Mathew Mithra Noel

Mathew Mithra Noel, Venkataraman Muthiah-Nakarajan, Yug D Oswal

Higher order artificial neurons whose outputs are computed by applying an activation function to a higher order multinomial function of the inputs have been considered in the past, but did not gain acceptance due to the extra parameters and computational cost. However, higher order neurons have significantly greater learning capabilities since the decision boundaries of higher order neurons can be complex surfaces instead of just hyperplanes. The boundary of a single quadratic neuron can be a general hyper-quadric surface allowing it to learn many nonlinearly separable datasets. Since quadratic forms can be represented by symmetric matrices, only $\frac{n(n+1)}{2}$ additional parameters are needed instead of $n^2$. A quadratic Logistic regression model is first presented. Solutions to the XOR problem with a single quadratic neuron are considered. The complete vectorized equations for both forward and backward propagation in feedforward networks composed of quadratic neurons are derived. A reduced parameter quadratic neural network model with just $ n $ additional parameters per neuron that provides a compromise between learning ability and computational cost is presented. Comparison on benchmark classification datasets are used to demonstrate that a final layer of quadratic neurons enables networks to achieve higher accuracy with significantly fewer hidden layer neurons. In particular this paper shows that any dataset composed of $\mathcal{C}$ bounded clusters can be separated with only a single layer of $\mathcal{C}$ quadratic neurons.

Mathew Mithra Noel, Arindam Banerjee, Yug D. Oswal et al.

The performance of supervised machine learning models is directly related to the quality of the training dataset. In particular, the presence of significantly many outliers in the training data can lead to low accuracy because popular loss function like the Mean Squared Error (MSE) assign very high importance to large errors. The Mean Absolute Error (MAE) loss assigns equal importance to all errors and is most robust to outliers, but suffers from being non-differentiable at the origin. MAE also has large derivative values close to its minimum leading to instability and oscillations during training. Thus differentiable approximations to MAE namely Huber and Log-Cosh losses were introduced for robust regression tasks. This paper introduces two new infinitely differentiable loss functions that more closely approximate the MAE loss and provide improved performance on regression tasks with significantly many outliers in the training dataset. A comparison of the performance of regression models with different loss functions on a wide variety of benchmarks and datasets is presented to demonstrate the superior performance of the Square Root Loss (SRL) and Smooth Mean Absolute Error (SMAE) losses proposed in this paper. The SRL loss is shown to be strictly convex and the SMAE loss is shown to be strictly quasi-convex. Given the fundamental importance of linear regression, two new robust linear regression models are presented.

Mathew Mithra Noel, Yug Oswal

This paper introduces a significantly better class of activation functions than the almost universally used ReLU like and Sigmoidal class of activation functions. Two new activation functions referred to as the Cone and Parabolic-Cone that differ drastically from popular activation functions and significantly outperform these on the CIFAR-10 and Imagenette benchmmarks are proposed. The cone activation functions are positive only on a finite interval and are strictly negative except at the end-points of the interval, where they become zero. Thus the set of inputs that produce a positive output for a neuron with cone activation functions is a hyperstrip and not a half-space as is the usual case. Since a hyper strip is the region between two parallel hyper-planes, it allows neurons to more finely divide the input feature space into positive and negative classes than with infinitely wide half-spaces. In particular the XOR function can be learn by a single neuron with cone-like activation functions. Both the cone and parabolic-cone activation functions are shown to achieve higher accuracies with significantly fewer neurons on benchmarks. The results presented in this paper indicate that many nonlinear real-world datasets may be separated with fewer hyperstrips than half-spaces. The Cone and Parabolic-Cone activation functions have larger derivatives than ReLU and are shown to significantly speedup training.

Mathew Mithra Noel, Arunkumar L, Advait Trivedi et al.

Convolutional neural networks have been successful in solving many socially important and economically significant problems. This ability to learn complex high-dimensional functions hierarchically can be attributed to the use of nonlinear activation functions. A key discovery that made training deep networks feasible was the adoption of the Rectified Linear Unit (ReLU) activation function to alleviate the vanishing gradient problem caused by using saturating activation functions. Since then, many improved variants of the ReLU activation have been proposed. However, a majority of activation functions used today are non-oscillatory and monotonically increasing due to their biological plausibility. This paper demonstrates that oscillatory activation functions can improve gradient flow and reduce network size. Two theorems on limits of non-oscillatory activation functions are presented. A new oscillatory activation function called Growing Cosine Unit(GCU) defined as $C(z) = z\cos z$ that outperforms Sigmoids, Swish, Mish and ReLU on a variety of architectures and benchmarks is presented. The GCU activation has multiple zeros enabling single GCU neurons to have multiple hyperplanes in the decision boundary. This allows single GCU neurons to learn the XOR function without feature engineering. Experimental results indicate that replacing the activation function in the convolution layers with the GCU activation function significantly improves performance on CIFAR-10, CIFAR-100 and Imagenette.