English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We introduce a convergent hierarchy of lower bounds on the minimum value of areal homogeneous polynomial over the sphere. The main practical advantage ofour hierarchy over the sum-of-squares (SOS) hierarchy is that the lower boundat each level of our hierarchy is obtained by a minimum eigenvalue computation,as opposed to the full semidefinite program (SDP) required at each level ofSOS. In practice, this allows us to go to much higher levels than arecomputationally feasible for the SOS hierarchy. For both hierarchies, theunderlying space at the $k$-th level is the set of homogeneous polynomials ofdegree $2k$. We prove that our hierarchy converges as $O(1/k)$ in the level$k$, matching the best-known convergence of the SOS hierarchy when the numberof variables $n$ is less than the half-degree $d$ (the best-known convergenceof SOS when $n \geq d$ is $O(1/k^2)$). More generally, we introduce aconvergent hierarchy of minimum eigenvalue computations for minimizing theinner product between a real tensor and an element of the sphericalSegre-Veronese variety, with similar convergence guarantees. As examples, weobtain hierarchies for computing the (real) tensor spectral norm, and forminimizing biquadratic forms over the sphere. Hierarchies of eigencomputationsfor more general constrained polynomial optimization problems are discussed.

A hierarchy of eigencomputations for polynomial optimization on the sphere