Probability, Convexity, and Harmonic Maps with Small Image I: Uniqueness and Fine Existence

Geometry of Covariance Matrices and Computation of Median

Yang, L., Arnaudon, M., Barbaresco, F., Ali Mohammad-djafari, A., Bercher, J., & Bessìère, P. (2011) · AIP conference proceedings · doi:10.1063/1.3573655 · Cited by 20

NEW LARGE SAMPLE AND BOOTSTRAP METHODS ON SHAPE SPACES IN HIGH LEVEL ANALYSIS OF NATURAL IMAGES

Patrangenaru, V. (2001) · Communication in Statistics- Theory and Methods · doi:10.1081/sta-100105692 · Cited by 20

Dimensionality Reduction and Clustering on Statistical Manifolds

Lee, S.-M., Abbott, A. L., & Araman, P. A. (2007) · IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings - IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings · doi:10.1109/cvpr.2007.383408 · Cited by 20

Convex hull properties of harmonic maps

Li, P., & Wang, J. (1998) · Journal of Differential Geometry · doi:10.4310/jdg/1214460861 · Cited by 20

Geometrical methods of inference

Krakowski, K. (2002) · UWA Profiles and Research Repository (UWA) · doi:10.26182/5d526805ae757 · Cited by 20

Barycenters of measures transported by stochastic flows

Arnaudon, M., & Li, X.-M. (2005) · The Annals of Probability · doi:10.1214/009117905000000071 · Cited by 19

Nonlinear Markov Operators, Discrete Heat Flow, and Harmonic Maps Between Singular Spaces

Sturm, K. (2002) · Potential Analysis · doi:10.1023/a:1014888715237 · Cited by 19

Barycentric Subspaces and Affine Spans in Manifolds

Pennec, X. (2015) · Lecture notes in computer science · doi:10.1007/978-3-319-25040-3_2 · Cited by 18

Brownian couplings, convexity, and shy-ness

Kendall, W. S. (2009) · Electronic Communications in Probability · doi:10.1214/ecp.v14-1417 · Cited by 17

Liouville properties of harmonic maps

Tam, L.-F. (1995) · Mathematical Research Letters · doi:10.4310/mrl.1995.v2.n6.a5 · Cited by 16

Brownian motion and the formation of singularities in the heat flow for harmonic maps

Anton Thalmaier, C. (1996) · Probability Theory and Related Fields · doi:10.1007/bf01192212 · Cited by 16

Limit Theorems for the Inductive Mean on Metric Trees

Basrak, B. (2010) · Journal of Applied Probability · doi:10.1017/s0021900200007427 · Cited by 15

Computing Bi-Invariant Pseudo-Metrics on Lie Groups for Consistent Statistics

Miolane, N., & Pennec, X. (2015) · Entropy · doi:10.3390/e17041850 · Cited by 15

Bi-invariant Means on Lie Groups with Cartan-Schouten Connections

Pennec, X. (2013) · Lecture notes in computer science · doi:10.1007/978-3-642-40020-9_5 · Cited by 15

Differentiable and analytic families of continuous martingales in manifolds with connection

Arnaudon, M. (1997) · Probability Theory and Related Fields · doi:10.1007/s004400050108 · Cited by 15

Statistical inference on the Hilbert sphere with application to random densities

Dai, X. (2022) · Electronic Journal of Statistics · doi:10.1214/21-ejs1942 · Cited by 14

Barycentres convexes et approximations des martingales continues dans les variétés

Arnaudon, M. (1995) · Lecture notes in mathematics · doi:10.1007/bfb0094201 · Cited by 14

Data analysis on nonstandard spaces

Huckemann, S., & Eltzner, B. (2020) · Wiley Interdisciplinary Reviews Computational Statistics · doi:10.1002/wics.1526 · Cited by 14

Remarks on harmonic maps into a cone from a stochastically complete manifold

Atsuji, A. (1999) · Proceedings of the Japan Academy Series A Mathematical Sciences · doi:10.3792/pjaa.75.105 · Cited by 14

A Deep Neural Network for Manifold-Valued Data with Applications to Neuroimaging

Chakraborty, R., Bouza, J., Manton, J. H., & Vemuri, B. C. (2019) · Lecture notes in computer science · doi:10.1007/978-3-030-20351-1_9 · Cited by 13

Parameter estimation and model-based clustering with spherical normal distribution on the unit hypersphere

You, K., & Suh, C. (2022) · Computational Statistics & Data Analysis · doi:10.1016/j.csda.2022.107457 · Cited by 13

Expectations and martingales in metric spaces

Christiansen, T., & Sturm, K. (2008) · Stochastics · doi:10.1080/17442500701433640 · Cited by 13

Intrinsic Grassmann Averages for Online Linear, Robust and Nonlinear Subspace Learning

Chakraborty, R., Yang, L., Hauberg, S., & Vemuri, B. C. (2020) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2020.2992392 · Cited by 13

Diffeomorphic Iterative Centroid Methods for Template Estimation on Large Datasets

Cury, C., Glaunès, J., & Colliot, O. (2014) · Signals and communication technology · doi:10.1007/978-3-319-05317-2_10 · Cited by 13

What does intrinsic mean in statistical estimation

García, G. H., & Oller, J. M. (2006) · LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Científicas) · Cited by 13

Stochastic calculus, statistical asymptotics, Taylor strings and phyla

Barndorff-Nielsen, O. E., Jupp, P. E., & Kendall, W. S. (1994) · Annales de la faculté des sciences de Toulouse Mathématiques · doi:10.5802/afst.773 · Cited by 13

Intrinsic Grassmann Averages for Online Linear and Robust Subspace Learning

Chakraborty, R., Hauberg, S., & Vemuri, B. C. (2017) · IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings - IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings · doi:10.1109/cvpr.2017.92 · Cited by 13

A diffusion model for bookstein triangle shape

Kendall, W. S. (1998) · Advances in Applied Probability · doi:10.1239/aap/1035228071 · Cited by 13

Curvature effects on the empirical mean in Riemannian and affine Manifolds: a non-asymptotic high concentration expansion in the small-sample regimePreprint

Pennec, X. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1906.07418 · Cited by 13

Backward stochastic differential equations on manifolds

Blache, F. (2004) · Probability Theory and Related Fields · doi:10.1007/s00440-004-0400-9 · Cited by 12

1 MEDIANS AND MEANS IN FINSLER GEOMETRY

Arnaudon, M., & Nielsen, F. (2012) · HAL · Cited by 12

Extrinsic Mean of Brownian Distributions on Compact Lie Groups

Said, S., & Manton, J. H. (2012) · IEEE Transactions on Information Theory · doi:10.1109/tit.2012.2185680 · Cited by 12

Liouville theorems for f-harmonic maps into Hadamard spaces

Hua, B., Liu, S., & Xia, C. (2017) · Pacific Journal of Mathematics · doi:10.2140/pjm.2017.290.381 · Cited by 12

Geometrical smeariness – a new phenomenon of Fréchet means

Eltzner, B. (2021) · Bernoulli · doi:10.3150/21-bej1340 · Cited by 12

Espérances conditionnelles et C-martingales dans les variétés

Arnaudon, M. (1994) · Lecture notes in mathematics · doi:10.1007/bfb0073853 · Cited by 12

Intrinsic statistical techniques for robust pose estimation

Dubbelman, G. (2011) · UvA-DARE (University of Amsterdam) · Cited by 12

Interpretable modelling and visualization of biomedical data

Ghosh, S., Baranowski, E. S., Biehl, M. L., Arlt, W., Peter Tiňo, B., & Bunte, K. (2025) · Neurocomputing · doi:10.1016/j.neucom.2025.129405 · Cited by 11

Road-Constrained Geometric Pose Estimation for Ground Vehicles

Jiang, R., Zhou, H., Wang, H., & Ge, S. S. (2019) · IEEE Transactions on Automation Science and Engineering · doi:10.1109/tase.2019.2942069 · Cited by 11

Bounded harmonic maps on a class of manifolds

Sung, C.-J. A., Tam, L.-F., & Wang, J. (1996) · Proceedings of the American Mathematical Society · doi:10.1090/s0002-9939-96-03246-7 · Cited by 11

The Manifold-Valued Dirichlet Problem for Symmetric Diffusions

Picard, J. (2001) · Potential Analysis · doi:10.1023/a:1026443030345 · Cited by 11

Gradient estimates for some diffusion semigroups

Picard, J. (2002) · Probability Theory and Related Fields · doi:10.1007/s004400100185 · Cited by 11

Statistics on Manifolds with Applications to Shape Spaces

Bhattacharya, R. N., & Bhattacharya, A. (2009) · Statistical science and interdisciplinary research · doi:10.1142/9789814273633_0003 · Cited by 11

A geometrically nonlinear Cosserat shell model for orientable and non-orientable surfaces: Discretization with geometric finite elements

Nebel, L. J., Sander, O., Bîrsan, M., & Neff, P. (2023) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2023.116309 · Cited by 10

Empirical geodesic graphs and CAT(k) metrics for data analysis

Kobayashi, K., & Wynn, H. P. (2019) · Statistics and Computing · doi:10.1007/s11222-019-09855-3 · Cited by 10

On Means and Their Asymptotics: Circles and Shape Spaces

Huckemann, S., & Hotz, T. (2013) · Journal of Mathematical Imaging and Vision · doi:10.1007/s10851-013-0462-3 · Cited by 10

On a Liouville type theorem for harmonic maps to convex spaces via Markov chains

Kuwae, K., & Sturm, K. (2008) · Cited by 10

Statistical Shape Analysis of Large Datasets Based on Diffeomorphic Iterative Centroids

Cury, C., Glaunès, J., Toro, R., Marie Chupin, A., Schumann, G., Frouin, V., Poline, J., & Colliot, O. (2018) · Frontiers in Neuroscience · doi:10.3389/fnins.2018.00803 · Cited by 10

Appearance-Based Re-identification of Humans in Low-Resolution Videos Using Means of Covariance Descriptors

Metzler, J. (2012) · IEEE International Conference on Advanced Video and Signal Based Surveillance (AVSS) · doi:10.1109/avss.2012.12 · Cited by 10

Statistics on the (compact) Stiefel manifold: Theory and ApplicationsPreprint

Chakraborty, R., & Vemuri, B. C. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1708.00045 · Cited by 10

Template Estimation in Computational Anatomy: Fréchet Means Top and Quotient Spaces Are Not Consistent

Devilliers, L., Allassonnière, S., Trouvé, A., & Pennec, X. (2017) · SIAM Journal on Imaging Sciences · doi:10.1137/16m1083931 · Cited by 9

Stability of the cut locus and a central limit theorem for Fréchet means of Riemannian manifolds

Eltzner, B., Galaz‐García, F., Huckemann, S., & Tuschmann, W. (2020) · Proceedings of the American Mathematical Society · doi:10.1090/proc/15429 · Cited by 9

Stochastic calculus and martingales on trees

Picard, J. (2004) · Annales de l Institut Henri Poincaré Probabilités et Statistiques · doi:10.1016/j.anihpb.2004.03.002 · Cited by 9

Manifold-valued martingales, changes of probabilities, and smoothness of finely harmonic maps

Arnaudon, M. (1999) · Annales de l Institut Henri Poincaré Probabilités et Statistiques · doi:10.1016/s0246-0203(99)00114-4 · Cited by 9

Meshless geometric subdivision

Moenning, C., Mémoli, F., Sapiro, G., Dyn, N., & Dodgson, N. A. (2006) · Graphical Models · doi:10.1016/j.gmod.2006.11.001 · Cited by 9

Barycentric coordinate neighbourhoods in Riemannian manifoldsPreprint

Dyer, R., Vegter, G., & Wintraecken, M. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1606.01585 · Cited by 9

Kriging Riemannian Data via Random Domain Decompositions

Menafoglio, A., Pigoli, D., & Secchi, P. (2020) · Journal of Computational and Graphical Statistics · doi:10.1080/10618600.2020.1853548 · Cited by 8

On an intrinsic analysis of statistical estimation

Oller, J. M. (1993) · Elsevier eBooks · doi:10.1016/b978-0-444-81531-6.50030-1 · Cited by 8

A Geometric Framework for Statistical Analysis of Trajectories with Distinct Temporal Spans

Chakraborty, R., Singh, V., Adluru, N., & Vemuri, B. C. (2017) · IEEE/CVF International Conference on Computer Vision (ICCV) · doi:10.1109/iccv.2017.28 · Cited by 8

Smoothness of harmonic maps for hypoelliptic diffusions

Picard, J. (2000) · The Annals of Probability · doi:10.1214/aop/1019160255 · Cited by 8

Area Comparison, Superharmonic Functions, and Applications

Yang, Y. H. (2001) · Acta Mathematica Sinica English Series · doi:10.1007/s101149900005 · Cited by 8

On the Wasserstein Median of Probability Measures

You, K., Shung, D. L., & Giuffrè, M. (2024) · Journal of Computational and Graphical Statistics · doi:10.1080/10618600.2024.2374580 · Cited by 7

Backward stochastic differential equations on manifolds II

Blache, F. (2005) · Probability Theory and Related Fields · doi:10.1007/s00440-005-0482-z · Cited by 7

Convex functions and $p$-barycenter on CAT(1)-spaces of small radii

Yokota, T. (2017) · Tsukuba Journal of Mathematics · doi:10.21099/tkbjm/1506353559 · Cited by 7

Calcul stochastique avec sauts sur une variete

Picard, J. (1991) · Lecture notes in mathematics · doi:10.1007/bfb0100857 · Cited by 7

Kernel Principal Geodesic Analysis

Awate, S. P., Yu, Y.-Y., & Whitaker, R. (2014) · Lecture notes in computer science · doi:10.1007/978-3-662-44848-9_6 · Cited by 6

Mean shapes, projections and intrinsic limiting distributions

Dryden, I. L., Le, H., Preston, S., & Wood, A. T. A. (2013) · Journal of Statistical Planning and Inference · doi:10.1016/j.jspi.2013.08.003 · Cited by 6

Tensor-Based Graph-Cut in Riemannian Metric Space and Its Application to Renal Artery Segmentation

Wang, C., Oda, M., Hayashi, Y., Yoshino, Y., Yamamoto, T., Frangi, A. F., & Mori, K. (2016) · Lecture notes in computer science · doi:10.1007/978-3-319-46726-9_41 · Cited by 6

Gaussian distributions on Riemannian symmetric spaces of nonpositive curvature

(2022) · Handbook of statistics · doi:10.1016/bs.host.2022.03.004 · Cited by 6

Medians and means in Finsler geometry

Arnaudon, M., & Nielsen, F. (2012) · LMS Journal of Computation and Mathematics · doi:10.1112/s1461157010000513 · Cited by 6

Form representions and means for landmarks: A survey and comparative study

Heo, G., & Small, C. G. (2006) · Computer Vision and Image Understanding · doi:10.1016/j.cviu.2006.01.003 · Cited by 6

Averaging complex subspaces via a Karcher mean approach

Hüper, K., Kleinsteuber, M., & Shen, H. (2012) · Signal Processing · doi:10.1016/j.sigpro.2012.08.020 · Cited by 6

On the computation of means on Grassmann manifolds

Uwe Helmke, B., & Herzberg, S. (2010) · Cited by 6

Online Change Point Detection on Riemannian Manifolds with Karcher Mean Estimates

Wang, X., Borsoi, R. A., & Richard, C. (2023) · European Signal Processing Conference (EUSIPCO) · doi:10.23919/eusipco58844.2023.10289744 · Cited by 6

Diffusion means in geometric spaces

Eltzner, B., Hansen, P., Huckemann, S., & Sommer, S. (2023) · Bernoulli · doi:10.3150/22-bej1578 · Cited by 6

Intrinsic Location Parameter of a Diffusion Process

Darling, R. (2002) · Electronic Journal of Probability · doi:10.1214/ejp.v7-102 · Cited by 6

A Stochastic Criterion for Yang—Mills Connections

Stafford, S. (1990) · Birkhäuser Boston eBooks · doi:10.1007/978-1-4684-0564-4_19 · Cited by 6

Parallel Transport with Pole Ladder: a Third Order Scheme in Affine Connection Spaces which is Exact in Affine Symmetric SpacesPreprint

Pennec, X. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1805.11436 · Cited by 6

Recent advances in principal component analysis for directional data

Nodehi, A., Moghimbeygi, M., & Ley, C. (2025) · Journal of Multivariate Analysis · doi:10.1016/j.jmva.2025.105528 · Cited by 5

The heat flows and harmonic maps from complete manifolds into regular balls

Li, J., & Wang, S. (1998) · Bulletin of the Australian Mathematical Society · doi:10.1017/s0004972700032159 · Cited by 5

Probabilistic Geodesic Models for Regression and Dimensionality Reduction on Riemannian Manifolds

Fletcher, P. T., & Zhang, M. (2015) · Springer eBooks · doi:10.1007/978-3-319-22957-7_5 · Cited by 5

A stability approach for solving multidimensional quadratic BSDEs

Harter, J., & Richou, A. (2019) · Electronic Journal of Probability · doi:10.1214/18-ejp260 · Cited by 5

Statistics on manifolds

Fletcher, T. (2019) · Elsevier eBooks · doi:10.1016/b978-0-12-814725-2.00009-1 · Cited by 5

Statistical Computing on Non-Linear Spaces for Computational Anatomy

Pennec, X., & Fillard, P. (2015) · Springer eBooks · doi:10.1007/978-0-387-09749-7_8 · Cited by 5

Analysis of Planar Shapes Using Geodesic Lengths on a Shape Manifold

Klassen, E. P., & Srivastava, A. (2002) · Cited by 5

On the meaning of mean shapePreprint

Huckemann, S. (2010) · arXiv (Cornell University) · doi:10.48550/arxiv.1002.0795 · Cited by 5

Correction note to “martingales on manifolds and harmonic maps”

Huang, H. R., & Kendallt, W. S. (1991) · Stochastics and stochastics reports · doi:10.1080/17442509108833740 · Cited by 5

Equivariant estimation of Fréchet means

McCormack, A., & Hoff, P. D. (2023) · Biometrika · doi:10.1093/biomet/asad014 · Cited by 4

A note on boundary regularity of subelliptic harmonic maps

Zhou, Z.-R. (2005) · Kodai Mathematical Journal · doi:10.2996/kmj/1134397766 · Cited by 4

Parabolicity, the divergence theorem for $\delta$-subharmonic functions and applications to the Liouville theorems for harmonic maps

Atsuji, A. (2005) · Tohoku Mathematical Journal · doi:10.2748/tmj/1128703002 · Cited by 4

A Kernel Regression Procedure in the 3D Shape Space with an Application to Online Sales of Children’s Wear

Quintana‐Ortí, G., & Simó, A. (2019) · Statistical Science · doi:10.1214/18-sts675 · Cited by 4

A study of backward stochastic differential equation on a Riemannian manifold

Chen, X., & Ye, W. (2021) · Electronic Journal of Probability · doi:10.1214/21-ejp649 · Cited by 4

Wavelet Sparse Regularization for Manifold-Valued Data

Storath, M., & Weinmann, A. (2020) · Multiscale Modeling and Simulation · doi:10.1137/19m1249801 · Cited by 4

Feature Patch Illumination Spaces and Karcher Compression for Face Recognition via Grassmannians

Chang, J.-M., Peterson, C., & Kirby, M. J. L. (2012) · Advances in Pure Mathematics · doi:10.4236/apm.2012.24033 · Cited by 4

Advances in Geometric Statistics for Manifold Dimension Reduction

Pennec, X. (2020) · Springer eBooks · doi:10.1007/978-3-030-31351-7_11 · Cited by 4

Distances and inference for covariance functionsPreprint

Pigoli, D., Aston, J. A. D., Dryden, I. L., & Secchi, P. (2012) · University of Warwick · doi:10.82444/warw.33679009 · Cited by 4

A Central Limit Theorem for Frechet MeansPreprint

Bhattacharya, R. N., & Lin, L. (2013) · arXiv (Cornell University) · Cited by 4

The regularity of harmonic maps into spheres and applications to Bernstein problemsPreprint

Jost, J., Xin, Y., & Yang, L. (2009) · arXiv (Cornell University) · doi:10.48550/arxiv.0912.0447 · Cited by 4

Barycentres and hurricane trajectories

Kendall, W. S. (2015) · Wiley series in probability and statistics · doi:10.1002/9781118866641.ch7 · Cited by 4

Continuous medial models in two-sample statistics of shape

Gerig, G., & Terriberry, T. B. (2006) · Cited by 4

Unscented kalman filtering on euclidean and riemannian manifolds

Menegaz, H. M. T. (2016) · doi:10.26512/2016.06.t.21617 · Cited by 4

Probability, Convexity, and Harmonic Maps. II. Smoothness via Probabilistic Gradient Inequalities

Kendall, W. S. (1994) · Journal of Functional Analysis · doi:10.1006/jfan.1994.1147 · Cited by 21

PROBABILITY, CONVEXITY, AND HARMONIC MAPS .2. SMOOTHNESS VIA PROBABILISTIC GRADIENT INEQUALITIES

provided, N. A. (2026) · University of Warwick · doi:10.82444/warw.33546871

PROBABILITY, CONVEXITY, AND HARMONIC MAPS WITH SMALL IMAGE .1. UNIQUENESS AND FINE EXISTENCE

provided, N. A. (2026) · University of Warwick · doi:10.82444/warw.33558067

Regularity and Uniqueness of p-harmonic Maps with Small RangePreprint

Fardoun, A., & Regbaoui, R. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1203.2101 · Cited by 2

On the Dirichlet problem

Villa‐Morales, J. (2012) · Expositiones Mathematicae · doi:10.1016/j.exmath.2012.08.008 · Cited by 3