Pattern Search Multidimensional Scaling

We present a novel view of nonlinear manifold learning using derivative-free optimization techniques. Specifically, we propose an extension of the classical multi-dimensional scaling (MDS) method, where instead of performing gradient descent, we sample and evaluate possible "moves" in a sphere of fixed radius for each point in the embedded space. A fixed-point convergence guarantee can be shown by formulating the proposed algorithm as an instance of General Pattern Search (GPS) framework. Evaluation on both clean and noisy synthetic datasets shows that pattern search MDS can accurately infer the intrinsic geometry of manifolds embedded in high-dimensional spaces. Additionally, experiments on real data, even under noisy conditions, demonstrate that the proposed pattern search MDS yields state-of-the-art results.Comment: 36 pages, Under review for JML

Bootstrapped Coordinate Search for Multidimensional Scaling

In this work, a unified framework for gradient-free Multidimensional Scaling (MDS) based on Coordinate Search (CS) is proposed. This family of algorithms is an instance of General Pattern Search (GPS) methods which avoid the explicit computation of derivatives but instead evaluate the objective function while searching on coordinate steps of the embedding space. The backbone element of CSMDS framework is the corresponding probability matrix that correspond to how likely is each corresponding coordinate to be evaluated. We propose a Bootstrapped instance of CSMDS (BS CSMDS) which enhances the probability of the direction that decreases the most the objective function while also reducing the corresponding probability of all the other coordinates. BS CSMDS manages to avoid unnecessary function evaluations and result to significant speedup over other CSMDS alternatives while also obtaining the same error rate. Experiments on both synthetic and real data reveal that BS CSMDS performs consistently better than other CSMDS alternatives under various experimental setups