Here, we propose an unsupervised fuzzy rule-based dimensionality reduction
method primarily for data visualization. It considers the following important
issues relevant to dimensionality reduction-based data visualization: (i)
preservation of neighborhood relationships, (ii) handling data on a non-linear
manifold, (iii) the capability of predicting projections for new test data
points, (iv) interpretability of the system, and (v) the ability to reject test
points if required. For this, we use a first-order Takagi-Sugeno type model. We
generate rule antecedents using clusters in the input data. In this context, we
also propose a new variant of the Geodesic c-means clustering algorithm. We
estimate the rule parameters by minimizing an error function that preserves the
inter-point geodesic distances (distances over the manifold) as Euclidean
distances on the projected space. We apply the proposed method on three
synthetic and three real-world data sets and visually compare the results with
four other standard data visualization methods. The obtained results show that
the proposed method behaves desirably and performs better than or comparable to
the methods compared with. The proposed method is found to be robust to the
initial conditions. The predictability of the proposed method for test points
is validated by experiments. We also assess the ability of our method to reject
output points when it should. Then, we extend this concept to provide a general
framework for learning an unsupervised fuzzy model for data projection with
different objective functions. To the best of our knowledge, this is the first
attempt to manifold learning using unsupervised fuzzy modeling