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IEEE SIGNAL PROCESSING MAGAZINE [147] MARCH 2015

dimensions; these may include space, time, frequency, trials,

classes, and dictionaries. A real-valued tensor of order N is denoted

by RA

II I

N12

## #g

and its entries by .a

,, ,ii i

N12

Then, an N 1#

vector a is considered a tensor of order one, and an NM# matrix

A a tensor of order two. Subtensors are parts of the original data

tensor, created when only a fixed subset of indices is used. Vector-

valued subtensors are called fibers, defined by fixing every index but

one, and matrix-valued subtensors are called slices, obtained by fix-

ing all but two indices (see Table 1). The manipulation of tensors

often requires their reformatting (reshaping); a particular case of

reshaping tensors to matrices is termed matrix unfolding or matri-

cization (see Figure 1). Note that a mode-

n multiplication of a ten-

sor A with a matrix B amounts to the multiplication of all

mode- n vector fibers with ,B and that, in linear algebra, the ten-

sor (or outer) product appears in the expression for a rank-1 mat-

rix:

.ab a b

Basic tensor notations are summarized in Table 1,

various product rules used in this article are given in Table 2, while

Figure 2 shows two particular ways to construct a tensor.

INTERPRETABLE COMPONENTS

IN TWO-WAY DATA ANALYSIS

The aim of BSS, FA, and latent variable analysis is to decompose

a data matrix

X R

I J

into the factor matrices [ ,A a

,,]aaR

I R

f !

and [ , , , ]B bb b R

f !=

XADB E Eab

m=+= +

,Eab

(1)

where (,, , )D diag

R12

fmm m= is a scaling (normalizing) matrix,

the columns of B represent the unknown source signals (factors or

latent variables depending on the tasks in hand), the columns of A

represent the associated mixing vectors (or factor loadings), while

E is noise due to an unmodeled data part or model error. In other

words, model (1) assumes that the data matrix X comprises hidden

components b

,, ,rR12f=

that are mixed together in an

unknown manner through coefficients ,A or, equivalently, that data

contain factors that have an associated loading for every data chan-

nel. Figure 3(a) depicts the model (1) as a dyadic decomposition,

whereby the terms

ab ab

rr r

are rank-1 matrices.

The well-known indeterminacies intrinsic to this model are:

1) arbitrary scaling of components and 2) permutation of the

rank-1 terms. Another indeterminacy is related to the physical

meaning of the factors: if the model in (1) is unconstrained, it

admits infinitely many combinations of

A and .B Standard

matrix factorizations in linear algebra, such as QR-factorization,

eigenvalue decomposition (EVD), and SVD, are only special

[FIG1] MWCA for a third-order tensor, assuming that the components are (a) principal and orthogonal in the first mode,

(b) nonnegative and sparse in the second mode, and (c) statistically independent in the third mode.

[TABLE 2] DEFINITION OF PRODUCTS.

BCA

MODE-n PRODUCT OF RA

II I

N12

## #g

AND B R

YIELDS RC

IIJI I

nnn N111

## ## ##gg

WITH ENTRIES

cab

i i ji i i i ii i ji

nnnN nnnNnn

111 111

gg gg

-+ -+

AND MATRIX REPRESENTATION CBA

() ()nn

;,,,BB BCA

() () ( )N12

FULL MULTILINEAR PRODUCT, BB BCA

() () ( )

## #g=

CAB=

TENSOR OR OUTER PRODUCT OF RA

II I

N12

## #g

AND RB

JJ J

M12

## #g

YIELDS RC

II IJJ J

NM12 1 2

## #### #gg

WITH

ENTRIES cab

iiijjj iii jjj

NNMM12 12 12 12

gg g g

aa aX

() () ()N12

%%%

TENSOR OR OUTER PRODUCT OF VECTORS a R

()nI

! (,

,)nN

1 f= YIELDS A RANK-1 TENSOR RX

II I

N12

## #g

WITH ENTRIES xaaa

() () ()

ii i

N12

CAB7=

KRONECKER PRODUCT OF A R

AND B R

YIELDS C R

IJ IJ

11 22

WITH ENTRIES

cab

() ,()i J j i J j ii jj11

1 1 1 2 2 2 12 12

-+ -+

CAB9=

KHATRI–RAO PRODUCT OF [, ,]aaA R

f !=

AND [,,]bbB R

f !=

YIELDS C R

IJ R

WITH COLUMNS

cab

rrr

THE WORLD’S NEWSSTAND

THE WORLD’S NEWSSTAND

...

(3)

(2)

(SVD/PCA)

X(:, :, k)

(1)

X(:, j, :)

(NMF/SCA)

(ICA)

= S

X(i, :, :)

(a)

(b)

(c)

Unfolding