Unsupervised Quadratic Discriminant Embeddings Using Gaussian Mixture Models

We address in this paper the problem of finding low-dimensional representation spaces for clustered high-dimensional data. The new embedding space proposed here, called the cluster space, is an unsupervised dimension reduction method that relies on the estimation of a Gaussian Mixture Model (GMM) parameters. This allows to capture information not only among data points, but also among clusters in the same embedding space. Points are represented in the cluster space by means of their a posteriori probability values estimated using the GMMs. We show the relationship between the cluster space and the Quadratic Discriminant Analysis (QDA), thus emphasizing the discriminant capability of the representation space proposed. The estimation of the parameters of the GMM in high dimensions is further discussed. Experiments on both artificial and real data illustrate the discriminative power of the cluster space compared with other known state-of-the-art embedding methods.