# Alternating Direction Method of Multipliers for Generalized Low-Rank Tensor Recovery
## 2 Notations and Preliminaries  
This section will briefly introduce mathematical notations and review tensor algebra. We denote tensors by boldface Euclid Math One, e.g., \(\mathbf{\mathcal{A}}\), matrices by boldface capital letters, e.g., \(\mathbf{\mathbf{A}}\), vectors by boldface letters, e.g., \(\mathbf{\mathbf{a}}\), and scalars by italic letters, e.g., \(a\) or \(A\). A tensor with order \(N\) indicates that its entries can be expressed via N indices. For an \(N\)th-order real tensor \(\mathbf{\mathcal{A}}\in\mathbb{R}^{\mathbf{h}\times\mathbf{l}_{2}\times\cdots\times\mathbf{l }_{N}}\), its \((\mathbf{i_{1}},\mathbf{i_{2}},\cdots,\mathbf{i_{N}})\) entry is denoted by \(\mathbf{a}_{\mathbf{i_{1}}\mathbf{i_{2}}\cdots\mathbf{i_{N}}}\). The \(n\)-mode matricization (unfolding) of tensor \(\mathbf{\mathcal{A}}\), denoted by \(\mathbf{\mathbf{A}}_{(n)}\), is an \(\mathbf{I}_{n}\times\prod_{\mathsf{j}\
eq n}\mathbf{\mathbf{j}}_{\mathsf{j}}\) matrix obtained by rearranging \(n\)-mode fibers to be the columns of the resulting matrix. Furthermore, we define an opposite operation \"fold\" as \(\mathbf{fold}_{n}(\mathbf{\mathbf{A}}_{(n)})=\mathbf{\mathcal{A}}\). Given another tensor \(\mathbf{B}\in\mathbb{R}^{\mathbf{\mathbf{h}}\times\mathbf{\mathbf{l}_{2}\times\cdots\times \mathbf{\mathbf{l}_{N}}}}\), the inner product between \(\mathbf{\mathcal{A}}\) and \(\mathbf{B}\) is defined as \(\langle\mathbf{\mathcal{A}},\mathbf{\mathcal{B}}\rangle=\sum_{i_{1}=1}^{\mathbf{\mathbf{l}} _{1}}\sum_{i_{2}=1}^{\mathbf{\mathbf{l}}_{2}}\cdots\sum_{i_{N}=1}^{\mathbf{\mathbf{l}} _{N}}\mathbf{a}_{\mathbf{i_{1}}\mathbf{i_{2}}\cdots\mathbf{i_{N}}}\mathbf{b}_{\mathbf{i_{1}}\mathbf{i_{2}} \cdots\mathbf{i_{N}}}\). Then, the Frobenius norm (or \(l_{2}\)-norm) of tensor \(\mathbf{\mathcal{A}}\) can be induced by the above inner product, that is, \(\left|\mathbf{\mathcal{A}}\right|\left|\right|_{\mathsf{F}}=\sqrt{\langle\mathbf{ \mathcal{A}},\mathbf{\mathcal{A}}\rangle}\). Moreover, we also give other norm definitions of a tensor, which will be needed in the following sections. The \(l_{1}\)-normof tensor \(\mathbf{\mathcal{A}}\) is expressed as \(\left|\left|\mathbf{\mathcal{A}}\right|\right|_{1}=\sum_{i=1}^{\mathbf{I}_{1}}\sum_{ i=2}^{\mathbf{I}_{2}}\cdots\sum_{i=N-1}^{\mathbf{I}_{N}}\left|\mathbf{a}_{i\mathbf{I}_{i} \mathbf{i}_{2}\cdots i\mathbf{}_{N}}\right|\) and the tensor nuclear norm is \(\left|\left|\mathbf{\mathcal{A}}\right|\right|_{\bullet}=\sum_{n=1}^{N}\mathbf{w}_{ n}\left|\left|\mathbf{A}_{(n)}\right|\right|_{\bullet}\), where \(\left|\left|\mathbf{A}_{(n)}\right|\right|_{\bullet}\), named the nuclear norm of \(\mathbf{A}_{(n)}\), is the sum of singular values of \(\mathbf{A}_{(n)}\), \(\mathbf{w}_{n}\geq\ \mathbf{0}\) and \(\sum_{n=1}^{N}\mathbf{w}_{n}=\mathbf{1}\). Due to the fact that the nuclear norm of one matrix is the tightest convex relaxation of the rank function on the unit ball in the spectral norm, it is easy to verify that the tensor nuclear norm \(\left|\left|\mathbf{\mathcal{A}}\right|\right|_{\bullet}\) is a convex function with respect to \(\mathbf{\mathcal{A}}\). The \(n\)-mode product of tensor \(\mathbf{\mathcal{A}}\) by a matrix \(\mathbf{U}\in\mathbb{R}^{\mathbf{I}_{n}\times\mathbf{I}_{n}}\), denoted as \(\mathbf{\mathcal{A}}\times_{\mathbf{n}}\mathbf{U}\), is an \(N\)th-order tensor with the dimensionality of \(\mathbf{I}_{1}\times\cdots\times\mathbf{I}_{n-1}\times\mathbf{I}_{n}\times \mathbf{I}_{n-1}\times\cdots\times\mathbf{I}_{N}\), and its \(\left(\mathbf{i}_{1},\cdots,\mathbf{i}_{n-1},\mathbf{j},\mathbf{i}_{n+1} \cdots,\mathbf{i}_{N}\right)-\ \mathbf{th}\) entry is calculated as \(\sum_{i_{n}=1}^{\mathbf{I}_{n}}\mathbf{a}_{i\mathbf{I}_{i}\mathbf{i}_{2}\cdots i _{N}}\mathbf{i}_{\mathbf{i}_{n}}\). The multi-linear rank of tensor \(\mathbf{\mathcal{A}}\) is stipulated as a vector \(\left(\mathbf{rank}(\mathbf{A}_{(1)}),\mathbf{rank}(\mathbf{A}_{(2)}),\ldots,\mathbf{rank}( \mathbf{A}_{(N)})\right)\), where \(\mathbf{rank}(\cdot)\) indicates the rank function of a matrix.