Fixed-point logic with rank (FPR) is an extension of fixed-point logic with counting (FPC) with operators for computing the rank of a matrix over a finite field. The expressive power of FPR properly extends that of FPC and is contained in PTime, but not known to be properly contained.We give a circuit characterization for FPR in terms of families of symmetric circuits with rank gates, along the lines of that for FPC given by [Anderson and Dawar 2017]. This requires the development of a broad framework of circuits in which the individual gates compute functions that are not symmetric (i.e., invariant under all permutations of their inputs). In the case of FPC, the proof of equivalence of circuits and logic rests heavily on the assumption that individual gates compute such symmetric functions and so novel techniques are required to make this work for FPR.
Fixed-point logic with rank (FPR) is an extension of fixed-point logic with counting (FPC) with operators for computing the rank of a matrix over a finit field. The expressive power of FPR properly extends that of FPC and is contained in P, but it is not known if that containment is proper. We give a circuit characterization for FPR in terms of families of symmetric circuits with rank gates, along the lines of that for FPC given by Anderson and Dawar in 2017. This requires the development of a broad framework of circuits in which the individual gates compute functions that are not symmetric (i.e., invariant under all permutations of their inputs). This framework also necessitates the development of novel techniques to prove the equivalence of circuits and logic. Both the framework and the techniques are of greater generality than the main result.
Abstract. Compressive Sensing (CS) allows for the sampling of signals at well below the Nyquist rate but does so, usually, at the cost of the suppression of lower amplitude signal components. Recent work suggests that important information essential for recognizing targets in the radar context is contained in the side-lobes as well, which are often suppressed by CS. In this paper we extend existing techniques and introduce new techniques both for improving the accuracy of CS reconstructions and for improving the separability of scenes reconstructed using CS. We investigate the Discrete Wavelet Transform (DWT), and show how the use of the DWT as a representation basis may improve the accuracy of reconstruction generally. Moreover, we introduce the concept of using multiple wavelet-based reconstructions of a scene, given only a single physical observation, to derive reconstructions that surpass even the best wavelet-based CS reconstructions. Lastly, we specifically consider the effect of the wavelet-based reconstruction on classification. This is done indirectly by comparing outputs of different algorithms using a variety of separability measures. We show that various wavelet-based CS reconstructions are substantially better than conventional CS approaches at inducing (or preserving) separability, and hence may be more useful in classification applications.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.