Course Drop-out Prediction on MOOC Platform via Clustering and Tensor Completion

We first throw light on the basics of tensors, and then present the tensor structure equipped with a fast low-rank tensor completion for drop-out prediction.

Tensor basics. A tensor is a high-dimensional data representation, whose expression is vector (1-mode) and matrix (2-mode). An $n$ -mode tensor can be defined as $X\in {\text{𝐑}}^{I_1\times I_2\times \mathrm{\dots }\times I_n}$ , where $I_n$ denotes the quantity of mode $n$ , and its elements are denoted as $x_{(I_1,\mathrm{\dots },I_k)}$ , where $1⩽k⩽n$ . The matriculating operator, which unfolds a tensor into a matrix, is defined as $\text{unfold}(X,n)=X_{(n)}$ , in which the tensor element $(I_1,I_2,\mathrm{\dots },I_n)$ is mapped to the matrix element $(I_n,J)$ , where

(1) $$J=\underset{m=1,m\ne n}{\overset{k-1}{\prod }}{I}_m$$

The reverse of the matriculation is defined as $\text{fold}(X_{(n)},n)=X$ in a similar way.

The inner product of two same-size tensors $A,B\in {\text{𝐑}}^{(I_1\times I_2\times \mathrm{\cdots }\times I_N)}$ is defined as the sum of the products of their entries:

(2) $$(A,B)=\underset{i_1}{\sum }\underset{i_2}{\sum }\mathrm{\dots }\underset{i_N}{\sum }{a}_{(i_1,i_2,\mathrm{\dots },i_n)}{b}_{(i_1,i_2,\mathrm{\dots },i_n)}$$

For any $1⩽n⩽N$ , the product of a matrix $M\in {\text{𝐑}}^{J\times I_n}$ with a tensor $A\in {\text{𝐑}}^{(I_1\times I_2\times \mathrm{\cdots }\times I_N)}$ is expressed as $A{\times }_{n}M$ , and transformed into the product of two matrixes.

(3) $$Y=A{\times }_{n}M\iff Y_{(n)}=M{A}_{(n)}$$

Denote ${\Vert X\Vert }_{\text{F}}=\sqrt{(X,X)}$ as the Frobenius norm of a tensor. It is clear that ${\Vert X\Vert }_{\text{F}}=\Vert X_{(k)}\Vert .$

Global tensor. Suppose that there are $m$ courses and $n$ students in the MOOC data. In order to employ the excellent explanatory power of the tensor structure, we construct the data into 3-dimension tensor $𝒯$ , as Fig. 2 shows.

10.26599/TST.2018.9010110.F2 Fig. 2Sketch of global tensor.

The abscissa axis and vertical axis represent the students, which means the scale of each matrix in the tensor is $n\times n$ , and the other dimension represents the $m$ course. Thus the size of $𝒯$ is $n\times n\times m$ , and we randomly select a slice of size $k$ . There are different value interpretations for ${𝒳}_{iik}$ : (1) The value ${𝒳}_{iik}=0$ represents "student $i$ does not select the course $k$ " ; (2) the value ${𝒳}_{iik}=1$ represents "student $i$ does select the course $k$ , and fails to finish it" ; (3) the value ${𝒳}_{iik}=2$ represents "student $i$ does select the course $k$ , and finishes it" ; (4) the value ${𝒳}_{ijk}=0$ represents "student $i$ and student $j$ do not select the same course $k$ " ; (5) the value ${𝒳}_{ijk}=1$ represents "student $i$ and student $j$ do select the same course $k$ , and they both fail to finish it" ; and (6) the value ${𝒳}_{ijk}=2$ represents "student $i$ and student $j$ select the same course $k$ , and they both finish it" . As we can see, the tensor can not only express the drop-out performance of all students in all courses, but also reveal the relationship between students in the specific course. Therefore, we put all data into one tensor to construct a global tensor as a step in generating a MOOP for drop-out prediction.