49 research outputs found
An algebraic characterization of injectivity in phase retrieval
A complex frame is a collection of vectors that span and
define measurements, called intensity measurements, on vectors in
. In purely mathematical terms, the problem of phase retrieval is
to recover a complex vector from its intensity measurements, namely the modulus
of its inner product with these frame vectors. We show that any vector is
uniquely determined (up to a global phase factor) from generic
measurements. To prove this, we identify the set of frames defining
non-injective measurements with the projection of a real variety and bound its
dimension.Comment: 11 page
A small frame and a certificate of its injectivity
We present a complex frame of eleven vectors in 4-space and prove that it
defines injective measurements. That is, any rank-one Hermitian
matrix is uniquely determined by its values as a Hermitian form on this
collection of eleven vectors. This disproves a recent conjecture of Bandeira,
Cahill, Mixon, and Nelson. We use algebraic computations and certificates in
order to prove injectivity.Comment: 4 pages, 3 figure