Abstract: The Inverse Eigenvalue Problem asks what eigenvalues a matrix with a fixed pattern of nonzero entries can have. For instance, a symmetric tridiagonal matrix must have distinct eigenvalues. We survey known results, current progress, and conjectures about this problem and its generalizations.
Abstract: Given a pair of graphs G1 and G2 and a vertex set of interest in G1, the vertex nomination problem seeks to find the corresponding vertices of interest in G2 (if they exist) and produce a rank list of the vertices in G2, with the corresponding vertices of interest in G2 concentrating, ideally, at the top of the rank list. We study the effect of an adversarial contamination model on the performance of a spectral graph embedding-based vertex nomination scheme. In both real and simulated examples, we demonstrate that this vertex nomination scheme performs effectively in the uncontaminated setting; adversarial network contamination adversely impacts the performance of our VN scheme; and network regularization successfully mitigates the impact of the contamination. In addition to furthering the theoretic basis of consistency in vertex nomination, the adversarial noise model is grounded in theoretical developments that allow us to frame the role of an adversary in terms of maximal vertex nomination consistency classes.
Abstract: Our talk develops a new approach to signal sampling, designed to deal with ultra-wide band (UWB) and adaptive frequency band (AFB) communication systems. These systems require either very high sampling or rapidly changing sampling rates. From a signal processing perspective, we have approached this problem by implementing an appropriate signal decomposition in the analog portion that provides parallel outputs for integrated digital conversion and processing. This naturally leads to an architecture with windowed time segmentation and parallel analog basis expansion. The method first windows the signal and then decomposes the signal into a basis via a continuous-time inner product operation, computing the basis coefficients in parallel. The windowing families are key, and we develop families that have variable partitioning length, variable roll-off and variable smoothness. We then show how these windowing families preserve orthogonality of any orthonormal systems between adjacent blocks, and use these to create bases in which do signal expansions in lapped transforms. We compute error bounds, demonstrating how to decrease error systematically by constructing more sophisticated basis systems. We also develop the method with a modified Gegenbauer system designed specifically for UWB signals.
The overarching goal of the theory developed in this talk is to develop a computable atomic decomposition of time-frequency space. The idea is to come up with a way of non-uniformly tiling time and frequency so that if the signal has a burst of high-frequency information, we tile quickly and efficiently in time and broadly in frequency, whereas if the signal has a relatively low-frequency segment, we can tile broadly in time and efficiently in frequency. Computability is key; systems are designed so that they can be implemented in circuitry.
Abstract: Fingerprints are commonly understood as traits that uniquely identify an individual, an object, or a message, and can be exploited to detect and prevent impersonation, fraud, or unlawful duplication. In this talk we consider the intentional introduction of fingerprints to provide security in wireless communications. This addresses the fingerprint design, and its embedding into a communications waveform, so that it has several desired properties including stealth, security, and predictable performance. The framework draws on communications, signal processing, cryptographic hashing, and information theory, enabling control of performance trade-offs by design. Privacy and security analysis quantify the limited ability of an eavesdropper to detect and estimate the fingerprint or to impersonate a legitimate user. Fingerprints provide a message, and a secret codebook design is described that enables secure side-channel communications through fingerprint coding.
Abstract: Our talk develops connections between some of the most powerful formulae in analysis – the Poisson summation formula, Cauchy’s integral and residue formulae, Jacobi interpola- tion, and the Selberg trace formula – to the Shannon sampling formula. These connections allow us to extend sampling in new directions, e.g., to unions of non-commensurate lattices, to the Riemann Sphere and to the Poincare disk. We close by discussing how to develop sampling for arbitrary Riemann surfaces using Ahlfors covering theory.
Abstract: Fuglede's conjecture, posed in 1974, arose from the work of von Neumann on momentum operators in quantum mechanics. For subsets E of Euclidean space with positive, finite measure, it says E tiles space by translations alone iff the square-integrable functions on E have an orthogonal basis of complex exponentials.
In 2004, Tao disproved the assertion by finding a counterexample to the analogous conjecture in a finite-dimensional vector space over a field with a prime number of elements. Which vector spaces of the type used by Tao lead to counterexamples, and which do not? In the first part of this talk, we give new counterexamples and proofs which yield an almost complete answer.
While Fuglede's conjecture fails, perhaps E having a Riesz basis of complex exponentials is the more natural property? Alas, it is not even known if the unit disk has such a basis. In the second part of this talk, to better understand this question, we examine tightness of exponential bases on subsets of finite abelian groups. Under certain hypotheses, we successfully estimate quantitative measures of optimal tightness in terms of discrete geometry information.