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We propose a new tool for visualizing complex, and potentially large and high-dimensional, data sets called Centroid-Encoder (CE). The architecture of the Centroid-Encoder is similar to the autoencoder neural network but it has a modified target, i.e., the class centroid in the ambient space. As such, CE incorporates label information and performs a supervised data visualization. The training of CE is done in the usual way with a training set whose parameters are tuned using a validation set. The evaluation of the resulting CE visualization is performed on a sequestered test set where the generalization of the model is assessed both visually and quantitatively. We present a detailed comparative analysis of the method using a wide variety of data sets and techniques, both supervised and unsupervised, including NCA, non-linear NCA, t-distributed NCA, t-distributed MCML, supervised UMAP, supervised PCA, Colored Maximum Variance Unfolding, supervised Isomap, Parametric Embedding, supervised Neighbor Retrieval Visualizer, and Multiple Relational Embedding. An analysis of variance using PCA demonstrates that a non-linear preprocessing by the CE transformation of the data captures more variance than PCA by dimension. 

We propose a sequential algorithm for learning sparse radial basis approximations for streaming data. The initial phase of the algorithm formulates the RBF training as a convex optimization problem with an objective function on the expansion weights while the data fitting problem imposed only as an ℓ∞-norm constraint. Each new data point observed is tested for feasibility, i.e., whether the data fitting constraint is satisfied. If so, that point is discarded and no model update is required. If it is infeasible, a new basic variable is added to the linear program. The result is a primal infeasible-dual feasible solution. The dual simplex algorithm is applied to determine a new optimal solution. A large fraction of the streaming data points does not require updates to the RBF model since they are similar enough to previously observed data and satisfy the data fitting constraints. The structure of the simplex algorithm makes the update to the solution particularly efficient given the inverse of the new basis matrix is easily computed from the old inverse. The second phase of the algorithm involves a non-convex refinement of the convex problem. Given the sparse nature of the LP solution, the computational expense of the non-convex algorithm is greatly reduced. We have also found that a small subset of the training data that includes the novel data identified by the algorithm can be used to train the non-convex optimization problem with substantial computation savings and comparable errors on the test data. We illustrate the method on the Mackey-Glass chaotic time-series, the monthly sunspot data, and a Fort Collins, Colorado weather data set. In each case we compare the results to artificial neural networks (ANN) and standard skew-RBFs.