Modern applications such as neuroscience, text mining, and large-scale social networks generate massive amounts of data with multiple aspects and high dimensionality. Tensors (i.e., multi-way arrays) provide a natural representation for such massive data. Consequently, tensor decompositions and factorizations are emerging as novel and promising tools for exploratory analysis of multidimensional...

Multi-way modeling has become an important tool in the analysis of large scale multi-modal data. An important class of multi-way models is given by the Tucker model which decomposes the data into components pertaining to each modality as well as a core array indicating how the components of the various modalities interact. Unfortunately, the Tucker model is not unique. Furthermore, establishing...

We propose two new multilinear operators for expressing the matrix compositions that are needed in the Tucker and PARAFAC (CANDECOMP) decompositions. The first operator, which we call the Tucker operator, is shorthand for performing an n-mode matrix multiplication for every mode of a given tensor and can be employed to consisely express the Tucker decomposition. The second operator, which we ca...

In recent years, sequential optimality conditions are frequently used for convergence of iterative methods to solve nonlinear constrained optimization problems. The sequential optimality conditions do not require any of the constraint qualications. In this paper, We present the necessary sequential complementary approximate Karush Kuhn Tucker (CAKKT) condition for a point to be a solution of a ...

We investigate the pinching problem for shrinking compact Ricci solitons. Firstly, we show that every $n$-dimensional $(n\ge 4)$ soliton $(M^n,g)$ is isometric to a finite quotient of $\mathbb S^n$ under an $L^{n/2}$-pinching condi