求∂‖A∘(Y−W⊤X)‖F2∂W和∂‖A∘(Y−W⊤X)‖F2∂X,其中,∘表示Hadamard product,各变量均为矩阵形式。 求解: 为方便起见,定义Z=A∘(Y−W⊤X),同时,将F-norm重写为Frobenius product(详见备注)的形式,则有: ∂‖A∘(Y−W⊤X)‖F2=∂Z:Z=2
Hence, the derivative simply is ∆ →A∆B 3.2. Frobenius Norm. Let f : R q×p →R be defined as f(B) = ||B|| 2 F . f(B +∆) = B +∆, B +∆ = B, B +2 B, ∆ + ∆, ∆ = f(B) +2 B, ∆ +||∆|| 2 ...
(unitary) matrix, (iii) The 2-norm and the Frobenius norm are invariant under multiplication by an orthogonal (unitary) matrix (See Section 2.6), and (iv) The error in multiplying a matrix by an orthogonal matrix is not magnified by the process of numerical matrix multiplication (See ...
U is an isometry with respect to the inner product determined by U. ■ U is a normal matrix with eigenvalues lying on the unit circle. Illustration ■ A 2-by-2 unitary matrix MatrixForm[A = {{0, I}, {I, 0}}] 0ii0 Inverse[A] == ConjugateTranspose[A] True MatrixForm[ConjugateTran...
The Frobenius norm is denoted by \Vert \cdot \Vert , and the Frobenius inner product by \langle \cdot , \cdot \rangle . The symbol \bullet denotes the entrywise or Hadmard product of matrices. For a given step size \tau we use the notation t_k = k \tau for any k with 2 k \...
Notice that the initial residual norm\deltain line 1 of Algorithm 1 can be computed at low cost by exploiting the properties of the Frobenius norm and the trace operator. In many cases the dimension of the final space{\mathcal {K}}_m, namely the number of columns ofV_m, turns out to...
The optimal value of the local variational parameters ξ ij can be computed by writing the expectation of the joint distribution in terms of ξ and setting its derivative to zero. In particular, L~(ξ)=∑i∑jR^ij(lnσ(ξij)−ξij2−12ξij(σ(ξij)−12)×(ξij2−E[(ui...
Y) to denote the inner product of X and Y. X F , X ∗ , b , σ i (X) and σ 2 i (X) denote the Frobenius norm, nuclear norm of the matrix X, the Euclidean norm of the vector b, the ith largest singular value of X and (σ...
(unitary) matrix, (iii) The 2-norm and the Frobenius norm are invariant under multiplication by an orthogonal (unitary) matrix (See Section 2.6), and (iv) The error in multiplying a matrix by an orthogonal matrix is not magnified by the process of numerical matrix multiplication (See ...
The scalar derivative of the product of two matrix time-functions is d(A(t)B(t))dt=A(t)dtB(t)+A(t)B(t)dt. This result is analogous to the derivative of a product of two scalar functions of a scalar, except caution must be used in reserving the order of the product. An importa...