A Robust and Scalable Solution for Interpolative Multidimensional Scaling with Weighting

Abstract

Abstract-Advances in modern bio-sequencing techniques have led to a proliferation of raw genomic data that enables an unprecedented opportunity for data mining. To analyze such large volume and high-dimensional scientific data, many high performance dimension reduction and clustering algorithms have been developed. Among the known algorithms, we use Multidimensional Scaling (MDS) to reduce the dimension of original data and Pairwise Clustering, and to classify the data. We have shown that an interpolative technique can be applied to get better performance on massive data. However, SMACOF MDS approach is only directly applicable to cases where all pairwise distances are used and where weight is one for each term. In this paper, we proposed a robust and scalable MDS and interpolation algorithm using Deterministic Annealing technique, to solve problems with either missing distances or a non-trivial weight function. We compared our method to three state-of-art techniques. By experimenting on three common types of bioinformatics dataset, the results illustrate that the precision of our algorithms are better than other algorithms, and the weighted solutions has a lower computational time cost as well.

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“…But DA-SMACOF assumes all weights are equal to one for all input distance matrices. So we previously added a weighting function to the SMACOF function, called WDA-SMACOF [31]. This uses Conjugate Gradient to avoid the cubic time complexity brought about by weighting and matrix inversion, so that it can converge under O(N 2 ) time.…”

“…WDA-MI-MDS is a robust iterative algorithm that can interpolate out-of-sample points into the target dimension space one by one [31]. For every out-of-sample point, the algorithm finds a majorizing function for equation (8), and by using the estimated value of ‫ݔ‬ ො in the previous iteration, it can guarantee a non-increasing STRESS value for ‫‬Ƹ as the number of iterations increases.…”

“…Note that the original pairwise distance matrix, denoted as ߂ must follow three rules: (1) Symmetric: ߜ ൌ ߜ ; (2) Positivity: ߜ Ͳ ; (3) Zero Diagonal: ߜ ൌ Ͳ. We use WDA-SMACOF [31] for our MDS cluster visualization since it can avoid local optima by using DA. DA [30] is an annealing process that finds the global optima of an optimization process instead of local optima by adding a computational temperature to the target object function.…”

“…Additionally, it can avoid possible local optima for the STRESS function by using DA. The detailed equations for this algorithm can be found in [31].…”

“…However, that Hadoop has a large overhead while r parallel applications, such as MDS applicatio avoid that extra computational cost, we use Tw is an iterative MapReduce framework for p WDA-SMACOF. The detailed parallelization [8] and [31]. Finally, since this entire workfl JAVA, it is easy to migrate it to either an HP Cloud environment.…”