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Unsupervised learning makes manifest the underlying structure of data without curated training and specific problem definitions. However, the inference of relationships between data points is frustrated by the `curse of dimensionality in high-dimensions. Inspired by replica theory from statistical mechanics, we consider replicas of the system to tune the dimensionality and take the limit as the number of replicas goes to zero. The result is the intensive embedding, which is not only isometric (preserving local distances) but allows global structure to be more transparently visualized. We develop the Intensive Principal Component Analysis (InPCA) and demonstrate clear improvements in visualizations of the Ising model of magnetic spins, a neural network, and the dark energy cold dark matter ({Lambda}CDM) model as applied to the Cosmic Microwave Background.
This chapter describes gene expression analysis by Singular Value Decomposition (SVD), emphasizing initial characterization of the data. We describe SVD methods for visualization of gene expression data, representation of the data using a smaller num
Performance of nuclear threat detection systems based on gamma-ray spectrometry often strongly depends on the ability to identify the part of measured signal that can be attributed to background radiation. We have successfully applied a method based
We show how to efficiently project a vector onto the top principal components of a matrix, without explicitly computing these components. Specifically, we introduce an iterative algorithm that provably computes the projection using few calls to any b
Principal Component Analysis (PCA) is a common multivariate statistical analysis method, and Probabilistic Principal Component Analysis (PPCA) is its probabilistic reformulation under the framework of Gaussian latent variable model. To improve the ro
We consider the problem of inferring the interactions between a set of N binary variables from the knowledge of their frequencies and pairwise correlations. The inference framework is based on the Hopfield model, a special case of the Ising model whe