Low-rank retractions: a survey and new results

P. A. Absil, I. V. Oseledets

    Research output: Contribution to journalArticlepeer-review

    25 Citations (Scopus)


    Retractions are a prevalent tool in Riemannian optimization that provides a way to smoothly select a curve on a manifold with given initial position and velocity. We review and propose several retractions on the manifold Mr of rank-r m×n matrices. With the exception of the exponential retraction (for the embedded geometry), which is clearly the least efficient choice, the retractions considered do not differ much in terms of run time and flop count. However, considerable differences are observed according to properties such as domain of definition, boundedness, first/second-order property, and symmetry.

    Original languageEnglish
    Pages (from-to)5-29
    Number of pages25
    JournalComputational Optimization and Applications
    Issue number1
    Publication statusPublished - 21 Sep 2015


    • Fixed-rank manifold
    • Geodesic
    • Lie–Trotter splitting
    • Low-rank manifold
    • Low-rank optimization
    • Orthographic retraction
    • Projective retraction
    • Quasi-geodesic
    • Retraction


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