Exponential decay of scattering coefficients

We study an aspect of the following general question: which properties of a signal can be characterized by its scattering transform? We show that the energy contained in high order scattering coefficients is upper bounded by the energy contained in the high frequencies of the signal. This result links the decay of the scattering coefficients of a signal with the decay of its Fourier transform. Additionally, it allows to generalize some results of Mallat (2012), by relaxing the admissibility condition on the wavelet family

Phase Recovery, MaxCut and Complex Semidefinite Programming

Phase retrieval seeks to recover a signal x from the amplitude |Ax| of linear measurements. We cast the phase retrieval problem as a non-convex quadratic program over a complex phase vector and formulate a tractable relaxation (called PhaseCut) similar to the classical MaxCut semidefinite program. We solve this problem using a provably convergent block coordinate descent algorithm whose structure is similar to that of the original greedy algorithm in Gerchberg-Saxton, where each iteration is a matrix vector product. Numerical results show the performance of this approach over three different phase retrieval problems, in comparison with greedy phase retrieval algorithms and matrix completion formulations.Comment: Submitted revisio

Shimura correspondence for level p2p^2 and the central values of LL-series II

Given a Hecke eigenform $f$ of weight $2$ and square-free level $N$, by the work of Kohnen, there is a unique weight $3/2$ modular form of level $4N$ mapping to $f$ under the Shimura correspondence. Furthermore, by the work of Waldspurger the Fourier coefficients of such a form are related to the quadratic twists of the form $f$. Gross gave a construction of the half integral weight form when $N$ is prime, and such construction was later generalized to square-free levels. However, in the non-square free case, the situation is more complicated since the natural construction is vacuous. The problem being that there are too many special points so that there is cancellation while trying to encode the information as a linear combination of theta series. In this paper, we concentrate in the case of level $p^2$, for $p>2$ a prime number, and show how the set of special points can be split into subsets (indexed by bilateral ideals for an order of reduced discriminant $p^2$) which gives two weight $3/2$ modular forms mapping to $f$ under the Shimura correspondence. Moreover, the splitting has a geometric interpretation which allows to prove that the forms are indeed a linear combination of theta series associated to ternary quadratic forms. Once such interpretation is given, we extend the method of Gross-Zagier to the case where the level and the discriminant are not prime to each other to prove a Gross-type formula in this situation