2.3. Benefit three: wider range of applications

There is no doubt that all the multi-view clustering methods can be applied to single-view data. However, many clustering tasks are impossible to implement by single-view clustering due to its limitations. For example, data with multiple modalities is becoming more and more common and heterogeneous information networks are gaining increasing popularity as well. These types of data naturally fit into multi-view learning, while cannot be settled by singleview learning methods appropriately. In all, the complementary property among multi-view data can overcome the limitations of single-view data and expand their application areas.

As can be seen, Eq. (1) adopts the Euclidean distance to measure the similarities between data samples. However, there are many data structures or data distributions in real world. Thus, it is not always suitable to apply this basic form of k-means to accurately identify the hidden patterns of datasets. What is more, some datasets may be not separable in the low-dimensional space. Recently, kernel method has been of wide concern in the field of machine learning. By introducing a kernel function, the original nonlinear datasets are mapped to a higher dimensional reproducing kernel Hilbert space. In the new space, the datasets become linearly separable. For this reason, the kernel k-means algorithm [16, 17] has been proposed. It is just a generalization of the standard k-means algorithm and has the following objective function:

> k∥2 <sup>2</sup>, v<sup>0</sup> <sup>k</sup> ¼

where ϕ : X ! H is a nonlinear transformation function. Define a kernel function

<sup>j</sup>¼<sup>1</sup> <sup>δ</sup>jk<sup>K</sup> <sup>x</sup>i; <sup>x</sup><sup>j</sup> � �

With the aid of the kernel function, there is no need to explicitly provide the transformation function ϕ. This is because, for certain kernel function, the corresponding transformation function is intractable. However, the inner products in the kernel space can be easily obtained

Both the k-means and the kernel k-means described above are designed for single-view data. To solve the multi-view clustering problem, some new objective functions should be developed. Assume that there are <sup>V</sup> views in total. Let <sup>X</sup> <sup>¼</sup> <sup>X</sup>ð Þ<sup>1</sup> ;Xð Þ<sup>2</sup> ; …;Xð Þ <sup>V</sup> n o denote the data of all the views. It is obvious that different views should have different contributions according to their conveyed information. To achieve this goal, it is straightforward to modify the standard k-means to make it applicable in the multi-view environment with a new objective function as follows:

> X V

v¼1

P<sup>N</sup> <sup>i</sup>¼<sup>1</sup> <sup>δ</sup>ik<sup>x</sup> ð Þv i

P<sup>N</sup> <sup>i</sup>¼<sup>1</sup> <sup>δ</sup>ik

where μ<sup>v</sup> is the weight factor for the v-th view, γ is a parameter used to control the weight distribution, and ε<sup>v</sup> corresponds to the objective function (i.e., loss function) of the v-th view:

Similarly, the objective function of the multi-view kernel k-means can be obtained, which is omitted here. Note that finding the optimal solution of Eq. (4) is an NP-hard problem; thus,

<sup>v</sup> εv, s:t:μ<sup>v</sup> ≥ 0,

P<sup>N</sup> <sup>j</sup>¼<sup>1</sup> <sup>δ</sup>jk P<sup>N</sup>

þ P<sup>N</sup> j¼1 P<sup>N</sup>

<sup>l</sup>¼<sup>1</sup> <sup>δ</sup>jkδlk !: (3)

<sup>i</sup>¼<sup>1</sup> <sup>δ</sup>ikϕð Þ <sup>x</sup><sup>i</sup> P<sup>N</sup> <sup>i</sup>¼<sup>1</sup> <sup>δ</sup>ik

� �. Then, Eq. (2) can be rewritten into the kernel form

P<sup>N</sup> j¼1 P<sup>N</sup>

<sup>l</sup>¼<sup>1</sup> <sup>δ</sup>jkδlk<sup>K</sup> <sup>x</sup>j; <sup>x</sup><sup>l</sup>

μ<sup>v</sup> ¼ 1, γ > 1, (4)

: (5)

� �

New Approaches in Multi-View Clustering http://dx.doi.org/10.5772/intechopen.75598 199

, (2)

<sup>ε</sup> <sup>¼</sup> <sup>X</sup> N

K : X � X ! IR with K xi; x<sup>j</sup>

<sup>ε</sup> <sup>¼</sup> <sup>X</sup> N

i¼1

according to the kernel function.

3.1.2. Basic form of multi-view k-means

X K

k¼1

as below:

i¼1

� � <sup>¼</sup> <sup>ϕ</sup>ð Þ <sup>x</sup><sup>i</sup>

δik K x<sup>i</sup> ð Þ� ; x<sup>i</sup> 2

<sup>ε</sup> <sup>¼</sup> <sup>X</sup> V

<sup>ε</sup><sup>v</sup> <sup>¼</sup> <sup>X</sup> N

i¼1

v¼1 μγ

> X K

δik∥x ð Þv <sup>i</sup> � <sup>v</sup>ð Þ<sup>v</sup> <sup>k</sup> <sup>∥</sup><sup>2</sup> <sup>2</sup>, <sup>v</sup>ð Þ<sup>v</sup> <sup>k</sup> ¼

k¼1

X K

δik∥ϕð Þ� x<sup>i</sup> v<sup>0</sup>

<sup>T</sup>ϕ x<sup>j</sup>

P<sup>N</sup>

k¼1
