Accurate Embedding-based Log Determinant Optimization

Many tangible and intangible objects are represented as itemsets; i.e., composition of individual items. In this paper, we address the problem of finding the embedding of such items so as to use those embeddings in tasks like missing item prediction. We approach this problem by means of determinantal point process (DPP) in order to reflect the diversity within each set. Doing so requires an optimization of a log determinant of a symmetric positive definite (SPD) matrix. The standard practice to achieve this is to perform a low-rank decomposition of the matrix and derive update rules for the low rank matrix. In this work, we propose to approach this problem by means of item embedding. That is, we will learn the SPD matrix by trying to find the right vector representations for the given data for a fixed kernel function. To this end, we propose a novel algorithm to accurately compute the gradients of the log determinant with respect to the embedding vectors. We also show that our approach outperforms Autodiff-based learning in terms of gradient direction and running time, and that other general log determinant optimization problems can be addressed.