Topic detection in cross-media: a semi-supervised co-clustering approach

Abstract

With the rapid development of social media, the topics emerge and propagate in a variety of media websites. Although much work has been done since NIST proposed the problem of topic detection and tracking (TDT), most of them focus on single media data and are mainly based on unsupervised clustering method, which does not use some side information to help detecting topics. Therefore, traditional TDT approaches are not competent for cross-media topic detection. To efficiently use the information contained in multi-modal data from different sources and the prior knowledge, we propose a semi-supervised co-clustering approach for cross-media topic detection by a constrained non-negative matrix factorization. The correctness and convergence of our approach are proved to demonstrate its mathematical rigorousness. Experiments on the cross-media dataset verify the effectiveness of our proposed approach.

Appendix

The proof of the correctness and convergence of Algorithm 1 is given in this section.

1.1 Correctness of the algorithm

We first prove the correctness of Algorithm1 and have the following theorem.

Theorem 1

If the update rule of ${\mathbf{F}}$, $\{{\mathbf{S}}^{(i)}\}_{1\le i\le h}$ and $\{{\mathbf{G}}^{(i)}\}_{1\le i\le h}$ in Algorithm 1 converges, then the final solution satisfies the KKT condition.

Proof

Following the standard theory of constrained optimization, we introduce the Lagrangian multipliers $\lambda _f$, $\lambda _{S^i}$ and $\lambda _{G^i}$ and minimize the Lagrangian function

$$\begin{aligned}&L({\mathbf{F}},\,{\mathbf{S}}^{(\mathrm{i})},\,{\mathbf{G}}^{(\mathrm{i})},\,\lambda _f ,\,\lambda _{S^1} ,\ldots ,\lambda _{S^h},\,\lambda _{G^1} ,\ldots ,\lambda _{G^h} )\\&\quad = \sum \limits _{i=1}^h {\left\| {{\mathbf{R}}^{(\mathrm{ci})}-{\mathbf{FS}}^{(\mathrm{i})}{\mathbf{G}}^{(\mathrm{i})}} \right\| _F^2 } +\alpha Tr[({\mathbf{F}}-{\mathbf{Y}})^T{\mathbf{U}}({\mathbf{F}}-{\mathbf{Y}})] \\&\qquad -Tr( {\lambda _f {\mathbf{F}}})-\sum \limits _{i=1}^h {\left[ {Tr({\lambda _{S^i} {\mathbf{S}}^{(\mathrm{i})}})+Tr( {\lambda _{G^i} {\mathbf{G}}^{(\mathrm{i})}})} \right] } \end{aligned}$$

(21)

The zero gradient condition gives

$$\begin{aligned}&\begin{aligned} \frac{\partial L}{\partial {\mathbf{F}}}&=-2\sum \limits _{i=1}^h {\left( {{\mathbf{R}}^{(\mathrm{ci})}{\mathbf{G}}^{{(\mathrm{i})}^{T}}{\mathbf{S}}^{{(\mathrm{i})}^{T}}+\alpha {\mathbf{UY}}}\right) }\\&\quad + 2\left( {\sum \limits _{i=1}^h{{\mathbf{FS}}^{(\mathrm{i})}{\mathbf{G}}^{(\mathrm{i})}} {\mathbf{G}}^{{(\mathrm{i})}^{T}}{\mathbf{S}}^{{(\mathrm{i})}^{T}}+\alpha {\mathbf{UF}}}\right) -\lambda _f =0, \end{aligned}\\&\frac{\partial L}{\partial {\mathbf{S}}^{(\mathrm{i})}}=-2{\mathbf{F}}^{T}{\mathbf{R}}^{(\mathrm{ci})}{\mathbf{G}}^{{(\mathrm{i})}^{T}}+2{\mathbf{F}}^{T}{\mathbf{FS}}^{(\mathrm{i})}{\mathbf{G}}^{(\mathrm{i})}{\mathbf{G}}^{{(\mathrm{i})}^{T}}-\lambda _{S^i} =0, \\&\frac{\partial L}{\partial {\mathbf{G}}^{(\mathrm{i})}}=-2{\mathbf{S}}^{{(\mathrm{i})}^{T}}{\mathbf{F}}^{T}{\mathbf{R}}^{(\mathrm{ci})}+2{\mathbf{S}}^{{(\mathrm{i})}^{T}}{\mathbf{F}}^{T}{\mathbf{FS}}^{(\mathrm{i})}{\mathbf{G}}^{(\mathrm{i})}-\lambda _{G^i} =0. \end{aligned}$$

From the complementary slackness condition, we obtain

$$\begin{aligned} \lambda _f {\mathbf{F}}_{pq}&=\left[ {-\sum \limits _{i=1}^h {( {{\mathbf{R}}^{(\mathrm{ci})}{\mathbf{G}}^{{(\mathrm{i})}^{T}}{\mathbf{S}}^{{(\mathrm{i})}^{T}}+\alpha {\mathbf{UY}}})} } \right. \\&\quad \left. {+\sum \limits _{i=1}^h {{\mathbf{FS}}^{(\mathrm{i})}{\mathbf{G}}^{(\mathrm{i})}} {\mathbf{G}}^{{(\mathrm{i})}^{T}}{\mathbf{S}}^{{(\mathrm{i})}^{T}}+\alpha {\mathbf{UF}}} \right] _{pq} {\mathbf{F}}_{pq} =0 \end{aligned}$$

(22)

$$\begin{aligned} \lambda _{S^i} {\mathbf{S}}^{(\mathrm{i})}_{pq}&=\left( {-{\mathbf{F}}^{T}{\mathbf{R}}^{(\mathrm{ci})}{\mathbf{G}}^{{(\mathrm{i})}^{T}}+{\mathbf{F}}^{T}{\mathbf{FS}}^{(\mathrm{i})}{\mathbf{G}}^{(\mathrm{i})}{\mathbf{G}}^{{(\mathrm{i})}^{T}}}\right) _{pq}\\&\quad \times {\mathbf{S}}^{(\mathrm{i})}_{pq} =0 \end{aligned}$$

(23)

$$\begin{aligned} \lambda _{G^i} {\mathbf{G}}^{(\mathrm{i})}_{pq}&=\left( {-{\mathbf{S}}^{{(\mathrm{i})}^{T}}{\mathbf{F}}^{T}{\mathbf{R}}^{(\mathrm{ci})}+{\mathbf{S}}^{{(\mathrm{i})}^{T}}{\mathbf{F}}^{T}{\mathbf{FS}}^{(\mathrm{i})}{\mathbf{S}}^{(\mathrm{i})}{\mathbf{G}}^{(\mathrm{i})}}\right) _{pq}\\&\quad \times {\mathbf{G}}^{(\mathrm{i})}_{pq} =0 \end{aligned}$$

(24)

Equations (22), (23) and (24) are the fixed point equations that the solution must satisfy at convergence. It is obvious that the limiting solutions of the update rules of (7), (8) and (9) satisfy these fixed point equations. For example, we have ${\mathbf{S}}^{{(\mathrm{i})}^{\infty }}={\mathbf{S}}^{{(\mathrm{i})}^{{\mathrm{t}+1}}}={\mathbf{S}}^{{(\mathrm{i})}^{\mathrm{t}}}$ at convergence, so the update rule of (7) becomes

$$\begin{aligned} {\mathbf{S}}_{pq}^{(i)} ={\mathbf{S}}_{pq}^{(i)} \frac{( {{\mathbf{F}}^T{\mathbf{R}}^{(\mathrm{ci})}{\mathbf{G}}^{{(\mathrm{i})}^{T}}})_{pq} }{( {{\mathbf{F}}^T{\mathbf{FS}}^{(\mathrm{i})}{\mathbf{G}}^{(\mathrm{i})}{\mathbf{G}}^{{(\mathrm{i})}^{T}}})_{pq} } \end{aligned}$$

then we obtain

$$\begin{aligned} ( {-{\mathbf{F}}^{T}{\mathbf{R}}^{(\mathrm{ci})}{\mathbf{G}}^{{(\mathrm{i})}^{T}}+{\mathbf{F}}^{T}{\mathbf{FS}}^{(\mathrm{i})}{\mathbf{G}}^{(\mathrm{i})}{\mathbf{G}}^{{(\mathrm{i})}^{T}}})_{pq} {\mathbf{S}}^{(\mathrm{i})}_{pq} =0 \end{aligned}$$

which is identical to Eq. (23). Similarly, we can see the update rule of (8) and (9) is identical to Eqs. (24) and (22), respectively. So the solution satisfies the KKT condition at convergence. The proof is completed. $\square$

1.2 Convergence of the algorithm

We now prove the convergence of the algorithm. First, we assume that $\{{\mathbf{S}}^{(i)}\}_{1\le i\le h}$ and $\{{\mathbf{G}}^{(i)}\}_{1\le i\le h}$ are fixed matrices, then the objective function (6) can be written as

$$\begin{aligned} \mathop {\min }\limits _{{\mathbf{F}}\ge 0} J_1 ({\mathbf{F}})&= Tr\left[ {-2\left( \, {\sum \limits _{i=1}^h {{\mathbf{R}}^{(\mathrm{ci})}{\mathbf{G}}^{{(\mathrm{i})}^{T}}{\mathbf{S}}^{{(\mathrm{i})}^{T}}+\alpha {\mathbf{UY}}} }\right) {\mathbf{F}}^T} \right] \\&\quad +\sum \limits _{i=1}^h {Tr( {{\mathbf{F}}^T{\mathbf{FS}}^{(\mathrm{i})}{\mathbf{G}}^{(\mathrm{i})}{\mathbf{G}}^{{(\mathrm{i})}^{T}}{\mathbf{S}}^{{(\mathrm{i})}^{T}}})}\\&\quad +\,\alpha Tr({{\mathbf{F}}^T{\mathbf{UF}}}) \end{aligned}$$

where we ignore the constant ${\mathbf{R}}^{{(\mathrm{ci})}^{T}}{\mathbf{R}}^{(\mathrm{ci})}$ and ${\mathbf{Y}}^T{\mathbf{UY}}$.

Theorem 2

The objective function $J_1 ({\mathbf{F}})$ is monotonically nonincreasing under the update rule (9).

To prove Theorem 2, we will make use of auxiliary function approach in [37]. We first introduce the definition of auxiliary function.

Definition 1

$Z(H,\widetilde{H})$ is an auxiliary function for $X(H)$ if the conditions

$$\begin{aligned} Z(H,\widetilde{H})\ge X(H),\quad \quad Z(H,H)=X(H) \end{aligned}$$

(25)

are satisfied.

Lemma 1

If $Z$ is an auxiliary, then $X$ is nonincreasing under the update

$$\begin{aligned} H^{t+1}=\mathop {\arg \min }\limits _H\, Z(H,H^t) \end{aligned}$$

(26)

Proof

$X(H^t)=Z(H^t,H^t)\ge Z(H^{t+1},H^t)\ge X(H^{t+1})$. The proof is completed. $\square$

So the key is to find an appropriate auxiliary function $Z({\mathbf{F}},\widetilde{{\mathbf{F}}})$ of $J_1 ({\mathbf{F}})$.

Proof of Theorem 2

Now we show that

$$\begin{aligned} Z({{\mathbf{F}}^{\mathrm{t}+1},{\mathbf{F}}^\mathrm{t}})&=-\sum \limits _{pq} {2\left( {\sum \limits _{i=1}^h {{\mathbf{R}}^{(\mathrm{ci})}{\mathbf{G}}^{{(\mathrm{i})}^{T}}{\mathbf{S}}^{{(\mathrm{i})}^{T}}+\alpha {\mathbf{UY}}} }\right) } _{pq} {\mathbf{F}}_{pq}^{\mathrm{t}+1} \\&\quad +\sum \limits _{i=1}^h {\sum \limits _{pq} {\frac{( {{\mathbf{F}}^\mathrm{t}{\mathbf{S}}^{(\mathrm{i})}{\mathbf{G}}^{(\mathrm{i})}{\mathbf{G}}^{{(\mathrm{i})}^{T}}{\mathbf{S}}^{{(\mathrm{i})}^{T}}})_{pq} ( {{\mathbf{F}}_{pq}^{\mathrm{t}+1} })^2}{{\mathbf{F}}_{pq}^\mathrm{t} }} }\\&\quad +\sum \limits _{pq} {\frac{( {{\mathbf{UF}}^\mathrm{t}})_{pq} ( {{\mathbf{F}}_{pq}^{\mathrm{t}+1} })^2}{{\mathbf{F}}_{pq}^\mathrm{t} }} \end{aligned}$$

(27)

is an auxiliary function of $J_1 ({\mathbf{F}}^{\mathrm{t}+1})$. First, it is obvious that the equality $J_1 ({\mathbf{F}}^\mathrm{t})=Z({\mathbf{F}}^\mathrm{t},{\mathbf{F}}^\mathrm{t})$ holds. Then the inequality $Z({\mathbf{F}}^{\mathrm{t}+1},{\mathbf{F}}^\mathrm{t})\ge J_1 ({\mathbf{F}}^{\mathrm{t}+1})$ holds because: the first term in $J_1 ({\mathbf{F}}^{\mathrm{t}+1})$ and $Z({\mathbf{F}}^{\mathrm{t}+1},{\mathbf{F}}^\mathrm{t})$ is equal, and the second and third term in $Z({\mathbf{F}}^{\mathrm{t}+1},{\mathbf{F}}^\mathrm{t})$ is always bigger than in $J_1 ({\mathbf{F}}^{\mathrm{t}+1})$ [27].

According to Eq. (26), we find the minimum of $Z({\mathbf{F}}^{\mathrm{t}+1},{\mathbf{F}}^\mathrm{t})$ fixing ${\mathbf{F}}^\mathrm{t}$. The minimum is obtained by

$$\begin{aligned} \frac{\partial Z( {{\mathbf{F}}^{\mathrm{t}+1},{\mathbf{F}}^\mathrm{t}})}{\partial {\mathbf{F}}_{pq}^{\mathrm{t}+1} }&= -2\left( \, {\sum \limits _{i=1}^h {{\mathbf{R}}^{(\mathrm{ci})}{\mathbf{G}}^{{(\mathrm{i})}^{T}}{\mathbf{S}}^{{(\mathrm{i})}^{T}}+\alpha {\mathbf{UY}}}}\right) _{pq} \\&\quad +\,2\sum \limits _{i=1}^h {\frac{( {{\mathbf{F}}^\mathrm{t}{\mathbf{S}}^{(\mathrm{i})}{\mathbf{G}}^{(\mathrm{i})}{\mathbf{G}}^{{(\mathrm{i})}^{T}}{\mathbf{S}}^{{(\mathrm{i})}^{T}}})_{pq} {\mathbf{F}}_{pq}^{\mathrm{t}+1} }{{\mathbf{F}}_{pq}^\mathrm{t} }}\\&\quad +\,2\frac{( {{\mathbf{UF}}^\mathrm{t}})_{pq} {\mathbf{F}}_{pq}^{\mathrm{t}+1} }{{\mathbf{F}}_{pq}^\mathrm{t} }=0 \end{aligned}$$

then we can get the update rule (9) by solving $F^{t+1}$:

$$\begin{aligned} \hbox {F}_{pq}^{\mathrm{t}+1} ={\mathbf{F}}_{pq}^\mathrm{t} \frac{( {\sum \nolimits _{i=1}^h {{\mathbf{R}}^{(\mathrm{ci})}{\mathbf{G}}^{{(\mathrm{i})}^{T}}} {\mathbf{S}}^{{(\mathrm{i})}^{T}}+\alpha {\mathbf{UY}}})_{pq} }{( {\sum \nolimits _{i=1}^h {{\mathbf{F}}^\mathrm{t}{\mathbf{S}}^{(\mathrm{i})}{\mathbf{G}}^{(\mathrm{i})}{\mathbf{G}}^{{(\mathrm{i})}^{T}}{\mathbf{S}}^{{(\mathrm{i})}^{T}}+\alpha {\mathbf{UF}}^\mathrm{t}} })_{pq} } \end{aligned}$$

So under this update rule, function $J_1 ({\mathbf{F}})$ is monotonically nonincreasing. The proof is completed. $\square$

So far we assume $\{{\mathbf{S}}^{(i)}\}_{1\le i\le h}$ and $\{{\mathbf{G}}^{(\mathrm{i})}\}_{1\le i\le h}$ are fixed matrices. Alternatively, we can also fix ${\mathbf{F}}$ and $\{{\mathbf{G}}^{(\mathrm{i})}\}_{1\le i\le h}$ or ${\mathbf{F}}$ and $\{{\mathbf{S}}^{(i)}\}_{1\le i\le h}$, and the proof for convergence of the update rules of (7) and (8) is similar.