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592
FINDING STRUCTURE WITH RANDOMNESS: PROBABILISTIC ALGORITHMS FOR CONSTRUCTING APPROXIMATE MATRIX DECOMPOSITIONS
"... Lowrank matrix approximations, such as the truncated singular value decomposition and the rankrevealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for ..."
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Cited by 253 (6 self)
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for performing lowrank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix
Restricted isometries for partial random circulant matrices
 APPL. COMPUT. HARMON. ANAL
, 2010
"... In the theory of compressed sensing, restricted isometry analysis has become a standard tool for studying how efficiently a measurement matrix acquires information about sparse and compressible signals. Many recovery algorithms are known to succeed when the restricted isometry constants of the sampl ..."
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Cited by 47 (8 self)
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of the sampling matrix are small. Many potential applications of compressed sensing involve a dataacquisition process that proceeds by convolution with a random pulse followed by (nonrandom) subsampling. At present, the theoretical analysis of this measurement technique is lacking. This paper demonstrates
Circulant and Toeplitz Matrices in Compressed Sensing
"... Compressed sensing seeks to recover a sparse vector from a small number of linear and nonadaptive measurements. While most work so far focuses on Gaussian or Bernoulli random measurements we investigate the use of partial random circulant and Toeplitz matrices in connection with recovery by ℓ1mini ..."
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Cited by 54 (10 self)
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minization. In contrast to recent work in this direction we allow the use of an arbitrary subset of rows of a circulant and Toeplitz matrix. Our recovery result predicts that the necessary number of measurements to ensure sparse reconstruction by ℓ1minimization with random partial circulant or Toeplitz
Kalman filtering with partial observation losses
 IEEE Trans. on Autom. Control
, 2004
"... We study the Kalman filtering problem when part or all of the observation measurements are lost in a random fashion. Pioneering work has recently addressed the Kalman filtering problem with intermittent observations, where the observation measurements are either received in full or completely lost. ..."
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Cited by 76 (10 self)
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the Kalman filter updates with partial observation measurements. We show that with these partial measurements the Kalman filter and its error covariance matrix iteration become stochastic, since they now depend on the random packet arrivals of the sensor measurements, which can be lost or delayed when
An Explicit Link between Gaussian Fields and Gaussian Markov random fields: the SPDE approach
 PREPRINTS IN MATHEMATICAL SCIENCES
, 2010
"... Continuously indexed Gaussian fields (GFs) is the most important ingredient in spatial statistical modelling and geostatistics. The specification through the covariance function gives an intuitive interpretation of its properties. On the computational side, GFs are hampered with the bign problem, ..."
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Cited by 115 (17 self)
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, since the cost of factorising dense matrices is cubic in the dimension. Although the computational power today is alltimehigh, this fact seems still to be a computational bottleneck in applications. Along with GFs, there is the class of Gaussian Markov random fields (GMRFs) which are discretely indexed
Estimating the Norms of Random Circulant and Toeplitz Matrices and Their Inverses ∗
"... We estimate the norms of standard Gaussian random Toeplitz and circulant matrices and their inverses, mostly by means of combining some basic techniques of linear algebra. In the case of circulant matrices we obtain sharp probabilistic estimates, which show that these matrices are expected to be ver ..."
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to be very well conditioned. Our probabilistic estimates for the norms of standard Gaussian random Toeplitz matrices are within a factor of 2 from those in the circulant case. We also achieve partial progress in estimating the norms of Toeplitz inverses. Namely we yield reasonable probabilistic upper
Finding structure with randomness: Stochastic algorithms for constructing approximate matrix decompositions
, 2009
"... Lowrank matrix approximations, such as the truncated singular value decomposition and the rankrevealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys recent research which demonstrates that randomization offers a powerful tool for performing l ..."
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Cited by 62 (4 self)
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. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed—either explicitly or implicitly
Randomized Matrix Computations III ∗
"... It is well known that random matrices tend to be well conditioned, and we employ this property to advance some fundamental matrix computations. We prove effectiveness of our novel techniques of randomized preconditioning, estimate the condition numbers of random Toeplitz and circulant matrices, nume ..."
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It is well known that random matrices tend to be well conditioned, and we employ this property to advance some fundamental matrix computations. We prove effectiveness of our novel techniques of randomized preconditioning, estimate the condition numbers of random Toeplitz and circulant matrices
Basic Concepts of Random Matrix Theory
"... Thesis presented in partial fulfilment of the requirements for the degree of ..."
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Thesis presented in partial fulfilment of the requirements for the degree of
Randomized Matrix Computations IV
, 2012
"... Random matrices tend to be well conditioned, and we employ this well known property to advance matrix computations. We prove that our algorithms employing Gaussian random matrices are efficient, but in our tests the algorithms have consistently remained as powerful where we used sparse and structure ..."
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Random matrices tend to be well conditioned, and we employ this well known property to advance matrix computations. We prove that our algorithms employing Gaussian random matrices are efficient, but in our tests the algorithms have consistently remained as powerful where we used sparse
Results 1  10
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592