Pointwise HSIC: A Linear-Time Kernelized Co-occurrence Norm for Sparse Linguistic Expressions

In this paper, we propose a new kernel-based co-occurrence measure that can be applied to sparse linguistic expressions (e.g., sentences) with a very short learning time, as an alternative to pointwise mutual information (PMI). As well as deriving PMI from mutual information, we derive this new measure from the Hilbert--Schmidt independence criterion (HSIC); thus, we call the new measure the pointwise HSIC (PHSIC). PHSIC can be interpreted as a smoothed variant of PMI that allows various similarity metrics (e.g., sentence embeddings) to be plugged in as kernels. Moreover, PHSIC can be estimated by simple and fast (linear in the size of the data) matrix calculations regardless of whether we use linear or nonlinear kernels. Empirically, in a dialogue response selection task, PHSIC is learned thousands of times faster than an RNN-based PMI while outperforming PMI in accuracy. In addition, we also demonstrate that PHSIC is beneficial as a criterion of a data selection task for machine translation owing to its ability to give high (low) scores to a consistent (inconsistent) pair with other pairs.