Multivariate Regression on the Grassmannian for Predicting Novel Domains

We study the problem of predicting how to recognise visual objects in novel domains with neither labelled nor unlabelled training data. Domain adaptation is now an established research area due to its value in ameliorating the issue of domain shift between train and test data. However, it is conventionally assumed that domains are discrete entities, and that at least unlabelled data is provided in testing domains. In this paper, we consider the case where domains are parametrised by a vector of continuous values (e.g., time, lighting or view angle). We aim to use such domain metadata to predict novel domains for recognition. This allows a recognition model to be pre-calibrated for a new domain in advance (e.g., future time or view angle) without waiting for data collection and re-training. We achieve this by posing the problem as one of multivariate regression on the Grassmannian, where we regress a domain's subspace (point on the Grassmannian) against an independent vector of domain parameters. We derive two novel methodologies to achieve this challenging task: a direct kernel regression from RM ! G, and an indirect method with better extrapolation properties. We evaluate our methods on two crossdomain visual recognition benchmarks, where they perform close to the upper bound of full data domain adaptation. This demonstrates that data is not necessary for domain adaptation if a domain can be parametrically described.