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IEEE SIGNAL PROCESSING MAGAZINE [159] MARCH 2015

EXAMPLE 5

We employed LWCA for classification based on common and dis-

tinct features of natural objects from the ETH-80 database (http://

www.d2.mpi-inf.mpg.de/Data sets/ETH80) whereby the discrimi-

nation among objects was performed using only the common fea-

tures. This data set consists of 3,280 images in eight categories,

each containing ten objects with 41 views per object. For each cat-

egory, the training data were organized in two distinct fourth-

order

()I128 128 3

### tensors, where . ,I p10 4105

##=

where p denotes the fraction of training data. LMWCA was applied

to these two tensors to find the common and individual features,

with the number of common features set to 80% of

In this

way, eight sets of common features were obtained for each cat-

egory. The test sample label was assigned to the category whose

common features matched the new sample best (evaluated by

canonical correlations) [110]. Figure 16 compares LMWCA with

the standard K-nearest neighbors (K-NNs) and LDA classifiers

(using 50 principal components as features), all averaged over 50

Monte Carlo runs. The enhanced classification results for LMWCA

are attributed to the fact that the classification makes use of only

the common components and is not hindered by components that

are not shared across objects or views.

SOFTWARE

The currently available software resources for tensor decompo-

sitions include:

■ The tensor toolbox, a versatile framework for basic opera-

tions on sparse and dense tensors, including CPD and Tucker

formats [111].

■ The TDALAB and TENSORBOX, which provide a user-

friendly interface and advanced algorithms for CPD, nonneg-

ative TKD, and MWCA [112], [113].

■ The Tensorlab toolbox builds upon the complex optimiza-

tion framework and offers numerical algorithms for comput-

ing the CPD, BTD, and TKD; the toolbox includes a library of

constraints (e.g., nonnegativity and orthogonality) and the

possibility to combine and jointly factorize dense, sparse, and

incomplete tensors [89].

PredictionTrainingData Acquisition

(c)(b)(a)

Motion Capture Tensorization

Marker

ECoG

Layout

91011

14 13 12

15161718

21 20 19

222324

27 26 25

282930

32 31

Ch1

Ch2

Ch3

Ch32

. . .

...

ECoG Recordings

Time (s)

Tensorization

Time

Frequency

Channel

Tensor (Regressions)

HOPLS

Model

Parameters

Marker

Time

X(t )

Y(t )

Z(t )

Tensor (Limb Trajectories)

Time

Frequency

Channel

Data Tensor

from New Recordings

Predictor

20 40 60

Time (s)

80 100

20 40 60

Time (s)

80 100

20 40 60

Time (s)

80 100

Trajectory

–1

–2

–3

–1

X-PositionY-PositionZ-Position

Prediction of 3-D Hand Trajectory

HOPLS PLS

Coordinates

[FIG14] The prediction of arm movement from brain electrical responses. (a) The experiment setup. (b) The construction of the

data and response tensors and training. (c) The new data tensor (bottom) and the predicted 3-D arm movement trajectories

,,(XYZ coordinates) obtained by tensor-based HOPLS and standard matrix-based PLS (top).

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