Global Distortions from Local Rewards: Neural Coding Strategies in Path-Integrating Neural Systems | NeurIPS
Acosta, F., Dinc, F., Redman, W., Madhav, M., Klindt, D., Miolane, N.
Grid cells in the mammalian brain are fundamental to spatial navigation, \emph{i.e.}, how animals perceive and interact with their environment. Traditionally, grid cells are thought to encode the physical position of an animal. 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 grid cells 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 the preserved hexagonal firing rates after fine-tuning may retain the spatial navigation abilities, whereas the global distortions emerging after training for location-specific reward information may encode dynamically changing environmental cues. By bridging the gap between computational models and biological reality of spatial navigation under reward information, we offer new insights into how neural systems prioritize environmental landmarks in their spatial navigation code.
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Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures | Submitted to Transactions on Pattern Analysis and Machine Intelligence (tPAMI)
Sanborn, A., Mathe, J., Papillon, M., Buracas, D., Lillemark, H., Shewmake, C., Bertics, A., Pennec, X., Miolane, N.
The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data that is inherently nonEuclidean. This data can exhibit intricate geometric, topological and algebraic structure: from the geometry of the curvature of space-time, to topologically complex interactions between neurons in the brain, to the algebraic transformations describing symmetries of physical systems. Extracting knowledge from such non-Euclidean data necessitates a broader mathematical perspective. Echoing the 19th-century revolutions that gave rise to non-Euclidean geometry, an emerging line of research is redefining modern machine learning with non-Euclidean structures. Its goal: generalizing classical methods to unconventional data types with geometry, topology, and algebra. In this review, we provide an accessible gateway to this fast-growing field and propose a graphical taxonomy that integrates recent advances into an intuitive unified framework. We subsequently extract insights into current challenges and highlight exciting opportunities for future development in this field.
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The Selective G-Bispectrum and its Inversion: Applications to G-Invariant Networks | NeurIPS
Mataigne, S., Mathe, J., Sanborn, S., Hillar, C., Miolane, N.
An important problem in signal processing and deep learning is to achieve invari- ance 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 character- istic 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—akin to a pooling mechanism, but with greater selectivity and robustness. However, the computational cost of the G-Bispectrum (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 O(|G|) complexity. We prove desirable mathematical properties of the selective G-Bispectrum and demonstrate how its integration in neu- ral networks enhances accuracy and robustness compared to traditional approaches, while enjoying considerable speeds-up compared to the full G-Bispectrum.
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Learning from landmarks, curves, surfaces, and shapes in Geomstats | Submitted to Transactions of Mathematical Software (TOMS)
Pereira, L., Le Brigant, A., Myers, A., Hartman, E., Khan, A., Tuerkoen, M., Dold, T., Gu, M., Suárez-Serrato, P., Miolane, N.
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.
On the Implementation of Geodesic Metric Spaces | Submitted to Journal of Machine Learning Research (JMLR)
Calissano, A., Pereira, L., Lueg, J., Miolane, N.
Analysis of non-Euclidean data such as graphs and trees requires (specific) mathematical machinery due to their less-rich structure when compared to Euclidean spaces or smooth Riemannian manifolds. These spaces can still leverage the rich structure of the latter. For example, graph space results from quotienting out matrices endowed with the Frobenius metric by the permutation group, Billera–Holmes–Vogtmann (BHV) space strata are Euclidean, and wald space is embedded in the space of symmetric positive definite (SPD) matrices. We present a Python package for the analysis of data living in geodesic metric spaces – topological spaces equipped with a metric and a geodesic function where the metric is the length of the shortest geodesic joining two points. We describe the package structure, based on a point, a point set, and a metric built using geodesic metric space theory, and we provide three implementation examples. The package is implemented as a plug-in of the Geomstats Python package, allowing users to access and adapt the available geometrical and data analysis tools for strongly non-Euclidean data in a theoretically consistent way. The code is unit-tested and documented.
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TopoBenchmarkX: A Framework for Benchmarking Topological Deep Learning | Submitted to NeurIPS
Telyatnikov, L., Bernardez, G., Montagna, M., Vasylenko, P., Zamzmi, G., Hajij, M., Schaub, M., Miolane, N., Scardapane, S., Papamarkou, T.
This work introduces TopoBenchmarkX, a modular open-source library designed to standardize benchmarking and accelerate research in Topological Deep Learning (TDL). TopoBenchmarkX maps the TDL pipeline into a sequence of independent and modular components for data loading and processing, as well as model training, optimization, and evaluation. This modular organization provides flexibility for modifications and facilitates the adaptation and optimization of various TDL pipelines. A key feature of TopoBenchmarkX is that it allows for the transformation and lifting between topological domains. This enables, for example, to obtain richer data representations and more fine-grained analyses by mapping the topology and features of a graph to higher-order topological domains such as simplicial and cell complexes. The range of applicability of TopoBenchmarkX is demonstrated by benchmarking several TDL architectures for various tasks and datasets. [Code]
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Not so griddy: Internal representations of RNNs path integrating more than one agent | NeurIPS
Redman W., Acosta, F., Acosta-Mendoza, A., Miolane, N.
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.
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Position: Topological Deep Learning is the New Frontier for Relational Learning | ICML
Papamarkou, T., Birdal, T., Bronstein, M., Carlsson, G., Curry, J., Gao, Y., Hajij, M., Kwitt, R., Liò, P., Di Lorenzo, P., Maroulas, V., Miolane, N., Nasrin, F., Natesan Ramamurthy, K., Rieck, B., Scardapane, S., T. Schaub, M., Veličković, P., Wang, B., Wang, Y., Wei, G., Zamzmi, G.
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 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.
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Bounds on Geodesic Distances on the Stiefel Manifold | MTNS
Mataigne, S., Absil, P.A., Miolane, N.
The past few years have seen the emergence of many applications for statistics on manifolds. The manifold-based statistical tools often rely on the notion of mean of two points, i.e., the midpoint of a minimizing geodesic between these two points. An important manifold for which we cannot guarantee the computation of a minimizing geodesic is the Stiefel manifold, which is the set of orthogonal p-frames in R^n. The goal of this master's thesis is to provide new insights into the Stiefel manifold to enhance the performances of minimizing geodesic computation algorithms. To this aim, we provide new results on the bilipschitz equivalence of a previously proposed one-parameter family of Riemannian metrics. We also show that the geodesic distances induced by this family of Riemannian metrics are equivalent to the easy-to compute Frobenius distance. Subsequently, we obtain tight bounds between the geodesic and the Frobenius distances. Many operations on the Stiefel manifold, and in particular the computation of a minimizing geodesic, require the computation of the matrix exponential of skew-symmetric matrices. To conclude this work, we present the SkewLinearAlgebra.jl package, a Julia library specialized in the efficient computation of the eigenvalue decomposition and the exponential of skew-symmetric matrices.
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On Accuracy and Speed of Geodesic Regression: Do Geometric Priors Improve Learning on Small Datasets? | CVPRW L3D-IVU
Myers, A., Miolane, N.
Image datasets in specialized fields of science, such as biomedicine, are typically smaller than traditional machine learning datasets. As such, they present a problem for training many models. To address this challenge, researchers often attempt to incorporate priors, i.e., external knowledge, to help the learning procedure. Geometric priors, for example, offer to restrict the learning process to the manifold to which the data belong. However, learning on manifolds is sometimes computationally intensive to the point of being prohibitive. Here, we ask a provocative question: is machine learning on manifolds really more accurate than its linear counterpart to the extent that it is worth sacrificing significant speedup in computation? We answer this question through an extensive theoretical and experimental study of one of the most common learning methods for manifold-valued data: geodesic regression.
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TopoX: A Suite of Python Packages for Machine Learning on Topological Domains | Submitted to the Journal of Machine Learning Research (JMLR)
Hajij, M., Papillon, M., Frantzen, F., Agerberg, J., AlJabea, I., Ballester, R., Battiloro, C., Bernárdez, G., Birdal, T., Brent, A., Chin, P., Escalera, S., Fiorellino, S., Hoff Gardaa, O., Gopalakrishnan, G., Govil, D., Hoppe, J., Reddy Karri, M., Khouja, J., Lecha, M., Livesay, N., Meißner, J., Mukherjee, S., Nikitin, A., Papamarkou, T., Prílepok, J., Natesan Ramamurthy, K., Rosen, P., Guzmán-Sáenz, A., Salatiello, A., N. Samaga, S., Scardapane, S., T. Schaub, M., Scofano, L., Spinelli, I., Telyatnikov, L., Truong, Q., Walters, R., Yang, M., Zaghen, O., Zamzmi, G., Zia, A., Miolane, N.
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 this https URL.
Towards Interpretable Cryo-EM: Disentangling Latent Spaces of Molecular Conformations | Frontiers in Molecular Biosciences
A. Klindt, D., Hyvärinen, A., Levy, A., Miolane, N., Poitevin, F.
Molecules are essential building blocks of life and their different conformations (i.e., shapes) crucially determine the functional role that they play in living organisms. Cryogenic Electron Microscopy (cryo-EM) allows for acquisition of large image datasets of individual molecules. Recent advances in computational cryo-EM have made it possible to learn latent variable models of conformation landscapes. However, interpreting these latent spaces remains a challenge as their individual dimensions are often arbitrary. The key message of our work is that this interpretation challenge can be viewed as an Independent Component Analysis (ICA) problem where we seek models that have the property of identifiability. That means, they have an essentially unique solution, representing a conformational latent space that separates the different degrees of freedom a molecule is equipped with in nature. Thus, we aim to advance the computational field of cryo-EM beyond visualizations as we connect it with the theoretical framework of (nonlinear) ICA and discuss the need for identifiable models, improved metrics, and benchmarks. Moving forward, we propose future directions for enhancing the disentanglement of latent spaces in cryo-EM, refining evaluation metrics and exploring techniques that leverage physics-based decoders of biomolecular systems. Moreover, we discuss how future technological developments in time-resolved single particle imaging may enable the application of nonlinear ICA models that can discover the true conformation changes of molecules in nature. The pursuit of interpretable conformational latent spaces will empower researchers to unravel complex biological processes and facilitate targeted interventions. This has significant implications for drug discovery and structural biology more broadly. More generally, latent variable models are deployed widely across many scientific disciplines. Thus, the argument we present in this work has much broader applications in AI for science if we want to move from impressive nonlinear neural network models to mathematically grounded methods that can help us learn something new about nature.
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An efficient algorithm for the Riemannian logarithm on the Stiefel manifold for a family of Riemannian metrics | Submitted to SIAM Journal on Matrix Analysis and Applications (SIMAX)
Mataigne, S., Zimmermann, R., Miolane, N.
Since its popularization in 1998, the Stiefel manifold has proven crucial for addressing numerous issues in optimization, statistics, and machine learning. In 2021, Hüper and collaborators introduced a one-parameter family of Riemannian metrics on the Stiefel manifold, encompassing both the renowned Euclidean and canonical metrics. Zimmermann, in 2017, developed an especially effective technique to compute the Riemannian Log for the canonical metric, utilizing a pure matrix-algebraic method. Here, we present an extension of this approach applicable to the entire one-parameter family of Riemannian metrics.
Unveiling cellular morphology: Statistical analysis using a Riemannian elastic metric in cancer cell image datasets | Published in Information Geometry
Li, W., Prasad, A., Miolane, N., Dao Duc, K.
Elastic metrics can provide a powerful tool to study the heterogeneity arising from cellular morphology. To assess their potential application (e.g. classifying cancer treated cells), we consider a specific instance of the elastic metric, the Square Root Velocity (SRV) metric and evaluate its performance against the linear met- ric for two datasets of osteosarcoma (bone cancer) cells including pharmacological treatments, and normal and cancerous breast cells. Our comparative statistical analysis shows superior performance of the SRV at capturing cell shape hetero- geneity when comparing distance to the mean shapes, with better separation and interpretation between different cell groups. Secondly, when using multidi-mensional scaling (MDS) to find a low-dimensional embedding for unrescaled contours, we observe that while the linear metric better preserves original pairwise distances, the SRV yields better classification.
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Relating Representational Geometry to Cortical Geometry in the Visual Cortex | NeurIPS Workshop on Unifying Representations in Neural Models
Acosta, F., Conwell, C., Sanborn, S., Klindt, D., Miolane, N.
A fundamental principle of neural representation is to minimize wiring length by spatially organizing neurons according to the frequency of their communication [Sterling and Laughlin, 2015]. A consequence is that nearby regions of the brain tend to represent similar content. This has been explored in the context of the visual cortex in recent works [Doshi and Konkle, 2023, Tong et al., 2023]. Here, we use the notion of cortical distance as a baseline to ground, evaluate, and interpret mea- sures of representational distance. We compare several popular methods—both second-order methods (Representational Similarity Analysis, Centered Kernel Alignment) and first-order methods (Shape Metrics)—and calculate how well the representational distance reflects 2D anatomical distance along the visual cortex (the anatomical stress score). We evaluate these metrics on a large-scale fMRI dataset of human ventral visual cortex [Allen et al., 2022b], and observe that the 3 types of Shape Metrics produce representational-anatomical stress scores with the smallest variance across subjects, (Z score = -1.5), which suggests that first-order representational scores quantify the relationship between representational and cortical geometry in a way that is more invariant across different subjects. Our work establishes a criterion with which to compare methods for quantifying representational similarity with implications for studying the anatomical organization of high-level ventral visual cortex.
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Visual Scene Representation with Hierarchical Equivariant Sparse Coding | PMLR
Shewmake, C., Buracas, D., Lillemark, H., Shin, J., Bekkers, E., Miolane, N., Olshausen, B.
We propose a hierarchical neural network architecture for unsupervised learning of equivariant part-whole decompositions of visual scenes. In contrast to the global equivariance of group-equivariant networks, the proposed architecture exhibits equivariance to part-whole transformations throughout the hierarchy, which we term hierarchical equivariance. The model achieves these structured internal representations via hierarchical Bayesian inference, which gives rise to rich bottom-up, top-down, and lateral information flows, hypothesized to underlie the mechanisms of perceptual inference in visual cortex. We demonstrate these useful properties of the model on a simple dataset of scenes with multiple objects under independent rotations and translations.
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Preface: NeurIPS Workshop on Symmetry and Geometry in Neural Representations | PMLR
Sanborn, S., Shewmake, C., Azeglio, S., Di Bernardo, A., Miolane, N.
The first annual NeurIPS Workshop on Symmetry and Geometry in Neural Representations (NeurReps) was conceived to bring together researchers at the nexus of applied geometry, deep learning, and neuroscience, with the goal of advancing this understanding and illuminating geometric principles for neural information processing. Ultimately, we hope that this venue and associated community will support the development of the geometric approach to understanding neural representations, while strengthening ties to the mathematics community. The Neural Information Processing Systems (NeurIPS) conference historically emerged from the field of theoretical neuroscience or “connectionism.” This venue thus further serves the workshop’s goal of reinforcing the bond between deep learning and neuroscience.
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ICML 2023 Topological Deep Learning Challenge: Design and Results | PMLR
Papillon, M., Miolane, N., et al.
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.
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A General Framework for Robust G-Invariance in G-Equivariant Networks | NeurIPS
Sanborn, S., Miolane, N.
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.
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Identifying Interpretable Visual Features in Artifical and Biological Neural Systems | arXiv
Klindt, D., Sanborn, S., Acosta, F., Poitevin, F., Miolane, N.
Single neurons in neural networks are often interpretable in that they represent individual, intuitively meaningful features. However, many neurons exhibit mixed selectivity, i.e., they represent multiple unrelated features. A recent hypothesis proposes that features in deep networks may be represented in superposition, i.e., on non-orthogonal axes by multiple neurons, since the number of possible interpretable features in natural data is generally larger than the number of neurons in a given network. Accordingly, we should be able to find meaningful directions in activation space that are not aligned with individual neurons. Here, we propose (1) an automated method for quantifying visual interpretability that is validated against a large database of human psychophysics judgments of neuron interpretability, and (2) an approach for finding meaningful directions in network activation space. We leverage these methods to discover directions in convolutional neural networks that are more intuitively meaningful than individual neurons, as we confirm and investigate in a series of analyses. Moreover, we apply the same method to three recent datasets of visual neural responses in the brain and find that our conclusions largely transfer to real neural data, suggesting that superposition might be deployed by the brain. This also provides a link with disentanglement and raises fundamental questions about robust, efficient and factorized representations in both artificial and biological neural systems.
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Geodesic Regression Characterizes 3D Shape Changes in the Female Brain During Menstruation | ICCV Computer Vision for Automated Medical Diagnosis
Myers, A., Taylor, C., Jacobs, E., Miolane, N. [Code]
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.
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CryoChains: Heterogeneous Reconstruction of Molecular Assembly of Semi-flexible Chains from Cryo-EM Images | ICML CompBio Workshop
Koo, B., Martel, J., Peck, A., Levy, A., Poitevin, F., Miolane, N.
We propose CryoChains to encode large deformations of biomolecules via rigid body transformation of their polymer instances (chains), while representing their finer shape variations with the normal mode analysis framework of biophysics. CryoChains gives a biophysically-grounded quantification of the heterogeneous conformations of biomolecules, while reconstructing their 3D molecular structures at an improved resolution compared to the current fastest, interpretable deep learning method.
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Septins Regulate Border Cell Shape and Surface Geometry ownstream of Rho | Developmental Cell
Gabbert, A., Mondo, J., Campanale, J., Mitchell, N., Myers, A., Streichan, S., Miolane, N., Montell, D.
Septins self-assemble into polymers that bind and deform membranes in vitro and regulate diverse cell behaviors in vivo. How their in vitro properties relate to their in vivo functions is under active investigation. Here we uncover requirements for septins in detachment and motility of border cell clusters in the Drosophila ovary.
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Topological Deep Learning: Going Beyond Graph Data | Submitted to JMLR: Journal of Machine Learning Research
Hajij, M., Zamzmi, G., Papamarkou, T., Miolane, N., Guzmán-Sáenz, A., Ramamurthy, K., Birdal, T., Dey, T., Mukherjee, S., Samaga, S., Livesay, N., Walters, R., Rosen, P., Schaub, M. [Code]
Topological deep learning is a rapidly growing field that pertains to the development of deep learning models for data supported on topological domains such as simplicial complexes, cell complexes, and hypergraphs, which generalize many domains encountered in scientific computations. In this paper, we present a unifying deep learning framework built upon a richer data structure that includes widely adopted topological domains.
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Architectures of Topological Deep Learning: A Survey on Topological Neural Networks | Submitted to PAMI: Transactions of Pattern Analysis and Machine Intelligence
Papillon, S., Sanborn, S., Hajij, M., Miolane, N. [Equations]
Topological Neural Networks (TNNs) are deep learning architectures that process signals defined on topological domains. The domains of topological deep learning generalize the binary relations of graphs to hierarchical relations and higher-order set-based relations. The additional flexibility and expressivity of these architectures permits the representation of complex natural systems such as proteins, neural activity, and many-body physical systems. In this concise review of the latest in topological deep learning, we offer a pedagogical introduction to the field that uses a unified mathematical language to describe the landscape of TNNs.
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Orthogonal Outlier Detection and Dimension Estimation for Improved MDS Embedding of Biological Datasets | Frontiers in Bioinformatics
Li, W., Mirone, J., Prasad, A., Miolane, N., Legrand, C. and Dao Duc, K.
Conventional dimensionality reduction methods are sensitive to orthogonal outliers, which yield significant defects in the embedding. We introduce a robust method to address this problem for cell morphology and human microbiomes datasets.
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Quantifying Extrinsic Curvatures of Neural Manifolds | CVPRW
Acosta, F., Sanborn, S., Dao Duc, K., Madhav, M., Miolane, N.
We leverage tools from Riemannian geometry and topologically-aware deep generative models to introduce a novel approach for studying the geometry of neural manifolds. This approach (1) computes an explicit parameterization of the manifolds and (2) estimates their local extrinsic curvature. [Code].
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Lipschitz Constants Between Riemannian Metrics on the Stiefel Manifold | Geometric Science of Information
Mataigne, S., Absil, P.-A., Miolane, N.
We give the best Lipschitz constants between the distances induced by any two members of a one-parameter family of Riemannian metrics on the Stiefel manifold of orthonormal p-frames in n dimensions.
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Using an Elastic Metric for Statistical Analysis of Tumor Cell Shape Heterogeneity | Geometric Science of Information
Li, W., Prasad, A., Miolane, N., Dao Duc, K. [Code]. Geometric Science of Information (GSI): Session Biological Shape Analysis. (Oral)
We propose a methodology grounded in geometric statistics to study and compare cellular morphologies from the contours they form on planar surfaces. We present findings on a dataset of images from osteocarcoma cells that includes different cancer treatments known to affect the cell morphology.
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Equivariant Sparse Coding | Geometric Science of Information
Shewmake, C., Miolane, N., Olshausen, B. Geometric Science of Information. (Oral)
We describe a sparse coding model of visual cortex that encodes image transformations in an equivariant manner. We present results on time-varying visual scenes.
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Differentially Private Fréchet Mean on the Manifold of Symmetric Positive Definite (SPD) Matrices | TMLR
Utpala, S., Vepakomma, P., Miolane, N.
Transactions of Machine Learning Research (TMLR).
We propose a novel, simple and fast mechanism - the Tangent Gaussian mechanism - to compute a differentially private Fréchet mean on the SPD manifold endowed with the log-Euclidean Riemannian metric. We show that our new mechanism obtains quadratic utility improvement in terms of data dimension over the current and only available baseline. Our mechanism is also simpler in practice as it does not require any expensive Markov Chain Monte Carlo (MCMC) sampling, and is computationally faster by multiple orders of magnitude - as confirmed by extensive experiments.
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Deep Generative Modeling for Volume Reconstruction in Cryo-Electron Microscopy | JSB
Donnat, C., Levy, A., Poitevin, F., Zhong, E., Miolane, N.
Journal of Structural Biology.
Advances in cryo-electron microscopy (cryo-EM) for high-resolution imaging of biomolecules in solution have provided new challenges and opportunities. Next-generation volume reconstruction algorithms that combine generative modelling with end-to-end unsupervised deep learning techniques have shown promise, but many technical and theoretical hurdles remain. In light of the proliferation of such methods, we propose here a critical review of recent advances in the field of deep generative modelling for cryo-EM reconstruction. [Code].
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Probabilistic Riemannian Functional Map Synchronization for 3D Shape Correspondence | arXiv
Huq, F., Dey, A., Yusuf, S., Bazazian, D., Birdal, T., Miolane, N.
We consider the problem of graph-matching on a network of 3D shapes with uncertainty quantification. We assume that the pairwise shape correspondences are efficiently represented as functional maps, that match real-valued functions defined over pairs of shapes. By modeling functional maps between nearly isometric shapes as elements of the Lie group SO(n), we employ synchronization to enforce cycle consistency of the collection of functional maps over the graph, hereby enhancing the accuracy of the individual maps.
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Introduction to Riemannian Geometry and Geometric Statistics: from theory to implementation with Geomstats | JFTML
Guigui, N., Miolane, N., Pennec, X.
Journal of Foundations and Trends in Machine Learning.
We give a self-contained exposition of the basic concepts of Riemannian geometry, providing illustrations and examples at each step and adopting a computational point of view. We cover the basics of differentiable manifolds, Riemannian manifolds, as well as quotient, homogeneous and symmetric spaces. Next, we demonstrate how these concepts are implemented in Geomstats, with details and code examples along the text. The culmination of this implementation is to be able to perform statistics and machine learning on manifolds, with as few lines of codes as in the wide-spread machine learning tool scikit-learn. [Code]
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Heterogeneous Reconstructions of Deformable Models in Cryo-Electron Microscopy | NeurIPS MLSB
Nashed, Y., Peck, A., Martel, J., Levy, A., Koo, B., Wetzstein, G., Miolane, N., Ratner, D., Poitevin, F.
NeurIPS Workshop of Machine Learning for Structural Biology.
Cryogenic electron microscopy (cryo-EM) provides a unique opportunity to study the structural heterogeneity of biomolecules. Being able to explain this heterogeneity with atomic models would help our understanding of their functional mechanisms but the size and ruggedness of the structural space (the space of atomic 3D cartesian coordinates) presents an immense challenge. Here, we describe a heterogeneous reconstruction method based on an atomistic representation whose deformation is reduced to a handful of collective motions through normal mode analysis.
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Testing Geometric Representation Hypotheses from Simulated Place Cells Recordings | NeurIPS NeurReps
Niederhauser, T., Lester, A., Miolane, N., Dao Duc, K., Madhav, M.
NeurIPS Workshop for Symmetry and Geometry in Neural Representations.
Hippocampal place cells can encode spatial locations of an animal in physical or task-relevant spaces. We simulated place cell populations that encoded either Euclidean- or graph-based positions of a rat navigating to goal nodes in a maze with a graph topology, and used manifold learning methods to analyze these neural population activities. [Code].
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Regression-Based Elastic Metric Learning on Shape Spaces of Elastic Curves | NeurIPS LMRL
Myers, A., Miolane, N.
NeurIPS Workshop on Learning Meaningful Representations of Life.
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. [Code].
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Challenge for Computational Geometry and Topology: Design and Results | ICLR GTRL
Myers, A., Utpala, ., Talbar, S., Sanborn, S., Shewmake, C., Donnat, C., Mathe, J., Lupo, U., Sonthalia, R., Cui, X., Szwagier, T., Pignet, A., Bergsson, A., Hauberg, S., Nielsen, D., Sommer, S., Klindt, D., Hermansen, E., Vaupel, M., Dunn, B., Xiong, J., Aharony, N., Pe'er, I., Ambellan, F., Hanik, M., Nava-Yazdani, E., von Tycowicz, C., Miolane, N.
ICLR Geometrical and Topological Representation Learning.
Proceedings of Machine Learning Research.
We present the computational challenge on differential geometry and topology that was hosted within the ICLR 2022 workshop "Geometric and Topological Representation Learning". The competition asked participants to provide implementations of machine learning algorithms on manifolds that would respect the API of the open-source software Geomstats (manifold part) and Scikit-Learn (machine learning part) or PyTorch. [Code].
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Defining an Action of SO(d)-Rotations on Projections of d-Dimensional Objects: Applications to Pose Inference with Geometric VAEs | GRETSI
Legendre, N, Dao Duc, K., Miolane, N.
GRETSI Conference (2022).
Recent advances in variational autoencoders (VAEs) have enabled learning latent manifolds as compact Lie groups, such as SO(d). Since this approach assumes that data lies on a subspace that is homeomorphic to the Lie group itself, we here investigate how this assumption holds in the context of images that are generated by projecting a d-dimensional volume with unknown pose in SO(d).
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CryoAI: Amortized Inference of Poses for Ab Initio Reconstruction of 3D Molecular Volumes from Real Cryo-EM Images | ECCV
Levy, A., Poitevin, F., Martel, J., Nashed, Y., Peck, A., Miolane, N., Ratner, D., Dunne, M., Wetzstein, G.
4th International Symposium on Cryo-3D Image Analysis (Best Poster Award).
ECCV European Conference on Computer Vision.
Cryo-electron microscopy (cryo-EM) has become a tool of fundamental importance in structural biology. The algorithmic challenge of cryo-EM is to jointly estimate the unknown 3D poses and the 3D electron scattering potential of a biomolecule from millions of extremely noisy 2D images. Existing reconstruction algorithms, however, cannot easily keep pace with the rapidly growing size of cryo-EM datasets due to their high computational and memory cost. We introduce cryoAI, an ab initio reconstruction algorithm for homogeneous conformations that uses direct gradient-based optimization of particle poses and the electron scattering potential from single-particle cryo-EM data. [Code].
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Parametric Information Geometry with the Package Geomstats | Transactions of Mathematical Software (TOMS)
Le Brigant, A., Deschamps, J., Collas, A., Miolane, N.
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. [Code].
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Intentional Choreography with Semi-Supervised Recurrent Variational Autoencoders | NeurIPS CAD
Papillon, M., Pettee, M., Miolane, N.
NeurIPS Workshop of Creativity and Design.
Given a small amount of dance sequences labeled with qualitative choreographic annotations, PirouNet conditionally generates dance sequences in the artistic style of the choreographer. [Code].
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Biological Shape Analysis with Geometric Statistics and Learning | Oberwolfach Snapshots
Utpala, S., Miolane, N.
Oberwolfach Snapshots of Modern Mathematics (2022).
The advances in biomedical imaging techniques have enabled us to access the 3D shapes of a variety of structures: organs, cells, proteins. Since biological shapes are related to physiological functions, shape data may hold the key to unlocking outstanding mysteries in biomedicine. This snapshot introduces the mathematical framework of geometric statistics and learning and its applications in biomedicine.
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PirouNet: Creating Dance through Artist-Centric Deep Learning | EAI ArtsIT
Papillon M., Pettee M., Miolane N.
EAI ArtsIT Conference. Best Paper Award (Oral).
Using Artificial Intelligence (AI) to create dance choreography with intention is still at an early stage. Methods that conditionally generate dance sequences remain limited in their ability to follow choreographer-specific creative direction, often relying on external prompts or supervised learning. In the same vein, fully annotated dance datasets are rare and labor intensive. To fill this gap and help leverage deep learning as a meaningful tool for choreographers, we propose "PirouNet": PirouNet allows dance professionals to annotate data with their own subjective creative labels and subsequently generate new bouts of choreography based on their aesthetic criteria. Thanks to the proposed semi-supervised approach, PirouNet only requires a small portion of the dataset to be labeled, typically on the order of 1%.. [Code].
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Higher-Order Attention Networks | ArXiv
Hajij, M., Zamzmi, G., Papamarkou, T., Miolane, N., Guzman-Saenz, A., Ramamurthy, K., N.
We introduce higher-order attention networks (HOANs), a novel class of attention-based neural networks defined on a generalized higher-order domain called a combinatorial complex (CC). Similar to hypergraphs, CCs admit arbitrary set-like relations between a collection of abstract entities. Simultaneously, CCs permit the construction of hierarchical higher-order relations analogous to those supported by cell complexes. Thus, CCs effectively generalize both hypergraphs and cell complexes and combine their desirable characteristics.
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Challenge for Computational Geometry & Topology: Design and Results | ICLR GTRL
Miolane, N., Caorsi, M., Lupo, U., Guerard, M., Guigui, N., Mathe, J., Cabanes, Y., Reise, W., Davies, T., Leitão, A., Mohapatra, S., Utpala, S., Shailja, S., Corso, G., Liu, G., Iuricich, F., Manolache, A., Nistor, M., Bejan, M., Mihai Nicolicioiu, A., Luchian, B.-A., Stupariu, M.-S., Michel, F., Dao Duc, K., Abdulrahman, B., Beketov, M., Maignant, E., Liu, Z., Černý, M., Bauw, M., Velasco-Forero, S., Angulo, J., Long Y.
ICLR Workshop on Geometrical and Topologic Representation Learning.
We present the computational challenge on differential geometry and topology that happened within the ICLR 2021 workshop "Geometric and Topological Representation Learning". The competition asked participants to provide creative contributions to the fields of computational geometry and topology through the open-source repositories Geomstats and Giotto-TDA. [Code].
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Geomstats: A Python Package for Riemannian Geometry in Machine Learning | JMLR
Miolane, N., Guigui, N., Le Brigant, A., Mathe, J., Hou, B., Thanwerdas, Y., Heyder, S., Peltre, O., Koep, N., Cabanes, Y., Chauchat, P., Zaatiti, H., Hajri, H., Gerald, T. , Shewmake, C., Brooks, D., Kainz, B., Donnat, C., Holmes, S., Pennec, X.
Journal of Machine Learning Research.
We introduce Geomstats, an open-source Python toolbox for computations and statistics on nonlinear manifolds, such as hyperbolic spaces, spaces of symmetric positive definite matrices, Lie groups of transformations, and many more. [Code].
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Introduction to Geometric Learning in Python with Geomstats | SciPy
Miolane, N., Guigui, N., Zaatiti, H., Shewmake, C., Hajri, H., Brooks, D., Le Brigant, A., Mathe, J. Hou, B., Thanwerdas, Y., Heyder, S., Peltre, O., Koep, N., Cabanes, Y., Gerald, T. Chauchat, P., Kainz, B., Donnat, C., Holmes, S., Pennec, X.
SciPy Conference on Scientific Computing in Python.
There is a growing interest in leveraging differential geometry in the machine learning community. Yet, the adoption of the associated geometric computations has been inhibited by the lack of a reference implementation. To address this gap, we present the open-source Python package geomstats and introduce hands-on tutorials for differential geometry and geometric machine learning algorithms. [Code].
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Learning Weighted Submanifolds With Riemannian Variational Autoencoders | CVPR
Miolane, N., Holmes, S.
CVPR Conference of Computer Vision and Pattern Recognition.
Manifold-valued data naturally arises in medical imaging. One of the challenges that naturally arises consists of finding a lower-dimensional subspace for representing such manifold-valued data. Traditional techniques, like principal component analysis, are ill-adapted to tackle non-Euclidean spaces. We introduce Riemannian Variational Autoencoders to perform weighted submanifold learning powered by amortized variational inference.
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Estimation of Orientation and Camera Parameters from Cryo-Electron Microscopy Images with Variational Autoencoders and Generative Adversarial Networks | CVPRW
Miolane, N., Poitevin, F., Li, Y.-T., Holmes, S.
CVPR Workshop on Computer Vision for Microscopy Imaging.
We combine variational autoencoders (VAEs) and generative adversarial networks (GANs) to learn a low-dimensional latent representation of cryo-EM images. Cryo-electron microscopy (cryo-EM) is capable of producing reconstructed 3D images of biomolecules at near-atomic resolution. As such, it represents one of the most promising imaging techniques in structural biology.
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A Bayesian Hierarchical Network for Combining Heterogeneous Data Sources in Medical Diagnoses | NeurIPS ML4H
Donnat, C., Miolane, N., Bunbury, F., Kreindler, J.
NeurIPS Workshop on Machine Learning for Health.
1st Prize: C3.ai Grand Covid Challenge (100,000$).
Computer-Aided Diagnosis has shown stellar performance in providing accurate medical diagnoses across multiple testing modalities (medical images, electrophysiological signals, etc.). While this field has typically focused on fully harvesting the signal provided by a single (and generally extremely reliable) modality, fewer efforts have utilized imprecise data lacking reliable ground truth labels. We devise a Stochastic Expectation-Maximization algorithm that allows the principled integration of heterogeneous, and potentially unreliable, data types. We showcase the practicality of this approach by deploying it on a real COVID-19 immunity study.
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Bias on Estimation in Quotient Space and Correction Methods | Elsevier
Miolane, N., Devilliers, L., Pennec, X.
Chapter in Riemannian Geometric Statistics in Medical Imaging. Statistics on Shape Spaces. Elsevier.
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Video Clip Selector for Medical Imaging and Diagnosis | Patent
Koepsell, K., Cadieu, D., Poilvert, N., Hong, H., Cannon, M., Bilenko, N., Romano, N., Mathe, J., Cheng, C., Miolane, N.
Caption Health.
United States Patent and Trademark Office.
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PVNet: A LRCN Architecture for Spatio-Temporal Photovoltaic Power Forecasting from Numerical Weather Prediction | ICML AICC
Mathe, J., Miolane, N., Sebastien, N., Lequeux, J.
ICML Workshop on AI for Climate Change.
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.
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Computing CNN Loss and Gradients for Pose Estimation with Riemannian Geometry | MICCAI
Hou, B., Miolane N., Khanal B., Lee M., Alansary A., McDonagh S., Hajnal J., Rueckert D., Glocker B., Kainz B.
MICCAI Conference on Medical Image Computing and Computer Assisted Intervention.
Pose estimation, i.e. predicting a 3D rigid transformation with respect to a fixed co-ordinate frame in, SE(3), is an omnipresent problem in medical image analysis with applications such as: image rigid registration, anatomical standard plane detection, tracking and device/camera pose estimation. Deep learning methods often parameterise a pose with a representation that separates rotation and translation. As commonly available frameworks do not provide means to calculate loss on a manifold, regression is usually performed using the L2-norm independently on the rotation's and the translation's parameterisations, which is a metric for linear spaces that does not take into account the Lie group structure of SE(3). In this paper, we propose a general Riemannian formulation of the pose estimation problem. We propose to train the CNN directly on SE(3) equipped with a left-invariant Riemannian metric, coupling the prediction of the translation and rotation defining the pose. At each training step, the ground truth and predicted pose are elements of the manifold, where the loss is calculated as the Riemannian geodesic distance. We then compute the optimisation direction by back-propagating the gradient with respect to the predicted pose on the tangent space of the manifold SE(3) and update the network weights. We thoroughly evaluate the effectiveness of our loss function by comparing its performance with popular and most commonly used existing methods, on tasks such as image-based localisation and intensity-based 2D/3D registration. We also show that hyper-parameters, used in our loss function to weight the contribution between rotations and translations, can be intrinsically calculated from the dataset to achieve greater performance margins.
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Topologically Constrained Template Estimation | SIAGA
Miolane, N., Holmes, S., Pennec, X.
SIAM Journal on Applied Algebra and Geometry.
In most neuroimaging studies, one builds a brain template that serves as a reference for normalizing the measurements of each individual subject into a common space. Such a template should be representative of the population under study, thus avoiding bias in subsequent statistical analyses. The template is often computed by iteratively registering all images to the current template and then averaging the intensities of the registered images. Geometrically, the procedure can be summarized as the computation of the template as the “Fréchet mean” of the images projected in a quotient space. It has been argued recently that this type of algorithm could actually be asymptotically biased and therefore inconsistent. In other words, even with an infinite number of brain images in the database, the template estimate may not converge to the brain anatomy it is meant to estimate. Our paper investigates this phenomenon. We present a methodology that spatially quantifies the brain template's asymptotic bias. We identify the main variables that influence inconsistency. This leads us to investigate the topology of the template's intensity level sets, represented by its Morse--Smale (MS) complex. We propose a topologically constrained adaptation of the template computation that constructs a hierarchical template with bounded bias. We apply our method to the analysis of a brain template of 136 T1 weighted MR images from the Open Access Series of Imaging Studies (OASIS) database.
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Template Shape Estimation in Computational Anatomy | SIIMS
Miolane, N., Holmes, S., Pennec, X.
SIAM Journal of Imaging Science.
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 the template shape estimation using the stratified geometry of the shape space. We give a Taylor expansion of the bias with respect to a parameter σ 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. This exhibits the parameters that control the bias’ magnitude. We illustrate our results on simulated and real shape data.
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Toward a Unified Geometric Bayesian Framework for Template Estimation in Computational Anatomy. | ISBA
Miolane, N., Pennec, X., Holmes, S.
ISBA World Meeting of the International Society for Bayesian Analysis. (Young Researcher Travel Award).
Computational Anatomy aims to model and analyze the variability of the human anatomy. Given a set of medical images of the same organ, the first step is the estimation of the mean organ’s shape. This mean anatomical shape is called the template in Computer vision or Medical imaging. The estimation of a template/atlas is central because it represents the starting point for all further processing or analyses. In view of the medical applications, evaluating the quality of this statistical estimate is crucial. How does the estimated template behave for varying amount of data, for small and large level of noise? We present a geometric Bayesian framework which unifies two estimation problems that are usually considered distinct: the template estimation problem and manifold learning problem - here associated to estimating the template’s orbit. We leverage this to evaluate the quality of the template estimator.
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Biased Estimators on Quotient spaces | GSI
Miolane, N., Pennec, X.
Conference on Geometric Sciences of Information. (Oral). [Video].
Usual statistics are defined, studied and implemented on Euclidean spaces. But what about statistics on other mathematical spaces, like manifolds with additional properties: Lie groups, Quotient spaces, Stratified spaces etc. How can we describe the interaction between statistics and geometry? The structure of Quotient space in particular is widely used to model data, for example every time one deals with shape data. These can be shapes of constellations in Astronomy, shapes of human organs in Computational Anatomy, shapes of skulls in Palaeontology, etc. Given this broad field of applications, statistics on shapes -and more generally on observations belonging to quotient spaces- have been studied since the 1980's. However, most theories model the variability in the shapes but do not take into account the noise on the observations themselves. In this paper, we show that statistics on quotient spaces are biased and even inconsistent when one takes into account the noise. In particular, some algorithms of template estimation in Computational Anatomy are biased and inconsistent. Our development thus gives a first theoretical geometric explanation of an experimentally observed phenomenon. A biased estimator is not necessarily a problem. In statistics, it is a general rule of thumb that a bias can be neglected for example when it represents less than 0.25 of the variance of the estimator. We can also think about neglecting the bias when it is low compared to the signal we estimate. In view of the applications, we thus characterize geometrically the situations when the bias can be neglected with respect to the situations when it must be corrected.
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Mathematical Structures for Extending 2D Neurogeometry to 3D Image Processing | MICCAI MCV
Miolane, N., Pennec, X.
MICCAI Workshop of Medical Computer Vision. [Code].
In the era of big data, one may apply generic learning algorithms for medical computer vision. But such algorithms are often "black-boxes" and as such, hard to interpret. We still need new constructive models, which could eventually feed the big data framework. Where can one find inspiration for new models in medical computer vision? The emerging field of Neurogeometry provides innovative ideas. Neurogeometry models the visual cortex through modern Differential Geometry: the neuronal architecture is represented as a sub-Riemannian manifold R2 x S1. On the one hand, Neurogeometry explains visual phenomena like human perceptual completion. On the other hand, it provides efficient algorithms for computer vision. Examples of applications are image completion (in-painting) and crossing-preserving smoothing. In medical image computer vision, Neurogeometry is less known although some algorithms exist. One reason is that one often deals with 3D images, whereas Neurogeometry is essentially 2D (our retina is 2D). Moreover, the generalization of (2D)-Neurogeometry to 3D is not straight-forward from the mathematical point of view. This article presents the theoretical framework of a 3D-Neurogeometry inspired by the 2D case. We survey the mathematical structures and a standard frame for algorithms in 3D- Neurogeometry. The aim of the paper is to provide a "theoretical toolbox" and inspiration for new algorithms in 3D medical computer vision.
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Computing Bi-Invariant Pseudo-Metrics on Lie Groups | Entropy
Miolane, N., Pennec, X.
Journal Entropy. [Code].
In computational anatomy, organ’s shapes are often modeled as deformations of a reference shape, i.e., as elements of a Lie group. To analyze the variability of the human anatomy in this framework, we need to perform statistics on Lie groups. A Lie group is a manifold with a consistent group structure. Statistics on Riemannian manifolds have been well studied, but to use the statistical Riemannian framework on Lie groups, one needs to define a Riemannian metric compatible with the group structure: a bi-invariant metric. However, it is known that Lie groups, which are not a direct product of compact and abelian groups, have no bi-invariant metric. However, what about bi-invariant pseudo-metrics? In other words: could we remove the assumption of the positivity of the metric and obtain consistent statistics on Lie groups through the pseudo-Riemannian framework? Our contribution is two-fold. First, we present an algorithm that constructs bi-invariant pseudo-metrics on a given Lie group, in the case of existence. Then, by running the algorithm on commonly-used Lie groups, we show that most of them do not admit any bi-invariant (pseudo-) metric. We thus conclude that the (pseudo-) Riemannian setting is too limited for the definition of consistent statistics on general Lie groups.
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Statistics on Lie Groups | MaxEnt
Miolane, N., Pennec, X.
MaxEnt Workshop on Bayesian Inference and Maximum Entropy Methods. (Oral). [Code].
Lie groups appear in many fields from Medical Imaging to Robotics. In Medical Imaging and particularly in Computational Anatomy, an organ’s shape is often modeled as the deformation of a reference shape, in other words: as an element of a Lie group. In this framework, if one wants to model the variability of the human anatomy, e.g. in order to help diagnosis of diseases, one needs to perform statistics on Lie groups. A Lie group 𝒢 is a manifold that carries an additional group structure. Statistics on Riemannian manifolds have been well studied with the pioneer work of Fréchet, Karcher and Kendall [1, 2, 3, 4] followed by others [5, 6, 7, 8, 9]. In order to use such a Riemannian structure for statistics on Lie groups, one needs to define a Riemannian metric that is compatible with the group structure, i.e a bi-invariant metric. However, it is well known that general Lie groups which cannot be decomposed into the direct product of compact and abelian groups do not admit a bi-invariant metric. One may wonder if removing the positivity of the metric, thus asking only for a bi-invariant pseudo-Riemannian metric, would be sufficient for most of the groups used in Computational Anatomy. In this paper, we provide an algorithmic procedure that constructs bi-invariant pseudo-metrics on a given Lie group 𝒢. The procedure relies on a classification theorem of Medina and Revoy. However in doing so, we prove that most Lie groups do not admit any bi-invariant (pseudo-) metric. We conclude that the (pseudo-) Riemannian setting is not the richest setting if one wants to perform statistics on Lie groups. One may have to rely on another framework, such as affine connection space.
Read MoreAnalyse Biométrique de l'Anneau Pelvien en 3 Dimensions | JRCOT
Darmante, H., Bugnas, B., Dompsure, R.B.D., Barresi, L., Miolane, N., Pennec, X., de Peretti, F., Bronsard, N.
Journal Revue de Chirurgie Orthopédique et Traumatologique.
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Defining a mean on Lie groups | Imperial College London
Miolane, N.
Statistics are mostly performed in vector spaces, flat structures where the computations are linear. When one wants to generalize this setting to the non-linear structure of Lie groups, it is necessary to redefine the core concepts. For instance, the linear definition of the mean as an integral can not be used anymore. In this thesis, we investigate three possible definitions depending of the Lie group geometric structure. First, we import on Lie groups the notion of Riemannian center of mass (CoM) which is used to define a mean on manifolds and investigate when it can define a mean which is compatible with the algebraic structure. It is the case only for a small class of Lie groups. Thus we extend the CoM’s definition with two others: the Riemannian exponential barycenter and the group exponential barycenter. This thesis investigates how they can define admissible means on Lie groups.
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