Measuring the galaxy power spectrum and scale-scale correlations with multiresolution-decomposed covariance -- I. method

We present a method of measuring galaxy power spectrum based on the multiresolution analysis of the discrete wavelet transformation (DWT). Since the DWT representation has strong capability of suppressing the off-diagonal components of the covariance for selfsimilar clustering, the DWT covariance for popular models of the cold dark matter cosmogony generally is diagonal, or $j$(scale)-diagonal in the scale range, in which the second scale-scale correlations are weak. In this range, the DWT covariance gives a lossless estimation of the power spectrum, which is equal to the corresponding Fourier power spectrum banded with a logarithmical scaling. In the scale range, in which the scale-scale correlation is significant, the accuracy of a power spectrum detection depends on the scale-scale or band-band correlations. This is, for a precision measurements of the power spectrum, a measurement of the scale-scale or band-band correlations is needed. We show that the DWT covariance can be employed to measuring both the band-power spectrum and second order scale-scale correlation. We also present the DWT algorithm of the binning and Poisson sampling with real observational data. We show that the alias effect appeared in usual binning schemes can exactly be eliminated by the DWT binning. Since Poisson process possesses diagonal covariance in the DWT representation, the Poisson sampling and selection effects on the power spectrum and second order scale-scale correlation detection are suppressed into minimum. Moreover, the effect of the non-Gaussian features of the Poisson sampling can be calculated in this frame.Comment: AAS Latex file, 44 pages, accepted for publication in Ap

Multiresolution analysis in statistical mechanics. II. The wavelet transform as a basis for Monte Carlo simulations on lattices

In this paper, we extend our analysis of lattice systems using the wavelet transform to systems for which exact enumeration is impractical. For such systems, we illustrate a wavelet-accelerated Monte Carlo (WAMC) algorithm, which hierarchically coarse-grains a lattice model by computing the probability distribution for successively larger block spins. We demonstrate that although the method perturbs the system by changing its Hamiltonian and by allowing block spins to take on values not permitted for individual spins, the results obtained agree with the analytical results in the preceding paper, and ``converge'' to exact results obtained in the absence of coarse-graining. Additionally, we show that the decorrelation time for the WAMC is no worse than that of Metropolis Monte Carlo (MMC), and that scaling laws can be constructed from data performed in several short simulations to estimate the results that would be obtained from the original simulation. Although the algorithm is not asymptotically faster than traditional MMC, because of its hierarchical design, the new algorithm executes several orders of magnitude faster than a full simulation of the original problem. Consequently, the new method allows for rapid analysis of a phase diagram, allowing computational time to be focused on regions near phase transitions.Comment: 11 pages plus 7 figures in PNG format (downloadable separately

An Inverse Problem for Localization Operators

A classical result of time-frequency analysis, obtained by I. Daubechies in 1988, states that the eigenfunctions of a time-frequency localization operator with circular localization domain and Gaussian analysis window are the Hermite functions. In this contribution, a converse of Daubechies' theorem is proved. More precisely, it is shown that, for simply connected localization domains, if one of the eigenfunctions of a time-frequency localization operator with Gaussian window is a Hermite function, then its localization domain is a disc. The general problem of obtaining, from some knowledge of its eigenfunctions, information about the symbol of a time-frequency localization operator, is denoted as the inverse problem, and the problem studied by Daubechies as the direct problem of time-frequency analysis. Here, we also solve the corresponding problem for wavelet localization, providing the inverse problem analogue of the direct problem studied by Daubechies and Paul.Comment: 18 pages, 1 figur

One-point Statistics of the Cosmic Density Field in Real and Redshift Spaces with A Multiresolutional Decomposition

In this paper, we develop a method of performing the one-point statistics of a perturbed density field with a multiresolutional decomposition based on the discrete wavelet transform (DWT). We establish the algorithm of the one-point variable and its moments in considering the effects of Poisson sampling and selection function. We also establish the mapping between the DWT one-point statistics in redshift space and real space, i.e. the algorithm for recovering the DWT one-point statistics from the redshift distortion of bulk velocity, velocity dispersion, and selection function. Numerical tests on N-body simulation samples show that this algorithm works well on scales from a few hundreds to a few Mpc/h for four popular cold dark matter models. Taking the advantage that the DWT one-point variable is dependent on both the scale and the shape (configuration) of decomposition modes, one can design estimators of the redshift distortion parameter (beta) from combinations of DWT modes. When the non-linear redshift distortion is not negligible, the beta estimator from quadrupole-to-monopole ratio is a function of scale. This estimator would not work without adding information about the scale-dependence, such as the power-spectrum index or the real-space correlation function of the random field. The DWT beta estimators, however, do not need such extra information. Numerical tests show that the proposed DWT estimators are able to determine beta robustly with less than 15% uncertainty in the redshift range 0 < z < 3.Comment: 39 pages, 12 figures, ApJ accepte

Multiscale 3D Shape Analysis using Spherical Wavelets

©2005 Springer. The original publication is available at www.springerlink.com: http://dx.doi.org/10.1007/11566489_57DOI: 10.1007/11566489_57Shape priors attempt to represent biological variations within a population. When variations are global, Principal Component Analysis (PCA) can be used to learn major modes of variation, even from a limited training set. However, when significant local variations exist, PCA typically cannot represent such variations from a small training set. To address this issue, we present a novel algorithm that learns shape variations from data at multiple scales and locations using spherical wavelets and spectral graph partitioning. Our results show that when the training set is small, our algorithm significantly improves the approximation of shapes in a testing set over PCA, which tends to oversmooth data

Discrepancy between sub-critical and fast rupture roughness: a cumulant analysis

We study the roughness of a crack interface in a sheet of paper. We distinguish between slow (sub-critical) and fast crack growth regimes. We show that the fracture roughness is different in the two regimes using a new method based on a multifractal formalism recently developed in the turbulence literature. Deviations from monofractality also appear to be different in both regimes

Numerical treatment of seismic accelerograms and of inelastic seismic structural responses using harmonic wavelets

The harmonic wavelet transform is employed to analyze various kinds of nonstationary signals common in aseismic design. The effectiveness of the harmonic wavelets for capturing the temporal evolution of the frequency content of strong ground motions is demonstrated. In this regard, a detailed study of important earthquake accelerograms is undertaken and smooth joint time-frequency spectra are provided for two near-field and two far-field records; inherent in this analysis is the concept of the mean instantaneous frequency. Furthermore, as a paradigm of usefulness for aseismic structural purposes, a similar analysis is conducted for the response of a 20-story steel frame benchmark building considering one of the four accelerograms scaled by appropriate factors as the excitation to simulate undamaged and severely damaged conditions for the structure. The resulting joint time-frequency representation of the response time histories captures the influence of nonlinearity on the variation of the effective natural frequencies of a structural system during the evolution of a seismic event. In this context, the potential of the harmonic wavelet transform as a detection tool for global structural damage is explored in conjunction with the concept of monitoring the mean instantaneous frequency of records of critical structural responses

Quantitative Regular Expressions for Arrhythmia Detection Algorithms

Motivated by the problem of verifying the correctness of arrhythmia-detection algorithms, we present a formalization of these algorithms in the language of Quantitative Regular Expressions. QREs are a flexible formal language for specifying complex numerical queries over data streams, with provable runtime and memory consumption guarantees. The medical-device algorithms of interest include peak detection (where a peak in a cardiac signal indicates a heartbeat) and various discriminators, each of which uses a feature of the cardiac signal to distinguish fatal from non-fatal arrhythmias. Expressing these algorithms' desired output in current temporal logics, and implementing them via monitor synthesis, is cumbersome, error-prone, computationally expensive, and sometimes infeasible. In contrast, we show that a range of peak detectors (in both the time and wavelet domains) and various discriminators at the heart of today's arrhythmia-detection devices are easily expressible in QREs. The fact that one formalism (QREs) is used to describe the desired end-to-end operation of an arrhythmia detector opens the way to formal analysis and rigorous testing of these detectors' correctness and performance. Such analysis could alleviate the regulatory burden on device developers when modifying their algorithms. The performance of the peak-detection QREs is demonstrated by running them on real patient data, on which they yield results on par with those provided by a cardiologist.Comment: CMSB 2017: 15th Conference on Computational Methods for Systems Biolog

Deep Regionlets for Object Detection

In this paper, we propose a novel object detection framework named "Deep Regionlets" by establishing a bridge between deep neural networks and conventional detection schema for accurate generic object detection. Motivated by the abilities of regionlets for modeling object deformation and multiple aspect ratios, we incorporate regionlets into an end-to-end trainable deep learning framework. The deep regionlets framework consists of a region selection network and a deep regionlet learning module. Specifically, given a detection bounding box proposal, the region selection network provides guidance on where to select regions to learn the features from. The regionlet learning module focuses on local feature selection and transformation to alleviate local variations. To this end, we first realize non-rectangular region selection within the detection framework to accommodate variations in object appearance. Moreover, we design a "gating network" within the regionlet leaning module to enable soft regionlet selection and pooling. The Deep Regionlets framework is trained end-to-end without additional efforts. We perform ablation studies and conduct extensive experiments on the PASCAL VOC and Microsoft COCO datasets. The proposed framework outperforms state-of-the-art algorithms, such as RetinaNet and Mask R-CNN, even without additional segmentation labels.Comment: Accepted to ECCV 201

On the efficient Monte Carlo implementation of path integrals

We demonstrate that the Levy-Ciesielski implementation of Lie-Trotter products enjoys several properties that make it extremely suitable for path-integral Monte Carlo simulations: fast computation of paths, fast Monte Carlo sampling, and the ability to use different numbers of time slices for the different degrees of freedom, commensurate with the quantum effects. It is demonstrated that a Monte Carlo simulation for which particles or small groups of variables are updated in a sequential fashion has a statistical efficiency that is always comparable to or better than that of an all-particle or all-variable update sampler. The sequential sampler results in significant computational savings if updating a variable costs only a fraction of the cost for updating all variables simultaneously or if the variables are independent. In the Levy-Ciesielski representation, the path variables are grouped in a small number of layers, with the variables from the same layer being statistically independent. The superior performance of the fast sampling algorithm is shown to be a consequence of these observations. Both mathematical arguments and numerical simulations are employed in order to quantify the computational advantages of the sequential sampler, the Levy-Ciesielski implementation of path integrals, and the fast sampling algorithm.Comment: 14 pages, 3 figures; submitted to Phys. Rev.