Effective and Sparse Count-Sketch via k-means clustering

Abstract

Count-sketch is a popular matrix sketching algorithm that can produce a sketch of an input data matrix X in O(nnz(X)) time where nnz(X) denotes the number of non-zero entries in X. The sketched matrix will be much smaller than X while preserving most of its properties. Therefore, count-sketch is widely used for addressing high-dimensionality challenge in machine learning. However, there are two main limitations of count-sketch: (1) The sketching matrix used count-sketch is generated randomly which does not consider any intrinsic data properties of X. This data-oblivious matrix sketching method could produce a bad sketched matrix which will result in low accuracy for subsequent machine learning tasks (e.g., classification); (2) For highly sparse input data, count-sketch could produce a dense sketched data matrix. This dense sketch matrix could make the subsequent machine learning tasks more computationally expensive than on the original sparse data X. To address these two limitations, we first show an interesting connection between count-sketch and k-means clustering by analyzing the reconstruction error of the countsketch method. Based on our analysis, we propose to reduce the reconstruction error of count-sketch by using k-means clustering algorithm to obtain the low-dimensional sketched matrix. In addition, we propose to solve k-mean clustering using gradient descent with -L1 ball projection to produce a sparse sketched matrix. Our experimental results based on six real-life classification datasets have demonstrated that our proposed method achieves higher accuracy than the original count-sketch and other popular matrix sketching algorithms. Our results also demonstrate that our method produces a sparser sketched data matrix than other methods and therefore the prediction cost of our method will be smaller than other matrix sketching methods.