Expressive Numbers of Two or More Hidden Layer ReLU Neural Networks One of the reasons why neural networks are used in machine learning is their high expressive power, that is, the ability to express functions. Expressive power of neural networks depends on its structures and is measured by some indices. In this paper, we focus on one of these measures named expressive number, which is based on the number of data that can be expressed. Expressive numbers enable us to see whether the size of a neural network is suitable for the given training data before we conduct machine learning. However, existing works on expressive numbers mainly target single hidden layer neural networks, and little is known about those with two or more hidden layers. In this paper, we give a lower bound of the maximum expressive number of two hidden layer neural networks and an upper bound of that of multilayer neural networks with ReLU activation function. This result shows the expressive number of two hidden layer neural networks is proportional to the product of the numbers of each hidden layeru0027s neurons.

Exploring Numerical Calculations with CalcNet Neural networks are not great generalizers outside their training range i.e. they are good at capturing bias but might miss the overall concept. Important issues with neural networks is that when testing data goes outside training range they fail to predict accurate results. Hence, they loose the ability to generalize a concept. For systematic numeric exploration neural accumulators (NAC) and neural arithmetic logic unit(NALU) are proposed which performs excellent for simple arithmetic operations. But, major limitation with these units is that they canu0027t handle complex mathematical operations u0026 equations. For example, NALU can predict accurate results for multiplication operation but not for factorial function which is essentially composition of multiplication operations only. It is unable to comprehend pattern behind an expression when composition of operations are involved. Hence, we propose a new neural network structure effectively which takes in complex compositional mathematical operations and produces best possible results with small NALU based neural networks as its pluggable modules which evaluates these expression at unitary level in a bottomup manner. We call this effective neural network as CalcNet, as it helps in predicting accurate calculations for complex numerical expressions even for values that are out of training range. As part of our study we applied this network on numerically approximating complex equations, evaluating biquadratic equations and tested reusability of these modules. We arrived at far better generalizations for complex arithmetic extrapolation tasks as compare to both only NALU layer based neural networks and simple feed forward neural networks. Also, we achieved even better results for our golden ratio based modified NAC and NALU structures for both interpolating and extrapolating tasks in all evaluation experiments. Finally, from reusability standpoint this model demonstrate strong invariance for making predictions on different tasks.