We provide the asymptotic minimax detection boundary for a bump, i.e. an abrupt change, in the mean function of a dependent Gaussian process. This will be characterized in terms of the asymptotic behaviour of the bump length and height as well as the dependency structure of the process. A major finding is that for stationary processes the asymptotic minimax detection boundary is generically determined by the value of is spectral density at zero. Hence, a positive long-run variance makes detection harder whereas a negative long-run variance eases bump detection. Finally, our asymptotic analysis is complemented by non-asymptotic results for the subclass of AR( @math ) processes and confirmed to serve as a good proxy for finite sample scenarios in a simulation study. Our proofs are based on laws of large numbers for non-independent and non-identically distributed arrays of random variables and the asymptotically sharp analysis of the precision matrix of the process.
Model can be seen as prototypical for the more complex situation when several bumps are to be detected. We do not intend to provide novel methodology for this situation in this paper, rather Theorem provides a benchmark for detecting such a bump which then can be used to evaluate the detection power of any method designed for this task. Minimax detection has a long history, see e.g. the seminal series of papers by or the monograph by . More recently, provided a general approach based on convex optimization. In case of independent observations, the problem of detecting a bump has been considered in , and our strategy of proof for the lower bound is adopted from @cite_0 . The additional difficulty we have to treat is due to the dependency structure @math in . We also mention for a model with a simultaneous bump in the variance, and for heterogeneous noise, however still restricted to independent observations.
@cite_0
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We present a novel probabilistic approach for fitting a statistical model to an image. A 3D Morphable Model (3DMM) of faces is interpreted as a generative (Top-Down) Bayesian model. Random Forests are used as noisy detectors (Bottom-Up) for the face and facial landmark positions. The Top-Down and Bottom-Up parts are then combined using a Data-Driven Markov Chain Monte Carlo Method (DDMCMC). As core of the integration, we use the Metropolis-Hastings algorithm which has two main advantages. First, the algorithm can handle unreliable detections and therefore does not need the detectors to take an early and possible wrong hard decision before fitting. Second, it is open for integration of various cues to guide the fitting process. Based on the proposed approach, we implemented a completely automatic, pose and illumination invariant face recognition application. We are able to train and test the building blocks of our application on different databases. The system is evaluated on the Multi-PIE database and reaches state of the art performance. The oldest and most robust technique to search for new particles is to look for bumps' in invariant mass spectra over smoothly falling backgrounds. This is a powerful technique, but only uses one-dimensional information. One can restrict the phase space to enhance a potential signal, but such tagging techniques require a signal hypothesis and training a classifier in simulation and applying it on data. We present a new method for using all of the available information (with machine learning) without using any prior knowledge about potential signals. Given the lack of new physics signals at the Large Hadron Collider (LHC), such model independent approaches are critical for ensuring full coverage to fully exploit the rich datasets from the LHC experiments. In addition to illustrating how the new method works in simple test cases, we demonstrate the power of the extended bump hunt on a realistic all-hadronic resonance search in a channel that would not be covered with existing techniques. We consider the non-parametric estimation of a function that is observed in white noise after convolution with a boxcar, the indicator of an interval ( a; a). In a recent paper Johnstone, Kerkyacharian, Picard and Raimondo (2004) have developed a wavelet deconvolution method (called WaveD) that can be used for " boxcar kernels. For example, WaveD can be tuned to achieve near optimal rates over Besov spaces when a is a Badly Approximable (BA) irrational number. While the set of all BA's contains quadratic irrationals, e.g., a = p 5, it has Lebesgue measure zero. In this paper we derive two tuning scenarios of WaveD that are valid for all" boxcar convolutions (i.e., when a 2 A where A is a full Lebesgue measure set). We propose (i) a tuning inspired from Minimax theory over Besov spaces; (ii) a tuning derived from Maxiset theory providing similar rates as for WaveD in the BA widths setting. Asymptotic theory nds that (i) in the worst case scenario, departures from the BA assumption aect WaveD convergence rates, at most, by log factors; (ii) the Maxiset tuning, which yields smaller thresholds, is superior to the Minimax tuning over a whole range of Besov sub-scales. Our asymptotic results are illustrated in an extensive simulation of boxcar convolution observed in white noise. Part-based models have demonstrated their merit in object detection. However, there is a key issue to be solved on how to integrate the inaccurate scores of part detectors when there are occlusions or large deformations. To handle the imperfectness of part detectors, this paper presents a probabilistic pedestrian detection framework. In this framework, a deformable part-based model is used to obtain the scores of part detectors and the visibilities of parts are modeled as hidden variables. Unlike previous occlusion handling approaches that assume independence among visibility probabilities of parts or manually define rules for the visibility relationship, a discriminative deep model is used in this paper for learning the visibility relationship among overlapping parts at multiple layers. Experimental results on three public datasets (Caltech, ETH and Daimler) and a new CUHK occlusion dataset1 specially designed for the evaluation of occlusion handling approaches show the effectiveness of the proposed approach.