Nonnegative Low Multi-Rank Transformed Tensor Approximation and its Applications

Nonnegative data from many applications are represented as multi-dimensional data which refers to as nonnegative tensor, e.g., video data and hyperspectral data are nonnegative tensors. Nonnegative low rank tensor approximation for nonnegative tensors plays a key role in many image processing and analysis applications. The main aim of nonnegative low rank tensor approximation is to identify latent features for objects representation and preserve nonnegative data constraints. The classification and recognition analysis can be performed by using latent features. In classical nonnegative tensor factorization, a nonnegative tensor is decomposed into the product of several components (nonnegative low rank tensors and matrices) for finding latent features.

In this project, we propose new models and algorithms for computing nonnegative low rank tensor approximation for nonnegative tensors. Our approach is to find a nonnegative low rank tensor such that the difference between nonnegative low rank tensor and the given nonnegative tensor is as small as possible. Here nonnegative constraints are preserved in low rank tensor approximation, but the underlying latent features are not necessary to be nonnegative. It is different from classical nonnegative tensor factorization where each factorization components are required to be nonnegative. In the proposed formulation, nonnegative constraints are less restricted. The low rank tensor approximation results and the resulting image analysis and recognition can be expected to be better. These results will be tested and demonstrated in the project.

On the other hand, low rank tensor structure can become more obvious after suitable transformations are applied to tensors like imaging data. We propose and investigate nonnegative low rank transformed tensor approximation for nonnegative tensors under the transformed tensor singular value decomposition. We develop alternating projection algorithms for solving such nonnegative tensor approximation problem. The convergence analysis will be conducted by analyzing nonnegative constraints in tensor entries (the original domain) and low rank structure in the transformed tensor entries (the transformed domain). Note that singular value decomposition calculation can be quite expensive especially when there are many singular value decompositions are required for nonnegative low rank transformed tensor approximation. In this project, we will study the tangent space of a point in the manifold in the transformed domain to approximate the projection onto the manifold in order to reduce the computational cost, and analyze the convergence results. Furthermore, we investigate into our theoretical results and numerical algorithms in the application of multi-dimensional image processing with missing and noisy observations.