arXiv subjects

Andrew Thompson

Publications and source records attributed to Andrew Thompson.

36 records · Page 2Linked to original sources

arXiv2019

CHIRRUP: a practical algorithm for unsourced multiple access

Unsourced multiple access abstracts grantless simultaneous communication of a large number of devices (messages) each of which transmits (is transmitted) infrequently. It provides a model for machine-to-machine communication in the Internet of Things (IoT), including the special case of radio-frequency identification (RFID), as well as neighbor discovery in ad hoc wireless networks. This paper presents a fast algorithm for unsourced multiple access that scales to $2^{100}$ devices (arbitrary $100$ bit messages). The primary building block is multiuser detection of binary chirps which are simply codewords in the second order Reed Muller code. The chirp detection algorithm originally presented by Howard et al. is enhanced and integrated into a peeling decoder designed for a patching and slotting framework. In terms of both energy per bit and number of transmitted messages, the proposed algorithm is within a factor of $2$ of state of the art approaches. A significant advantage of our algorithm is its computational efficiency. We prove that the worst-case complexity of the basic chirp reconstruction algorithm is $\mathcal{O}[nK(\log_2 n + K)]$, where $n$ is the codeword length and $K$ is the number of active users, and we report computing times for our algorithm. Our performance and computing time results represent a benchmark against which other practical algorithms can be measured.

eess.SP↗

arXiv2019

Sparse non-negative super-resolution -- simplified and stabilised

The convolution of a discrete measure, $x=\sum_{i=1}^ka_iδ_{t_i}$, with a local window function, $ϕ(s-t)$, is a common model for a measurement device whose resolution is substantially lower than that of the objects being observed. Super-resolution concerns localising the point sources $\{a_i,t_i\}_{i=1}^k$ with an accuracy beyond the essential support of $ϕ(s-t)$, typically from $m$ samples $y(s_j)=\sum_{i=1}^k a_iϕ(s_j-t_i)+η_j$, where $η_j$ indicates an inexactness in the sample value. We consider the setting of $x$ being non-negative and seek to characterise all non-negative measures approximately consistent with the samples. We first show that $x$ is the unique non-negative measure consistent with the samples provided the samples are exact, i.e. $η_j=0$, $m\ge 2k+1$ samples are available, and $ϕ(s-t)$ generates a Chebyshev system. This is independent of how close the sample locations are and {\em does not rely on any regulariser beyond non-negativity}; as such, it extends and clarifies the work by Schiebinger et al. and De Castro et al., who achieve the same results but require a total variation regulariser, which we show is unnecessary. Moreover, we characterise non-negative solutions $\hat{x}$ consistent with the samples within the bound $\sum_{j=1}^mη_j^2\le δ^2$. Any such non-negative measure is within ${\mathcal O}(δ^{1/7})$ of the discrete measure $x$ generating the samples in the generalised Wasserstein distance, converging to one another as $δ$ approaches zero. We also show how to make these general results, for windows that form a Chebyshev system, precise for the case of $ϕ(s-t)$ being a Gaussian window. The main innovation of these results is that non-negativity alone is sufficient to localise point sources beyond the essential sensor resolution.

math.OC↗

arXiv2019

Matrix rigidity and the ill-posedness of Robust PCA and matrix completion

Robust Principal Component Analysis (PCA) (Candes et al., 2011) and low-rank matrix completion (Recht et al., 2010) are extensions of PCA to allow for outliers and missing entries respectively. It is well-known that solving these problems requires a low coherence between the low-rank matrix and the canonical basis, since in the extreme cases -- when the low-rank matrix we wish to recover is also sparse -- there is an inherent ambiguity. However, the well-posedness issue in both problems is an even more fundamental one: in some cases, both Robust PCA and matrix completion can fail to have any solutions at due to the set of low-rank plus sparse matrices not being closed, which in turn is equivalent to the notion of the matrix rigidity function not being lower semicontinuous (Kumar et al., 2014). By constructing infinite families of matrices, we derive bounds on the rank and sparsity such that the set of low-rank plus sparse matrices is not closed. We also demonstrate numerically that a wide range of non-convex algorithms for both Robust PCA and matrix completion have diverging components when applied to our constructed matrices. An analogy can be drawn to the case of sets of higher order tensors not being closed under canonical polyadic (CP) tensor rank, rendering the best low-rank tensor approximation unsolvable (De Silva and Lim, 2008) and hence encourage the use of multilinear tensor rank (De Lathauwer, 2000).

math.NA↗

arXiv2019

The dual approach to non-negative super-resolution: impact on primal reconstruction accuracy

We study the problem of super-resolution, where we recover the locations and weights of non-negative point sources from a few samples of their convolution with a Gaussian kernel. It has been recently shown that exact recovery is possible by minimising the total variation norm of the measure. An alternative practical approach is to solve its dual. In this paper, we study the stability of solutions with respect to the solutions to the dual problem. In particular, we establish a relationship between perturbations in the dual variable and the primal variables around the optimiser. This is achieved by applying a quantitative version of the implicit function theorem in a non-trivial way.

math.OC↗

arXiv2019

Sparse Inverse Problems Over Measures: Equivalence of the Conditional Gradient and Exchange Methods

We study an optimization program over nonnegative Borel measures that encourages sparsity in its solution. Efficient solvers for this program are in increasing demand, as it arises when learning from data generated by a `continuum-of-subspaces' model, a recent trend with applications in signal processing, machine learning, and high-dimensional statistics. We prove that the conditional gradient method (CGM) applied to this infinite-dimensional program, as proposed recently in the literature, is equivalent to the exchange method (EM) applied to its Lagrangian dual, which is a semi-infinite program. In doing so, we formally connect such infinite-dimensional programs to the well-established field of semi-infinite programming. On the one hand, the equivalence established in this paper allows us to provide a rate of convergence for EM which is more general than those existing in the literature. On the other hand, this connection and the resulting geometric insights might in the future lead to the design of improved variants of CGM for infinite-dimensional programs, which has been an active research topic. CGM is also known as the Frank-Wolfe algorithm.

math.OC↗

arXiv2018

Extreme asteroids in the Pan-STARRS 1 Survey

Using the first 18 months of the Pan-STARRS 1 survey we have identified 33 candidate high-amplitude objects for follow-up observations and carried out observations of 22 asteroids. 4 of the observed objects were found to have observed amplitude $A_{obs}\geq 1.0$ mag. We find that these high amplitude objects are most simply explained by single rubble pile objects with some density-dependent internal strength, allowing them to resist mass shedding even at their highly elongated shapes. 3 further objects although below the cut-off for 'high-amplitude' had a combination of elongation and rotation period which also may require internal cohesive strength, depending on the density of the body. We find that none of the 'high-amplitude asteroids' identified here require any unusual cohesive strengths to resist rotational fission. 3 asteroids were sufficiently observed to allow for shape and spin pole models to be determined through light curve inversion. 45864 was determined to have retrograde rotation with spin pole axes $λ=218\pm 10^{\circ}, β=-82\pm 5^{\circ}$ and asteroid 206167 was found to have best fit spin pole axes $λ= 57 \pm 5^{\circ}$, $β=-67 \pm 5^{\circ}$. An additional object not initially measured with $A_{obs}>1.0$ mag, 49257, was determined to have a shape model which does suggest a high-amplitude object. Its spin pole axes were best fit for values $λ=112\pm 6^{\circ}, β=6\pm 5^{\circ}$. In the course of this project to date no large super-fast rotators ($P_{rot} < 2.2$ h) have been identified.

astro-ph.EP↗

arXiv2018

Sketching for Sequential Change-Point Detection

We study sequential change-point detection procedures based on linear sketches of high-dimensional signal vectors using generalized likelihood ratio (GLR) statistics. The GLR statistics allow for an unknown post-change mean that represents an anomaly or novelty. We consider both fixed and time-varying projections, derive theoretical approximations to two fundamental performance metrics: the average run length (ARL) and the expected detection delay (EDD); these approximations are shown to be highly accurate by numerical simulations. We further characterize the relative performance measure of the sketching procedure compared to that without sketching and show that there can be little performance loss when the signal strength is sufficiently large, and enough number of sketches are used. Finally, we demonstrate the good performance of sketching procedures using simulation and real-data examples on solar flare detection and failure detection in power networks.

cs.LG↗

arXiv2018

Compressed Neighbour Discovery using Sparse Kerdock Matrices

We study the network-wide neighbour discovery problem in wireless networks in which each node in a network must discovery the network interface addresses (NIAs) of its neighbours. We work within the rapid on-off division duplex framework proposed by Guo and Zhang (2010) in which all nodes are assigned different on-off signatures which allow them listen to the transmissions of neighbouring nodes during their off slots, leading to a compressed sensing problem at each node with a collapsed codebook determined by a given node's transmission signature. We propose sparse Kerdock matrices as codebooks for the neighbour discovery problem. These matrices share the same row space as certain Delsarte-Goethals frames based upon Reed Muller codes, whilst at the same time being extremely sparse. We present numerical experiments using two different compressed sensing recovery algorithms, One Step Thresholding (OST) and Normalised Iterative Hard Thresholding (NIHT). For both algorithms, a higher proportion of neighbours are successfully identified using sparse Kerdock matrices compared to codebooks based on Reed Muller codes with random erasures as proposed by Zhang and Guo (2011). We argue that the improvement is due to the better interference cancellation properties of sparse Kerdock matrices when collapsed according to a given node's transmission signature. We show by explicit calculation that the coherence of the collapsed codebooks resulting from sparse Kerdock matrices remains near-optimal.

cs.IT↗

arXiv2017

Adaptive DCTNet for Audio Signal Classification

In this paper, we investigate DCTNet for audio signal classification. Its output feature is related to Cohen's class of time-frequency distributions. We introduce the use of adaptive DCTNet (A-DCTNet) for audio signals feature extraction. The A-DCTNet applies the idea of constant-Q transform, with its center frequencies of filterbanks geometrically spaced. The A-DCTNet is adaptive to different acoustic scales, and it can better capture low frequency acoustic information that is sensitive to human audio perception than features such as Mel-frequency spectral coefficients (MFSC). We use features extracted by the A-DCTNet as input for classifiers. Experimental results show that the A-DCTNet and Recurrent Neural Networks (RNN) achieve state-of-the-art performance in bird song classification rate, and improve artist identification accuracy in music data. They demonstrate A-DCTNet's applicability to signal processing problems.

cs.SD↗

arXiv2016

The Cascading Haar Wavelet algorithm for computing the Walsh-Hadamard Transform

We propose a novel algorithm for computing the Walsh-Hadamard Transform (WHT) which consists entirely of Haar wavelet transforms. We prove that the algorithm, which we call the Cascading Haar Wavelet (CHW) algorithm, shares precisely the same serial complexity as the popular divide-and-conquer algorithm for the WHT. We also propose a natural way of parallelizing the algorithm which has a number of attractive features.

cs.DS↗

arXiv2016

Compressive imaging using fast transform coding

We propose deterministic sampling strategies for compressive imaging based on Delsarte-Goethals frames. We show that these sampling strategies result in multi-scale measurements which can be related to the 2D Haar wavelet transform. We demonstrate the effectiveness of our proposed strategies through numerical experiments.

cs.IT↗

arXiv2016

Quantitative recovery conditions for tree-based compressed sensing

As shown in [Blumensath and Davies 2009, Baraniuk et al. 2010], signals whose wavelet coefficients exhibit a rooted tree structure can be recovered using specially-adapted compressed sensing algorithms from just n=O(k) measurements, where k is the sparsity of the signal. Motivated by these results, we introduce a simplified proportional-dimensional asymptotic framework which enables the quantitative evaluation of recovery guarantees for tree-based compressed sensing. In the context of Gaussian matrices, we apply this framework to existing worst-case analysis of the Iterative Tree Projection (ITP) algorithm which makes use of the tree-based Restricted Isometry Property (RIP). Within the same framework, we then obtain quantitative results based on a new method of analysis, recently introduced in [Cartis and Thompson, 2015], which considers the fixed points of the algorithm. By exploiting the realistic average-case assumption that the measurements are statistically independent of the signal, we obtain significant quantitative improvements when compared to the tree-based RIP analysis. Our results have a refreshingly simple interpretation, explicitly determining a bound on the number of measurements that are required as a multiple of the sparsity. For example we prove that exact recovery of binary tree-based signals from noiseless Gaussian measurements is asymptotically guaranteed for ITP with constant stepsize provided n>50k. All our results extend to the more realistic case in which measurements are corrupted by noise.

cs.IT↗

arXiv2016

DCTNet and PCANet for acoustic signal feature extraction

We introduce the use of DCTNet, an efficient approximation and alternative to PCANet, for acoustic signal classification. In PCANet, the eigenfunctions of the local sample covariance matrix (PCA) are used as filterbanks for convolution and feature extraction. When the eigenfunctions are well approximated by the Discrete Cosine Transform (DCT) functions, each layer of of PCANet and DCTNet is essentially a time-frequency representation. We relate DCTNet to spectral feature representation methods, such as the the short time Fourier transform (STFT), spectrogram and linear frequency spectral coefficients (LFSC). Experimental results on whale vocalization data show that DCTNet improves classification rate, demonstrating DCTNet's applicability to signal processing problems such as underwater acoustics.

cs.SD↗

arXiv2015

Data Representation using the Weyl Transform

The Weyl transform is introduced as a rich framework for data representation. Transform coefficients are connected to the Walsh-Hadamard transform of multiscale autocorrelations, and different forms of dyadic periodicity in a signal are shown to appear as different features in its Weyl coefficients. The Weyl transform has a high degree of symmetry with respect to a large group of multiscale transformations, which allows compact yet discriminative representations to be obtained by pooling coefficients. The effectiveness of the Weyl transform is demonstrated through the example of textured image classification.

cs.CV↗

arXiv2014

A new and improved quantitative recovery analysis for iterative hard thresholding algorithms in compressed sensing

We present a new recovery analysis for a standard compressed sensing algorithm, Iterative Hard Thresholding (IHT) (Blumensath and Davies, 2008), which considers the fixed points of the algorithm. In the context of arbitrary measurement matrices, we derive a sufficient condition for convergence of IHT to a fixed point and a necessary condition for the existence of fixed points. These conditions allow us to perform a sparse signal recovery analysis in the deterministic noiseless case by implying that the original sparse signal is the unique fixed point and limit point of IHT, and in the case of Gaussian measurement matrices and noise by generating a bound on the approximation error of the IHT limit as a multiple of the noise level. By generalizing the notion of fixed points, we extend our analysis to the variable stepsize Normalised IHT (N-IHT) (Blumensath and Davies, 2010). For both stepsize schemes, we obtain lower bounds on asymptotic phase transitions in a proportional-dimensional framework, quantifying the sparsity/undersampling trade-off for which recovery is guaranteed. Exploiting the reasonable average-case assumption that the underlying signal and measurement matrix are independent, comparison with previous results within this framework shows a substantial quantitative improvement.

math.NA↗

arXiv2014

On block coherence of frames

Block coherence of matrices plays an important role in analyzing the performance of block compressed sensing recovery algorithms (Bajwa and Mixon, 2012). In this paper, we characterize two block coherence metrics: worst-case and average block coherence. First, we present lower bounds on worst-case block coherence, in both the general case and also when the matrix is constrained to be a union of orthobases. We then present deterministic matrix constructions based upon Kronecker products which obtain these lower bounds. We also characterize the worst-case block coherence of random subspaces. Finally, we present a flipping algorithm that can improve the average block coherence of a matrix, while maintaining the worst-case block coherence of the original matrix. We provide numerical examples which demonstrate that our proposed deterministic matrix construction performs well in block compressed sensing.

cs.IT↗

arXiv2013

An exact tree projection algorithm for wavelets

We propose a dynamic programming algorithm for projection onto wavelet tree structures. In contrast to other recently proposed algorithms which only give approximate tree projections for a given sparsity, our algorithm is guaranteed to calculate the projection exactly. We also prove that our algorithm has O(Nk) complexity, where N is the signal dimension and k is the sparsity of the tree approximation.

math.NA↗

arXiv2010

Phase Transitions for Greedy Sparse Approximation Algorithms

A major enterprise in compressed sensing and sparse approximation is the design and analysis of computationally tractable algorithms for recovering sparse, exact or approximate, solutions of underdetermined linear systems of equations. Many such algorithms have now been proven to have optimal-order uniform recovery guarantees using the ubiquitous Restricted Isometry Property (RIP). However, it is unclear when the RIP-based sufficient conditions on the algorithm are satisfied. We present a framework in which this task can be achieved; translating these conditions for Gaussian measurement matrices into requirements on the signal's sparsity level, length, and number of measurements. We illustrate this approach on three of the state-of-the-art greedy algorithms: CoSaMP, Subspace Pursuit (SP), and Iterative Hard Thresholding (IHT). Designed to allow a direct comparison of existing theory, our framework implies that, according to the best known bounds, IHT requires the fewest number of compressed sensing measurements and has the lowest per iteration computational cost of the three algorithms compared here.

cs.IT↗