\mathbb{Z}_n\times S^1,$0

If the filter $G_n=\mathbb{Z}_n\times S^1,$1 has kernel size $G_n=\mathbb{Z}_n\times S^1,$2, it is embedded in $G_n=\mathbb{Z}_n\times S^1,$3 by zero-padding for $G_n=\mathbb{Z}_n\times S^1,$4 (Gardaa et al., 22 Aug 2025).

For multi-channel inputs $G_n=\mathbb{Z}_n\times S^1,$5 and a filter $G_n=\mathbb{Z}_n\times S^1,$6 with components $G_n=\mathbb{Z}_n\times S^1,$7, the framework defines

$G_n=\mathbb{Z}_n\times S^1,$8

A bank $G_n=\mathbb{Z}_n\times S^1,$9 then yields an operator $(l,w)\cdot x(q)=w\cdot x(q-l)$0 (Gardaa et al., 22 Aug 2025). This construction is the mechanism through which the two symmetries are fused in a single linear map.

Rotation equivariance follows from complex linearity. A planar rotation by angle $(l,w)\cdot x(q)=w\cdot x(q-l)$1 is multiplication by $(l,w)\cdot x(q)=w\cdot x(q-l)$2, and

$(l,w)\cdot x(q)=w\cdot x(q-l)$3

Shift equivariance is expressed through the shift operator

$(l,w)\cdot x(q)=w\cdot x(q-l)$4

for which

$(l,w)\cdot x(q)=w\cdot x(q-l)$5

The paper states the resulting joint equivariance proposition: for all $(l,w)\cdot x(q)=w\cdot x(q-l)$6 and $(l,w)\cdot x(q)=w\cdot x(q-l)$7,

$(l,w)\cdot x(q)=w\cdot x(q-l)$8

(Gardaa et al., 22 Aug 2025). In hidden layers, all linear maps are realized via circular convolution, so they commute with every $(l,w)\cdot x(q)=w\cdot x(q-l)$9. This suggests that the architecture achieves exact symmetry handling at the operator level rather than approximately through data preprocessing.

3. Equivariant network components

A typical RotaTouille block is repeated $l\in\mathbb{Z}_n$0 times. Given $l\in\mathbb{Z}_n$1, the first step is equivariant convolution,

$l\in\mathbb{Z}_n$2

where $l\in\mathbb{Z}_n$3 is a bank of $l\in\mathbb{Z}_n$4 filters, each of length $l\in\mathbb{Z}_n$5 (Gardaa et al., 22 Aug 2025).

The second step is an equivariant element-wise non-linearity. The admissible form is

$l\in\mathbb{Z}_n$6

The paper lists three common choices: $l\in\mathbb{Z}_n$7

$l\in\mathbb{Z}_n$8

$l\in\mathbb{Z}_n$9

Because these maps preserve phase while modulating magnitude, they maintain the $w\in S^1$0-equivariance required by the representation (Gardaa et al., 22 Aug 2025).

The third step is equivariant coarsening, or downsampling. Two modes are described. The first is strided pooling with stride $w\in S^1$1 and an $w\in S^1$2-equivariant aggregator $w\in S^1$3 over a block of $w\in S^1$4 equispaced points: $w\in S^1$5 The second is coset pooling, defined by $w\in S^1$6 and $w\in S^1$7, which is described as truly equivariant with respect to full $w\in S^1$8 (Gardaa et al., 22 Aug 2025). Example aggregators include the mean $w\in S^1$9 and max-magnitude argmax, with the requirement that they satisfy $q\in\mathbb{Z}_n$0.

Taken together, these components constitute a full equivariant toolbox: equivariant convolutions, equivariant non-linearities, and equivariant pooling. The framework is therefore not limited to a single symmetry-preserving layer type, but extends equivariance through the principal architectural stages of a deep network.

4. Invariant representations and downstream outputs

RotaTouille separates equivariant hidden processing from invariant readout. After the final block, $q\in\mathbb{Z}_n$1 is passed to a global pooling map

$q\in\mathbb{Z}_n$2

defined componentwise by

$q\in\mathbb{Z}_n$3

This pooling produces a real-valued invariant representation for downstream tasks (Gardaa et al., 22 Aug 2025).

By design, every hidden tensor $q\in\mathbb{Z}_n$4 is $q\in\mathbb{Z}_n$5-equivariant, while the final global pooling $q\in\mathbb{Z}_n$6 is $q\in\mathbb{Z}_n$7-invariant and yields a real vector $q\in\mathbb{Z}_n$8. For classification or regression, $q\in\mathbb{Z}_n$9 is then fed into a real-valued fully connected head. The paper also notes a multi-scale variant in which one applies $\mathbb{Z}_n$0 after each block and concatenates the resulting features, giving richer features without extra parameters (Gardaa et al., 22 Aug 2025).

This division between equivariant intermediate representations and invariant terminal features is important for interpreting the framework. It means that RotaTouille can preserve pose information when needed inside the network, but discard it at the output when the task depends only on contour identity or geometry up to the specified symmetries.

5. Experimental evaluation

The reported experiments cover shape classification, shape reconstruction, and contour regression (Gardaa et al., 22 Aug 2025). In shape classification, the datasets are FashionMNIST (contour only), ModelNet (4-channel contour slices), and RotatedMNIST (contour plus optional radial histogram). The baselines are a 2D-CNN on binary contour-images, a GCN on a cycle-graph, and ContourCNN. The numerical results are summarized below.

Setting	Baselines	RotaTouille
FashionMNIST accuracy	2D-CNN (filled) $\mathbb{Z}_n$1; ContourCNN $\mathbb{Z}_n$2	$\mathbb{Z}_n$3
ModelNet accuracy	ContourCNN $\mathbb{Z}_n$4	$\mathbb{Z}_n$5
RotatedMNIST error (\%)	CNN $\mathbb{Z}_n$6; P4CNN $\mathbb{Z}_n$7	contours only $\mathbb{Z}_n$8; $\mathbb{Z}_n$9RH feature $S^1$0

For shape reconstruction on auto-encoding PCST cell shapes, both image-AE and contour-AE achieve $S^1$1. The contour-AE is reported to guarantee valid simple contours and to exhibit better sharp-corner fidelity (Gardaa et al., 22 Aug 2025). For curvature regression on synthetic Fourier-mode contours, the baselines are finite-difference $S^1$2, 3-point circle-fit $S^1$3, and a real 1D-CNN $S^1$4, while RotaTouille achieves $S^1$5.

The paper’s conclusion is that RotaTouille yields consistent performance gains, often a few percent, on contour-based tasks by explicitly encoding the full rotation$S^1$6shift symmetry via complex circular convolution, equivariant nonlinearities, pooling, and global invariants (Gardaa et al., 22 Aug 2025). At the same time, the RotatedMNIST results show that the contour-only variant is not uniformly superior to all equivariant image models.

6. Scope, interpretation, and common misunderstandings

A frequent simplification in contour learning is to focus on rotation equivariance alone. RotaTouille is built around the stronger claim that the arbitrary choice of starting point in an ordered contour representation makes cyclic-shift equivariance equally fundamental (Gardaa et al., 22 Aug 2025). In the framework’s formulation, these are not separate engineering concerns but a joint symmetry encoded by $S^1$7.

Another possible misunderstanding is that any complex-valued architecture on contours would inherit the same guarantees. The framework does not make that claim. Its exact equivariance depends on specific architectural choices: circular convolution for all linear maps, element-wise non-linearities of the form $S^1$8, coarsening operators with $S^1$9-equivariant aggregators, and an invariant global pooling layer (Gardaa et al., 22 Aug 2025). Removing or altering these components would not automatically preserve the stated symmetry properties.

The empirical results also constrain interpretation. The reported gains are strongest where contour structure is central and where exact rotation$G_n$0shift symmetry is well aligned with the task. On RotatedMNIST, RotaTouille with contours only reports $G_n$1 error, compared with $G_n$2 for P4CNN, while the addition of the RH feature improves RotaTouille to $G_n$3 (Gardaa et al., 22 Aug 2025). This suggests that contour-only representations may omit useful information for some datasets, even when the symmetry handling itself is exact.

Taken together, these points place RotaTouille within a specific methodological niche: it is a contour-native framework for settings in which closed planar curves are the primary data type and in which both planar rotations and cyclic re-indexings are structural symmetries of the learning problem.