Optimization Algorithms on Matrix Manifolds

By Pierre-Antoine Absil, Robert E. Mahony, Rodolphe Sepulchre. 2007

1,098 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/238176f85df700e0679ad3bacc8b2c5b1114cc58

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In Computational Theory and Mathematics

In Numerical Analysis

Citations in the corpus

Wasserstein Riemannian geometry of Gaussian densities (2018, background): “[1] and [5], are source of promising researches.”
Log-Euclidean Metric Learning on Symmetric Positive Definite Manifold with Application to Image Set Classification (2015, background): “As indicated in (Absil et al., 2008) , if and only if the tangent map DF (S) : TSS+ → TF (S)S+ is an injection (respectively, surjection) for every S ∈ S+, then the mapping F : S+ → S+ from manifold to manifold is an immersion, i.”
Matrix Completion from Noisy Entries (2009, methodology): “adient descent algorithm is applied to the function Fe: M(m,n) ≡ G(m,r) × G(n,r) → R. For further details on optimization by gradient descent on matrix manifolds we refer to Edelman et al. (1999) and Absil et al. (2008). 1.3 Some Notations The matrix M to be reconstructed takes the form (1) where U ”
Domain Adaptation by Joint Distribution Invariant Projections (2020, methodology): “As a matter of fact, Ws and Wt are now points on the Stiefel manifold [22], [41], ie, Ws, Wt ∈ St(d, D).”
A review of classification algorithms for EEG-based brain–computer interfaces: a 10 year update (2018, background): “Furthermore, optimization problems can be formulated specifically on such spaces, which is leading to several new optimization methods and to the solution of new problems [2].”
Computing the Karcher mean of symmetric positive definite matrices (2013, methodology): “Another approach is to use optimization algorithms on the objective function (1), like the Newton method or the steepest descent in a Riemannian manifold (see [1]).”
Elastic Functional Coding of Riemannian Trajectories (2016, methodology): “h is sufﬁcient for our purpose. Further, since 2 , it has an inverse that can be used in further analysis. The minimization over is solved for using dynamic programming. Here one samples the interval [0;1] using Tdiscrete points and then restricts to only piecewise linear that passes through the Tgr”
Data Augmentation with Manifold Exploring Geometric Transformations for Increased Performance and Robustness (2019, methodology): “ and the calculated u is the direction towards the decision boundary for step i. To improve the accuracy and convergence speed during the calculation of u a manifold optimization technique similar to [10] has been adopted: u(i) = i J+ x(i) rf(x (i)) jjJ+ x(i) rf(x (i))jj + u(i 1); (2) where iis th”
Orthogonal Deep Neural Networks (2019, background): “In literature, optimization of a differentiable cost function on such a matrix manifold and its convergence analysis have been intensively studied [1], [9].”
Geometric Structures in Tensor Representations (Final Release) (2013, neutral): “Then, for each µ ∈ T D \{ D } such that S ( µ ) 6 = ∅ , thanks to Proposition 2.10, we have Consider { u , which is the set of matrices in R r µ × ( Q β ∈ S ( α ) r β ) whose rows are linearly independent (see 3.1.5 in [1]).”
The geometry of algorithms using hierarchical tensors (2013, background): “, [41] for an overview),which generalizes standard optimization techniques tomanifolds.”
Information Geometry of the Gaussian Distribution in View of Stochastic Optimization (2015, methodology): “The methods used here are actually first order optimization methods on Riemannian manifolds, see [1] for the spe-”