Suppose that x is a discrete-time signal of length N that can be expressed with only m digital fr... more Suppose that x is a discrete-time signal of length N that can be expressed with only m digital frequencies where m ≪ N: x[t] = 1 N m∑
The problem of finding heavy hitters and approximating the frequencies of items is at the heart o... more The problem of finding heavy hitters and approximating the frequencies of items is at the heart of many problems in data stream analysis. It has been observed that several proposed solutions to this problem can outperform their worst-case guarantees on real data. This leads to the question of whether some stronger bounds can be guaranteed. We answer this in the positive by showing that a class of counter-based algorithms (including the popular and very space-efficient Frequent and SpacesSaving algorithms) provides much stronger approximation guarantees than previously known. Specifically, we show that errors in the approximation of individual elements do not depend on the frequencies of the most frequent elements, but only on the frequency of the remaining tail. This shows that counter-based methods are the most space-efficient (in fact, space-optimal) algorithms having this strong error bound. This tail guarantee allows these algorithms to solve the sparse recovery problem. Here, t...
Order statistics, i.e., quantiles, are frequently used in databases both at the database server a... more Order statistics, i.e., quantiles, are frequently used in databases both at the database server as well as the application level. For example, they are useful in selectivity estimation during query optimization, in partitioning large relations, in estimating query result sizes when building user interfaces, and in characterizing the data distribution of evolving datasets in the process of data mining.
Several recent papers have shown how to approximate the difference ∑i |ai −bi | or ∑|ai −bi | 2 b... more Several recent papers have shown how to approximate the difference ∑i |ai −bi | or ∑|ai −bi | 2 between two functions, when the function values ai and bi are given in a data stream, and their order is chosen by an adversary. These algorithms use little space (much less than would be needed to store the entire stream) and little time to process each item in the stream. They approximate with small relative error. Using different techniques, we show how to approximate the L p-difference ∑i |ai − bi | p for any rational-valued p ∈ (0,2], with comparable efficiency and error. We also show how to approximate ∑i |ai − bi | p for larger values of p but with a worse error guarantee. Our results fill in gaps left by recent work, by providing an algorithm that is precisely tunable for the application at hand. These results can be used to assess the difference between two chronologically or physically separated massive data sets, making one quick pass over each data set, without buffering the d...
Suppose that x is a discrete-time signal of length N that can be expressed with only m digital fr... more Suppose that x is a discrete-time signal of length N that can be expressed with only m digital frequencies where m ≪ N: ... We study the problem of identifying the unknown frequencies ω1,...,ωm that partici-pate and their coefficients a1,..., am. Conceptually, the easiest way ...
Proceedings of the Eleventh Annual Acm Siam Symposium on Discrete Algorithms, 2000
We consider the tasks of testing and spot-checking for data streams. These testers and spot-check... more We consider the tasks of testing and spot-checking for data streams. These testers and spot-checkers are potentially useful in real-time or near real-time applications that process huge datasets. Crucial aspects of the computational model include the space complexity of the testers and spot-checkers (ideally much lower than the size of the input stream) and the number of passes that the
Acoustics Speech and Signal Processing 1988 Icassp 88 1988 International Conference on, Mar 1, 2008
In this paper we explore several fundamental bounds for a compressive sampling system which uses ... more In this paper we explore several fundamental bounds for a compressive sampling system which uses the Fourier sampling algorithm of Gilbert et al. Beginning with the theoretical bounds on the number of samples necessary to reconstruct high fidelity approximations of input signals, we refine those theoretical bounds with empirical values in several practical input models. We show that the performance is consistent with traditional sampling systems, and, in certain cases, much better.
Suppose that x is a discrete-time signal of length N that can be expressed with only m digital fr... more Suppose that x is a discrete-time signal of length N that can be expressed with only m digital frequencies where m ≪ N: x[t] = 1 N m∑
The problem of finding heavy hitters and approximating the frequencies of items is at the heart o... more The problem of finding heavy hitters and approximating the frequencies of items is at the heart of many problems in data stream analysis. It has been observed that several proposed solutions to this problem can outperform their worst-case guarantees on real data. This leads to the question of whether some stronger bounds can be guaranteed. We answer this in the positive by showing that a class of counter-based algorithms (including the popular and very space-efficient Frequent and SpacesSaving algorithms) provides much stronger approximation guarantees than previously known. Specifically, we show that errors in the approximation of individual elements do not depend on the frequencies of the most frequent elements, but only on the frequency of the remaining tail. This shows that counter-based methods are the most space-efficient (in fact, space-optimal) algorithms having this strong error bound. This tail guarantee allows these algorithms to solve the sparse recovery problem. Here, t...
Order statistics, i.e., quantiles, are frequently used in databases both at the database server a... more Order statistics, i.e., quantiles, are frequently used in databases both at the database server as well as the application level. For example, they are useful in selectivity estimation during query optimization, in partitioning large relations, in estimating query result sizes when building user interfaces, and in characterizing the data distribution of evolving datasets in the process of data mining.
Several recent papers have shown how to approximate the difference ∑i |ai −bi | or ∑|ai −bi | 2 b... more Several recent papers have shown how to approximate the difference ∑i |ai −bi | or ∑|ai −bi | 2 between two functions, when the function values ai and bi are given in a data stream, and their order is chosen by an adversary. These algorithms use little space (much less than would be needed to store the entire stream) and little time to process each item in the stream. They approximate with small relative error. Using different techniques, we show how to approximate the L p-difference ∑i |ai − bi | p for any rational-valued p ∈ (0,2], with comparable efficiency and error. We also show how to approximate ∑i |ai − bi | p for larger values of p but with a worse error guarantee. Our results fill in gaps left by recent work, by providing an algorithm that is precisely tunable for the application at hand. These results can be used to assess the difference between two chronologically or physically separated massive data sets, making one quick pass over each data set, without buffering the d...
Suppose that x is a discrete-time signal of length N that can be expressed with only m digital fr... more Suppose that x is a discrete-time signal of length N that can be expressed with only m digital frequencies where m ≪ N: ... We study the problem of identifying the unknown frequencies ω1,...,ωm that partici-pate and their coefficients a1,..., am. Conceptually, the easiest way ...
Proceedings of the Eleventh Annual Acm Siam Symposium on Discrete Algorithms, 2000
We consider the tasks of testing and spot-checking for data streams. These testers and spot-check... more We consider the tasks of testing and spot-checking for data streams. These testers and spot-checkers are potentially useful in real-time or near real-time applications that process huge datasets. Crucial aspects of the computational model include the space complexity of the testers and spot-checkers (ideally much lower than the size of the input stream) and the number of passes that the
Acoustics Speech and Signal Processing 1988 Icassp 88 1988 International Conference on, Mar 1, 2008
In this paper we explore several fundamental bounds for a compressive sampling system which uses ... more In this paper we explore several fundamental bounds for a compressive sampling system which uses the Fourier sampling algorithm of Gilbert et al. Beginning with the theoretical bounds on the number of samples necessary to reconstruct high fidelity approximations of input signals, we refine those theoretical bounds with empirical values in several practical input models. We show that the performance is consistent with traditional sampling systems, and, in certain cases, much better.
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