In this paper we discuss the use of graph deep learning in solving quadratic assignment problems (QAP). The quadratic assignment problem is an NP hard optimization problem. We shall analyze an approach using Graph Convolutional Networks (GCN). We prove that a specially designed GCN produces the optimal solution for a broad class of assignment problems. By appropriate training, the class of problems correctly solved is thus enlarged. Numerical examples compare this method with other simpler methods.
In this paper we study the property of phase retrievability by redundant systems of vectors under perturbations of the frame set. Specifically we show that if a set F of m vectors in the complex Hilbert space of dimension n allows for vector reconstruction from magnitudes of its coefficients, then there is a perturbation bound ρ so that any frame set within ρ from F has the same property. In particular this proves the recent construction in<sup>15</sup> is stable under perturbations. By the same token we reduce the critical cardinality conjectured in<sup>11</sup> to proving a stability result for non phase-retrievable frames.
We derive fast algorithms for doing signal reconstruction without phase. This type of problem is important in
signal processing, especially speech recognition technology, and has relevance for state tomography in quantum
theory. We show that a generic frame gives reconstruction from the absolute value of the frame coefficients in
polynomial time. An improved efficiency of reconstruction is obtained with a family of sparse frames or frames
associated with complex projective 2-designs.
We analyze a fundamental question in Hilbert space frame theory: What is the <i>optimal</i> decomposition of a Parseval frame? We will see that this question impacts several famous unsolved problems in different areas of mathematics. As a step towards the solution of this question, we give a new identity which holds for all Parseval frames.
We will construct new classes of Parseval frames for a Hilbert space which allow signal reconstruction from the absolute value of the frame coefficients. As a consequence, signal reconstruction can be done without using phase or its estimation.
This paper compares wavelet and short time Fourier transform based techniques for single channel speech signal noise reduction. Despite success of wavelet denoising of images, it has not yet been widely used for removal of noise in speech signals. We explored how to extend this technique to speech denoising, and discovered some problems in this endeavor. Experimental comparison with large amount test data has been performed. Our results have shown that although the Fourier domain methods still has the superiority, wavelet based alternatives can be very close, and enormous different configurations can still be tried out for possible better solutions.
In this paper we present topology aspects in non- localization results. A well-known such no-go result in the Balian-Low theorem that states the generator of a Weyl- Heisenberg Riesz basis cannot be well time-frequency localized. More general, the statement applies to multi WH Riesz bases, or super frames as well. These results turn out to be connected to non-triviality of a complex vector bundle. Another class of problem is related to optimality of coherent approximations of stochastic signals. More specific, for a given deficit ((alpha) (beta) >1), find the best Riesz sequence generator optimal to respect to the mean square approximation error. A topological obstruction turns out to be responsible for ill-localization of the optimal generator.
In this paper we intend to present the concept of superframes and its use, primarily, in multiplexing techniques. The signal are supposed band limited and three multiplexing schemes are considered: time division multiple access, frequency division multiple access and frequency hoping multiple access (FHMA). The first two schemes give rise to tight superframes, whereas for FHMA, the associated superframes are more complex. For some such superframes the dual superframe is obtained in closed form. An example of a FHMA scheme is also presented.