Notes on Optimization on Stiefel Manifolds

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Full-Capacity Unitary Recurrent Neural Networks (2016, methodology): “We consider the Stiefel manifold of all N × N complex-valued matrices whose columns are N orthonormal vectors in C N [14].”
Cascaded Projection: End-To-End Network Compression and Acceleration (2019, methodology): “There are several optimization methods on Grassmann manifolds, most of which include iterative optimization and retraction methods [7, 1, 48, 2, 51].”
Orthogonal Weight Normalization: Solution to Optimization over Multiple Dependent Stiefel Manifolds in Deep Neural Networks (2017, methodology): “…OMDSM problem, one straightforward idea is to use Reimannian optimization method that is extensively used for single manifold or multiple independent manifolds problem, either in optimization communities [1, 32, 2, 36, 5] or in applications to the hidden-to-hidden transformation of RNNs [37, 35].”