TensorFlow ManOpt

A library for manifold-constrained optimization in TensorFlow.

Installation

To install the latest development version from GitHub:

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

To install a package from PyPI:

pip install tensorflow-manopt

Features

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.

S = manopt.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

• 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

Optimizers

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

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

Examples

• 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.

References

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

@software{tfmanopt2020smirnov,
  author       = {Oleg Smirnov},
  title        = {{TensorFlow ManOpt: a library for manifold-
                   constrained optimization in TensorFlow}},
  month        = oct,
  year         = 2020,
  publisher    = {Zenodo},
  version      = {v0.1.0},
  doi          = {10.5281/zenodo.4094937},
  url          = {https://doi.org/10.5281/zenodo.4094937}
}

Acknowledgment

TensorFlow ManOpt 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

License

The code is MIT-licensed.