Co-clustering with augmented matrix

Abstract

Clustering plays an important role in data mining as many applications use it as a preprocessing step for data analysis. Traditional clustering focuses on the grouping of similar objects, while two-way co-clustering can group dyadic data (objects as well as their attributes) simultaneously. Most co-clustering research focuses on single correlation data, but there might be other possible descriptions of dyadic data that could improve co-clustering performance. In this research, we extend ITCC (Information Theoretic Co-Clustering) to the problem of co-clustering with augmented matrix. We proposed CCAM (Co-Clustering with Augmented Matrix) to include this augmented data for better co-clustering. We apply CCAM in the analysis of on-line advertising, where both ads and users must be clustered. The key data that connect ads and users are the user-ad link matrix, which identifies the ads that each user has linked; both ads and users also have their feature data, i.e. the augmented matrix. To evaluate the proposed method, we use two measures: classification accuracy and K-L divergence. The experiment is done using the advertisements and user data from Morgenstern, a financial social website that focuses on the advertisement agency. The experiment results show that CCAM provides better performance than ITCC since it considers the use of augmented matrix during clustering.

Appendices

Appendix A: Proof of Lemma 1

For a fixed co-clustering $(\hat{A}, \hat{U})$, we can re-write the loss in mutual information as K-L divergence or relative entropy measure as

(29)

Proof

By definition,

where in the second step, $g(a \mid\hat{a})=\frac{g(a)}{g(\hat{a})}$ solidifies the followed equality.

where in the second step, $h(u \mid\hat{u})=\frac{h(u)}{h(\hat{u})}$ solidifies the followed equality. □

Appendix B: Proof of Lemma 2

Proof

Since Eq. (18) is proved in [9], we focus on the rest two equations.

(30)

Similarly, we could use the same argument to prove Eq. (20).

(31)

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Appendix C: Proof of Theorem 2

The CCAM algorithm could monotonically decreases the objective function Eq. (6). Since

$$ q^{(t)} (\hat{A} , \hat{U}) \geq q^{(t+1)} (\hat{A} , \hat{U}) $$

(32)

Proof

Let $\varPhi=\varphi\cdot D(h(U, L) \,||\,\hat{h}^{(t+1)}(U, L))$. For t=1,3,…,2T+1.

The inequality follows from step 1 since $C_{A}^{t+1}(a)$ is selected to minimize the objective function.

By using an identical argument, we can prove Eq. (32) for t=2,4,…,2T+2. Let $\varLambda=\lambda\cdot D(f(A, S) \,||\,\hat {f}^{(t+1)}(A, S))$.

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