A compressed sensing method consists of a rectangular measurement matrix, @math with @math , together with an associated recovery algorithm, @math . Compressed sensing methods aim to construct a high quality approximation to any given input vector @math using only @math as input. In particular, we focus herein on instance optimal nonlinear approximation error bounds for @math and @math of the form @math for @math , where @math is the best possible @math -term approximation to @math . In this paper we develop a compressed sensing method whose associated recovery algorithm, @math , runs in @math -time, matching a lower bound up to a @math factor. This runtime is obtained by using a new class of sparse binary compressed sensing matrices of near optimal size in combination with sublinear-time recovery techniques motivated by sketching algorithms for high-volume data streams. The new class of matrices is constructed by randomly subsampling rows from well-chosen incoherent matrix constructions which already have a sub-linear number of rows. As a consequence, fewer random bits than previously required are needed in order to select the rows utilized by the fast reconstruction algorithms considered herein.
A Euclidean approximate sparse recovery system consists of parameters k,N, an m-by-N measurement matrix, Φ, and a decoding algorithm, D. Given a vector, x, the system approximates x by ^x=D(Φ x), which must satisfy ||x - x||2≤ C ||x - xk||2, where xk denotes the optimal k-term approximation to x. (The output ^x may have more than k terms). For each vector x, the system must succeed with probability at least 3 4. Among the goals in designing such systems are minimizing the number m of measurements and the runtime of the decoding algorithm, D. In this paper, we give a system with m=O(k log(N k)) measurements--matching a lower bound, up to a constant factor--and decoding time k log O(1) N, matching a lower bound up to log(N) factors. We also consider the encode time (i.e., the time to multiply Φ by x), the time to update measurements (i.e., the time to multiply Φ by a 1-sparse x), and the robustness and stability of the algorithm (adding noise before and after the measurements). Our encode and update times are optimal up to log(k) factors. The columns of Φ have at most O(log2(k)log(N k)) non-zeros, each of which can be found in constant time. Our full result, an FPRAS, is as follows. If x=xk+ν1, where ν1 and ν2 (below) are arbitrary vectors (regarded as noise), then, setting ^x = D(Φ x + ν2), and for properly normalized ν, we get [||^x - x||22 ≤ (1+e)||ν1||22 + e||ν2||22,] using O((k e)log(N k)) measurements and (k e)logO(1)(N) time for decoding. In sparse approximation theory, the fundamental problem is to reconstruct a signal A∈ℝn from linear measurements 〈Aψi〉 with respect to a dictionary of ψi's. Recently, there is focus on the novel direction of Compressed Sensing [9] where the reconstruction can be done with very few—O(k logn)—linear measurements over a modified dictionary if the signal is compressible, that is, its information is concentrated in k coefficients with the original dictionary. In particular, these results [9, 4, 23] prove that there exists a single O(k logn) ×n measurement matrix such that any such signal can be reconstructed from these measurements, with error at most O(1) times the worst case error for the class of such signals. Compressed sensing has generated tremendous excitement both because of the sophisticated underlying Mathematics and because of its potential applications In this paper, we address outstanding open problems in Compressed Sensing. Our main result is an explicit construction of a non-adaptive measurement matrix and the corresponding reconstruction algorithm so that with a number of measurements polynomial in k, logn, 1 e, we can reconstruct compressible signals. This is the first known polynomial time explicit construction of any such measurement matrix. In addition, our result improves the error guarantee from O(1) to 1 + e and improves the reconstruction time from poly(n) to poly(k logn) Our second result is a randomized construction of O(kpolylog (n)) measurements that work for each signal with high probability and gives per-instance approximation guarantees rather than over the class of all signals. Previous work on Compressed Sensing does not provide such per-instance approximation guarantees; our result improves the best known number of measurements known from prior work in other areas including Learning Theory [20, 21], Streaming algorithms [11, 12, 6] and Complexity Theory [1] for this case Our approach is combinatorial. In particular, we use two parallel sets of group tests, one to filter and the other to certify and estimate; the resulting algorithms are quite simple to implement 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 e2 to 1 e in factor.  Compressed Sensing is a new paradigm for acquiring the compressible signals that arise in many applications. These signals can be approximated using an amount of information much smaller than the nominal dimension of the signal. Traditional approaches acquire the entire signal and process it to extract the information. The new approach acquires a small number of nonadaptive linear measurements of the signal and uses sophisticated algorithms to determine its information content. Emerging technologies can compute these general linear measurements of a signal at unit cost per measurement. This paper exhibits a randomized measurement ensemble and a signal reconstruction algorithm that satisfy four requirements: 1. The measurement ensemble succeeds for all signals, with high probability over the random choices in its construction. 2. The number of measurements of the signal is optimal, except for a factor polylogarithmic in the signal length. 3. The running time of the algorithm is polynomial in the amount of information in the signal and polylogarithmic in the signal length. 4. The recovery algorithm offers the strongest possible type of error guarantee. Moreover, it is a fully polynomial approximation scheme with respect to this type of error bound. Emerging applications demand this level of performance. Yet no otheralgorithm in the literature simultaneously achieves all four of these desiderata. Most database management systems maintain statistics on the underlying relation. One of the important statistics is that of the "hot items" in the relation: those that appear many times (most frequently, or more than some threshold). For example, end-biased histograms keep the hot items as part of the histogram and are used in selectivity estimation. Hot items are used as simple outliers in data mining, and in anomaly detection in networking applications.We present a new algorithm for dynamically determining the hot items at any time in the relation that is undergoing deletion operations as well as inserts. Our algorithm maintains a small space data structure that monitors the transactions on the relation, and when required, quickly outputs all hot items, without rescanning the relation in the database. With user-specified probability, it is able to report all hot items. Our algorithm relies on the idea of "group testing", is simple to implement, and has provable quality, space and time guarantees. Previously known algorithms for this problem that make similar quality and performance guarantees can not handle deletions, and those that handle deletions can not make similar guarantees without rescanning the database. Our experiments with real and synthetic data shows that our algorithm is remarkably accurate in dynamically tracking the hot items independent of the rate of insertions and deletions.