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

Forward Attention in Sequence-to-sequence Acoustic Modelling for Speech Synthesis

This paper proposes a forward attention method for the sequenceto- sequence acoustic modeling of speech synthesis. This method is motivated by the nature of the monotonic alignment from phone sequences to acoustic sequences. Only the alignment paths that satisfy the monotonic condition are taken into consideration at each decoder timestep. The modified attention probabilities at each timestep are computed recursively using a forward algorithm. A transition agent for forward attention is further proposed, which helps the attention mechanism to make decisions whether to move forward or stay at each decoder timestep. Experimental results show that the proposed forward attention method achieves faster convergence speed and higher stability than the baseline attention method. Besides, the method of forward attention with transition agent can also help improve the naturalness of synthetic speech and control the speed of synthetic speech effectively.Comment: 5 pages, 3 figures, 2 tables. Published in IEEE International Conference on Acoustics, Speech and Signal Processing 2018 (ICASSP2018