In this paper, we propose three new tensor decompositions for even-order tensors corresponding respectively to the rank-one decompositions of some unfolded matrices. Consequently such new decompositions lead to three new notions of (even-order) tensor ranks, to be called the M-rank, the symmetric M-rank, and the strongly symmetric M-rank in this paper. We discuss the bounds between these new te...

Although Zipf's law is widespread in natural and social data, one often encounters situations where one or both ends of the ranked data deviate from the power-law function. Previously we proposed the Beta rank function to improve the fitting of data which does not follow a perfect Zipf's law. Here we show that when the two parameters in the Beta rank function have the same value, the Lavalette ...

In this paper, we discuss the connection of the kernel versions of the ELM classifier with infinite Single-hidden Layer Feedforward Neural networks and show that the original ELM kernel definition can be adopted for the calculation of the ELM kernel matrix for two of the most common activation functions, i.e., the RBF and the sigmoid functions. In addition, we show that a low-rank decomposition...

Several recent works have introduced highly compact representations of single-particle Green's functions in the imaginary time and Matsubara frequency domains, as well efficient interpolation grids used to recover representations. In particular, intermediate representation with sparse sampling discrete Lehmann (DLR) make use low-rank compression techniques obtain optimal approximations controll...

Receptor activator of NF-κB (RANK) plays a critical role in osteoclastogenesis, an essential process for the initiation of bone remodeling to maintain healthy bone mass and structure. Although the signaling and function of RANK have been investigated extensively, much less is known about the negative regulatory mechanisms of its signaling. We demonstrate in this paper that RANK trafficking, sig...

Multidimensional persistence studies topological features of shapes by analyzing the lower level sets of vector-valued functions. The rank invariant completely determines the multidimensional analogue of persistent homology groups. We prove that multidimensional rank invariants are stable with respect to function perturbations. More precisely, we construct a distance between rank invariants suc...