The Expected Norm of a Sum of Independent Random Matrices: An Elementary Approach

Tropp, Joel A.

doi:10.1007/978-3-319-40519-3_8

Cited by 29 publications

(22 citation statements)

References 36 publications

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“…In other words, the remaining part Ψ J of the original matrix plays a negligible role in determining the restricted singular value. We perform this estimate using the Gaussian Minimax Theorem [Gor85], some convex duality arguments [TOH15], and some coarse results from nonasymptotic random matrix theory [Ver12,Tro15c].…”

Section: Theorem Ii: Universality For the Restricted Minimum Singularmentioning

confidence: 99%

Universality laws for randomized dimension reduction, with applications

Oymak

Tropp

2017

Information and Inference: A Journal of the IMA

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Dimension reduction is the process of embedding high-dimensional data into a lower dimensional space to facilitate its analysis. In the Euclidean setting, one fundamental technique for dimension reduction is to apply a random linear map to the data. This dimension reduction procedure succeeds when it preserves certain geometric features of the set. The question is how large the embedding dimension must be to ensure that randomized dimension reduction succeeds with high probability.This paper studies a natural family of randomized dimension reduction maps and a large class of data sets. It proves that there is a phase transition in the success probability of the dimension reduction map as the embedding dimension increases. For a given data set, the location of the phase transition is the same for all maps in this family. Furthermore, each map has the same stability properties, as quantified through the restricted minimum singular value. These results can be viewed as new universality laws in high-dimensional stochastic geometry.Universality laws for randomized dimension reduction have many applications in applied mathematics, signal processing, and statistics. They yield design principles for numerical linear algebra algorithms, for compressed sensing measurement ensembles, and for random linear codes. Furthermore, these results have implications for the performance of statistical estimation methods under a large class of random experimental designs.

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Section: Theorem Ii: Universality For the Restricted Minimum Singularmentioning

confidence: 99%

Universality laws for randomized dimension reduction, with applications

Oymak

Tropp

2017

Information and Inference: A Journal of the IMA

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“…where (41) follows from the fact that Z k 0, see for instance [17,Sec. 2], (42) follows from NT k=1Z k = I s = 1 and (43) follows from (37).…”

Section: Lemma 61: [Matrix Bernstein Inequalitymentioning

confidence: 99%

“…Using this eigenvalue bound in (23) leads to the bounds in (8)- (11). We now consider the second set of bounds given in (14)- (17). We first consider the probability that p k ≥ γĒ 0…”

Section: Lemma 61: [Matrix Bernstein Inequalitymentioning

confidence: 99%

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Performance bounds for remote estimation with an energy harvesting sensor

Özçelikkale

McKelvey

Viberg

2016

2016 IEEE International Symposium on Information Theory (ISIT)

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Remote estimation with an energy harvesting sensor with a limited data buffer is considered. The sensor node observes an unknown correlated circularly wide-sense stationary (c.w.s.s.) Gaussian field and communicates its observations to a remote fusion center using the energy it harvested. The fusion center employs minimum mean-square error (MMSE) estimation to reconstruct the unknown field. The distortion minimization problem under the online scheme, where the sensor has only access to the statistical information for the future energy packets is considered. We provide performance bounds on the achievable distortion under a slotted block transmission scheme, where at each transmission time slot, the data and the energy buffer is completely emptied. Our bounds provide insight to the tradeoff between the buffer sizes and the achievable distortion. These trade-offs illustrate the insensitivity of the performance to the buffer size for signals with low degree of freedom and suggest performance improvements with increasing buffer size for signals with relatively higher degree of freedom.

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“…The proof of concentration inequality (A.33) follows, for example, Theorem 1.6 of Tropp (2011). The proof of inequality (A.34) follows, for example, Theorem 1 of Tropp (2015).…”

Section: A·9 Concentration Inequalitiesmentioning

confidence: 92%

Multisample estimation of bacterial composition matrices in metagenomics data

Cao

Zhang

2019

Biometrika

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Metagenomics sequencing is routinely applied to quantify bacterial abundances in microbiome studies, where the bacterial composition is estimated based on the sequencing read counts. Due to limited sequencing depth and DNA dropouts, many rare bacterial taxa might not be captured in the final sequencing reads, which results in many zero counts. Naive composition estimation using count normalization leads to many zero proportions, which tend to result in inaccurate estimates of bacterial abundance and diversity. This paper takes a multi-sample approach to the estimation of bacterial abundances in order to borrow information across samples and across species. Empirical results from real data sets suggest that the composition matrix over multiple samples is approximately low rank, which motivates a regularized maximum likelihood estimation with a nuclear norm penalty. An efficient optimization algorithm using the generalized accelerated proximal gradient and Euclidean projection onto simplex space is developed. The theoretical upper bounds and the minimax lower bounds of the estimation errors, measured by the Kullback-Leibler divergence and the Frobenius norm, are established. Simulation studies demonstrate that the proposed estimator outperforms the naive estimators. The method is applied to an analysis of a human gut microbiome dataset.

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The Expected Norm of a Sum of Independent Random Matrices: An Elementary Approach

Cited by 29 publications

References 36 publications

Universality laws for randomized dimension reduction, with applications

Universality laws for randomized dimension reduction, with applications

Performance bounds for remote estimation with an energy harvesting sensor

Multisample estimation of bacterial composition matrices in metagenomics data

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