On two Random Models in Data Analysis

Abstract

In this thesis, we study two random models with various applications in data analysis. For our first model, we investigate subspaces spanned by biased random vectors. The underlying random model is motivated by applications in computational biology, where one aims at computing a low-rank matrix factorization involving a binary factor. In a random model with adjustable expected sparsity of the binary factor, we show for a large class of random binary factors that the corresponding factorization problem is uniquely solvable with high probability. In data analysis, such uniqueness results are of particular interest; ambiguous solutions often lack interpretability and do not give an insight into the structure of the underlying data. For proving uniqueness in this random model, small ball probability estimates are a key ingredient. Since to the best of our knowledge, there are no such estimate suitable for our application, we prove an extension of the famous Lemma of Littlewood and Offord. Hereby, we also discover a connection between the matrix factorization problem at hand and the notion of Sperner families. In the second part of this thesis, we will investigate a model for randomized ultrasonic data in nondestructive testing. Here, we aim at accelerating the data acquisition process by superposing ultrasonic measurements with random time shifts. To this end, we will first study the effects of randomized ultrasonic measurements in the context of the Synthetic Aperture Focusing Technique (SAFT), a widely used defect imaging method. By adapting SAFT to our random data model, we will significantly improve its performance for randomized data. In this way, for sparse defects and with high probability, we achieve better defect reconstructions as with SAFT applied to deterministic ultrasonic data acquired in the same amount of time.