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7.2 Temporal Analysis on Semantic Graph using Three-Way Tensor Decomposition 155Fig. 7.8. Three-way DEDECOM model [15]7.2.2 AlgorithmsNotationsTo exactly formulate the mathematical problems in the three-way model, the following similarnotations that are discussed Chapter 2 are introduced. Scalars are denoted by lowercase letters,e.g. a. Vectors are denoted by boldface lowercase letter, e.g. a, the ith element of a is denotedby a i . Matrices are denoted by boldface capital letters, e.g. A. The jth column of A is denotedby a j and the entry (i, j) by a ij . The three-way array is denoted by a tensor symbol, e.g. X,the element (i, j,k) of a three-way array X is denoted by x ijk , and the kth frontal slice of X isdenoted by X k [15].The symbol ⊗ denotes the matrix Kroneck product, and the symbol ∗ denotes theHadamard matrix product. The Frobenius norm of a matrix, ‖Y‖ F , is the square root of thesum of squares of its all elements [15].AlgorithmAs discussed before, the proposed algorithm is in fact to find the best approximation of theoriginal data, X k . In this manner, the solution to the three-way DEDICOM model is equivalentto an optimization problem, i.e. finding a best approximation to X. To fit the three-wayDEDICOM model, we need to tackle the following minimization problem,m∥min ∑∥ ∥X k − AD k RD k A T 2∥ (7.8)A,R,D k=1FTo solve Eq.7.8, there are a few algorithms. Here Bader et al. [15] proposed an alternatingoptimization algorithm, which is called ASALSAN (for Alternating Simultaneous Approximation,Least Square, and Newton). The improvement of the algorithm claimed by the authorsis the capability of dealing with large and sparse arrays.The algorithm is composed of one initialization step and three updating steps. The initializationstep starts with randomly selecting A, R and D k = I. Then the algorithm updates A, Rand D in an alternating manner as follows.• Updating A: First obtain all front slices of X by stacking the data side by side, i.e. X k ,k = 1,···,m; then update A with the following formula:A ←[ m∑k=1(X k AD k R T D k + X T k AD kRD k) ] normalized(7.9)