  Cost volume is an essential component of recent deep models for optical flow
estimation and is usually constructed by calculating the inner product between
two feature vectors. However, the standard inner product in the commonly-used
cost volume may limit the representation capacity of flow models because it
neglects the correlation among different channel dimensions and weighs each
dimension equally. To address this issue, we propose a learnable cost volume
(LCV) using an elliptical inner product, which generalizes the standard inner
product by a positive definite kernel matrix. To guarantee its positive
definiteness, we perform spectral decomposition on the kernel matrix and
re-parameterize it via the Cayley representation. The proposed LCV is a
lightweight module and can be easily plugged into existing models to replace
the vanilla cost volume. Experimental results show that the LCV module not only
improves the accuracy of state-of-the-art models on standard benchmarks, but
also promotes their robustness against illumination change, noises, and
adversarial perturbations of the input signals.
