Non-equispaced B-spline wavelets

This paper has three main contributions. The first is the construction of wavelet transforms from B-spline scaling functions defined on a grid of non-equispaced knots. The new construction extends the equispaced, biorthogonal, compactly supported Cohen-Daubechies-Feauveau wavelets. The new construction is based on the factorisation of wavelet transforms into lifting steps. The second and third contributions are new insights on how to use these and other wavelets in statistical applications. The second contribution is related to the bias of a wavelet representation. It is investigated how the fine scaling coefficients should be derived from the observations. In the context of equispaced data, it is common practice to simply take the observations as fine scale coefficients. It is argued in this paper that this is not acceptable for non-interpolating wavelets on non-equidistant data. Finally, the third contribution is the study of the variance in a non-orthogonal wavelet transform in a new framework, replacing the numerical condition as a measure for non-orthogonality. By controlling the variances of the reconstruction from the wavelet coefficients, the new framework allows us to design wavelet transforms on irregular point sets with a focus on their use for smoothing or other applications in statistics.Comment: 42 pages, 2 figure

Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks

We consider the approximate sparse recovery problem in Wireless Sensor Networks (WSNs) using Compressed Sensing/Compressive Sampling (CS). The goal is to recover the n \mbox{-}dimensional data values by querying only $m \ll n$ sensors based on some linear projection of sensor readings. To solve this problem, a two-tiered sampling model is considered and a novel distributed compressive sparse sampling (DCSS) algorithm is proposed based on sparse binary CS measurement matrix. In the two-tiered sampling model, each sensor first samples the environment independently. Then the fusion center (FC), acting as a pseudo-sensor, samples the sensor network to select a subset of sensors ($m$ out of $n$) that directly respond to the FC for data recovery purpose. The sparse binary matrix is designed using unbalanced expander graph which achieves the state-of-the-art performance for CS schemes. This binary matrix can be interpreted as a sensor selection matrix-whose fairness is analyzed. Extensive experiments on both synthetic and real data set show that by querying only the minimum amount of $m$ sensors using the DCSS algorithm, the CS recovery accuracy can be as good as dense measurement matrices (e.g., Gaussian, Fourier Scrambles). We also show that the sparse binary measurement matrix works well on compressible data which has the closest recovery result to the known best k\mbox{-}term approximation. The recovery is robust against noisy measurements. The sparsity and binary properties of the measurement matrix contribute, to a great extent, the reduction of the in-network communication cost as well as the computational burden

Sketching via hashing: from heavy hitters to compressed sensing to sparse fourier transform

Sketching via hashing is a popular and useful method for processing large data sets. Its basic idea is as follows. Suppose that we have a large multi-set of elements S=[formula], and we would like to identify the elements that occur “frequently" in S. The algorithm starts by selecting a hash function h that maps the elements into an array c[1…m]. The array entries are initialized to 0. Then, for each element a ∈ S, the algorithm increments c[h(a)]. At the end of the process, each array entry c[j] contains the count of all data elements a ∈ S mapped to j

Measurement Bounds for Sparse Signal Ensembles via Graphical Models

In compressive sensing, a small collection of linear projections of a sparse signal contains enough information to permit signal recovery. Distributed compressive sensing (DCS) extends this framework by defining ensemble sparsity models, allowing a correlated ensemble of sparse signals to be jointly recovered from a collection of separately acquired compressive measurements. In this paper, we introduce a framework for modeling sparse signal ensembles that quantifies the intra- and inter-signal dependencies within and among the signals. This framework is based on a novel bipartite graph representation that links the sparse signal coefficients with the measurements obtained for each signal. Using our framework, we provide fundamental bounds on the number of noiseless measurements that each sensor must collect to ensure that the signals are jointly recoverable.Comment: 11 pages, 2 figure