A law of large numbers for nearest neighbour statistics

Evans, Dafydd

doi:10.1098/rspa.2008.0235

Cited by 18 publications

(23 citation statements)

References 24 publications

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“…Denoting as the distance from to its th nearest neighbor in , we express the (conditional) mean distance to the th nearest neighbor using (27) as (31) where, as before, the expectation is taken over . Evans et al [29] derived a more general expression for the distance to the th nearest neighbor as (32) where using the approximation (33) one obtains the same relation as in (31). Note that in our case, we consider a fixed search radius of which is chosen independent of the image patches.…”

Section: B Relationship Between Denoising Bounds and Entropymentioning

confidence: 99%

Practical Bounds on Image Denoising: From Estimation to Information

Chatterjee

Milanfar

2011

IEEE Trans. on Image Process.

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Abstract-Recently, in a previous work, we proposed a way to bound how well any given image can be denoised. The bound was computed directly from the noise-free image that was assumed to be available. In this work, we extend the formulation to the more practical case where no ground truth is available. We show that the parameters of the bounds, namely the cluster covariances and level of redundancy for patches in the image, can be estimated directly from the noise corrupted image. Further, we analyze the bounds formulation to show that these two parameters are interdependent and they, along with the bounds formulation as a whole, have a nice information-theoretic interpretation as well. The results are verified through a variety of well-motivated experiments.Index Terms-Bayesian Cramér-Rao lower bound, image clustering, image denoising, image patch model, mutual information, Rényi entropy, Shannon entropy.

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Section: B Relationship Between Denoising Bounds and Entropymentioning

confidence: 99%

Practical Bounds on Image Denoising: From Estimation to Information

Chatterjee

Milanfar

2011

IEEE Trans. on Image Process.

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“…An edge effect means that points located close to the domain borders are problematic because a part of the circle inside which points are supposed to be counted is outside the domain. Result of ignoring this effect is underestimating K. D. Evans [14] studies a class of random variables defined to be functions of a given point and its nearest neighbors. If the sample points are independent and identically distributed, the associated random variables will also be identically distributed but not independent.…”

Section: Introductionmentioning

confidence: 99%

Near neighbor distribution in fractal and finite sets

Jirina

2011

2011 International Conference of Soft Computing and Pattern Recognition (SoCPaR)

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Distances of several nearest neighbors of a given point in a multidimensional space play important role in some tasks of data mining. Here we analyze these distances analyzed as random variables defined to be functions of a given point and its k-th nearest neighbor. We prove that if there is a constant q such that the mean k-th neighbor distance to this constant power is proportional to the near neighbor index k then its distance to this constant power converges to Erlang distribution of order k. We also show that constant q is the scaling exponent known from the theory of multifractals.

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“…Evans et.al. [19] establish an upper bound on the rates of decay of the variance, while the authors of [9,10] provide upper bounds on the ℓ 1 rate of convergence.…”

Section: Introductionmentioning

confidence: 99%

k-nearest neighbor estimation of entropies with confidence

Sricharan

Raich

Hero

2011

2011 IEEE International Symposium on Information Theory Proceedings

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Abstract-We analyze a k-nearest neighbor (k-NN) class of plug-in estimators for estimating Shannon entropy and Rényi entropy. Based on the statistical properties of k-NN balls, we derive explicit rates for the bias and variance of these plug-in estimators in terms of the sample size, the dimension of the samples and the underlying probability distribution. In addition, we establish a central limit theorem for the plug-in estimator that allows us to specify confidence intervals on the entropy functionals. As an application, we use our theory in anomaly detection problems to specify thresholds for achieving desired false alarm rates.

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A law of large numbers for nearest neighbour statistics

Cited by 18 publications

References 24 publications

Practical Bounds on Image Denoising: From Estimation to Information

Practical Bounds on Image Denoising: From Estimation to Information

Near neighbor distribution in fractal and finite sets

k-nearest neighbor estimation of entropies with confidence

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