We introduce the shape module of the Python package Geomstats to analyze shapes of objects represented as landmarks, curves, and surfaces across fields of natural sciences and engineering. The shape module first implements widely used shape spaces, such as the Kendall shape space, as well as elastic spaces of discrete curves and surfaces. The shape module further implements the abstract mathematical structures of group actions, fiber bundles, quotient spaces, and associated Riemannian metrics which allow users to build their own shape spaces. The Riemannian geometry tools enable users to compare, average, interpolate between shapes inside a given shape space. These essential operations can then be leveraged to perform statistics and machine learning on shape data. We present the object-oriented implementation of the shape module along with illustrative examples and show how it can be used to perform statistics and machine learning on shape spaces.

Short-term memory is essential for cognitive processing, yet our understanding of its neural mechanisms remains unclear. Neuroscience has long focused on how sequential activity patterns, where neurons fire one after another within large networks, can explain how information is maintained. While recurrent connections were shown to drive sequential dynamics, a mechanistic understanding of this process still remains unknown. In this work, we introduce two unique mechanisms that can support this form of short-term memory: slow-point manifolds generating direct sequences or limit cycles providing temporally localized approximations. Using analytical models, we identify fundamental properties that govern the selection of each mechanism. Precisely, on short-term memory tasks (delayed cue-discrimination tasks), we derive theoretical scaling laws for critical learning rates as a function of the delay period length, beyond which no learning is possible. We empirically verify these results by training and evaluating approximately 80,000 recurrent neural networks (RNNs), which are publicly available for further analysis. Overall, our work provides new insights into short-term memory mechanisms and proposes experimentally testable predictions for systems neuroscience.

Graph Neural Networks (GNNs) effectively learn from relational data by leveraging graph symmetries. However, many real-world systems—such as biological or social networks—feature multi-way interactions that GNNs fail to capture. Topological Deep Learning (TDL) addresses this by modeling and leveraging higher-order structures, with Combinatorial Complex Neural Networks (CCNNs) offering a general and expressive approach that has been shown to outperform GNNs. However, TDL lacks the principled and standardized frameworks that underpin GNN development, restricting its accessibility and applicability. To address this issue, we introduce Generalized CCNNs (GCCNs), a simple yet powerful family of TDL models that can be used to systematically transform any (graph) neural network into its TDL counterpart. We prove that GCCNs generalize and subsume CCNNs, while extensive experiments on a diverse class of GCCNs show that these architectures consistently match or outperform CCNNs, often with less model complexity. In an effort to accelerate and democratize TDL, we introduce TopoTune, a lightweight software for defining, building, and training GCCNs with unprecedented flexibility and ease.

What features neural networks learn, and how, remains an open question. In this paper, we introduce Alternating Gradient Flows (AGF), an algorithmic framework that describes the dynamics of feature learning in two-layer networks trained from small initialization. Prior works have shown that gradient flow in this regime exhibits a staircase-like loss curve, alternating between plateaus where neurons slowly align to useful directions and sharp drops where neurons rapidly grow in norm. AGF approximates this behavior as an alternating two-step process: maximizing a utility function over dormant neurons and minimizing a cost function over active ones. AGF begins with all neurons dormant, corresponding to an initialization at the origin. At each iteration, a dormant neuron activates, triggering the acquisition of a feature and a drop in the loss. AGF quantifies the order, timing, and magnitude of these drops, matching experiments across several commonly studied architectures. We show that AGF unifies and extends existing saddle-to-saddle analyses in fully connected linear networks and attention-only linear transformers, where the learned features are singular modes and principal components, respectively. In diagonal linear networks, we prove AGF converges to gradient flow in the limit of vanishing initialization. Applying AGF to quadratic networks trained to perform modular addition, we give the first complete characterization of the training dynamics, revealing that networks learn Fourier features in decreasing order of coefficient magnitude. Altogether, AGF offers a promising step towards understanding feature learning in neural networks.

Topological deep learning (TDL) is a rapidly evolving field that uses topological features to understand and design deep learning models. This paper posits that TDL is the new frontier for relational learning. TDL may complement graph representation learning and geometric deep learning by incorporating topological concepts, and can thus provide a natural choice for various machine learning settings. To this end, this paper discusses open problems in TDL, ranging from practical benefits to theoretical foundations. For each problem, it outlines potential solutions and future research opportunities. At the same time, this paper serves as an invitation to the scientific community to actively participate in TDL research to unlock the potential of this emerging field.

Grid cells in the mammalian brain are fundamental to spatial navigation, and therefore crucial to how animals perceive and interact with their environment. Traditionally, grid cells are thought support path integration through highly symmetric hexagonal lattice firing patterns. However, recent findings show that their firing patterns become distorted in the presence of significant spatial landmarks such as rewarded locations. This introduces a novel perspective of dynamic, subjective, and action-relevant interactions between spatial representations and environmental cues. Here, we propose a practical and theoretical framework to quantify and explain these interactions. To this end, we train path-integrating recurrent neural networks (piRNNs) on a spatial navigation task, whose goal is to predict the agent's position with a special focus on rewarded locations. Grid-like neurons naturally emerge from the training of piRNNs, which allows us to investigate how the two aspects of the task, space and reward, are integrated in their firing patterns. We find that geometry, but not topology, of the grid cell population code becomes distorted. Surprisingly, these distortions are global in the firing patterns of the grid cells despite local changes in the reward. Our results indicate that after training with location-specific reward information, the preserved representational topology supports successful path integration, whereas the emergent heterogeneity in individual responses due to global distortions may encode dynamically changing environmental cues. By bridging the gap between computational models and the biological reality of spatial navigation under reward information, we offer new insights into how neural systems prioritize environmental landmarks in their spatial navigation code.

An important problem in signal processing and deep learning is to achieve *invariance* to nuisance factors not relevant for the task. Since many of these factors are describable as the action of a group $G$ (e.g. rotations, translations, scalings), we want methods to be $G$-invariant. The $G$-Bispectrum extracts every characteristic of a given signal up to group action: for example, the shape of an object in an image, but not its orientation. Consequently, the $G$-Bispectrum has been incorporated into deep neural network architectures as a computational primitive for $G$-invariance\textemdash akin to a pooling mechanism, but with greater selectivity and robustness. However, the computational cost of the $G$-Bispectrum ($\mathcal{O}(|G|^2)$, with $|G|$ the size of the group) has limited its widespread adoption. Here, we show that the $G$-Bispectrum computation contains redundancies that can be reduced into a *selective $G$-Bispectrum* with $\mathcal{O}(|G|)$ complexity. We prove desirable mathematical properties of the selective $G$-Bispectrum and demonstrate how its integration in neural networks enhances accuracy and robustness compared to traditional approaches, while enjoying considerable speeds-up compared to the full $G$-Bispectrum.

Success in collaborative and competitive environments, where agents must work with or against each other, requires individuals to encode the position and trajectory of themselves and others. Decades of neurophysiological experiments have shed light on how brain regions [e.g., medial entorhinal cortex (MEC), hippocampus] encode the self's position and trajectory. However, it has only recently been discovered that MEC and hippocampus are modulated by the positions and trajectories of others. To understand how encoding spatial information of multiple agents shapes neural representations, we train a recurrent neural network (RNN) model that captures properties of MEC to path integrate trajectories of two agents simultaneously navigating the same environment. We find significant differences between these RNNs and those trained to path integrate only a single agent. At the individual unit level, RNNs trained to path integrate more than one agent develop weaker grid responses, stronger border responses, and tuning for the relative position of the two agents. At the population level, they develop more distributed and robust representations, with changes in network dynamics and manifold topology. Our results provide testable predictions and open new directions with which to study the neural computations supporting spatial navigation.

We introduce TopoX, a Python software suite that provides reliable and user-friendly building blocks for computing and machine learning on topological domains that extend graphs: hypergraphs, simplicial, cellular, path and combinatorial complexes. TopoX consists of three packages: TopoNetX facilitates constructing and computing on these domains, including working with nodes, edges and higher-order cells; TopoEmbedX provides methods to embed topological domains into vector spaces, akin to popular graph-based embedding algorithms such as node2vec; TopoModelX is built on top of PyTorch and offers a comprehensive toolbox of higher-order message passing functions for neural networks on topological domains. The extensively documented and unit-tested source code of TopoX is available under MIT license at https://pyt-team.github.io.

We introduce a general method for achieving robust group-invariance in group-equivariant convolutional neural networks ($G$-CNNs), which we call the $G$-triple-correlation ($G$-TC) layer. The approach leverages the theory of the triple-correlation on groups, which is the unique, lowest-degree polynomial invariant map that is also \textit{complete}. Many commonly used invariant maps\textemdash such as the \texttt{max}\textemdash are incomplete: they remove both group and signal structure. A complete invariant, by contrast, removes only the variation due to the actions of the group, while preserving all information about the structure of the signal. The completeness of the triple correlation endows the $G$-TC layer with strong robustness, which can be observed in its resistance to invariance-based adversarial attacks. In addition, we observe that it yields measurable improvements in classification accuracy over standard Max $G$-Pooling in $G$-CNN architectures. We provide a general and efficient implementation of the method for any discretized group, which requires only a table defining the group's product structure. We demonstrate the benefits of this method for $G$-CNNs defined on both commutative and non-commutative groups\textemdash $SO(2)$, $O(2)$, $SO(3)$, and $O(3)$ (discretized as the cyclic $C8$, dihedral $D16$, chiral octahedral $O$ and full octahedral $O_h$ groups)\textemdash acting on $\mathbb{R}^2$ and $\mathbb{R}^3$ on both $G$-MNIST and $G$-ModelNet10 datasets.

We introduce the information geometry module of the Python package Geomstats. The module first implements Fisher–Rao Riemannian manifolds of widely used parametric families of probability distributions, such as normal, gamma, beta, Dirichlet distributions, and more. The module further gives the Fisher–Rao Riemannian geometry of any parametric family of distributions of interest, given a parameterized probability density function as input. The implemented Riemannian geometry tools allow users to compare, average, interpolate between distributions inside a given family. Importantly, such capabilities open the door to statistics and machine learning on probability distributions. We present the object-oriented implementation of the module along with illustrative examples and show how it can be used to perform learning on manifolds of parametric probability distributions.

Manifold-valued data naturally arises in medical imaging. In cognitive neuroscience for instance, brain connectomes base the analysis of coactivation patterns between different brain regions on the analysis of the correlations of their functional Magnetic Resonance Imaging (fMRI) time series - an object thus constrained by construction to belong to the manifold of symmetric positive definite matrices. One of the challenges that naturally arises in these studies consists in finding a lower-dimensional subspace for representing such manifold-valued and typically high-dimensional data. Traditional techniques, like principal component analysis, are ill-adapted to tackle non-Euclidean spaces and may fail to achieve a lower-dimensional representation of the data - thus potentially pointing to the absence of lower-dimensional representation of the data. However, these techniques are restricted in that: (i) they do not leverage the assumption that the connectomes belong on a pre-specified manifold, therefore discarding information; (ii) they can only fit a linear subspace to the data. In this paper, we are interested in variants to learn potentially highly curved submanifolds of manifold-valued data. Motivated by the brain connectomes example, we investigate a latent variable generative model, which has the added benefit of providing us with uncertainty estimates - a crucial quantity in the medical applications we are considering. While latent variable models have been proposed to learn linear and nonlinear spaces for Euclidean data, or geodesic subspaces for manifold data, no intrinsic latent variable model exists to learn non-geodesic subspaces for manifold data. This paper fills this gap and formulates a Riemannian variational autoencoder with an intrinsic generative model of manifold-valued data. We evaluate its performances on synthetic and real datasets, by introducing the formalism of weighted Riemannian submanifolds.

We use tools from geometric statistics to analyze the usual estimation procedure of a template shape. This applies to shapes from landmarks, curves, surfaces, images, etc. We demonstrate the asymptotic bias of template shape estimation using the stratified geometry of the shape space. We give a Taylor expansion of the bias with respect to a parameter $\sigma$ describing the measurement error on the data. We propose two bootstrap procedures that quantify the bias and correct it if needed; they are applicable for any type of shape data. We give a rule of thumb to provide intuition on whether the bias has to be corrected, which reveals the parameters that control the bias' magnitude. We illustrate our results on simulated and real shape data.