What is the direct definition for the trace norm of a tensor? By direct I mean without matricization.
Edit 1: By tensor, I mean a multi-way array, a generalization of vectors and matrices. (The nuclear norm of a matrix, which is a special case of a tensor, is sum of its singular values.)
In particular, I am interested on tensors $\cal T$ over $\mathbb{R}$. $${\cal T}\in \mathbb{R}^{d_1\times d_2 \times \dots \times d_n} $$
$\|\cal T\|_* = \|\cal T\|_{tr} = ?$
Edit 2: Could anyone guide me where this property is defined for the first time, i.e. what is the historical origin of the definition?