  In this paper, we propose an extension of trace ratio based Manifold learning
methods to deal with multidimensional data sets. Based on recent progress on
the tensor-tensor product, we present a generalization of the trace ratio
criterion by using the properties of the t-product. This will conduct us to
introduce some new concepts such as Laplacian tensor and we will study formally
the trace ratio problem by discuting the conditions for the exitence of
solutions and optimality. Next, we will present a tensor Newton QR
decomposition algorithm for solving the trace ratio problem. Manifold learning
methods such as Laplacian eigenmaps, linear discriminant analysis and locally
linear embedding will be formulated in a tensor representation and optimized by
the proposed algorithm. Lastly, we will evaluate the performance of the
different studied dimension reduction methods on several synthetic and real
world data sets.
