There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

https://minerva-access.unimelb.edu.au/bitstream/handle/11343/197439/1207.0203v1.pdf

Internet of Things (IoT): A vision, architectural elements, and future directions

It is now well understood that (1) it is possible to reconstruct sparse signals exactly from what appear to be highly incomplete sets of linear measurements and (2) that this can be done by constrained l1 minimization. In this paper, we study a novel method for sparse signal recovery that in many situations outperforms l1 minimization in the sense that substantially fewer measurements are needed for exact recovery. The algorithm consists of solving a sequence of weighted l1-minimization problems where the weights used for the next iteration are computed from the value of the current solution. We present a series of experiments demonstrating the remarkable performance and broad applicability of this algorithm in the areas of sparse signal recovery, statistical estimation, error correction and image processing. Interestingly, superior gains are also achieved when our method is applied to recover signals with assumed near-sparsity in overcomplete representations—not by reweighting the l1 norm of the coefficient sequence as is common, but by reweighting the l1 norm of the transformed object. An immediate consequence is the possibility of highly efficient data acquisition protocols by improving on a technique known as Compressive Sensing.

/pdf/enhancing-sparsity-by-reweighted-l-1-minimization-znwoz4koxr.pdf

Enhancing Sparsity by Reweighted ℓ 1 Minimization

Millimeter wave (mmWave) signals experience orders-of-magnitude more pathloss than the microwave signals currently used in most wireless applications and all cellular systems. MmWave systems must therefore leverage large antenna arrays, made possible by the decrease in wavelength, to combat pathloss with beamforming gain. Beamforming with multiple data streams, known as precoding, can be used to further improve mmWave spectral efficiency. Both beamforming and precoding are done digitally at baseband in traditional multi-antenna systems. The high cost and power consumption of mixed-signal devices in mmWave systems, however, make analog processing in the RF domain more attractive. This hardware limitation restricts the feasible set of precoders and combiners that can be applied by practical mmWave transceivers. In this paper, we consider transmit precoding and receiver combining in mmWave systems with large antenna arrays. We exploit the spatial structure of mmWave channels to formulate the precoding/combining problem as a sparse reconstruction problem. Using the principle of basis pursuit, we develop algorithms that accurately approximate optimal unconstrained precoders and combiners such that they can be implemented in low-cost RF hardware. We present numerical results on the performance of the proposed algorithms and show that they allow mmWave systems to approach their unconstrained performance limits, even when transceiver hardware constraints are considered.

Spatially Sparse Precoding in Millimeter Wave MIMO Systems

High-rate data communication over a multipath wireless channel often requires that the channel response be known at the receiver. Training-based methods, which probe the channel in time, frequency, and space with known signals and reconstruct the channel response from the output signals, are most commonly used to accomplish this task. Traditional training-based channel estimation methods, typically comprising linear reconstruction techniques, are known to be optimal for rich multipath channels. However, physical arguments and growing experimental evidence suggest that many wireless channels encountered in practice tend to exhibit a sparse multipath structure that gets pronounced as the signal space dimension gets large (e.g., due to large bandwidth or large number of antennas). In this paper, we formalize the notion of multipath sparsity and present a new approach to estimating sparse (or effectively sparse) multipath channels that is based on some of the recent advances in the theory of compressed sensing. In particular, it is shown in the paper that the proposed approach, which is termed as compressed channel sensing (CCS), can potentially achieve a target reconstruction error using far less energy and, in many instances, latency and bandwidth than that dictated by the traditional least-squares-based training methods.

/pdf/compressed-channel-sensing-a-new-approach-to-estimating-xka10ywc3v.pdf

Compressed Channel Sensing: A New Approach to Estimating Sparse Multipath Channels

This article describes a very different approach to the decentralized compression of networked data. Considering a particularly salient aspect of this struggle that revolves around large-scale distributed sources of data and their storage, transmission, and retrieval. The task of transmitting information from one point to another is a common and well-understood exercise. But the problem of efficiently transmitting or sharing information from and among a vast number of distributed nodes remains a great challenge, primarily because we do not yet have well developed theories and tools for distributed signal processing, communications, and information theory in large-scale networked systems.

/pdf/compressed-sensing-for-networked-data-2exg8trfse.pdf

Compressed Sensing for Networked Data

Compressed sensing (CS) has recently emerged as a powerful signal acquisition paradigm. In essence, CS enables the recovery of high-dimensional sparse signals from relatively few linear observations in the form of projections onto a collection of test vectors. Existing results show that if the entries of the test vectors are independent realizations of certain zero-mean random variables, then with high probability the unknown signals can be recovered by solving a tractable convex optimization. This work extends CS theory to settings where the entries of the test vectors exhibit structured statistical dependencies. It follows that CS can be effectively utilized in linear, time-invariant system identification problems provided the impulse response of the system is (approximately or exactly) sparse. An immediate application is in wireless multipath channel estimation. It is shown here that time-domain probing of a multipath channel with a random binary sequence, along with utilization of CS reconstruction techniques, can provide significant improvements in estimation accuracy compared to traditional least-squares based linear channel estimation strategies. Abstract extensions of the main results are also discussed, where the theory of equitable graph coloring is employed to establish the utility of CS in settings where the test vectors exhibit more general statistical dependencies.

/pdf/toeplitz-compressed-sensing-matrices-with-applications-to-3m3324v0d1.pdf

Toeplitz Compressed Sensing Matrices With Applications to Sparse Channel Estimation

Compressive sampling is an emerging theory that is based on the fact that a relatively small number of random projections of a signal can contain most of its salient information. In this paper, we introduce the concept of compressive wireless sensing for sensor networks in which a fusion center retrieves signal field information from an ensemble of spatially distributed sensor nodes. Energy and bandwidth are scarce resources in sensor networks and the relevant metrics of interest in our context are 1) the latency involved in information retrieval; and 2) the associated power-distortion trade-off. It is generally recognized that given sufficient prior knowledge about the sensed data (e.g., statistical characterization, homogeneity etc.), there exist schemes that have very favorable power-distortion-latency trade-offs. We propose a distributed matched source-channel communication scheme, based in part on recent results in compressive sampling theory, for estimation of sensed data at the fusion center and analyze, as a function of number of sensor nodes, the trade-offs between power, distortion and latency. Compressive wireless sensing is a universal scheme in the sense that it requires no prior knowledge about the sensed data. This universality, however, comes at the cost of optimality (in terms of a less favorable power-distortion-latency trade-off) and we quantify this cost relative to the case when sufficient prior information about the sensed data is assumed.

/pdf/compressive-wireless-sensing-16cfx54z5h.pdf

Compressive wireless sensing

The problem of recovering a sparse signal x Rn from a relatively small number of its observations of the form y = Ax Rk, where A is a known matrix and k « n, has recently received a lot of attention under the rubric of compressed sensing (CS) and has applications in many areas of signal processing such as data cmpression, image processing, dimensionality reduction, etc. Recent work has established that if A is a random matrix with entries drawn independently from certain probability distributions then exact recovery of x from these observations can be guaranteed with high probability. In this paper, we show that Toeplitz-structured matrices with entries drawn independently from the same distributions are also sufficient to recover x from y with high probability, and we compare the performance of such matrices with that of fully independent and identically distributed ones. The use of Toeplitz matrices in CS applications has several potential advantages: (i) they require the generation of only O(n) independent random variables; (ii) multiplication with Toeplitz matrices can be efficiently implemented using fast Fourier transform, resulting in faster acquisition and reconstruction algorithms; and (iii) Toeplitz-structured matrices arise naturally in certain application areas such as system identification.

http://www2.ece.umn.edu/users/jdhaupt/publications/ssp07_toep_cs_matrices.pdf

Waheed U. Bajwa

Papers

Compressed Channel Sensing: A New Approach to Estimating Sparse Multipath Channels

Compressed Sensing for Networked Data

Toeplitz Compressed Sensing Matrices With Applications to Sparse Channel Estimation

Compressive wireless sensing

Toeplitz-Structured Compressed Sensing Matrices