IEEE Signal Processing - July 2018 - 18
where A l ! R n 1 # n 2 is the lth measurement matrix and w l ! R
is a noise term. We may rewrite these equations more compactly in a matrix form as
y = A (X ) + w,
(6)
where A : R n 1 # n 2 " R m is the linear measurement operator
defined by [A (X )] l = G A l, X H and w = [w 1, f, w m] T ! R m
is the vector of noise terms. Denote by A * the conjugate
operator of A, where A * ( y) = R ml = 1 y l A l .
As we are primarily concerned with estimating X from
m % n 1 n 2 measurements, direct approximation via SVD and
the Eckart-Young theorem are impossible. Instead, we need to
develop alternative methods to find an (approximate) low-rank
solution that best fits the set of noisy underdetermined linear
equations (6). We further categorize the low-rank matrix estimation problem into two main types based on the structure of
the measurement operator:
■ Low-rank matrix sensing: One observes linear combinations of the entries of X, where each measurement matrix
A l defining the linear combinations is typically dense.
■ Low-rank matrix completion: One directly observes a subset of the entries of X and aims to interpolate the missing
entries. In this case, each A l is a sparse matrix with a single entry equal to 1 at the corresponding observed index.
For matrix completion, it is convenient to write the measurements in a matrix form as
Y = PX (X ) + W,
Convex relaxation via nuclear norm minimization
We begin by deriving the nuclear norm minimization algorithm as a convex relaxation for rank minimization. Recall
our linear measurement model in (6), to which we seek a lowrank solution. A natural approach is to find the matrix with
the minimum rank that is consistent with these measurements, which can be formulated as an optimization problem:
min rank (X )
X ! R n1 # n2
y = A (X ).
subject to
(8)
The rank, however, is a nonconvex function of X, and rank
minimization (8) is known to be NP-hard in general. To develop a tractable formulation, one observes that the rank of X is
equal to the number of its nonzero singular values. Therefore,
analogously to using the , 1 norm as a convex surrogate of
sparsity, we may replace the rank of X by the sum of its singular values, a quantity known as the nuclear norm:
min {n 1, n 2}
X
/
* _
v i.
i=1
Then, instead of solving (8) directly, one solves for a matrix
that minimizes the nuclear norm:
Xt = arg min X
(7)
where X 1 {1, 2, f, n 1} # {1, 2, f, n 2} is the collection of
indices of the observed entries, PX : R n 1 # n 2 " R n 1 # n 2 is the
entry-wise partial observation operator defined by
[PX (X )] ij = '
mating low-rank matrices. In this section, we provide a
survey of this algorithmic approach and the associated theoretical results.
X ! R n1 # n2
*
s.t. y = A (X ).
(9)
In the case where the measurements y are noisy, one seeks a
matrix with a small nuclear norm that is approximately consistent with the measurements, which can be formulated
either as a regularized optimization problem,
X ij, (i, j) ! X,
0, otherwise,
Xt = arg min 1 y - A (X )
X ! Rn #n 2
1
2
2
2
+ x X *,
(10)
and W ! R n 1 # n 2 is the noise matrix supported on X. With
this notation, matrix completion is the problem of (approximately) recovering X given Y and X.
or as a constrained optimization problem,
Theory and algorithms for low-rank matrix
estimation via convex optimization
where x and c are tuning parameters. Note that the nuclear
norm can be represented as the solution to a semidefinite program [25],
The development of efficient algorithms for low-rank estimation owes much of its inspiration to the success of compressed
sensing [1], [2]. There, the convex relaxation approach based on
, 1 -minimization is widely used for recovering sparse signals. For low-rank problems, the role of the , 1 norm is
replaced by its matrix counterpart, namely the nuclear norm
(also known as the trace norm), which is a convex surrogate
for the rank. This idea gives rise to convex optimization
approaches for low-rank estimation based on nuclear norm
minimization, an approach put forth by Fazel et al. in the
seminal work [25]. This approach has since been extensively
developed and expanded and remains the most mature and
well-understood method (though not the only one) for esti18
Xt = arg min
X!R
n1 # n2
X
y - A (X )
2
2
subject to
X
*
= min 1 ^Tr (W 1) + Tr (W 2)h
W 1, W 2 2
W1 X
E * 0.
subject to ; T
X W2
# c,
(11)
*
(12)
Consequently, the optimization problems (9)-(11) are convex,
semidefinite programs.
Guarantees for matrix sensing via the restricted
isometry property
For there to be any hope of recovering X from the output of
the sensing process (6), the sensing operator A needs to
IEEE Signal Processing Magazine
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July 2018
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Table of Contents for the Digital Edition of IEEE Signal Processing - July 2018
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