We present a greedy-based approach to construct an efficient single hidden layer neural network with the ReLU activation that approximates a target function. In our approach we obtain a shallow network by utilizing a greedy algorithm with the prescribed dictionary provided by the available training data and a set of possible inner weights. To facilitate the greedy selection process we employ an integral representation of the network, based on the ridgelet transform, that significantly reduces the cardinality of the dictionary and hence promotes feasibility of the greedy selection. Our approach allows for the construction of efficient architectures which can be treated either as improved initializations to be used in place of random-based alternatives, or as fully-trained networks in certain cases, thus potentially nullifying the need for backpropagation training. Numerical experiments demonstrate the tenability of the proposed concept and its advantages compared to the conventional techniques for selecting architectures and initializations for neural networks.
Another related approach to ours is the Random Vector Functional-Link (or RVFL) network @cite_2 where the inner weights @math are generated randomly and the outer weights @math are found via a least square regularization. The random selection can be interpreted as a Monte-Carlo sampling for an integral representation. However the authors do not use any integral representation explicitly, only the fact of existence of such representation to justify their approach.
@cite_2
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Abstract With the direct input–output connections, a random vector functional link (RVFL) network is a simple and effective learning algorithm for single-hidden layer feedforward neural networks (SLFNs). RVFL is a universal approximator for continuous functions on compact sets with fast learning property. Owing to its simplicity and effectiveness, RVFL has attracted significant interest in numerous real-world applications. In reality, the performance of RVFL is often challenged by randomly assigned network parameters. In this paper, we propose a novel unsupervised network parameter learning method for RVFL, named sparse pre-trained random vector functional link (SP-RVFL for short) network. The proposed SP-RVFL uses a sparse autoencoder with l 1 -norm regularization to adaptively learn superior network parameters for specific learning tasks. By doing so, the learned network parameters in SP-RVFL are embedded with the valuable information of input data, which alleviate the randomly generated parameter issue and improve the algorithmic performance. Experiments and comparisons on 16 diverse benchmarks from different domains confirm the effectiveness of the proposed SP-RVFL. The corresponding results also demonstrate that RVFL outperforms extreme learning machine (ELM). Learning a function of many arguments is viewed from the perspective of high– dimensional numerical quadrature. It is shown that many of the popular ensemble learning procedures can be cast in this framework. In particular randomized methods, including bagging and random forests, are seen to correspond to random Monte Carlo integration methods each based on particular importance sampling strategies. Non random boosting methods are seen to correspond to deterministic quasi Monte Carlo integration techniques. This view helps explain some of their properties and suggests modifications to them that can substantially improve their accuracy while dramatically improving computational performance. Deep neural network-based methods have recently achieved excellent performance in visual tracking task. As very few training samples are available in visual tracking task, those approaches rely heavily on extremely large auxiliary dataset such as ImageNet to pretrain the model. In order to address the discrepancy between the source domain (the auxiliary data) and the target domain (the object being tracked), they need to be finetuned during the tracking process. However, those methods suffer from sensitivity to the hyper-parameters such as learning rate, maximum number of epochs, size of mini-batch, and so on. Thus, it is worthy to investigate whether pretraining and fine tuning through conventional back-prop is essential for visual tracking. In this paper, we shed light on this line of research by proposing convolutional random vector functional link (CRVFL) neural network, which can be regarded as a marriage of the convolutional neural network and random vector functional link network, to simplify the visual tracking system. The parameters in the convolutional layer are randomly initialized and kept fixed. Only the parameters in the fully connected layer need to be learned. We further propose an elegant approach to update the tracker. In the widely used visual tracking benchmark, without any auxiliary data, a single CRVFL model achieves 79.0 with a threshold of 20 pixels for the precision plot. Moreover, an ensemble of CRVFL yields comparatively the best result of 86.3 . 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.