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IEEE SIGNAL PROCESSING MAGAZINE [153] MARCH 2015

■ Low-rank tensor approximation via a rank-2 CPD was used

to estimate A as the third factor matrix, which was then

inverted to yield the sources. The accuracy of CPD was com-

promised as the components of tensor

X cannot be repre-

sented by rank-1 terms.

■ Low multilinear rank approximation via TKD for the mul-

tilinear rank (4, 4, 2) was able to retrieve the column space of

the mixing matrix but could not find the individual mixing

vectors because of the nonuniqueness of TKD.

■ BTD in multilinear rank-(2, 2, 1) terms matched the data

structure [78]; it is remarkable that the sources were recov-

ered using as few as six samples in the noise-free case.

HIGHER-ORDER COMPRESSED SENSING (HO-CS)

The aim of CS is to provide a faithful reconstruction of a signal of

interest, even when the set of available measurements is (much)

smaller than the size of the original signal [80]–[83]. Formally, we

have available

M (compressive) data samples ,y R

! which are

assumed to be linear transformations of the original signal x R

().MI1 In other words,

,U= where the sensing matrix

is usually random. Since the projections are of a lower

dimension than the original data, the reconstruction is an ill-posed

inverse problem whose solution requires knowledge of the physics

of the problem converted into constraints. For example, a two-

dimensional image

X R

can be vectorized as a long vector

()Xx vec R

!= )(III

= that admits sparse representation in a

known dictionary B R

so that ,Bxg= where the matrix B

may be a wavelet or discrete cosine transform dictionary. Then,

faithful recovery of the original signal

x requires finding the spars-

est vector g such that

, ,WWB,yg gKwith

# U==(13)

where ·

is the

, -norm (number of nonzero entries) and

.KI%

Since the

, -norm minimization is not practical, alternative

solutions involve iterative refinements of the estimates of vector g

using greedy algorithms such as the orthogonal matching pur-

suit (OMP) algorithm, or the

, -norm minimization algorithms

[83]. Low coherence of the composite dictionary

matrix W is a prerequisite for a satisfactory recovery of g (and

hence )x —we need to choose U and B so that the correlation

between the columns of W is minimum [83].

When extending the CS framework to tensor data, we face

two obstacles:

■ loss of information, such as spatial and contextual relation-

ships in data, when a tensor RX

II I

N12

## #g

is vectorized.

0.05 0.1 0.15 0.2

−0.3

−0.2

−0.1

0.1

Time (s)

0.05 0.1 0.15 0.2

Time (s)

0.05 0.1 0.15 0.2

Time (s)

−0.2

−0.1

0.1

PCA

ICA

CPD

TKD

BTD

CPD

TKD

BTD

−0.2

−0.1

0.1

0.2

0.3

0 10 20 30 40

SNR (dB)

(a) (b)

(c)

(d)

SAE (dB)

PCA ICA CPD TKD BTD

[FIG6] The blind separation of the mixture of a pure sine wave and an exponentially modulated sine wave using PCA, ICA, CPD, TKD,

and BTD. The sources s

and s

are correlated and of short duration; the symbols s

and s

denote the estimated sources. (a)–(c)

Sources ()ts

and ( )ts

and their estimates using PCA, ICA, CPD, TKD, and BTD; (d) average squared angular errors (SAE) in estimation

of the sources.

THE WORLD’S NEWSSTAND

THE WORLD’S NEWSSTAND