Table 1 The summarization of notations

\(M \in {\mathbb {R}}^{m \times n}\)

Matrix with size \(m \times n\)

\(\sigma _{i}(M) = \sigma _{i}\)

ith largest singular value of M

\(u_{i}(M) = u_{i}\)

ith left singular vector of M

\(v_{i}(M) = v_{i}\)

ith right singular vector of M

\(\Arrowvert M \Arrowvert _{*} = \sum _{i = 1}^{\min \{m, n\}}\sigma _{i}(M)\)

Nuclear norm of M

\(\Arrowvert M \Arrowvert _{F} = \sqrt{Tr(M^{T}M)}\)

Frobenius norm of M

\(\Arrowvert M \Arrowvert _{p} = \left( \sum _{i = 1}^{\min \{m, n\}}\sigma _{i}(M)^{p}\right) ^{1/p}\)

Schatten p-norm of M

\(\langle M_{1}, M_{2} \rangle = Tr(M_{1}^{T}M_{2})\)

Standard inner product

\(\nabla f\)

Gradient of a differentiable function f