IEEE Signal Processing - July 2018 - 17
Lifting for quadratic and bilinear optimization problems
2
y l = G a l, x H = G a l a l*, xx * H.
(2)
Due to the nonlinear nature of these equations, it is difficult
to solve them directly, particularly when the problem size is
large. A popular approach for solving such equations is called
lifting: one rewrites the previous equations in terms of the
matrix variable M = xx * and casts this problem as recovering the rank-1 matrix M from a set of linear measurements
[15]. A similar formulation has been used for the blind deconvolution problem; cf. [16].
The lifting approach can be applied to other classes of
quadratic equations, whose lifting formulations may lead to
low-rank matrices of rank larger than one. For instance, in the
problem of sensor network localization [17], the goal is to determine the locations of a set of n points/sensors {x i} in= 1 lying in
an r-dimensional Euclidean space, where r % n, given a subset
of their pairwise distances. The complete set of pairwise Euclidean distances can be arranged as a matrix E = [E ij] ! R n # n,
2
where E ij = x i - x j 2, 1 # i, j # n. Interestingly, each pairwise distance (i.e., an entry of E) is in fact a linear function
of the rank-r, positive semidefinite (PSD) matrix M = XX T ,
where X = [x 1, x 2, f, x n] T ! R n # r; more precisely, one has
E ij = X T (e i - e j)
2
2
= (e i - e j) T M (e i - e j).
In quantum state tomography, the density matrix of a pure
or nearly pure quantum state is approximately low rank,
which can be exploited in the problem of state reconstruction from a small number of Pauli measurements [21].
■ In a sparse graphical model with latent variables, one can
show, using the Schur complement, that the inverse marginal covariance matrix of the observed variables can be
approximated by a matrix with rank equal to the number of
latent variables [22].
■ Matrices with certain monotonicity properties can be wellapproximated by a matrix with rank much smaller than the
ambient dimension. Such matrices arise, for example, when
measuring the pairwise comparison scores of a set of objects
that possess an underlying ordering [23].
■ The pairwise affinity matrix of a set of objects is often approximately low rank due to the presence of clustering/community
structures [24] (cf. the section "Cluster Matrices").
The full list is much longer. The ubiquity of these structures, either as a physical property or as an engineering choice,
is what makes low-rank models useful and motivates the extensive study of the low-rank matrix estimation problem.
■
Another important source of low-rank structures is solving
quadratic/bilinear optimization problems. As an example, consider the phase retrieval problem [14], an important routine in
X-ray crystallography and optical imaging, where the goal is
to recover a vector x in C n or R n given only the magnitudes
of its linear measurements, i.e.,
Low-rank matrix estimation from
incomplete observations
In this section, we formally define the problem of low-rank
matrix estimation, i.e., recovery of a low-rank matrix from a
number of measurements much smaller than the dimension
of the matrix. Let X ! R n 1 # n 2 be the matrix-valued signal of
interest. (Our discussions can be extended complex-valued
matrices straightforwardly.) Denote the SVD of X by
(3)
Therefore, the problem of determining the locations X of
the sensors is equivalent to recovering the low-rank lifted
matrix M from a set of linear measurements in the form
of (3); see [17] for a more detailed treatment of this powerful reformulation.
X = URV T =
min {n 1, n 2}
/
T
vi ui vi ,
i=1
where the singular values v 1 $ v 2 $ g are organized in an
nonincreasing order. The best rank-r approximation of X is
defined as
X r _ arg min X - G F .
Other sources of low-rank structures
rank (G) # r
There are many potential sources of low-rank structures. Next,
we provide a few further examples drawn from different science and engineering domains.
■ In system identification and time-series analysis, finding
the minimum-order linear time-invariant system is equivalent to minimizing the rank of Hankel structured matrices
[18] (cf. the section "Hankel Matrix Completion").
■ In recommendation systems [19], the matrix of user ratings
for a set of items is often approximately low rank, as user
preferences typically depend on a small number of underlying factors and, hence, their ratings correlate with each other.
■ The background of a video usually changes slowly from
frame to frame; therefore, stacking the frames as columns
leads to an approximately low-rank matrix [20]. Similar
low-rank structures arise from the smoothness properties of
other visual and physical objects [5].
By the Eckart-Young theorem, the optimal approximation X r
is given by
Xr =
r
/ v i u i v Ti .
(4)
i=1
Correspondingly, the rank-r approximation error is given by
X - X r F , and we say that the matrix X is approximately
low-rank if its rank-r approximation error is small for some
r % min {n 1, n 2}.
As mentioned previously, in many modern applications, one
does not directly observe X but rather is given an underdetermined set of indirect noisy measurements of it. Here we assume
that one has access to a set of linear measurements in the form
IEEE Signal Processing Magazine
y l = G A l, X H + w l, l = 1, f, m,
|
July 2018
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(5)
17
Table of Contents for the Digital Edition of IEEE Signal Processing - July 2018
Contents
IEEE Signal Processing - July 2018 - Cover1
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