A sparsity-inducing formulation for evolutionary co-clustering

Traditional co-clustering methods identify block structures from static data matrices. However, the data matrices in many applications are dynamic; that is, they evolve smoothly over time. Consequently, the hidden block structures embedded into the matrices are also expected to vary smoothly along the temporal dimension. It is therefore desirable to encourage smoothness between the block structures identified from temporally adjacent data matrices. In this paper, we propose an evolutionary co-clustering formulation for identifying co-cluster structures from time-varying data. The proposed formulation encourages smoothness between temporally adjacent blocks by employing the fused Lasso type of regularization. Our formulation is very flexible and allows for imposing smoothness constraints over only one dimension of the data matrices, thereby enabling its applicability to a large variety of settings. The optimization problem for the proposed formulation is non-convex, non-smooth, and non-separable. We develop an iterative procedure to compute the solution. Each step of the iterative procedure involves a convex, but non-smooth and non-separable problem. We propose to solve this problem in its dual form, which is convex and smooth. This leads to a simple gradient descent algorithm for computing the dual optimal solution. We evaluate the proposed formulation using the Allen Developing Mouse Brain Atlas data. Results show that our formulation consistently outperforms methods without the temporal smoothness constraints.