MR-NTD: Manifold Regularization Nonnegative Tucker Decomposition for Tensor Data Dimension Reduction and Representation

Abstract

With the advancement of data acquisition techniques, tensor (multidimensional data) objects are increasingly accumulated and generated, for example, multichannel electroencephalographies, multiview images, and videos. In these applications, the tensor objects are usually nonnegative, since the physical signals are recorded. As the dimensionality of tensor objects is often very high, a dimension reduction technique becomes an important research topic of tensor data. From the perspective of geometry, high-dimensional objects often reside in a low-dimensional submanifold of the ambient space. In this paper, we propose a new approach to perform the dimension reduction for nonnegative tensor objects. Our idea is to use nonnegative Tucker decomposition (NTD) to obtain a set of core tensors of smaller sizes by finding a common set of projection matrices for tensor objects. To preserve geometric information in tensor data, we employ a manifold regularization term for the core tensors constructed in the Tucker decomposition. An algorithm called manifold regularization NTD (MR-NTD) is developed to solve the common projection matrices and core tensors in an alternating least squares manner. The convergence of the proposed algorithm is shown, and the computational complexity of the proposed method scales linearly with respect to the number of tensor objects and the size of the tensor objects, respectively. These theoretical results show that the proposed algorithm can be efficient. Extensive experimental results have been provided to further demonstrate the effectiveness and efficiency of the proposed MR-NTD algorithm.

Maximum Discriminant Difference Criterion for Dimensionality Reduction of Tensor Data

“…In this section, MDD-TDR is compared with the GLTD [38], MMF [35], GNMF [34], NTD [23], LRRHTD [36], TLRDE [47] and STDA [48] algorithms mentioned in the paper to perform clustering and classification comparison experiments on 5 real-world datasets to verify the effectiveness of the MDD-TDR algorithm. Among these algorithms, GLTD and MMF are two classical nonnegative matrix factorization algorithms.…”

“…In 2017, Li et al then put forward a manifold canonical nonnegative Tucker decomposition algorithm (MR-NTD) [35] which optimized the projection matrix and compression features corresponding to each tensor one by one. The minimization objective function is constructed as follows…”

Sparse‐graph manifold learning method for bioluminescence tomography

“…Recently, manifold learning has become one of the hottest research fields in machine learning. As a geometrically motivated framework, many graph‐based manifold learning methods have been constructed in face recognition, image classification, medical imaging, and image reconstruction . In , a novel Laplacian manifold regularization method has been proposed to improve the reconstruction performance in both spatial aggregation and location accuracy for fluorescence molecular tomography.…”

Robust Tensor Factorization for Color Image and Grayscale Video Recovery

“…Low-rank tensor completion (LRTC) has become a research hotspot in machine learning and computer vision, because it can recover an ideal tensor by using only a part of the known entries under low-rank constraints. LRTC has achieved state-of-the-art performance in recommendation systems [1], computer vision [2], [3], machine learning [4], [5], multienergy computed tomography [6], image processing [7], [8], etc.…”