Controlling singular values with semidefinite programming

Shahar Z. Kovalsky, Noam Aigerman, Ronen Basri, Yaron Lipman

Research output: Contribution to journalConference articlepeer-review


Controlling the singular values of n-dimensional matrices is often required in geometric algorithms in graphics and engineering. This paper introduces a convex framework for problems that involve singular values. Specifically, it enables the optimization of functionals and constraints expressed in terms of the extremal singular values of matrices. Towards this end, we introduce a family of convex sets of matrices whose singular values are bounded. These sets are formulated using Linear Matrix Inequalities (LMI), allowing optimization with standard convex Semidefinite Programming (SDP) solvers. We further show that these sets are optimal, in the sense that there exist no larger convex sets that bound singular values. A number of geometry processing problems are naturally described in terms of singular values. We employ the proposed framework to optimize and improve upon standard approaches. We experiment with this new framework in several applications: volumetric mesh deformations, extremal quasi-conformal mappings in three dimensions, non-rigid shape registration and averaging of rotations. We show that in all applications the proposed approach leads to algorithms that compare favorably to state-of-art algorithms.

Original languageEnglish
Article number68
JournalACM Transactions on Graphics
Issue number4
StatePublished - 2014
Event41st International Conference and Exhibition on Computer Graphics and Interactive Techniques, ACM SIGGRAPH 2014 - Vancouver, BC, Canada
Duration: 10 Aug 201414 Aug 2014

All Science Journal Classification (ASJC) codes

  • Computer Graphics and Computer-Aided Design


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