arXiv subjects

Nina Miolane

Publications and source records attributed to Nina Miolane.

At least 37 records · Page 2Linked to original sources

arXiv2025

On the approximation of the Riemannian barycenter

We present a method for computing an approximate Riemannian barycenter of a collection of points lying on a Riemannian manifold. Our approach relies on the use of theoretically proven under- and over-approximations of the Riemannian distance function. We compare it to Riemannian steepest descent on the exact objective function of the Riemannian barycenter and to an approach that approximates the Riemannian logarithm using lifting maps. Experiments are conducted on the Stiefel manifold.

math.DG↗

arXiv2025

Dynamical phases of short-term memory mechanisms in RNNs

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.

q-bio.NC↗

arXiv2025

Ordered Topological Deep Learning: a Network Modeling Case Study

Computer networks are the foundation of modern digital infrastructure, facilitating global communication and data exchange. As demand for reliable high-bandwidth connectivity grows, advanced network modeling techniques become increasingly essential to optimize performance and predict network behavior. Traditional modeling methods, such as packet-level simulators and queueing theory, have notable limitations --either being computationally expensive or relying on restrictive assumptions that reduce accuracy. In this context, the deep learning-based RouteNet family of models has recently redefined network modeling by showing an unprecedented cost-performance trade-off. In this work, we revisit RouteNet's sophisticated design and uncover its hidden connection to Topological Deep Learning (TDL), an emerging field that models higher-order interactions beyond standard graph-based methods. We demonstrate that, although originally formulated as a heterogeneous Graph Neural Network, RouteNet serves as the first instantiation of a new form of TDL. More specifically, this paper presents OrdGCCN, a novel TDL framework that introduces the notion of ordered neighbors in arbitrary discrete topological spaces, and shows that RouteNet's architecture can be naturally described as an ordered topological neural network. To the best of our knowledge, this marks the first successful real-world application of state-of-the-art TDL principles --which we confirm through extensive testbed experiments--, laying the foundation for the next generation of ordered TDL-driven applications.

cs.LG↗

arXiv2025

From superposition to sparse codes: interpretable representations in neural networks

Understanding how information is represented in neural networks is a fundamental challenge in both neuroscience and artificial intelligence. Despite their nonlinear architectures, recent evidence suggests that neural networks encode features in superposition, meaning that input concepts are linearly overlaid within the network's representations. We present a perspective that explains this phenomenon and provides a foundation for extracting interpretable representations from neural activations. Our theoretical framework consists of three steps: (1) Identifiability theory shows that neural networks trained for classification recover latent features up to a linear transformation. (2) Sparse coding methods can extract disentangled features from these representations by leveraging principles from compressed sensing. (3) Quantitative interpretability metrics provide a means to assess the success of these methods, ensuring that extracted features align with human-interpretable concepts. By bridging insights from theoretical neuroscience, representation learning, and interpretability research, we propose an emerging perspective on understanding neural representations in both artificial and biological systems. Our arguments have implications for neural coding theories, AI transparency, and the broader goal of making deep learning models more interpretable.

cs.LG↗

arXiv2025

Latent computing by biological neural networks: A dynamical systems framework

Although individual neurons and neural populations exhibit the phenomenon of representational drift, perceptual and behavioral outputs of many neural circuits can remain stable across time scales over which representational drift is substantial. These observations motivate a dynamical systems framework for neural network activity that focuses on the concept of \emph{latent processing units,} core elements for robust coding and computation embedded in collective neural dynamics. Our theoretical treatment of these latent processing units yields five key attributes of computing through neural network dynamics. First, neural computations that are low-dimensional can nevertheless generate high-dimensional neural dynamics. Second, the manifolds defined by neural dynamical trajectories exhibit an inherent coding redundancy as a direct consequence of the universal computing capabilities of the underlying dynamical system. Third, linear readouts or decoders of neural population activity can suffice to optimally subserve downstream circuits controlling behavioral outputs. Fourth, whereas recordings from thousands of neurons may suffice for near optimal decoding from instantaneous neural activity patterns, experimental access to millions of neurons may be necessary to predict neural ensemble dynamical trajectories across timescales of seconds. Fifth, despite the variable activity of single cells, neural networks can maintain stable representations of the variables computed by the latent processing units, thereby making computations robust to representational drift. Overall, our framework for latent computation provides an analytic description and empirically testable predictions regarding how large systems of neurons perform robust computations via their collective dynamics.

q-bio.NC↗

arXiv2025

Understanding and controlling the geometry of memory organization in RNNs

Training recurrent neural networks (RNNs) is a high-dimensional process that requires updating numerous parameters. Therefore, it is often difficult to pinpoint the underlying learning mechanisms. To address this challenge, we propose to gain mechanistic insights into the phenomenon of \emph{abrupt learning} by studying RNNs trained to perform diverse short-term memory tasks. In these tasks, RNN training begins with an initial search phase. Following a long period of plateau in accuracy, the values of the loss function suddenly drop, indicating abrupt learning. Analyzing the neural computation performed by these RNNs reveals geometric restructuring (GR) in their phase spaces prior to the drop. To promote these GR events, we introduce a temporal consistency regularization that accelerates (bioplausible) training, facilitates attractor formation, and enables efficient learning in strongly connected networks. Our findings offer testable predictions for neuroscientists and emphasize the need for goal-agnostic secondary mechanisms to facilitate learning in biological and artificial networks.

q-bio.NC↗

arXiv2024

An efficient algorithm for the Riemannian logarithm on the Stiefel manifold for a family of Riemannian metrics

Since the popularization of the Stiefel manifold for numerical applications in 1998 in a seminal paper from Edelman et al., it has been exhibited to be a key to solve many problems from optimization, statistics and machine learning. In 2021, Hüper et al. proposed a one-parameter family of Riemannian metrics on the Stiefel manifold, subsuming the well-known Euclidean and canonical metrics. Since then, several methods have been proposed to obtain a candidate for the Riemannian logarithm given any metric from the family. Most of these methods are based on the shooting method or rely on optimization approaches. For the canonical metric, Zimmermann proposed in 2017 a particularly efficient method based on a pure matrix-algebraic approach. In this paper, we derive a generalization of this algorithm that works for the one-parameter family of Riemannian metrics. The algorithm is proposed in two versions, termed backward and forward, for which we prove that it conserves the local linear convergence previously exhibited in Zimmermann's algorithm for the canonical metric.

math.NA↗

arXiv2024

TopoX: A Suite of Python Packages for Machine Learning on Topological Domains

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/}{https://pyt-team.github.io/.

cs.LG↗

arXiv2024

The Selective G-Bispectrum and its Inversion: Applications to G-Invariant Networks

An important problem in signal processing and deep learning is to achieve \textit{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 \textit{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.

cs.LG↗

arXiv2024

ICML Topological Deep Learning Challenge 2024: Beyond the Graph Domain

This paper describes the 2nd edition of the ICML Topological Deep Learning Challenge that was hosted within the ICML 2024 ELLIS Workshop on Geometry-grounded Representation Learning and Generative Modeling (GRaM). The challenge focused on the problem of representing data in different discrete topological domains in order to bridge the gap between Topological Deep Learning (TDL) and other types of structured datasets (e.g. point clouds, graphs). Specifically, participants were asked to design and implement topological liftings, i.e. mappings between different data structures and topological domains --like hypergraphs, or simplicial/cell/combinatorial complexes. The challenge received 52 submissions satisfying all the requirements. This paper introduces the main scope of the challenge, and summarizes the main results and findings.

cs.LG↗

arXiv2024

Position: Topological Deep Learning is the New Frontier for Relational Learning

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.

cs.LG↗

arXiv2024

Bounds on the geodesic distances on the Stiefel manifold for a family of Riemannian metrics

We give bounds on geodesic distances on the Stiefel manifold, derived from new geometric insights. The considered geodesic distances are induced by the one-parameter family of Riemannian metrics introduced by Hüper et al. (2021), which contains the well-known Euclidean and canonical metrics. First, we give the best Lipschitz constants between the distances induced by any two members of the family of metrics. Then, we give a lower and an upper bound on the geodesic distance by the easily computable Frobenius distance. We give explicit families of pairs of matrices that depend on the parameter of the metric and the dimensions of the manifold, where the lower and the upper bound are attained. These bounds aim at improving the theoretical guarantees and performance of minimal geodesic computation algorithms by reducing the initial velocity search space. In addition, these findings contribute to advancing the understanding of geodesic distances on the Stiefel manifold and their applications.

math.DG↗

arXiv2024

Learning from landmarks, curves, surfaces, and shapes in Geomstats

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.

cs.GR↗

arXiv2024

Attending to Topological Spaces: The Cellular Transformer

Topological Deep Learning seeks to enhance the predictive performance of neural network models by harnessing topological structures in input data. Topological neural networks operate on spaces such as cell complexes and hypergraphs, that can be seen as generalizations of graphs. In this work, we introduce the Cellular Transformer (CT), a novel architecture that generalizes graph-based transformers to cell complexes. First, we propose a new formulation of the usual self- and cross-attention mechanisms, tailored to leverage incidence relations in cell complexes, e.g., edge-face and node-edge relations. Additionally, we propose a set of topological positional encodings specifically designed for cell complexes. By transforming three graph datasets into cell complex datasets, our experiments reveal that CT not only achieves state-of-the-art performance, but it does so without the need for more complex enhancements such as virtual nodes, in-domain structural encodings, or graph rewiring.

cs.LG↗

arXiv2024

Architectures of Topological Deep Learning: A Survey of Message-Passing Topological Neural Networks

The natural world is full of complex systems characterized by intricate relations between their components: from social interactions between individuals in a social network to electrostatic interactions between atoms in a protein. Topological Deep Learning (TDL) provides a comprehensive framework to process and extract knowledge from data associated with these systems, such as predicting the social community to which an individual belongs or predicting whether a protein can be a reasonable target for drug development. TDL has demonstrated theoretical and practical advantages that hold the promise of breaking ground in the applied sciences and beyond. However, the rapid growth of the TDL literature for relational systems has also led to a lack of unification in notation and language across message-passing Topological Neural Network (TNN) architectures. This presents a real obstacle for building upon existing works and for deploying message-passing TNNs to new real-world problems. To address this issue, we provide an accessible introduction to TDL for relational systems, and compare the recently published message-passing TNNs using a unified mathematical and graphical notation. Through an intuitive and critical review of the emerging field of TDL, we extract valuable insights into current challenges and exciting opportunities for future development.

cs.LG↗

arXiv2024

A General Framework for Robust G-Invariance in G-Equivariant Networks

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 complete. Many commonly used invariant maps--such as the max--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--$SO(2)$, $O(2)$, $SO(3)$, and $O(3)$ (discretized as the cyclic $C8$, dihedral $D16$, chiral octahedral $O$ and full octahedral $O_h$ groups)--acting on $\mathbb{R}^2$ and $\mathbb{R}^3$ on both $G$-MNIST and $G$-ModelNet10 datasets.

cs.LG↗

arXiv2024

ICML 2023 Topological Deep Learning Challenge : Design and Results

This paper presents the computational challenge on topological deep learning that was hosted within the ICML 2023 Workshop on Topology and Geometry in Machine Learning. The competition asked participants to provide open-source implementations of topological neural networks from the literature by contributing to the python packages TopoNetX (data processing) and TopoModelX (deep learning). The challenge attracted twenty-eight qualifying submissions in its two-month duration. This paper describes the design of the challenge and summarizes its main findings.

cs.LG↗

arXiv2024

PVNet: A LRCN Architecture for Spatio-Temporal Photovoltaic PowerForecasting from Numerical Weather Prediction

Photovoltaic (PV) power generation has emerged as one of the lead renewable energy sources. Yet, its production is characterized by high uncertainty, being dependent on weather conditions like solar irradiance and temperature. Predicting PV production, even in the 24-hour forecast, remains a challenge and leads energy providers to left idling - often carbon emitting - plants. In this paper, we introduce a Long-Term Recurrent Convolutional Network using Numerical Weather Predictions (NWP) to predict, in turn, PV production in the 24-hour and 48-hour forecast horizons. This network architecture fully leverages both temporal and spatial weather data, sampled over the whole geographical area of interest. We train our model on an NWP dataset from the National Oceanic and Atmospheric Administration (NOAA) to predict spatially aggregated PV production in Germany. We compare its performance to the persistence model and state-of-the-art methods.

cs.LG↗