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Singular Value Decomposition
The transaction item matrix is a matrix for which each row corresponds to a transaction and each column corresponds to an item. The entries of the matrix are zeros and ones. If an item occurs in a transaction, the corresponding row and column entry is one. Otherwise, the row and column entry is zero. Because the transaction item matrix usually contains more values of zero than one, it is called a sparse matrix.
The partial singular value decomposition approximates the transaction item matrix using three matrices: U, S, and V‘. The relationship between these matrices is defined as follows:
Transaction Item Matrix U * S * V‘
Define nTran as the number of transactions (rows) in the transaction item matrix, nItem as the number of items (columns) in the transaction item matrix, and nVec as the specified number of singular vectors. Note that nVec must be less than or equal to min(nTran, nItem). It follows that U is an nTran by nVec matrix that contains the left singular vectors of the transaction item matrix. S is a diagonal matrix of dimension nVec. The diagonal entries in S are the singular values in the transaction item matrix. V‘ is an nVec by nItem matrix. The rows in V‘ (or columns in V) are the right singular vectors.
The transaction item matrix is centered, scaled, and divided by nTran minus 1 before the singular value decomposition is carried out. This analysis is equivalent to a PCA of the correlation matrix of the transaction item matrix. The SVD implementation takes advantage of the sparsity of the transaction item matrix.

Help created on 3/19/2020