Adaptive sampling theory has shown that, with proper assumptions on the signal class, algorithms exist to reconstruct a signal in @math with an optimal number of samples. We generalize this problem to the case of spatial signals, where the sampling cost is a function of both the number of samples taken and the distance traveled during estimation. This is motivated by our work studying regions of low oxygen concentration in the Great Lakes. We show that for one-dimensional threshold classifiers, a tradeoff between the number of samples taken and distance traveled can be achieved using a generalization of binary search, which we refer to as quantile search. We characterize both the estimation error after a fixed number of samples and the distance traveled in the noiseless case, as well as the estimation error in the case of noisy measurements. We illustrate our results in both simulations and experiments and show that our method outperforms existing algorithms in the majority of practical scenarios.
More sophisticated active learning algorithms have been widely studied, achieving optimal rates for piecewise constant functions in @cite_25 and for the linear support vector machine in @cite_35 . In both cases, the algorithm begins by uniformly sampling the entire feature space. Again considering the problem of interest, the central basin of Lake Erie has an area of approximately 14,000 km @math , making this approach infeasible. In contrast, the algorithm studied in @cite_25 was used in @cite_24 to measure the hydrodynamics of Lake Wingra in Madison, WI, which has an area of 1.3 km @math .
@cite_24 @cite_35 @cite_25
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Abstract Visual tracking algorithms based on structured output support vector machine (SOSVM) have demonstrated excellent performance. However, sampling methods and optimization strategies of SOSVM undesirably increase the computational overloads, which hinder real-time application of these algorithms. Moreover, due to the lack of high-dimensional features and dense training samples, SOSVM-based algorithms are unstable to deal with various challenging scenarios, such as occlusions and scale variations. Recently, visual tracking algorithms based on discriminative correlation filters (DCF), especially the combination of DCF and features from deep convolutional neural networks (CNN), have been successfully applied to visual tracking, and attains surprisingly good performance on recent benchmarks. The success is mainly attributed to two aspects: the circular correlation properties of DCF and the powerful representation capabilities of CNN features. Nevertheless, compared with SOSVM, DCF-based algorithms are restricted to simple ridge regression which has a weaker discriminative ability. In this paper, a novel circular and structural operator tracker (CSOT) is proposed for high performance visual tracking, it not only possesses the powerful discriminative capability of SOSVM but also efficiently inherits the superior computational efficiency of DCF. Based on the proposed circular and structural operators, a set of primal confidence score maps can be obtained by circular correlating feature maps with their corresponding structural correlation filters. Furthermore, an implicit interpolation is applied to convert the multi-resolution feature maps to the continuous domain and make all primal confidence score maps have the same spatial resolution. Then, we exploit an efficient ensemble post-processor based on relative entropy, which can coalesce primal confidence score maps and create an optimal confidence score map for more accurate localization. The target is localized on the peak of the optimal confidence score map. Besides, we introduce a collaborative optimization strategy to update circular and structural operators by iteratively training structural correlation filters, which significantly reduces computational complexity and improves robustness. Experimental results demonstrate that our approach achieves state-of-the-art performance in mean AUC scores of 71.5 and 69.4 on the OTB2013 and OTB2015 benchmarks respectively, and obtains a third-best expected average overlap (EAO) score of 29.8 on the VOT2017 benchmark. This paper demonstrates that the computational effort required to develop numerical solutions to continuous-state dynamic programs can be reduced significantly when cubic piecewise polynomial functions, rather than tensor product linear interpolants, are used to approximate the value function. Tensor product cubic splines, represented in either piecewise polynomial or B-spline form, and multivariate Hermite polynomials are considered. Computational savings are possible because of the improved accuracy of higher-order functions and because the smoothness of higher-order functions allows efficient quasi-Newton methods to be used to compute optimal decisions. The use of the more efficient piecewise polynomial form of the spline was slightly superior to the use of Hermite polynomials for the test problem and easier to program. In comparison to linear interpolation, use of splines in piecewise polynomial form reduced the CPU time to obtain results of equivalent accuracy by a factor of 250–330 for a stochastic 4-dimensional water supply reservoir problem with a smooth objective function, and factors ranging from 25–400 for a sequence of 2-, 3-, 4-, and 5-dimensional problems. As a result, a problem that required two hours to solve with linear interpolation was solved in a less than a minute with spline interpolation with no loss of accuracy. We explore how active learning with Support Vector Machines works well for a non-trivial task in natural language processing. We use Japanese word segmentation as a test case. In particular, we discuss how the size of a pool affects the learning curve. It is found that in the early stage of training with a larger pool, more labeled examples are required to achieve a given level of accuracy than those with a smaller pool. In addition, we propose a novel technique to use a large number of unlabeled examples effectively by adding them gradually to a pool. The experimental results show that our technique requires less labeled examples than those with the technique in previous research. To achieve 97.0 accuracy, the proposed technique needs 59.3 of labeled examples that are required when using the previous technique and only 17.4 of labeled examples with random sampling. We consider the problem of the reconstruction of a continuous-time function f(x) spl isin H from the samples of the responses of m linear shift-invariant systems sampled at 1 m the reconstruction rate. We extend Papoulis' generalized sampling theory in two important respects. First, our class of admissible input signals (typ. H=L sub 2 ) is considerably larger than the subspace of band-limited functions. Second, we use a more general specification of the reconstruction subspace V( spl psi ), so that the output of the system can take the form of a band-limited function, a spline, or a wavelet expansion. Since we have enlarged the class of admissible input functions, we have to give up Shannon and Papoulis' principle of an exact reconstruction. Instead, we seek an approximation f spl isin V( spl psi ) that is consistent in the sense that it produces exactly the same measurements as the input of the system. This leads to a generalization of Papoulis' sampling theorem and a practical reconstruction algorithm that takes the form of a multivariate filter. In particular, we show that the corresponding system acts as a projector from H onto V( spl psi ). We then propose two complementary polyphase and modulation domain interpretations of our solution. The polyphase representation leads to a simple understanding of our reconstruction algorithm in terms of a perfect reconstruction filter bank. The modulation analysis, on the other hand, is useful in providing the connection with Papoulis' earlier results for the band-limited case. Finally, we illustrate the general applicability of our theory by presenting new examples of interlaced and derivative sampling using splines.