The 0-1 integer linear programming feasibility problem is an important NP-complete problem. This paper proposes a continuous-time dynamical system for solving that problem without getting trapped in non-solution local minima. First, the problem is transformed to an easier form in linear time. Then, we propose an "impulse algorithm" to escape from local traps and show its performance is better than randomization for escaping traps. Second, we present the time-to-solution distribution of the impulse algorithm and compare it with exhaustive search to see its advantages. Third, we show that the fractional size of the basin of attraction of the global minimum is significantly larger than @math , the corresponding discrete probability for exhaustive search. Finally, we conduct a case study to show that the location of the basin is independent of different dimensions. These findings reveal a better way to solve the 0-1 integer linear programming feasibility problem continuously and show that its cost could be less than discrete methods in average cases.
In recent years many discrete problems have been solved by continuous methods, especially decision problems, because continuous-time dynamical systems operate similarly to our brain. The state of neurons change continuously but their outputs are usually binary --- fire or not --- which depends on the real-time difference between the present state and a threshold @cite_19 . Most of continuous-time complex systems can be treated as neural networks, and particularly when the neurons are connected with feedback paths and the input is a time sequence, this kind of network is called a Recurrent Neural Network (RNN). RNNs have been applied in many fields, such as neuroscience @cite_4 and chaotic physical system @cite_13 @cite_17 .
@cite_13 @cite_19 @cite_4 @cite_17
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Recurrent neural networks (RNNs) have been widely used for processing sequential data. However, RNNs are commonly difficult to train due to the well-known gradient vanishing and exploding problems and hard to learn long-term patterns. Long short-term memory (LSTM) and gated recurrent unit (GRU) were developed to address these problems, but the use of hyperbolic tangent and the sigmoid action functions results in gradient decay over layers. Consequently, construction of an efficiently trainable deep network is challenging. In addition, all the neurons in an RNN layer are entangled together and their behaviour is hard to interpret. To address these problems, a new type of RNN, referred to as independently recurrent neural network (IndRNN), is proposed in this paper, where neurons in the same layer are independent of each other and they are connected across layers. We have shown that an IndRNN can be easily regulated to prevent the gradient exploding and vanishing problems while allowing the network to learn long-term dependencies. Moreover, an IndRNN can work with non-saturated activation functions such as relu (rectified linear unit) and be still trained robustly. Multiple IndRNNs can be stacked to construct a network that is deeper than the existing RNNs. Experimental results have shown that the proposed IndRNN is able to process very long sequences (over 5000 time steps), can be used to construct very deep networks (21 layers used in the experiment) and still be trained robustly. Better performances have been achieved on various tasks by using IndRNNs compared with the traditional RNN and LSTM. Deep Recurrent Neural Network architectures, though remarkably capable at modeling sequences, lack an intuitive high-level spatio-temporal structure. That is while many problems in computer vision inherently have an underlying high-level structure and can benefit from it. Spatiotemporal graphs are a popular tool for imposing such high-level intuitions in the formulation of real world problems. In this paper, we propose an approach for combining the power of high-level spatio-temporal graphs and sequence learning success of Recurrent Neural Networks (RNNs). We develop a scalable method for casting an arbitrary spatio-temporal graph as a rich RNN mixture that is feedforward, fully differentiable, and jointly trainable. The proposed method is generic and principled as it can be used for transforming any spatio-temporal graph through employing a certain set of well defined steps. The evaluations of the proposed approach on a diverse set of problems, ranging from modeling human motion to object interactions, shows improvement over the state-of-the-art with a large margin. We expect this method to empower new approaches to problem formulation through high-level spatio-temporal graphs and Recurrent Neural Networks. Recurrent Neural Networks (RNNs) have become the state-of-the-art choice for extracting patterns from temporal sequences. However, current RNN models are ill-suited to process irregularly sampled data triggered by events generated in continuous time by sensors or other neurons. Such data can occur, for example, when the input comes from novel event-driven artificial sensors that generate sparse, asynchronous streams of events or from multiple conventional sensors with different update intervals. In this work, we introduce the Phased LSTM model, which extends the LSTM unit by adding a new time gate. This gate is controlled by a parametrized oscillation with a frequency range that produces updates of the memory cell only during a small percentage of the cycle. Even with the sparse updates imposed by the oscillation, the Phased LSTM network achieves faster convergence than regular LSTMs on tasks which require learning of long sequences. The model naturally integrates inputs from sensors of arbitrary sampling rates, thereby opening new areas of investigation for processing asynchronous sensory events that carry timing information. It also greatly improves the performance of LSTMs in standard RNN applications, and does so with an order-of-magnitude fewer computes at runtime. Deep Recurrent Neural Network architectures, though remarkably capable at modeling sequences, lack an intuitive high-level spatio-temporal structure. That is while many problems in computer vision inherently have an underlying high-level structure and can benefit from it. Spatio-temporal graphs are a popular tool for imposing such high-level intuitions in the formulation of real world problems. In this paper, we propose an approach for combining the power of high-level spatio-temporal graphs and sequence learning success of Recurrent Neural Networks (RNNs). We develop a scalable method for casting an arbitrary spatio-temporal graph as a rich RNN mixture that is feedforward, fully differentiable, and jointly trainable. The proposed method is generic and principled as it can be used for transforming any spatio-temporal graph through employing a certain set of well defined steps. The evaluations of the proposed approach on a diverse set of problems, ranging from modeling human motion to object interactions, shows improvement over the state-of-the-art with a large margin. We expect this method to empower new approaches to problem formulation through high-level spatio-temporal graphs and Recurrent Neural Networks.