Multilinear classifiers

Tenenbuam and Freeman proposed bilinear classifiers as a tool to classify observations influenced by both the variable to classify (“content”) and a nuisance factor (“style”). In most real-world problem however, observations are influenced by a number of such factors. Gait identification is a typical such situation. To tackle their influence, it is natural to resort to multilinear or tensor decompositions: approaches such as Higher-Order SVD and Nonnegative Tensor Factorization have indeed been formulated to address the problem of decomposing a tensor into its constituent factors. These efforts, however, have mainly focussed on the generalization of PCAstyle dimensionality reduction to tensorial observations (MPCA), while no proposals have been brought forward for exploiting tensor decomposition to classify (vectorial) observations depending on a number of covariates. In this paper we show how HOSVD can be exploited to formulate a natural generalization of bilinear classifiers, which we call “multilinear classifiers”, able to classify observations depending on one content label and several style labels. A set of style-specific linear maps are learned by HOSVD of the training set, represented as a tensor. When a new observation in a different combination of styles is presented, EM is applied to alternative learn a new style matrix and classify the content of the observation. This approach is validated on the UCF gait ID dataset, demonstrating how explicitly modeling the different nuisance factors delivers superior performances.