Approximate Message Passing with Consistent Parameter Estimation and Applications to Sparse Learning

We consider the estimation of an i.i.d. (possibly non-Gaussian) vector \xbf \in \R^n from measurements \ybf \in \R^m obtained by a general cascade model consisting of a known linear transform followed by a probabilistic componentwise (possibly nonlinear) measurement channel. A novel method, called adaptive generalized approximate message passing (Adaptive GAMP), that enables joint learning of the statistics of the prior and measurement channel along with estimation of the unknown vector \xbf is presented. The proposed algorithm is a generalization of a recently-developed EM-GAMP that uses expectation-maximization (EM) iterations where the posteriors in the E-steps are computed via approximate message passing. The methodology can be applied to a large class of learning problems including the learning of sparse priors in compressed sensing or identification of linear-nonlinear cascade models in dynamical systems and neural spiking processes. We prove that for large i.i.d. Gaussian transform matrices the asymptotic componentwise behavior of the adaptive GAMP algorithm is predicted by a simple set of scalar state evolution equations. In addition, we show that when a certain maximum-likelihood estimation can be performed in each step, the adaptive GAMP method can yield asymptotically consistent parameter estimates, which implies that the algorithm achieves a reconstruction quality equivalent to the oracle algorithm that knows the correct parameter values. Remarkably, this result applies to essentially arbitrary parametrizations of the unknown distributions, including ones that are nonlinear and non-Gaussian. The adaptive GAMP methodology thus provides a systematic, general and computationally efficient method applicable to a large range of complex linear-nonlinear models with provable guarantees.Comment: 14 pages, 3 figure

Accuracy of the LEP Spectrometer Beam Orbit Monitors

At the LEP e+/e- collider, a spectrometer is used to determine the beam energy with a target accuracy of 10-4. The spectrometer measures the lattice dipole bending angle of the beam using six beam position monitors (BPMs). The required calibration error imposes a BPM accuracy of a 10-6 m corresponding to a relative electrical signal variation of 2. 10-5. The operating parameters have been compared with beam simulator results and non-linearBPM response simulations. The relative beam current variations between 0.02 and 0.03 and position changes of 0.1 mm during the fills of last year lead to uncertainties in the orbit measurements of well below 10-6 m. For accuracy tests absolute beam currents were varied by a factor of three. The environment magnetical field is introduced to correct orbit readings. The BPM linearity and calibration was checked using moveable supports and wire position sensors. The BPM triplet quantity is used to determine the orbit position monitors accuracy. The BPM triplet changed during the fills between 1 and 2 10-6 m RMS, which indicates a single BPM orbit determination accuracy between 1 and 1.5 10-6 m

Smoothness-Increasing Accuracy-Conserving (SIAC) filtering and quasi interpolation: A unified view

Filtering plays a crucial role in postprocessing and analyzing data in scientific and engineering applications. Various application-specific filtering schemes have been proposed based on particular design criteria. In this paper, we focus on establishing the theoretical connection between quasi-interpolation and a class of kernels (based on B-splines) that are specifically designed for the postprocessing of the discontinuous Galerkin (DG) method called Smoothness-Increasing Accuracy-Conserving (SIAC) filtering. SIAC filtering, as the name suggests, aims to increase the smoothness of the DG approximation while conserving the inherent accuracy of the DG solution (superconvergence). Superconvergence properties of SIAC filtering has been studied in the literature. In this paper, we present the theoretical results that establish the connection between SIAC filtering to long-standing concepts in approximation theory such as quasi-interpolation and polynomial reproduction. This connection bridges the gap between the two related disciplines and provides a decisive advancement in designing new filters and mathematical analysis of their properties. In particular, we derive a closed formulation for convolution of SIAC kernels with polynomials. We also compare and contrast cardinal spline functions as an example of filters designed for image processing applications with SIAC filters of the same order, and study their properties

Operation of the DC Current Transformer intensity monitors at FNAL during Run II

Circulating beam intensity measurements at FNAL are provided by five DC current transformers (DCCT), one per machine. With the exception of the DCCT in the Recycler, all DCCT systems were designed and built at FNAL. This paper presents an overview of both DCCT systems, including the sensor, the electronics, and the front-end instrumentation software, as well as their performance during Run II

Performance of BPM Electronics for the LEP Spectrometer

At the LEP e+/e- collider at CERN, Geneva, a Spectrometer is used to determine the beam energy with a relative accuracy of 10-4. The Spectrometer measures the change in bending angle in a well-characterised dipole magnet as LEP is ramped. The beam trajectory is obtained using three beam position monitors (BPMs) on each side of the magnet. The error on each BPM measurement should not exceed 1 micron if the desired accuracy on the bending angle is to be reached. The BPMs used consist of an aluminium block with an elliptical aperture and four capacitive button pickup electrodes. The button signals are fed to customised electronics supplied by Bergoz. The electronics use time multiplexing of individual button signals through a single processing chain to optimise for long-term stability. We report on our experience of the performance of these electronics, describing measurements made with test signals and with beam. We have implemented a beam-based calibration procedure and have monitored the reproducibility of the measurements obtained over time. Measurements show that a relative accuracy better than 300 nm is achievable over a period of 1 hr

An Efficient Numerical Method for General

Reconstruction algorithms for fluorescence tomography have to address two crucial issues : 1) the ill-posedness of the reconstruction problem, 2) the large scale of numerical problems arising from imaging of 3-D samples. Our contribution is the design and implementation of a reconstruction algorithm that incorporates general Lp regularization (p ≥ 1). The originality of this work lies in the application of general Lp constraints to fluorescence tomography, combined with an efficient matrix-free strategy that enables the algorithm to deal with large reconstruction problems at reduced memory and computational costs. In the experimental part, we specialize the application of the algorithm to the case of sparsity promoting constraints (L1). We validate the adequacy of L1 regularization for the investigation of phenomena that are well described by a sparse model, using data acquired during phantom experiments

Centered Pyramids

Quadtree-like pyramids have the advantage of resulting in a multiresolution representation where each pyramid node has four unambiguous parents. Such a centered topology guarantees a clearly defined up-projection of labels. This concept has been successfully and extensively used in applications of contour detection, object recognition and segmentation. Unfortunately, the quadtree-like type of pyramid has poor approximation powers because of the employed piecewise-constant image model. This paper deals with the construction of improved centered image pyramids in terms of general approximation functions. The advantages of the centered topology such a symmetry, consistent boundary conditions and accurate up-projection of labels are combined with a more faithful image representation at coarser pyramid levels. We start by introducing a general framework for the design of least squares pyramids using the standard filtering and decimation tools. We give the most general explicit formulas for the computation of the filter coefficients by any (well behaving) approximation function in both the continuous $(L _{ \infty } )$ and the discrete $(l _{ \infty } )$ norm. We then define centered pyramids and provide the filter coefficients for odd spline approximation functions. Finally, we compare the centered pyramid to the ordinary one and highlight some applications

Multiresolution Approximation Using Shifted Splines

We consider the construction of least squares pyramids using shifted polynomial spline basis functions. We derive the pre- and post-filters as a function of the degree n and the shift parameter Δ. We show that the underlying projection operator is entirely specified by two transfer functions acting on the even and odd signal samples, respectively. We introduce a measure of shift-invariance and show that the most favorable configuration is obtained when the knots of the splines are centered with respect to the grid points (i.e., Δ=1/2 when n is odd, and Δ=0 when n is even). The worst case corresponds to the standard multiresolution setting where the spline spaces are nested

Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential

Coupled-mode systems are used in physical literature to simplify the nonlinear Maxwell and Gross-Pitaevskii equations with a small periodic potential and to approximate localized solutions called gap solitons by analytical expressions involving hyperbolic functions. We justify the use of the one-dimensional stationary coupled-mode system for a relevant elliptic problem by employing the method of Lyapunov--Schmidt reductions in Fourier space. In particular, existence of periodic/anti-periodic and decaying solutions is proved and the error terms are controlled in suitable norms. The use of multi-dimensional stationary coupled-mode systems is justified for analysis of bifurcations of periodic/anti-periodic solutions in a small multi-dimensional periodic potential.Comment: 18 pages, no figure