A figure showing the domains of topological deep learning.

Topological Deep Learning: Going Beyond Graph Data | arXiv

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. [Paper] [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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A figure summarizing the landscape of topological neural network architectures on hypergraphs, simplicial, cellular, & combinatorial complexes in a unified graphical notation.

Architectures of Topological Deep Learning: A Survey on Topological Neural Networks | arXiv

Papillon, S., Sanborn, S., Hajij, M., Miolane, N. [Paper] [Equations]

Submitted to Nature Machine Intelligence.

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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cancer cellmorphology

Orthogonal Outlier Detection and Dimension Estimation for Improved MDS Embedding of Biological Datasets | BioRxiv

Li, W., Mirone, J., Prasad, A., Miolane, N., Legrand, C. and Dao Duc, K. [Paper]

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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so3 sketch

Interpolation of Animations on Lie Groups | Submitted

Kushner, S., Modi, V., Miolane, N.

Submitted to the International Conference on Computer Vision (ICCV).

We propose a family of Riemannian geodesic interpolation techniques to perform upsampling of low frame rate animations to high frame rates. 


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Quantifying Extrinsic Curvatures of Neural Manifolds | CVPRW

Acosta, F., Sanborn, S., Dao Duc, K., Madhav, M., Miolane, N.

CVPR Workshop Topology Algebra and Geometry (TAG) for Pattern Recognition with Applications.

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. Geometric Science of Information.

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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cancer cell

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

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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cryo volumes

Deep Generative Modeling for Volume Reconstruction in Cryo-Electron Microscopy | JSB

Donnat, C., Levy, A., Poitevin, F., 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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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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Probabilistic Riemannian Functional Map Synchronization for 3D Shape Correspondence | Submitted

Huq, F., Dey, A., Yusuf, S., Bazazian, D., Birdal, T., Miolane, N.

Submitted to Transactions of Machine Learning Research (TMLR).

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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Computations on the Stiefel Manifold and the Special Orthogonal Group | In Prep.

Mataigne, S., Absil, P.-A., Miolane, N.

This master thesis presents new theoretical and computational developments of differential geometry on the Stiefel manifold of orthonormal p-frames in n-dimensions, and the Special Orthogonal group of rotations in n-dimensions.

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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.

Submitted to Developmental Cell.

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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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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cryo em image noiseless

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 | Submitted

Le Brigant, A., Deschamps, J., Collas, A., Miolane, N.

Submitted to Transactions of Mathematical Software (TOMS).

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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cell green

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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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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PirouNet: Creating Dance through Artist-Centric Deep Learning | EAI ArtsIT

Papillon M., Pettee M., Miolane N. 

EAI ArtsIT ConferenceBest 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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long cell

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: 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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Intelligence-based Medicine | Elsevier

Chang, A. et al. (including Miolane, N.).

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Exploring Cryo-EM Latent Space with Variational Autoencoders | Stanford Bio-X

Miolane, N., Poitevin, F., Holmes, S.

Stanford Bio-X Workshop on Cryo-Electron Microscopy.

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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. [Patent].

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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.

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Convenience Tools to Explore Variability in Cryo-EM Data | Stanford Bio-X

Poitevin, F., Li, Y.T., Miolane, N., Gati, C., Levitt, M.

Stanford Bio-X Workshop on Cryo-Electron Microscopy.

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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.

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Topologically Constrained Template Estimation | SIAGA

Miolane, N., Holmes, S., Pennec, X.

SIAM Journal on Applied Algebra and Geometry.

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Template Shape Estimation in Computational Anatomy | SIIMS

Miolane, N., Holmes, S., Pennec, X.

SIAM Journal of Imaging Science.

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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).

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Biased Estimators on Quotient spaces | GSI

Miolane, N., Pennec, X.

Conference on Geometric Sciences of Information(Oral). [Video].

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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].

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Computing Bi-Invariant Pseudo-Metrics on Lie Groups | Entropy

Miolane, N., Pennec, X.

Journal Entropy. [Code].

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Statistics on Lie Groups for Computational Anatomy | MICCAI

Miolane, N., Khanal, B.

Video for the 2014 Educational Challenge at MIT. (1st Prize). [Video].

MICCAI Conference on Medical Image Computing and Computer Assisted Intervention.

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Statistics on Lie Groups | MaxEnt

Miolane, N., Pennec, X.

MaxEnt Workshop on Bayesian Inference and Maximum Entropy Methods(Oral). [Code].

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Analyse 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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Statistics on Lie Groups: Can We Obtain a Consistent Framework with Pseudo-Riemannian Metrics? | GMV

Miolane, N. 

Institut Henri Poincaré Workshop on Geometrical Models in Vision.

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