In this paper we consider the problem of estimating the parameters of a Poisson arrival process where the rate function is assumed to lie in the span of a known basis. Our goal is to estimate the basis expansions coefficients given a realization of this process. We establish novel guarantees concerning the accuracy achieved by the maximum likelihood estimate. Our initial result is near optimal, with the exception of an undesirable dependence on the dynamic range of the rate function. We then show how to remove this dependence through a process of "noise regularization", which results in an improved bound. We conjecture that a similar guarantee should be possible when using a more direct (deterministic) regularization scheme. We conclude with a discussion of practical applications and an empirical examination of the proposed regularization schemes.
In @cite_23 , a variation of the LASSO estimator (similar to that used in @cite_4 ) is proposed that is suitable for solving the Poisson counting inference problem. Performance bounds for constrained penalized maximum-likelihood estimators are provided in @cite_7 @cite_11 @cite_17 . Of course, these bounds apply only to Poisson counting problems -- nevertheless, since our results also imply bounds for the counting model, it is instructive to compare our bounds with the existing work in this domain. Among the prior work which is most relevant is that of @cite_11 , from which we specifically draw inspiration in portions of our analysis. We provide a detailed comparison of our results with those in @cite_11 in .
@cite_11 @cite_4 @cite_7 @cite_23 @cite_17
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We consider the problem of learning a coefficient vector xο ∈ RN from noisy linear observation y = Axo + ∈ Rn. In many contexts (ranging from model selection to image processing), it is desirable to construct a sparse estimator x. In this case, a popular approach consists in solving an l1-penalized least-squares problem known as the LASSO or basis pursuit denoising. For sequences of matrices A of increasing dimensions, with independent Gaussian entries, we prove that the normalized risk of the LASSO converges to a limit, and we obtain an explicit expression for this limit. Our result is the first rigorous derivation of an explicit formula for the asymptotic mean square error of the LASSO for random instances. The proof technique is based on the analysis of AMP, a recently developed efficient algorithm, that is inspired from graphical model ideas. Simulations on real data matrices suggest that our results can be relevant in a broad array of practical applications. We consider the problem of fitting the parameters of a high-dimensional linear regression model. In the regime where the number of parameters p is comparable to or exceeds the sample size n, a successful approach uses an l1-penalized least squares estimator, known as Lasso. Unfortunately, unlike for linear estimators (e.g. ordinary least squares), no well-established method exists to compute confidence intervals or p-values on the basis of the Lasso estimator. Very recently, a line of work [8], [7], [13] has addressed this problem by constructing a debiased version of the Lasso estimator. We propose a special debiasing method that is well suited for random designs with sparse inverse covariance. Our approach improves over the state of the art in that it yields nearly optimal average testing power if sample size n asymptotically dominates s0(logp)2, with s0 being the sparsity level (number of non-zero coefficients). Earlier work achieved similar performances only for much larger sample size, namely it requires n to asymptotically dominates (s0 log p)2. We evaluate our method on synthetic data, and compare it with earlier proposals. Sparse linear inverse problems appear in a variety of settings, but often the noise contaminating observations cannot accurately be described as bounded by or arising from a Gaussian distribution. Poisson observations in particular are a feature of several real-world applications. Previous work on sparse Poisson inverse problems encountered several limiting technical hurdles. This paper describes a novel alternative analysis approach for sparse Poisson inverse problems that (a) sidesteps the technical challenges in previous work, (b) admits estimators that can readily be computed using off-the-shelf LASSO algorithms, and (c) hints at a general framework for broad classes of noise in sparse linear inverse problems. At the heart of this new approach lies a weighted LASSO estimator for which data-dependent weights are based on Poisson concentration inequalities. Unlike previous analyses of the weighted LASSO, the proposed analysis depends on conditions which can be checked or shown to hold in general settings with high probability. We consider the problem of fitting the parameters of a high-dimensional linear regression model. In the regime where the number of parameters @math is comparable to or exceeds the sample size @math , a successful approach uses an @math -penalized least squares estimator, known as Lasso. Unfortunately, unlike for linear estimators (e.g., ordinary least squares), no well-established method exists to compute confidence intervals or p-values on the basis of the Lasso estimator. Very recently, a line of work javanmard2013hypothesis, confidenceJM, GBR-hypothesis has addressed this problem by constructing a debiased version of the Lasso estimator. In this paper, we study this approach for random design model, under the assumption that a good estimator exists for the precision matrix of the design. Our analysis improves over the state of the art in that it establishes nearly optimal testing power if the sample size @math asymptotically dominates @math , with @math being the sparsity level (number of non-zero coefficients). Earlier work obtains provable guarantees only for much larger sample size, namely it requires @math to asymptotically dominate @math . In particular, for random designs with a sparse precision matrix we show that an estimator thereof having the required properties can be computed efficiently. Finally, we evaluate this approach on synthetic data and compare it with earlier proposals.