Hierarchical tucker tensor regression: Application to brain imaging data analysis

We present a novel generalized linear tensor regression model, which takes tensor-variate inputs as covariates and finds low-rank (almost) best approximation of regression coefficient arrays using hierarchical Tucker decomposition. With limited sample size, our model is highly compact and extremely efficient as it requires only O(dr3 + dpr) parameters for order d tensors of mode size p and rank r, which avoids the exponential growth in d, in contrast to O(rd + dpr) parameters of Tucker regression modeling. Our model also maintains the flexibility like classical Tucker regression by allowing distinct ranks on different modes according to a dimension tree structure. We evaluate our new model on both synthetic data and real-life MRI images to show its effectiveness.