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A Fast Approximation to Multidimensional Scaling

We present an approximation algorithm for Multidimensional Scaling (MDS) for use with large datasets and interactive applications. MDS describes a class of dimensionality reduction techniques that takes a dissimilarity matrix as input. It is often used as a tool for understanding relative measurements when absolute measurements are not available. MDS is also used for visualizing high-dimensional datasets. At the core of MDS is an eigendecomposition on an n×n symmetric matrix. For large n, this eigendecomposition becomes unwieldy. Our method employs a divide-and-conquer approach, dividing the matrix into submatrices of reasonable size to perform MDS, and then stitching the subproblem solutions back together for a complete solution for the n × n matrix. It requires Θ(n lg n) steps and is easily parallelized.