Low-rank tensor as a Dimension Reduction Tool in Complex Data Analysis

Description

In recent years there has been an increasing research interest on dealing with structured data sets. We are herein interested in the case tensors are involved in the modelling stage. Typically this means the predictors, the responses and/or the parameters are tensors.Tensors are natural generalizations of matrices. In the literature there have already been a lot of works on statistical estimation involving matrices. For multivariate (multi-response) linear model, one can apply reduced-rank regression assuming that the coefficient matrix has a low rank. Such an assumption reduced the effective number of parameters to be estimated and thus when the assumption is (at least approximately) true, the efficiency gain leads to a better estimate.Works on tensors are relatively more recent. Regression models with tensor predictors and/or tensor responses have been investigated in recent years. In this proposal, we consider several families of regression problems where the observations are scalars or vectors as in classical situations and these problems were all previously studied by non-tensor approaches. In the concrete problems we examine here, we make the key observation that the model parameters naturally organize themselves into tensors. Although tensors can be reshaped as matrices or vectors, this destroys the tensor structure. There have been a lot of works in the statistical literature that demonstrated that when parameters are matrices, one can often design models to take into account the matrix structure which performs better than using methods designed for vectors after vectorization. We expect the same phenomenon would be true for tensors. By treating parameters as tensors, more efficient and flexible dimension reduction schemes can be constructed, which is particularly useful when the number of parameters is large while the sample size is limited. In this proposal, we investigate three specific models and rely on the concept of low-rankness of tensor to reduce dimension.