Universality laws for randomized dimension reduction, with applications

Dimension reduction is the process of embedding high-dimensional data into a lower dimensional space to facilitate its analysis. In the Euclidean setting, one fundamental technique for dimension reduction is to apply a random linear map to the data. This dimension reduction procedure succeeds when it preserves certain geometric features of the set. The question is how large the embedding dimension must be to ensure that randomized dimension reduction succeeds with high probability. This paper studies a natural family of randomized dimension reduction maps and a large class of data sets. It proves that there is a phase transition in the success probability of the dimension reduction map as the embedding dimension increases. For a given data set, the location of the phase transition is the same for all maps in this family. Furthermore, each map has the same stability properties, as quantified through the restricted minimum singular value. These results can be viewed as new universality laws in high-dimensional stochastic geometry. Universality laws for randomized dimension reduction have many applications in applied mathematics, signal processing, and statistics. They yield design principles for numerical linear algebra algorithms, for compressed sensing measurement ensembles, and for random linear codes. Furthermore, these results have implications for the performance of statistical estimation methods under a large class of random experimental designs

Universality laws for randomized dimension reduction, with applications

Dimension reduction is the process of embedding high-dimensional data into a lower dimensional space to facilitate its analysis. In the Euclidean setting, one fundamental technique for dimension reduction is to apply a random linear map to the data. This dimension reduction procedure succeeds when it preserves certain geometric features of the set. The question is how large the embedding dimension must be to ensure that randomized dimension reduction succeeds with high probability. This paper studies a natural family of randomized dimension reduction maps and a large class of data sets. It proves that there is a phase transition in the success probability of the dimension reduction map as the embedding dimension increases. For a given data set, the location of the phase transition is the same for all maps in this family. Furthermore, each map has the same stability properties, as quantified through the restricted minimum singular value. These results can be viewed as new universality laws in high-dimensional stochastic geometry. Universality laws for randomized dimension reduction have many applications in applied mathematics, signal processing, and statistics. They yield design principles for numerical linear algebra algorithms, for compressed sensing measurement ensembles, and for random linear codes. Furthermore, these results have implications for the performance of statistical estimation methods under a large class of random experimental designs

From Review to Rating: Exploring Dependency Measures for Text Classification

Various text analysis techniques exist, which attempt to uncover unstructured information from text. In this work, we explore using statistical dependence measures for textual classification, representing text as word vectors. Student satisfaction scores on a 3-point scale and their free text comments written about university subjects are used as the dataset. We have compared two textual representations: a frequency word representation and term frequency relationship to word vectors, and found that word vectors provide a greater accuracy. However, these word vectors have a large number of features which aggravates the burden of computational complexity. Thus, we explored using a non-linear dependency measure for feature selection by maximizing the dependence between the text reviews and corresponding scores. Our quantitative and qualitative analysis on a student satisfaction dataset shows that our approach achieves comparable accuracy to the full feature vector, while being an order of magnitude faster in testing. These text analysis and feature reduction techniques can be used for other textual data applications such as sentiment analysis.Comment: 8 page