Anisotropic adaptive kernel deconvolution

Comte, Fabienne; Lacour, Claire

doi:10.1214/11-aihp470

Cited by 76 publications

(96 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, d. It is easy to check that whatever the value of ⃗ β and ⃗ µ, the corresponding set of parameters belongs to the dense zone. The lower bound for the L 2 -risk in the deconvolution model was recently obtained in Comte and Lacour (2013) but under more restrictive assumption imposed on the density g. It is worth noting that in this case the asymptotics found in Theorem 1 is the minimax rate of convergence, see Comte and Lacour (2013).…”

Section: Resultsmentioning

confidence: 86%

See 1 more Smart Citation

Lower bounds in the convolution structure density model

Lepski¹,

Willer²

2017

Bernoulli

View full text Add to dashboard Cite

Abstract:The aim of the paper is to establish asymptotic lower bounds for the minimax risk in two generalized forms of the density deconvolution problem. The observation consists of an independent and identically distributed (i.i.d.) sample of n random vectors in R d . Their common probability distribution function p can be written aswhere f is the unknown function to be estimated, g is a known function, α is a known proportion, and ⋆ denotes the convolution product. The bounds on the risk are established in a very general minimax setting and for moderately ill posed convolutions. Our results show notably that neither the ill-posedness nor the proportion α play any role in the bounds whenever α ∈ [0, 1), and that a particular inconsistency zone appears for some values of the parameters. Moreover, we introduce an additional boundedness condition on f and we show that the inconsistency zone then disappears.AMS 2000 subject classifications: 62G05, 62G20.

show abstract

Section: Resultsmentioning

confidence: 86%

“…First there are several results on pointwise estimation, see for example Carroll and Hall (1988), Fan (1991), Goldenshluger (1999), Butucea and Tsybakov (2008a), or the already cited paper Comte and Lacour (2013). Secondly, note that other types of ill-posedness can appear.…”

Section: Introductionmentioning

confidence: 99%

Lower bounds in the convolution structure density model

Lepski¹,

Willer²

2017

Bernoulli

View full text Add to dashboard Cite

show abstract

“…The choice of the method for estimating (µ l ) l should be investigated with great care. One possible avenue is to use Lepski's method [20] and its many recent variants and improvements (see for eg., [11]). Next, it is natural to extend the techniques developed here to the multi-dimensional setting as multivariate signals are of high importance in many applications such as DNA sequencing and Mass Spectrometry.…”

Section: Future Directionsmentioning

confidence: 99%

Multi-kernel Unmixing and Super-Resolution Using the Modified Matrix Pencil Method

Chrétien

Tyagi

2020

J Fourier Anal Appl

View full text Add to dashboard Cite

Consider L groups of point sources or spike trains, with the l th group represented by x l (t). For a function g : R → R, let g l (t) = g(t/µ l ) denote a point spread function with scale µ l > 0, and with µ 1 < · · · < µ L . With y(t) = L l=1 (g l x l )(t), our goal is to recover the source parameters given samples of y, or given the Fourier samples of y. This problem is a generalization of the usual super-resolution setup wherein L = 1; we call this the multi-kernel unmixing superresolution problem. Assuming access to Fourier samples of y, we derive an algorithm for this problem for estimating the source parameters of each group, along with precise non-asymptotic guarantees. Our approach involves estimating the group parameters sequentially in the order of increasing scale parameters, i.e., from group 1 to L. In particular, the estimation process at stage 1 ≤ l ≤ L involves (i) carefully sampling the tail of the Fourier transform of y, (ii) a deflation step wherein we subtract the contribution of the groups processed thus far from the obtained Fourier samples, and (iii) applying Moitra's modified Matrix Pencil method on a deconvolved version of the samples in (ii).

show abstract

“…The literature on this problem is large with very strong theoretical results, with a small sample including Carroll and Hall (1988), Stefanski and Carroll (1990), Fan (1991), Masry (1991, Li and Vuong (1998) and Comte et al (2013). In this model, W = X + U, where X and U are independent, have distribution functions F X and F U respectively, and where F X is unknown but, as is often assumed in the deconvolution literature, F U is known: there are also papers where this last assumption is weakened.…”

Section: The Case Of Nonparametric Deconvolutionmentioning

confidence: 99%

Clustering in General Measurement Error Models

Su¹,

Reedy²,

Carroll³

2018

STAT SINICA

View full text Add to dashboard Cite

This paper is dedicated to the memory of Peter G. Hall. It concerns a deceptively simple question: if one observes variables corrupted with measurement error of possibly very complex form, can one recreate asymptotically the clusters that would have been found had there been no measurement error? We show that the answer is yes, and that the solution is surprisingly simple and general. The method itself is to simulate, by computer, realizations with the same distribution as that of the true variables , and then to apply clustering to these realizations. Technically, we show that if one uses K-means clustering or any other risk minimizing clustering, and a multivariate deconvolution device with certain smoothness and convergence properties, then, in the limit, the cluster means based on our method converge to the same cluster means as if there is no measurement error. Along with the method and its technical justification, we analyze two important nutrition data sets, finding patterns that make sense nutritionally.Some Key Words: Clustering; Deconvolution; K-means; Measurement error; Mixtures of distributions. Short title: Clustering and Measurement ErrorStatistica Sinica: Newly accepted Paper (accepted version subject to English editing) Dedication to the memory of Peter G. HallThe last author, Raymond J. Carroll, was very fond of Peter and visited him many times.His facets included his brilliance, dedication, kindness, sense of humor, graciousness to young researchers, puzzle solving, madcap driving to take photos of trains, discussions about airplanes, love of cats and photographic advice. As Peter said in his Statistical Science interview (Delaigle and Wand, 2016), I always like working with Ray, because I felt I could contribute something from the problem solving side, the theoretical side, whereas he is more an applied person, ... in working with Ray we bring to the table things that don't overlap, and which complemented each other very well.In his interview, Peter also mentioned that a lot of his joint work with Raymond grew out of nutrition research, and hence this paper is an appropriate contribution to this special issue. It involves a deceptively simple question: if one observes variables corrupted with measurement error of possibly complex form, such as occurs in nutritional and radiation applications, can one recreate the clusters that would have been found had there been no measurement error? We show that the answer is yes, and that the solution is surprisingly simple and general.

show abstract

Anisotropic adaptive kernel deconvolution

Cited by 76 publications

References 28 publications

Lower bounds in the convolution structure density model

Lower bounds in the convolution structure density model

Multi-kernel Unmixing and Super-Resolution Using the Modified Matrix Pencil Method

Clustering in General Measurement Error Models

Contact Info

Product

Resources

About