Nonlinear Dimensionality Reduction for Data Visualization: An Unsupervised Fuzzy Rule-Based Approach

In this article, we propose a general framework for the unsupervised fuzzy rule-based dimensionality reduction primarily for data visualization. This framework has the following important characteristics relevant to the dimensionality reduction for visualization: preserves neighborhood relationships; effectively handles data on nonlinear manifolds; capable of projecting out-of-sample test points; can reject test points, when it is appropriate; and interpretable to a reasonable extent. We use the first-order Takagi–Sugeno model. Typically, fuzzy rules are either provided by experts or extracted using an input–output training set. Here, neither the output data nor experts are available. This makes the problem challenging. We estimate the rule parameters minimizing a suitable objective function that preserves the interpoint geodesic distances (distances over the manifold) as Euclidean distances on the projected space. In this context, we propose a new variant of the geodesic
$c$
-means clustering algorithm. The proposed method is tested on several synthetic and real-world datasets and compared with the results of six state-of-the-art data visualization methods. The proposed method is the only method that performs equally well on all the datasets tried. Our method is found to be robust to the initial conditions. The predictability of the method is validated by suitable experiments. We also assess the ability of our method to reject test points when it should. The scalability issue of the scheme is also discussed. Due to the general nature of the framework, we can use different objective functions to obtain projections satisfying different goals. To the best of our knowledge, this is the first attempt to manifold learning using unsupervised fuzzy rule-based modeling.