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research
Low rank matrix recovery from rank one measurements
Authors
Richard Kueng
Holger Rauhut
Ulrich Terstiege
Publication date
25 October 2014
Publisher
View
on
arXiv
Abstract
We study the recovery of Hermitian low rank matrices
X
β
C
n
Γ
n
X \in \mathbb{C}^{n \times n}
X
β
C
n
Γ
n
from undersampled measurements via nuclear norm minimization. We consider the particular scenario where the measurements are Frobenius inner products with random rank-one matrices of the form
a
j
a
j
β
a_j a_j^*
a
j
β
a
j
β
β
for some measurement vectors
a
1
,
.
.
.
,
a
m
a_1,...,a_m
a
1
β
,
...
,
a
m
β
, i.e., the measurements are given by
y
j
=
t
r
(
X
a
j
a
j
β
)
y_j = \mathrm{tr}(X a_j a_j^*)
y
j
β
=
tr
(
X
a
j
β
a
j
β
β
)
. The case where the matrix
X
=
x
x
β
X=x x^*
X
=
x
x
β
to be recovered is of rank one reduces to the problem of phaseless estimation (from measurements,
y
j
=
β£
β¨
x
,
a
j
β©
β£
2
y_j = |\langle x,a_j\rangle|^2
y
j
β
=
β£
β¨
x
,
a
j
β
β©
β£
2
via the PhaseLift approach, which has been introduced recently. We derive bounds for the number
m
m
m
of measurements that guarantee successful uniform recovery of Hermitian rank
r
r
r
matrices, either for the vectors
a
j
a_j
a
j
β
,
j
=
1
,
.
.
.
,
m
j=1,...,m
j
=
1
,
...
,
m
, being chosen independently at random according to a standard Gaussian distribution, or
a
j
a_j
a
j
β
being sampled independently from an (approximate) complex projective
t
t
t
-design with
t
=
4
t=4
t
=
4
. In the Gaussian case, we require
m
β₯
C
r
n
m \geq C r n
m
β₯
C
r
n
measurements, while in the case of
4
4
4
-designs we need
m
β₯
C
r
n
log
β‘
(
n
)
m \geq Cr n \log(n)
m
β₯
C
r
n
lo
g
(
n
)
. Our results are uniform in the sense that one random choice of the measurement vectors
a
j
a_j
a
j
β
guarantees recovery of all rank
r
r
r
-matrices simultaneously with high probability. Moreover, we prove robustness of recovery under perturbation of the measurements by noise. The result for approximate
4
4
4
-designs generalizes and improves a recent bound on phase retrieval due to Gross, Kueng and Krahmer. In addition, it has applications in quantum state tomography. Our proofs employ the so-called bowling scheme which is based on recent ideas by Mendelson and Koltchinskii.Comment: 24 page
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