Quaternion Ordinal Embedding

Ordinal embedding (OE) aims to project objects into a low-dimensional space while preserving their ordinal constraints as well as possible. Generally speaking, a reasonable OE algorithm should simultaneously capture a) semantic meaning and b) the ordinal relationship of the objects. However, most of the existing methods merely focus on b). To address this issue, our goal in this paper is to seek a generic OE method to embrace the two features simultaneously. We argue that different dimensions of vector-based embedding are naturally entangled with each other. To realize a), we expect to decompose the D dimensional embedding space into D different semantic subspaces, where each subspace is associated with a matrix representation. Unfortunately, introducing a matrix-based representation requires far more complex parametric space than its vector-based counterparts. Thanks to the algebraic property of quaternions, we are able to find a more efficient way to represent a matrix with quaternions. For b), inspired by the classic chordal Grassmannian distance, a new distance function is defined to measure the distance between different quaternions/matrices, on top of which we construct a generic OE loss function. Experimental results for different tasks on both simulated and real-world datasets verify the effectiveness of our proposed method.