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=2Z:dZ=2Z:d(A∘...
MATRIX DERIVATIVE 3 (3) Let f : R n →R m and g : R n →R m with derivatives A, B at x 0 . Inner Product Define h : R n →R such that h(x) = f(x) T g(x). Then the derivative of h is x 0 is f(x 0 ) T B +g(x 0 ) T A Outer Product Define h ...
where ∥⋅∥F‖⋅‖F is the Frobenius norm. DMF can make up for the deficiency of the shallow NMF method because its multi-layer decomposition can capture the hierarchical structure of data to improve the performance of low-dimensional data representation and clustering. In order to tackle ...
Matrix Multiplication (Inner Product) Matrix multiplication is not an element-by-element operation like addition or multiplication by a scalar. Instead, it is a more complicated operation in which each element of the product is formed by combining elements of a row of the first operand with corre...
Also talked about the definition of anorm(which can be obtained from an inner product if you have one, but can also be defined by itself), and why a norm is necessary to define a derivative: it is embedded in the definition of what a higher-order term o(δx) means. (Although there...
However, the presence of the extra terms containing the discrete first order derivative operator does not allow for the memory-saving strategy described in Sect. 4.2. Nevertheless, the structure of K_2^{\text {cd}} can be exploited to easily include the boundary conditions in the matrix ...
(x) over x and theLagrange multiplierν₁. The ν₁ partial derivative of L ensures h₁(x)=0, in which case L=f0 and the remaining derivatives extremize f0 along the constraint surface. Noted that ∇L=0 then enforces ∇f₀=0 in the direction parallel to the constraint, ...
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...
That is, the eigenvectors v˜i need to be normalized with respect to the real inner product. Defining V:=V˜D˜ with D˜=diag(v˜1Tv˜1,…,v˜kTv˜k)−1 yields,VTV=D˜V˜TV˜D˜=Ik⇔D˜V˜T=VT=V−1=D˜−1V˜−1, with D˜−1=diag(v...
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...