Random Matrix Theory and Its Innovative Applications

Recently more and more disciplines of science and engineering have found Random Matrix Theory valuable. Some disciplines use the limiting densities to indicate the cutoff between "noise" and "signal." Other disciplines are finding eigenvalue repulsions a compelling model of reality. This survey introduces both the theory behind these applications and MATLAB experiments allowing a reader immediate access to the ideas. Keywords: computational science; dynamic blocking problems; elliptic curves; mathematical modeling; random matrix theor

The GSVD: Where are the ellipses?, Matrix Trigonometry, and more

This paper provides an advanced mathematical theory of the Generalized Singular Value Decomposition (GSVD) and its applications. We explore the geometry of the GSVD which provides a long sought for ellipse picture which includes a horizontal and a vertical multiaxis. We further propose that the GSVD provides natural coordinates for the Grassmann manifold. This paper proves a theorem showing how the finite generalized singular values do or do not relate to the singular values of $AB^\dagger$. We then turn to the applications arguing that this geometrical theory is natural for understanding existing applications and recognizing opportunities for new applications. In particular the generalized singular vectors play a direct and as natural a mathematical role for certain applications as the singular vectors do for the SVD. In the same way that experts on the SVD often prefer not to cast SVD problems as eigenproblems, we propose that the GSVD, often cast as a generalized eigenproblem, is rather best cast in its natural setting. We illustrate this theoretical approach and the natural multiaxes (with labels from technical domains) in the context of applications where the GSVD arises: Tikhonov regularization (unregularized vs regularization), Genome Reconstruction (humans vs yeast), Signal Processing (signal vs noise), and stastical analysis such as ANOVA and discriminant analysis (between clusters vs within clusters.) With the aid of our ellipse figure, we encourage in the future the labelling of the natural multiaxes in any GSVD problem.Comment: 28 page

An Optimal Separation Between Two Property Testing Models for Bounded Degree Directed Graphs

We revisit the relation between two fundamental property testing models for bounded-degree directed graphs: the bidirectional model in which the algorithms are allowed to query both the outgoing edges and incoming edges of a vertex, and the unidirectional model in which only queries to the outgoing edges are allowed. Czumaj, Peng and Sohler [STOC 2016] showed that for directed graphs with both maximum indegree and maximum outdegree upper bounded by d, any property that can be tested with query complexity O_{?,d}(1) in the bidirectional model can be tested with n^{1-?_{?,d}(1)} queries in the unidirectional model. In particular, {if the proximity parameter ? approaches 0, then the query complexity of the transformed tester in the unidirectional model approaches n}. It was left open if this transformation can be further improved or there exists any property that exhibits such an extreme separation. We prove that testing subgraph-freeness in which the subgraph contains k source components, requires ?(n^{1-1/k}) queries in the unidirectional model. This directly gives the first explicit properties that exhibit an O_{?,d}(1) vs ?(n^{1-f(?,d)}) separation of the query complexities between the bidirectional model and unidirectional model, where f(?,d) is a function that approaches 0 as ? approaches 0. Furthermore, our lower bound also resolves a conjecture by Hellweg and Sohler [ESA 2012] on the query complexity of testing k-star-freeness