Projection-like Retractions on Matrix Manifolds

By Pierre-Antoine Absil, Jérôme Malick. SIAM Journal on Optimization, 2012

376 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/6aa8b611d67af1d32bcb3b7af8ca82510f6b1949

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Citations in the corpus

A new, globally convergent Riemannian conjugate gradient method (2013, background): “For this R, the differentiated retraction T R defined by (2.8) is written out as T Rη (ξ) = 1√ (x + η)T (x + η) ( I − (x + η)(x + η) T (x + η)T (x + η) ) ξ, η, ξ ∈ Tx Sn−1, x ∈ Sn−1.”
Efficient Riemannian Optimization on the Stiefel Manifold via the Cayley Transform (2020, methodology): “For example, Absil & Malick (2012); Gawlik & Leok (2018); Manton (2002) use a projection-based method to map the gradient back to the Stiefel manifold that rely on computational expensive SVD.”
Linear dimensionality reduction: survey, insights, and generalizations (2014, background): “Absil and Malick (2012) discuss projective retractions compared with geodesics/exponential maps.”
A unifying framework for the analysis of projection-free first-order methods under a sufficient slope condition (2020, background): “[1], [2], [26], [46]), in which case it can be proved that they are differentiable retractions [1].”
Orthogonal Weight Normalization: Solution to Optimization over Multiple Dependent Stiefel Manifolds in Deep Neural Networks (2017, methodology): “In optimization community, there exist methods to solve the optimization problem over matrix manifolds with orthogonal constraints [32, 2, 36, 1, 5].”
Label Propagation via Teaching-to-Learn and Learning-to-Teach (2017, background): “Second, a retraction mapping [36] [see the red arrow in Fig.”
Global rates of convergence for nonconvex optimization on manifolds (2016, positive): “Indeed, retractions for submanifolds obtained as (certain types of) projections—arguably one of the most natural classes of retractions for submanifolds—are second order (Absil and Malick, 2012, Thm. 22).”
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].”
SpinQuant: LLM quantization with learned rotations (2024, neutral): “Detailed GPU latency results are provided in the Appendix.”
Online optimization in closed loop on the power flow manifold (2017, background): “This notion is inspired by a similar notion in manifold optimization [11], [12], yet our use of the term is relatively informal.”