X-code: MDS array codes with optimal encoding

We present a new class of MDS (maximum distance separable) array codes of size n×n (n a prime number) called X-code. The X-codes are of minimum column distance 3, namely, they can correct either one column error or two column erasures. The key novelty in X-code is that it has a simple geometrical construction which achieves encoding/update optimal complexity, i.e., a change of any single information bit affects exactly two parity bits. The key idea in our constructions is that all parity symbols are placed in rows rather than columns

Greedy weights for matroids

We introduce greedy weights of matroids, inspired by those for linear codes. We show that a Wei duality holds for two of these types of greedy weights for matroids. Moreover we show that in the cases where the matroids involved are associated to linear codes, our definitions coincide with those for codes. Thus our Wei duality is a generalization of that for linear codes given by Schaathun. In the last part of the paper we show how some important chains of cycles of the matroids appearing, correspond to chains of component maps of minimal resolutions of the independence complex of the corresponding matroids. We also relate properties of these resolutions to chainedness and greedy weights of the matroids, and in many cases codes, that appear.Comment: 17 page

A mathematical theory of semantic development in deep neural networks

An extensive body of empirical research has revealed remarkable regularities in the acquisition, organization, deployment, and neural representation of human semantic knowledge, thereby raising a fundamental conceptual question: what are the theoretical principles governing the ability of neural networks to acquire, organize, and deploy abstract knowledge by integrating across many individual experiences? We address this question by mathematically analyzing the nonlinear dynamics of learning in deep linear networks. We find exact solutions to this learning dynamics that yield a conceptual explanation for the prevalence of many disparate phenomena in semantic cognition, including the hierarchical differentiation of concepts through rapid developmental transitions, the ubiquity of semantic illusions between such transitions, the emergence of item typicality and category coherence as factors controlling the speed of semantic processing, changing patterns of inductive projection over development, and the conservation of semantic similarity in neural representations across species. Thus, surprisingly, our simple neural model qualitatively recapitulates many diverse regularities underlying semantic development, while providing analytic insight into how the statistical structure of an environment can interact with nonlinear deep learning dynamics to give rise to these regularities

Symmetry reduction in convex optimization with applications in combinatorics

This dissertation explores different approaches to and applications of symmetry reduction in convex optimization. Using tools from semidefinite programming, representation theory and algebraic combinatorics, hard combinatorial problems are solved or bounded. The first chapters consider the Jordan reduction method, extend the method to optimization over the doubly nonnegative cone, and apply it to quadratic assignment problems and energy minimization on a discrete torus. The following chapter uses symmetry reduction as a proving tool, to approach a problem from queuing theory with redundancy scheduling. The final chapters propose generalizations and reductions of flag algebras, a powerful tool for problems coming from extremal combinatorics

Applications of Locality and Asymmetry to Quantum Fault-Tolerance

Quantum computing sounds like something out of a science-fiction novel. If we can exert control over unimaginably small systems, then we can harness their quantum mechanical behavior as a computational resource. This resource allows for astounding computational feats, and a new perspective on information-theory as a whole. But there's a caveat. The events we have to control are so fast and so small that they can hardly be said to have occurred at all. For a long time after Feynman's proposal and even still, there are some who believe that the barriers to controlling such events are fundamental. While we have yet to find anything insurmountable, the road is so pockmarked with challenges both experimental and theoretical that it is often difficult to see the road at all. Only a marriage of both engineering and theory in concert can hope to find the way forward. Quantum error-correction, and more broadly quantum fault-tolerance, is an unfinished answer to this question. It concerns the scaling of these microscopic systems into macroscopic regimes which we can fully control, straddling practical and theoretical considerations in its design. We will explore and prove several results on the theory of quantum fault-tolerance, but which are guided by the ultimate goal of realizing a physical quantum computer. In this thesis, we demonstrate applications of locality and asymmetry to quantum fault-tolerance. We introduce novel code families which we use to probe the behavior of thresholds in quantum subsystem codes. We also demonstrate codes in this family that are well-suited to efficiently correct asymmetric noise models, and determine their parameters. Next we show that quantum error-correcting encodings are incommensurate with transversal implementations of universal classical-reversible computation. Along the way, we resolve an open question concerning almost information-theoretically secure quantum fully homomorphic encryption, showing that it is impossible. Finally, we augment a framework for transversally mapping between stabilizer subspace codes, and discuss prospects for fault-tolerance.PHDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/145948/1/mgnewman_1.pd