A Symmetry Preserving Algorithm for Matrix Scaling

Abstract

International audienceWe present an iterative algorithm which asymptotically scales the $\infty$-norm of each row and each column of a matrix to one. This scaling algorithm preserves symmetry of the original matrix and shows fast linear convergence with an asymptotic rate of $1/2$. We discuss extensions of the algorithm to the one-norm, and by inference to other norms. For the 1-norm case, we show again that convergence is linear, with the rate dependent on the spectrum of the scaled matrix. We demonstrate experimentally that the scaling algorithm improves the conditioning of the matrix and that it helps direct solvers by reducing the need for pivoting. In particular, for symmetric matrices the theoretical and experimental results highlight the potential of the proposed algorithm over existing alternatives.Nous décrivons un algorithme itératif qui, asymptotiquement, met une matrice à l'échelle de telle sorte que chaque ligne et chaque colonne est de taille 1 dans la norme infini. Cet algorithme préserve la symétrie. De plus, il converge assez rapidement avec un taux asymptotique de 1/2. Nous discutons la généralisation de l'algorithme à la norme 1 et, par inférence, à d'autres normes. Pour le cas de la norme 1, nous établissons que l'algorithme converge avec un taux linéaire. Nous démontrons expérimentalement que notre algorithme améliore le conditionnement de la matrice et qu'il aide les méthodes directes de résolution en réduisant le pivotage. Particulièrement pour des matrices symétriques, nos résultats théoriques et expérimentaux mettent en valeur l'intérêt de notre algorithme par rapport aux algorithmes existants

Bipartite Matching Heuristics with Quality Guarantees on Shared Memory Parallel Computers

“…Any nonnegative matrix A with total support can be scaled with two (unique) positive diagonal matrices D R and D C such that D R AD C is doubly stochastic (that is, the sum of entries in any row and in any column of D R AD C is equal to one). If A has a support but not a total support then A can be scaled to a doubly stochastic matrix but not with two positive diagonal matrices (see [30] or more recent treatments [22,23,29]). …”

“…Another scaling algorithm is proposed by Ruiz [29], parallelized for distributed [1] and shared memory [7] parallel systems, and its properties are investigated [23]. This algorithm also builds a sequence of matrices converging to a double stochastic matrix.…”

Two approximation algorithms for bipartite matching on multicore architectures

“…Any nonnegative matrix A with total support can be scaled with two (unique) positive diagonal matrices D R and D C such that D R AD C is doubly stochastic (that is, the sum of entries in any row and in any column of D R AD C is equal to one). If A has support but not total support, then A can be scaled to a doubly stochastic matrix but not 170 with two positive diagonal matrices [24]-this fact is also seen in more recent treatments [25,26,27]). …”

“…We use a parallel implementation of the Sinkhorn-Knopp scaling method, shown in Algorithm 1, but other doubly stochastic scaling methods [25,26,27,180 28] can also be used. In Algorithm 1, A i * and A * j are the sets of column and row indices of the nonzeros at the ith row and jth column of A, respectively.…”

“…This can be relaxed in two ways to render the overall approach practical in any bipartite graph. The first relaxation is that we do not need to run the scaling algorithms until convergence (as in some other uses of similar algorithms [26]). If i∈A * j s ij ≥ α instead of i∈A * j s ij = 1 for all j ∈ 530 {1, .…”