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IEEE SIGNAL PROCESSING MAGAZINE [152] MARCH 2015

Then, the singular values of X

()n

are the Frobenius norms of

the corresponding slices of the core tensor :S ()

,nrr

R =

(:,:, , , :, ,:) ,rS

ff with slices in the same mode being

mutually orthogonal, i.e., their inner products are zero. The col-

umns of

may thus be seen as multilinear singular vectors,

while the norms of the slices of the core are multilinear singular

values [15]. As in the matrix case, the multilinear singular values

govern the multilinear rank, while the multilinear singular vectors

allow, for each mode separately, an interpretation as in PCA [8].

LOW MULTILINEAR RANK APPROXIMATION

Analogous to PCA, a large-scale data tensor

X can be approxi-

mated by discarding the multilinear singular vectors and slices of

the core tensor that correspond to small multilinear singular val-

ues, i.e., through truncated matrix SVDs. Low multilinear rank

approximation is always well posed; however, the truncation is not

necessarily optimal in the LS sense, although a good estimate can

often be made as the approximation error corresponds to the

degree of truncation. When it comes to finding the best approxi-

mation, the ALS-type algorithms exhibit similar advantages and

drawbacks to those used for CPD [8], [70]. Optimization-based

algorithms exploiting second-order information have also been

proposed [71], [72].

CONSTRAINTS AND TUCKER-BASED

MULTIWAY COMPONENT ANALYSIS

Besides orthogonality, constraints that may help to find unique

basis vectors in a Tucker representation include statistical inde-

pendence, sparsity, smoothness, and nonnegativity [21], [73], [74].

Components of a data tensor seldom have the same properties in

its modes, and for physically meaningful representation, different

constraints may be required in different modes so as to match the

properties of the data at hand. Figure 1 illustrates the concept of

multiway component analysis (MWCA) and its flexibility in choos-

ing the modewise constraints; a Tucker representation of MWCA

naturally accommodates such diversities in different modes.

OTHER APPLICATIONS

We have shown that TKD may be considered a multilinear

extension of PCA [8]; it therefore generalizes signal subspace

techniques, with applications including classification, feature

extraction, and subspace-based harmonic retrieval [27], [39],

[75], [76]. For instance, a low multilinear rank approximation

achieved through TKD may yield a higher SNR than the SNR in

the original raw data tensor, making TKD a very natural tool for

compression and signal enhancement [7], [8], [26].

BLOCK TERM DECOMPOSITIONS

We have already shown that CPD is unique under quite mild con-

ditions. A further advantage of tensors over matrices is that it is

even possible to relax the rank-1 constraint on the terms, thus

opening completely new possibilities in, e.g., BSS. For clarity, we

shall consider the third-order case, whereby, by replacing the

rank-1 matrices

bb bb

() () () ()

rr rr

T12 12

% = in (3) by low-rank matrices

,AB

the tensor X can be represented as [Figure 5(a)]

().AB cX

(11)

Figure 5(b) shows that we can even use terms that are only

required to have a low multilinear rank (see the “Tucker Decom-

position” section) to give

.ABCXG

rrr

123

###=

(12)

These so-called block term decompositions (BTDs) in (11) and

(12) admit the modeling of more complex signal components

than CPD and are unique under more restrictive but still fairly

natural conditions [77]–[79].

EXAMPLE 2

To compare some standard and tensor approaches for the separa-

tion of short duration correlated sources, BSS was performed on

five linear mixtures of the sources

() ( )sinst t6

r= and

() ( ) ( ),exp sinst t t10 20

r= which were contaminated by white

Gaussian noise, to give the mixtures ,X AS E R

560

!=+

where

() [ (), ()]S ssttt

= and A ! R

52#

was a random matrix whose

columns (mixing vectors) satisfy .,aa 01

= .aa1

The 3-Hz sine wave did not complete a full period over the 60 sam-

ples so that the two sources had a correlation degree of

(| |)/( ) . .ss s s 035

= The tensor approaches, CPD, TKD,

and BTD employed a third-order tensor X of size 24 # 37 # 5

generated from five Hankel matrices whose elements obey

(, , )ijkX = (, )X ki j 1+- (see the section “Tensorization—

Blessing of Dimensionality”). The average squared angular error

(SAE) was used as the performance measure. Figure 6 shows the

simulation results, illustrating the following.

■ PCA failed since the mixing vectors were not orthogonal

and the source signals were correlated, both violating the

assumptions for PCA.

■ The ICA [using the joint approximate diagonalization of

eigenmatrices (JADE) algorithm [10]] failed because the sig-

nals were not statistically independent, as assumed in ICA.

(I × J × K )

(I × L

)

× J )

+ ··· +

(I × L

)

× J )

(K )(K )

(K × N

)

(I × J × K )

(I × L

)

× J )

+ ··· +

× M

× N

)

(a)

(b)

[FIG5] BTDs find data components that are structurally more

complex than the rank-1 terms in CPD. (a) Decomposition into

terms with multilinear rank

( , , ) .LL1

(b) Decomposition into

terms with multilinear rank ( , , ) .LMN

rrr

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