TensorFlow RiemOpt

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A library for manifold-constrained optimization in TensorFlow.


To install the latest development version from GitHub:

pip install git+https://github.com/master/tensorflow-riemopt.git

To install a package from PyPI:

pip install tensorflow-riemopt


The core package implements concepts in differential geometry, such as manifolds and Riemannian metrics with associated exponential and logarithmic maps, geodesics, retractions, and transports. For manifolds, where closed-form expressions are not available, the library provides numerical approximations.

import tensorflow_riemopt as riemopt

S = riemopt.manifolds.Sphere()

x = S.projx(tf.constant([0.1, -0.1, 0.1]))
u = S.proju(x, tf.constant([1., 1., 1.]))
v = S.proju(x, tf.constant([-0.7, -1.4, 1.4]))

y = S.exp(x, v)

u_ = S.transp(x, y, u)
v_ = S.transp(x, y, v)


  • manifolds.Cholesky - manifold of lower triangular matrices with positive diagonal elements
  • manifolds.Euclidian - unconstrained manifold with the Euclidean metric
  • manifolds.Grassmannian - manifold of p-dimensional linear subspaces of the n-dimensional space
  • manifolds.Hyperboloid - manifold of n-dimensional hyperbolic space embedded in the n+1-dimensional Minkowski space
  • manifolds.Poincare - the Poincaré ball model of the hyperbolic space
  • manifolds.Product - Cartesian product of manifolds
  • manifolds.SPDAffineInvariant - manifold of symmetric positive definite (SPD) matrices endowed with the affine-invariant metric
  • manifolds.SPDLogCholesky - SPD manifold with the Log-Cholesky metric
  • manifolds.SPDLogEuclidean - SPD manifold with the Log-Euclidean metric
  • manifolds.SpecialOrthogonal - manifold of rotation matrices
  • manifolds.Sphere - manifold of unit-normalized points
  • manifolds.StiefelEuclidean - manifold of orthonormal p-frames in the n-dimensional space endowed with the Euclidean metric
  • manifolds.StiefelCanonical - Stiefel manifold with the canonical metric
  • manifolds.StiefelCayley - Stiefel manifold the retraction map via an iterative Cayley transform


Constrained optimization algorithms work as drop-in replacements for Keras optimizers for sparse and dense updates in both Eager and Graph modes.

  • optimizers.RiemannianSGD - Riemannian Gradient Descent
  • optimizers.RiemannianAdam - Riemannian Adam and AMSGrad
  • optimizers.ConstrainedRMSProp - Constrained RMSProp


  • layers.ManifoldEmbedding - constrained keras.layers.Embedding layer


  • SPDNet - Huang, Zhiwu, and Luc Van Gool. "A Riemannian network for SPD matrix learning." Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence. AAAI Press, 2017.
  • LieNet - Huang, Zhiwu, et al. "Deep learning on Lie groups for skeleton-based action recognition." Proceedings of the IEEE conference on computer vision and pattern recognition. 2017.
  • GrNet - Huang, Zhiwu, Jiqing Wu, and Luc Van Gool. "Building Deep Networks on Grassmann Manifolds." AAAI. AAAI Press, 2018.
  • Hyperbolic Neural Network - Ganea, Octavian, Gary Bécigneul, and Thomas Hofmann. "Hyperbolic neural networks." Advances in neural information processing systems. 2018.
  • Poincaré GloVe - Tifrea, Alexandru, Gary Becigneul, and Octavian-Eugen Ganea. "Poincaré Glove: Hyperbolic Word Embeddings." International Conference on Learning Representations. 2018.


If you find TensorFlow RiemOpt useful in your research, please cite:

      title={TensorFlow RiemOpt: a library for optimization on Riemannian manifolds},
      author={Oleg Smirnov},


TensorFlow RiemOpt was inspired by many similar projects:

  • Manopt, a matlab toolbox for optimization on manifolds
  • Pymanopt, a Python toolbox for optimization on manifolds
  • Geoopt: Riemannian Optimization in PyTorch
  • Geomstats, an open-source Python package for computations and statistics on nonlinear manifolds


The code is MIT-licensed.

Tensorflow Riemopt

A library for optimization on Riemannian manifolds

Tensorflow Riemopt Info

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🔗 Source Codegithub.com
🕒 Last Update10 months ago
🕒 Created2 years ago
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