Adele Myers

Adele Myers, Caitlin Taylor, Emily Jacobs et al.

Women are at higher risk of Alzheimer's and other neurological diseases after menopause, and yet research connecting female brain health to sex hormone fluctuations is limited. We seek to investigate this connection by developing tools that quantify 3D shape changes that occur in the brain during sex hormone fluctuations. Geodesic regression on the space of 3D discrete surfaces offers a principled way to characterize the evolution of a brain's shape. However, in its current form, this approach is too computationally expensive for practical use. In this paper, we propose approximation schemes that accelerate geodesic regression on shape spaces of 3D discrete surfaces. We also provide rules of thumb for when each approximation can be used. We test our approach on synthetic data to quantify the speed-accuracy trade-off of these approximations and show that practitioners can expect very significant speed-up while only sacrificing little accuracy. Finally, we apply the method to real brain shape data and produce the first characterization of how the female hippocampus changes shape during the menstrual cycle as a function of progesterone: a characterization made (practically) possible by our approximation schemes. Our work paves the way for comprehensive, practical shape analyses in the fields of bio-medicine and computer vision. Our implementation is publicly available on GitHub: https://github.com/bioshape-lab/my28brains.

Johan Mathe, Adele Myers, Simon Mataigne et al.

Many machine learning tasks are invariant under the action of a group $G$ of transformations: signal classification can be invariant under translations, image classification under 2D rotations, and spherical-image classification under 3D rotations. The $G$-bispectrum is a principled complete invariant of a signal (retaining all all signal's information up to the group action) with proven benefits in machine learning and as a pooling layer in deep networks. However, its deployment has been hampered by high computational cost and a patchwork of group-specific implementations. We present bispectrum, an open-source, fully unit-tested PyTorch library that implements selective $G$-bispectra for seven different group actions, as differentiable modules that can be directly incorporated into machine learning pipelines and deep learning architectures. For finite groups $G$, selectivity reduces the computational cost from $O(|G|^2)$ to $O(|G|)$. For planar rotations, we leverage the disk bispectrum. For spherical 3D rotations, we introduce an augmented selective bispectrum at band-limit $L$ which reduces the cost from $O(L^3)$ to $Θ(L^2)$ coefficients. We profile the entire library (for which we implemented various compute optimizations), showing that it delivers near-exact $G$-invariance with its selective $G$-bispectra computed in sub-millisecond time on GPU (up to commonly used bandlimits). We evaluate the benefits of incorporating $G$-bispectra as pooling layers into deep learning architectures on three classical benchmark datasets --comparing against norm pooling, gated pooling, Fourier-ELU pooling, max pooling, and (non-equivariant) data-augmented convolutional baselines. Results show that $G$-bispectra consistently outperform alternatives in the low-data, moderate-capacity regime.

Adele Myers, Nina Miolane

We propose a metric learning paradigm, Regression-based Elastic Metric Learning (REML), which optimizes the elastic metric for geodesic regression on the manifold of discrete curves. Geodesic regression is most accurate when the chosen metric models the data trajectory close to a geodesic on the discrete curve manifold. When tested on cell shape trajectories, regression with REML's learned metric has better predictive power than with the conventionally used square-root-velocity (SRV) metric.

Giovanni Luca Marchetti, Daniel Kunin, Adele Myers et al.

How do neural networks trained over sequences acquire the ability to perform structured operations, such as arithmetic, geometric, and algorithmic computation? To gain insight into this question, we introduce the sequential group composition task. In this task, networks receive a sequence of elements from a finite group encoded in a real vector space and must predict their cumulative product. The task can be order-sensitive and requires a nonlinear architecture to be learned. Our analysis isolates the roles of the group structure, encoding statistics, and sequence length in shaping learning. We prove that two-layer networks learn this task one irreducible representation of the group at a time in an order determined by the Fourier statistics of the encoding. These networks can perfectly learn the task, but doing so requires a hidden width exponential in the sequence length $k$. In contrast, we show how deeper models exploit the associativity of the task to dramatically improve this scaling: recurrent neural networks compose elements sequentially in $k$ steps, while multilayer networks compose adjacent pairs in parallel in $\log k$ layers. Overall, the sequential group composition task offers a tractable window into the mechanics of deep learning.

Adele Myers, Nina Miolane

In many computer vision and shape analysis tasks, practitioners are interested in learning from the shape of the object in an image, while disregarding the object's orientation. To this end, it is valuable to define a rotation-invariant representation of images, retaining all information about that image, but disregarding the way an object is rotated in the frame. To be practical for learning tasks, this representation must be computationally efficient for large datasets and invertible, so the representation can be visualized in image space. To this end, we present the selective disk bispectrum: a fast, rotation-invariant representation for image shape analysis. While the translational bispectrum has long been used as a translational invariant representation for 1-D and 2-D signals, its extension to 2-D (disk) rotational invariance on images has been hindered by the absence of an invertible formulation and its cubic complexity. In this work, we derive an explicit inverse for the disk bispectrum, which allows us to define a "selective" disk bispectrum, which only uses the minimal number of coefficients needed for faithful shape recovery. We show that this representation enables multi-reference alignment for rotated images-a task previously intractable for disk bispectrum methods. These results establish the disk bispectrum as a practical and theoretically grounded tool for learning on rotation-invariant shape data.