Achievable Angles Between two Compressed Sparse Vectors Under Norm/Distance Constraints Imposed by the Restricted Isometry Property: A Plane Geometry Approach

The angle between two compressed sparse vectors subject to the norm/distance constraints imposed by the restricted isometry property (RIP) of the sensing matrix plays a crucial role in the studies of many compressive sensing (CS) problems. Assuming that (i) u and v are two sparse vectors separated by an angle thetha, and (ii) the sensing matrix Phi satisfies RIP, this paper is aimed at analytically characterizing the achievable angles between Phi*u and Phi*v. Motivated by geometric interpretations of RIP and with the aid of the well-known law of cosines, we propose a plane geometry based formulation for the study of the considered problem. It is shown that all the RIP-induced norm/distance constraints on Phi*u and Phi*v can be jointly depicted via a simple geometric diagram in the two-dimensional plane. This allows for a joint analysis of all the considered algebraic constraints from a geometric perspective. By conducting plane geometry analyses based on the constructed diagram, closed-form formulae for the maximal and minimal achievable angles are derived. Computer simulations confirm that the proposed solution is tighter than an existing algebraic-based estimate derived using the polarization identity. The obtained results are used to derive a tighter restricted isometry constant of structured sensing matrices of a certain kind, to wit, those in the form of a product of an orthogonal projection matrix and a random sensing matrix. Follow-up applications to three CS problems, namely, compressed-domain interference cancellation, RIP-based analysis of the orthogonal matching pursuit algorithm, and the study of democratic nature of random sensing matrices are investigated.Comment: submitted to IEEE Trans. Information Theor

How to interpret a discovery or null result of the 0ν2β0\nu 2\beta decay

The Majorana nature of massive neutrinos will be crucially probed in the next-generation experiments of the neutrinoless double-beta ($0\nu 2\beta$) decay. The effective mass term of this process, $\langle m\rangle^{}_{ee}$, may be contaminated by new physics. So how to interpret a discovery or null result of the $0\nu 2\beta$ decay in the foreseeable future is highly nontrivial. In this paper we introduce a novel three-dimensional description of $|\langle m\rangle_{ee}^{}|$, which allows us to see its sensitivity to the lightest neutrino mass and two Majorana phases in a transparent way. We take a look at to what extent the free parameters of $|\langle m\rangle_{ee}^{}|$ can be well constrained provided a signal of the $0\nu 2\beta$ decay is observed someday. To fully explore lepton number violation, all the six effective Majorana mass terms $\langle m\rangle_{\alpha\beta}^{}$ (for $\alpha, \beta = e, \mu, \tau$) are calculated and their lower bounds are illustrated with the two-dimensional contour figures. The effect of possible new physics on the $0\nu 2\beta$ decay is also discussed in a model-independent way. We find that the result of $|\langle m\rangle_{ee}^{}|$ in the normal (or inverted) neutrino mass ordering case modified by the new physics effect may somewhat mimic that in the inverted (or normal) mass ordering case in the standard three-flavor scheme. Hence a proper interpretation of a discovery or null result of the $0\nu 2\beta$ decay may demand extra information from some other measurements.Comment: 13 pages, 6 figures, Figures and references update