WU et al.: OSC WITH BLOCK-DIAGONAL PRIORS 3211
the orthogonal linear subspaces assumption [28]. BD-SSC and
BD-LRR are formulated as follows:
min
Z
X − XZ
2
F
+ λZ
1
s.t. diag(Z) = 0, Z ∈ K (4)
and
min
Z
X − XZ
2,1
+ λZ
∗
s.t. Z ∈ K (5)
where K is defined as
K =
{
Z | rank(L
W
) = n − K
}
where L
W
is the Laplacian matrix depending on the affinity
matrix W of Z. A common choice is W = (|Z|+|Z
T
|)/2 and
L
W
( j, j
) =
−W( j, j
) if j = j
l=j
W( j, l) otherwise.
(6)
Using K as the favorable penalty originates from the important
conclusion in [11] which says the rank of the Laplacian matrix
L
W
is complementary to the number of blocks in the corre-
sponding affinity matrix W.SoZ ∈ K enforces the coefficient
matrix Z to form a block-diagonal affinity matrix.
The above block-diagonal priori equips SSQP, BD-SSC,
and BD-LRR with improved clustering performance. However,
those models do not make use of sequential property of data
such as time sequential or spatial adjacent data.
For sequential data clustering, SpatSC method [25], [26]
extends the standard SSC method by incorporating an extra
penalty term as follows:
min
Z
X − XZ
2
F
+ λ
1
Z
1
+ λ
2
ZR
1
s.t. diag(Z) = 0(7)
where λ
1
and λ
2
are the balance weights. R is defined as
follows, to enforce the consistency of the sparse representation
of the sequential data:
R =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
−10 0··· 0
1 −10··· 0
01−1 ··· 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
000
.
.
.
−1
000··· 1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
n×(n−1)
.
The penalty term ZR
1
does encourage the sequential con-
sistency of data coefficient matrix, but at level of individual
data components. Instead of using the
1
measurement for
the sequential error term ZR in SpatSC, OSC method [27]
adopts the
2
/
1
error measurement to encourage consistency
in representation at data level. The model is defined as follows:
min
Z
X − XZ
2
F
+ λ
1
Z
1
+ λ
2
ZR
2,1
s.t. diag(Z) = 0. (8)
As both SpatSC and OSC methods properly encourage the
sequential property of data representation, they all have better
clustering results than their original versions SSC or LRR.
III. O
RDERED SUBSPACE CLUSTERING WITH
BLOCK-DIAGONAL PRIOR
It has been seen in the last section that incorporating data
intrinsic properties and sparsity priori information improves
the performance of clustering methods, as demonstrated by
BD-SSC and BD-LRR over their corresponding counterpart
SSC and LRR, respectively. We also note the performance
improvement of SpatSC and OSC over sequential cluster-
ing problems, by introducing a constraint term for promoting
order relations in representation for sequential data. It would
be much beneficial for one to combine two approaches by
incorporating both sparsity prior and sequential information
for sequential data in one model. In this paper, we propose a
model in which certain intrinsic properties of the data, such
as the correlation of the data and its sparse representation,
and extra prior information, e.g., the block-diagonal prior of
the affinity matrix, are integrated for the purpose of clustering
sequential data.
Inspired by the SSQP method [29], first we replace the
sparsity inducing
1
norm in (8)[orsimilarlyin(7)] with a
quadratic constrained term, which favors not only the coef-
ficient sparsity but also the block-diagonal property in the
affinity matrix. The new clustering model for sequential data
is defined by
min
Z
X − XZ
2
F
+ λ
1
Z
T
Z
1
+ λ
2
ZR
2,1
s.t. diag(Z) = 0, Z ≥ 0. (9)
We call this model the OSC method with the quadratic con-
straint term, abbreviated as QOSC. Similar to SSQP, QOSC
encourages a representation coefficient matrix Z with possi-
ble block-diagonal structure, a status which may be achieved
if the data are restrictively drawn from the orthogonal linear
subspaces [29]. However, it is hard for this condition to be
satisfied in practical scenarios. So we further propose a block-
diagonal constrained OSC method by integrating the general
block-diagonal constraint in (4)asthefollowingform:
min
Z
X − XZ
2
F
+ λ
1
Z
T
Z
1
+ λ
2
ZR
2,1
s.t. diag(Z) = 0, Z ≥ 0, Z ∈ K. (10)
We name the model as the OSC method with block-diagonal
prior, denoted by BD-QOSC. This model will have more
flexibility than QOSC when the data drawn from different
subspaces.
For a given data set X, we can find the data representa-
tion coefficient matrix Z by solving the optimization defined
in either (9)or(10). How to solve these problems will be dis-
cussed in the next section. Under the data self-representative
principle used in the models, the element z
ij
∈ Z repre-
sents the correlation between datum i and j. So a natural
way to define the affinity matrix for model (9)or(10)is
W = (|Z|+|Z
T
|)/2. This affinity matrix can be used in
a generic spectral clustering method for data clustering or
classification. As a state-of-the-art spectral clustering method
with good performance in subspace segmentation problem,
Ncut [20] is adopted for the spectral clustering in this paper.
The whole clustering procedure of the proposed BD-QOSC is
summarized as Algorithm 1.
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