Stochastic Proximity Embedding Research Articles

A modified update rule for stochastic proximity embedding

Recently, we described a fast self-organizing algorithm for embedding a set of objects into a low-dimensional Euclidean space in a way that preserves the intrinsic dimensionality and metric structure of the data [Proc. Natl. Acad. Sci. U.S.A. 99 (2002) 15869–15872]. The method, called stochastic proximity embedding (SPE), attempts to preserve the geodesic distances between the embedded objects, and scales linearly with the size of the data set. SPE starts with an initial configuration, and iteratively refines it by repeatedly selecting pairs of objects at random, and adjusting their coordinates so that their distances on the map match more closely their respective proximities. Here, we describe an alternative update rule that drastically reduces the number of calls to the random number generator and thus improves the efficiency of the algorithm.

Stochastic proximity embedding.

We introduce stochastic proximity embedding (SPE), a novel self-organizing algorithm for producing meaningful underlying dimensions from proximity data. SPE attempts to generate low-dimensional Euclidean embeddings that best preserve the similarities between a set of related observations. The method starts with an initial configuration, and iteratively refines it by repeatedly selecting pairs of objects at random, and adjusting their coordinates so that their distances on the map match more closely their respective proximities. The magnitude of these adjustments is controlled by a learning rate parameter, which decreases during the course of the simulation to avoid oscillatory behavior. Unlike classical multidimensional scaling (MDS) and nonlinear mapping (NLM), SPE scales linearly with respect to sample size, and can be applied to very large data sets that are intractable by conventional embedding procedures. The method is programmatically simple, robust, and convergent, and can be applied to a wide range of scientific problems involving exploratory data analysis and visualization.

A geodesic framework for analyzing molecular similarities.

A fast self-organizing algorithm for extracting the minimum number of independent variables that can fully describe a set of observations was recently described (Agrafiotis, D. K.; Xu, H. Proc. Natl. Acad. Sci.U.S.A. 2002, 99, 15869-15872). The method, called stochastic proximity embedding (SPE), attempts to generate low-dimensional Euclidean maps that best preserve the similarities between a set of related objects. Unlike conventional multidimensional scaling (MDS) and nonlinear mapping (NLM), SPE preserves only local relationships and, by doing so, reveals the intrinsic dimensionality and metric structure of the data. Its success depends critically on the choice of the neighborhood radius, which should be consistent with the local curvature of the underlying manifold. Here, we describe a procedure for determining that radius by examining the tradeoff between the stress function and the number of connected components in the neighborhood graph and show that it can be used to produce meaningful maps in any embedding dimension. The power of the algorithm is illustrated in two major areas of computational drug design: conformational analysis and diversity profiling of large chemical libraries.