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Mini-Workshop: Algebraic, Geometric, and Combinatorial Methods in Frame Theory
Get PDFFrames are collections of vectors in a Hilbert space which have reconstruction properties similar to orthonormal bases and applications in areas such as signal and image processing, quantum information theory, quantization, compressed sensing, and phase retrieval. Further desirable properties of frames for robustness in these applications coincide with structures that have appeared independently in other areas of mathematics, such as special matroids, Gel’Fand-Zetlin polytopes, and combinatorial designs. Within the past few years, the desire to understand these structures has led to many new fruitful interactions between frame theory and fields in pure mathematics, such as algebraic and symplectic geometry, discrete geometry, algebraic combinatorics, combinatorial design theory, and algebraic number theory. These connections have led to the solutions of several open problems and are ripe for further exploration. The central goal of our mini-workshop was to attack open problems that were amenable to an interdisciplinary approach combining certain subfields of frame theory, geometry, and combinatorics
Mini-Workshop: Algebraic, Geometric, and Combinatorial Methods in Frame Theory
No full textFrames are collections of vectors in a Hilbert space which have reconstruction properties similar to orthonormal bases and applications in areas such as signal and image processing, quantum information theory, quantization, compressed sensing, and phase retrieval. Further desirable properties of frames for robustness in these applications coincide with structures that have appeared independently in other areas of mathematics, such as special matroids, Gel’Fand-Zetlin polytopes, and combinatorial designs. Within the past few years, the desire to understand these structures has led to many new fruitful interactions between frame theory and fields in pure mathematics, such as algebraic and symplectic geometry, discrete geometry, algebraic combinatorics, combinatorial design theory, and algebraic number theory. These connections have led to the solutions of several open problems and are ripe for further exploration. The central goal of our mini-workshop was to attack open problems that were amenable to an interdisciplinary approach combining certain subfields of frame theory, geometry, and combinatorics
Mini-Workshop: Algebraic, Geometric, and Combinatorial Methods in Frame Theory
No full textFrames are collections of vectors in a Hilbert space which have reconstruction properties similar to orthonormal bases and applications in areas such as signal and image processing, quantum information theory, quantization, compressed sensing, and phase retrieval. Further desirable properties of frames for robustness in these applications coincide with structures that have appeared independently in other areas of mathematics, such as special matroids, Gel’Fand-Zetlin polytopes, and combinatorial designs. Within the past few years, the desire to understand these structures has led to many new fruitful interactions between frame theory and fields in pure mathematics, such as algebraic and symplectic geometry, discrete geometry, algebraic combinatorics, combinatorial design theory, and algebraic number theory. These connections have led to the solutions of several open problems and are ripe for further exploration. The central goal of our mini-workshop was to attack open problems that were amenable to an interdisciplinary approach combining certain subfields of frame theory, geometry, and combinatorics
Phase Retrieval for Finitely-Supported Complex Measures via the Fourier Transform
No full textWe study the recovery of a finitely-supported complex measure from the magnitudes of linear measurements. The distribution is completely determined by the amplitude vector and the support set , where is assumed to be known. We show that by using magnitudes of point evaluations of the Fourier transform of at , along with magnitudes of differences of modulated point evaluations of , we can construct injective maps over the space of all such measures of support length at most . We follow a measurement design by Alexeev et al. \cite{alexeev_bandeira_fickus_mixon_2014} whereby point evaluations of are encoded as vertices of a graph, and edges of this graph correspond to interference measurements. In particular, if \Lambda\Omega\frac{6(1+\sfrac{6}{\ln{(\sfrac{s}{\Lambda\Omega})}})s}{1-\sfrac{2\sqrt{d-1}}{d}} vertices, then a set of magnitude measurements associated with is sufficient for identifying up to an overall unimodular multiplicative constant.
Under some additional assumptions, we provide two recovery algorithms. The first algorithm is based on phase propagation and the Prony method. We show that the reconstruction problem can be reduced to applying linear inverses and finding roots of a polynomial in the case of exact measurements for almost every signal of the above form. In the second algorithm, at the cost of introducing a truncation error, we follow the technique presented by Cand\`es and Fernandez-Granda in \cite{cand'es_fernandez-granda_2013} to show that the solution to a total-variation norm minimization problem defined by the given intensity measurements yields an approximation of . We give explicit error bounds for recovery using this method depending on the number of given samples, and discuss the effect of noise in this approach.Mathematics, Department o
Doppler-only Multistatic Radar
Get PDFIn order to estimate the position and velocity of a target, most multistatic radar systems require multiple independent target measurements, such as angle-of-arrival, time-of-arrival, and Doppler information. Though inexpensive and reliable, Doppler-only systems have not been widely implemented due to the inherent nonlinear problem of determining a target’s position and velocity from their measurements. We solve this problem. In particular, we first establish the lack of observability in the Doppler-only bistatic system, thereby demonstrating the need for multiple transmitters and/or receivers. Next, for a multistatic system with a sufficient number of transmitter-receiver pairs, we invoke classical optimization techniques, such as gradient-descent and Newton’s method, to quickly and reliably find a numerical solution to the system of nonlinear Doppler equations. Finally, we indicate a best design for the transmitter-receiver constellation to be employed in the aforementioned optimization
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Get PDFWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
Stable Phase Retrieval Using Low-Redundancy Frames of Polynomials
No full textIn many applications, measurements of a signal consist of the magnitudes of linear functionals while the phase information of these functionals is unavailable. Examples of these type of measurements occur in optics, quantum mechanics, speech recognition, and x-ray crystallography. The main topic of this thesis is the recovery of the phase information of a signal using a small number of these magnitude measurements. This is called phase retrieval. We provide a choice of 4d − 4 magnitude measurements that uniquely determines any d dimensional signal, up to a unimodular constant. Then we provide a choice of 6d − 3 magnitude measurements that admits a stable polynomial time algorithm to recover the signal under the influence of noise. We also explore the behavior of pathological signals in this algorithm, as well as the mean squared error. Finally, we show that if the signal is known to be s sparse, then we only need a suitable choice of O(s log d/s) such measurements for the stable algorithm to successfully recover the signal.Mathematics, Department o
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