Deep Quaternion Networks

The field of deep learning has seen significant advancement in recent years. However, much of the existing work has been focused on real-valued numbers. Recent work has shown that a deep learning system using the complex numbers can be deeper for a fixed parameter budget compared to its real-valued counterpart. In this work, we explore the benefits of generalizing one step further into the hyper-complex numbers, quaternions specifically, and provide the architecture components needed to build deep quaternion networks. We develop the theoretical basis by reviewing quaternion convolutions, developing a novel quaternion weight initialization scheme, and developing novel algorithms for quaternion batch-normalization. These pieces are tested in a classification model by end-to-end training on the CIFAR-10 and CIFAR-100 data sets and a segmentation model by end-to-end training on the KITTI Road Segmentation data set. These quaternion networks show improved convergence compared to real-valued and complex-valued networks, especially on the segmentation task, while having fewer parametersComment: IJCNN 2018, 8 pages, 1 figur

On The Douglas-Kazakov Phase Transition

We give a rigorous proof of the fact that a phase transition discovered by Douglas and Kazakov in 1993 in the context of two-dimensional gauge theories occurs. This phase transition can be formulated in terms of the Brownian bridge on the unitary group U(N) when N tends to infinity. We explain how it can be understood by considering the asymptotic behaviour of the eigenvalues of the unitary Brownian bridge, and how it can be technically approached by means of Fourier analysis on the unitary group. Moreover, we advertise some more or less classical methods for solving certain minimisation problems which play a fundamental role in the study of the phase transition

Convergence to Equilibrium in the Free Fokker-Planck Equation With a Double-Well Potential

We consider the one-dimensional free Fokker-Planck equation $\frac{\partial \mu\_t}{\partial t} = \frac{\partial}{\partial x} \left[ \mu\_t \left( \frac12 V' - H\mu\_t \right) \right]$, where $H$ denotes the Hilbert transform and $V$ is a particular double-well quartic potential, namely $V(x) = \frac14 x^4 + \frac{c}{2} x^2$, with $-2 \le c < 0$. We prove that the solution $(\mu\_t)\_{t \ge 0}$ of this PDE converges to the equilibrium measure $\mu\_V$ as $t$ goes to infinity, which provides a first result of convergence in a non-convex setting. The proof involves free probability and complex analysis techniques

FIRST LANGUAGE ACQUISITION FOR CHILDREN AGED 3-4 YEARS IN SEBERANG ULU 1 KERTAPATI PALEMBANG

This study aims to describe the acquisition of first language in children aged 3-4 years at Seberang Ulu 1 Kertapati Palembang with a total of 4 children who have different ages in the phonology and morphology study level. The research method used is descriptive qualitative method. Data collection techniques used in this study were observation techniques, recording techniques, listening techniques, interview techniques and note taking techniques. The data analysis technique used is descriptive analysis technique. Therefore, the difficulty that often arises in analyzing novels is having to read Keywords : First language acquisition, phonology, morphology, Palembang, and language acquisition studie