34 research outputs found
Length minimizing Hamiltonian paths for symplectically aspherical manifolds
In this paper we consider the length minimizing properties of Hamiltonian
paths generated by quasi-autonomous Hamiltonians on symplectically aspherical
manifolds. Motivated by the work of L. Polterovich and M. Schwarz, we study the
role of the fixed global extrema in the Floer complex of the generating
Hamiltonian. Our main result determines a natural condition on a fixed global
maximum of a Hamiltonian which implies that the corresponding path minimizes
the positive Hofer length. We use this to prove that a quasi-autonomous
Hamiltonian generates a length minimizing path if it has under-twisted fixed
global extrema and no periodic orbits with period one and action greater than
the fixed extrema. This, in turn, allows us to produce new examples of
autonomous Hamiltonian flows which are length minimizing for all times. These
constructions are based on the geometry of coisotropic submanifolds. Finally,
we give a new proof of the recent theorem of D. McDuff which states that
quasi-autonomous Hamiltonians generate length minimizing paths over short time
intervals.Comment: 23 pages, references added and final revisions made for publicatio
Subspace Evolution and Transfer (SET) for Low-Rank Matrix Completion
We describe a new algorithm, termed subspace evolution and transfer (SET),
for solving low-rank matrix completion problems. The algorithm takes as its
input a subset of entries of a low-rank matrix, and outputs one low-rank matrix
consistent with the given observations. The completion task is accomplished by
searching for a column space on the Grassmann manifold that matches the
incomplete observations. The SET algorithm consists of two parts -- subspace
evolution and subspace transfer. In the evolution part, we use a gradient
descent method on the Grassmann manifold to refine our estimate of the column
space. Since the gradient descent algorithm is not guaranteed to converge, due
to the existence of barriers along the search path, we design a new mechanism
for detecting barriers and transferring the estimated column space across the
barriers. This mechanism constitutes the core of the transfer step of the
algorithm. The SET algorithm exhibits excellent empirical performance for both
high and low sampling rate regimes