Suppose x is an unknown vector in Ropfm (a digital image or signal); we plan to measure n general linear functionals of x and then reconstruct. If x is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements n can be dramatically smaller than the size m. Thus, certain natural classes of images with m pixels need only n=O(m1/4log5/2(m)) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual m pixel samples. More specifically, suppose x has a sparse representation in some orthonormal basis (e.g., wavelet, Fourier) or tight frame (e.g., curvelet, Gabor)-so the coefficients belong to an lscrp ball for 0<ples1. The N most important coefficients in that expansion allow reconstruction with lscr2 error O(N1/2-1p/). It is possible to design n=O(Nlog(m)) nonadaptive measurements allowing reconstruction with accuracy comparable to that attainable with direct knowledge of the N most important coefficients. Moreover, a good approximation to those N important coefficients is extracted from the n measurements by solving a linear program-Basis Pursuit in signal processing. The nonadaptive measurements have the character of "random" linear combinations of basis/frame elements. Our results use the notions of optimal recovery, of n-widths, and information-based complexity. We estimate the Gel'fand n-widths of lscrp balls in high-dimensional Euclidean space in the case 0<ples1, and give a criterion identifying near- optimal subspaces for Gel'fand n-widths. We show that "most" subspaces are near-optimal, and show that convex optimization (Basis Pursuit) is a near-optimal way to extract information derived from these near-optimal subspaces

Compressed sensing

This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal f/spl isin/C/sup N/ and a randomly chosen set of frequencies /spl Omega/. Is it possible to reconstruct f from the partial knowledge of its Fourier coefficients on the set /spl Omega/? A typical result of this paper is as follows. Suppose that f is a superposition of |T| spikes f(t)=/spl sigma//sub /spl tau//spl isin/T/f(/spl tau/)/spl delta/(t-/spl tau/) obeying |T|/spl les/C/sub M//spl middot/(log N)/sup -1/ /spl middot/ |/spl Omega/| for some constant C/sub M/>0. We do not know the locations of the spikes nor their amplitudes. Then with probability at least 1-O(N/sup -M/), f can be reconstructed exactly as the solution to the /spl lscr//sub 1/ minimization problem. In short, exact recovery may be obtained by solving a convex optimization problem. We give numerical values for C/sub M/ which depend on the desired probability of success. Our result may be interpreted as a novel kind of nonlinear sampling theorem. In effect, it says that any signal made out of |T| spikes may be recovered by convex programming from almost every set of frequencies of size O(|T|/spl middot/logN). Moreover, this is nearly optimal in the sense that any method succeeding with probability 1-O(N/sup -M/) would in general require a number of frequency samples at least proportional to |T|/spl middot/logN. The methodology extends to a variety of other situations and higher dimensions. For example, we show how one can reconstruct a piecewise constant (one- or two-dimensional) object from incomplete frequency samples - provided that the number of jumps (discontinuities) obeys the condition above - by minimizing other convex functionals such as the total variation of f.

/pdf/robust-uncertainty-principles-exact-signal-reconstruction-32lo4m4643.pdf

Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information

Machine Learning is the study of methods for programming computers to learn. Computers are applied to a wide range of tasks, and for most of these it is relatively easy for programmers to design and implement the necessary software. However, there are many tasks for which this is difficult or impossible. These can be divided into four general categories. First, there are problems for which there exist no human experts. For example, in modern automated manufacturing facilities, there is a need to predict machine failures before they occur by analyzing sensor readings. Because the machines are new, there are no human experts who can be interviewed by a programmer to provide the knowledge necessary to build a computer system. A machine learning system can study recorded data and subsequent machine failures and learn prediction rules. Second, there are problems where human experts exist, but where they are unable to explain their expertise. This is the case in many perceptual tasks, such as speech recognition, hand-writing recognition, and natural language understanding. Virtually all humans exhibit expert-level abilities on these tasks, but none of them can describe the detailed steps that they follow as they perform them. Fortunately, humans can provide machines with examples of the inputs and correct outputs for these tasks, so machine learning algorithms can learn to map the inputs to the outputs. Third, there are problems where phenomena are changing rapidly. In finance, for example, people would like to predict the future behavior of the stock market, of consumer purchases, or of exchange rates. These behaviors change frequently, so that even if a programmer could construct a good predictive computer program, it would need to be rewritten frequently. A learning program can relieve the programmer of this burden by constantly modifying and tuning a set of learned prediction rules. Fourth, there are applications that need to be customized for each computer user separately. Consider, for example, a program to filter unwanted electronic mail messages. Different users will need different filters. It is unreasonable to expect each user to program his or her own rules, and it is infeasible to provide every user with a software engineer to keep the rules up-to-date. A machine learning system can learn which mail messages the user rejects and maintain the filtering rules automatically. Machine learning addresses many of the same research questions as the fields of statistics, data mining, and psychology, but with differences of emphasis. Statistics focuses on understanding the phenomena that have generated the data, often with the goal of testing different hypotheses about those phenomena. Data mining seeks to find patterns in the data that are understandable by people. Psychological studies of human learning aspire to understand the mechanisms underlying the various learning behaviors exhibited by people (concept learning, skill acquisition, strategy change, etc.).

Machine learning

Conventional approaches to sampling signals or images follow Shannon's theorem: the sampling rate must be at least twice the maximum frequency present in the signal (Nyquist rate). In the field of data conversion, standard analog-to-digital converter (ADC) technology implements the usual quantized Shannon representation - the signal is uniformly sampled at or above the Nyquist rate. This article surveys the theory of compressive sampling, also known as compressed sensing or CS, a novel sensing/sampling paradigm that goes against the common wisdom in data acquisition. CS theory asserts that one can recover certain signals and images from far fewer samples or measurements than traditional methods use.

An Introduction To Compressive Sampling

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

https://www.cse.unsw.edu.au/~cs9314/07s1/lectures/Lin_CS9314_References/cm-latin.pdf

An improved data stream summary: the count-min sketch and its applications

We introduce a new sublinear space data structure—the Count-Min Sketch— for summarizing data streams. Our sketch allows fundamental queries in data stream summarization such as point, range, and inner product queries to be approximately answered very quickly; in addition, it can be applied to solve several important problems in data streams such as finding quantiles, frequent items, etc. The time and space bounds we show for using the CM sketch to solve these problems significantly improve those previously known — typically from 1/e 2 to 1/e in factor.

/pdf/an-improved-data-stream-summary-the-count-min-sketch-and-its-3iuw85bqwc.pdf

An improved data stream summary: The count-min sketch and its applications

(MATH) We give an algorithm for finding a Fourier representation R of B terms for a given discrete signal signal A of length N, such that $\|\signal-\repn\|_2^2$ is within the factor (1 +e) of best possible $\|\signal-\repn_\opt\|_2^2$. Our algorithm can access A by reading its values on a sample set T ⊆[0,N), chosen randomly from a (non-product) distribution of our choice, independent of A. That is, we sample non-adaptively. The total time cost of the algorithm is polynomial in B log(N)log(M)e (where M is the ratio of largest to smallest numerical quantity encountered), which implies a similar bound for the number of samples.

/pdf/near-optimal-sparse-fourier-representations-via-sampling-508hnitpdb.pdf

Near-optimal sparse fourier representations via sampling

(MATH) A vector A of length N is defined implicitly, via a stream of updates of the form "add 5 to A3." We give a sketching algorithm, that constructs a small sketch from the stream of updates, and a reconstruction algorithm, that produces a B-bucket piecewise-constant representation (histogram) H for A from the sketch, such that ||A—H||≤(1+e)||A—Hopt||, where the error ||A—H|| is either $\ell_1$ (absolute) or $\ell_2$ (root-mean-square) error. The time to process a single update, time to reconstruct the histogram, and size of the sketch are each bounded by poly(B,log(N),log||A,1/e. Our result is obtained in two steps. First we obtain what we call a robust histogram approximation for A, a histogram such that adding a small number of buckets does not help improve the representation quality significantly. From the robust histogram, we cull a histogram of desired accruacy and B buckets in the second step. This technique also provides similar results for Haar wavelet representations, under $\ell_2$ error. Our results have applications in summarizing data distributions fast and succinctly even in distributed settings.

http://www.math.lsa.umich.edu/~annacg/papers/GGIKMS02.hist.stoc.pdf

Fast, small-space algorithms for approximate histogram maintenance

We are given a collection D of text documents d1,…,dk, with ∑i = n, which may be preprocessed. In the document listing problem, we are given an online query comprising of a pattern string p of length m and our goal is to return the set of all documents that contain one or more copies of p. In the closely related occurrence listing problem, we output the set of all positions within the documents where pattern p occurs. In 1973, Weiner [24] presented an algorithm with O(n) time and space preprocessing following which the occurrence listing problem can be solved in time O(m + output) where output is the number of positions where p occurs; this algorithm is clearly optimal. In contrast, no optimal algorithm is known for the closely related document listing problem, which is perhaps more natural and certainly well-motivated.We provide the first known optimal algorithm for the document listing problem. More generally, we initiate the study of pattern matching problems that require retrieving documents matched by the patterns; this contrasts with pattern matching problems that have been studied more frequently, namely, those that involve retrieving all occurrences of patterns. We consider document retrieval problems that are motivated by online query processing in databases, Information Retrieval systems and Computational Biology. We present very efficient (optimal) algorithms for our document retrieval problems. Our approach for solving such problems involve performing "local" encodings whereby they are reduced to range query problems on geometric objects --- points and lines --- that have color. We present improved algorithms for these colored range query problems that arise in our reductions using the structural properties of strings. This approach is quite general and yields simple, efficient, implementable algorithms for all the document retrieval problems in this paper.

S. Muthukrishnan

Papers

An improved data stream summary: the count-min sketch and its applications

An improved data stream summary: The count-min sketch and its applications

Near-optimal sparse fourier representations via sampling

Fast, small-space algorithms for approximate histogram maintenance

Efficient algorithms for document retrieval problems