The Van der Pol Fourier representation of the Riemann Zeta function in the critical strip is evaluated using the Fast Fourier Transform (FFT) algorithm and the Method of Stationary Phase. The Method of Stationary Phase approximation allows us to introduce a new representation for the zeta function. The new representation is used to propose new analytic predictions about non-trivial zeros.

We propose an exact new sampling algorithm using classical statistical theory. The algorithm is powered by white noise and non-local information. The proposed method will process an infinite number of rejected samples in parallel before accepting one exact single sample from a target distribution using an optimization mechanism. The algorithm is applied to sampling from general distributions in addition to sampling from the wave function density distribution which lies at the heart of the measurement problem in quantum physics.

Using data augmentation techniques, unsupervised representation learning methods extract features from data by training artificial neural networks to recognize that different views of an object are just different instances of the same object. We extend current unsupervised representation learning methods to networks that can self-organize data representations into two-dimensional (2D) maps. The proposed method combines ideas from Kohonen’s original self-organizing maps (SOM) and recent development in unsupervised representation learning. A ResNet backbone with an added 2D Softmax output layer is used to organize the data representations. A new loss function with linear complexity is proposed to enforce SOM requirements of winner-take-all (WTA) and competition between neurons while explicitly avoiding collapse into trivial solutions. We show that enforcing SOM topological neighborhood requirement can be achieved by a fixed radial convolution at the 2D output layer without having to resort to actual radial activation functions which prevented the original SOM algorithm from being extended to nowadays neural network architectures. We demonstrate that when combined with data augmentation techniques, self-organization is a simple emergent property of the 2D output layer because of neighborhood recruitment combined with WTA competition between neurons. The proposed methodology is demonstrated on SVHN and CIFAR10 data sets. The proposed algorithm is the first end-to-end unsupervised learning method that combines data self-organization and visualization as integral parts of unsupervised representation learning.