Integral representations for elliptic functions

We derive new integral representations for constituents of the classical theory of elliptic functions: the Eisenstein series, and Weierstrass' Weierstrass and ζ functions. The derivations proceed from the Laplace-Mellin representation of multipoles, and an elementary lemma on the summation of 2D geometric series. In addition, we present results concerning the analytic continuation of the Eisenstein series to an entire function in the complex plane, and the value of the conditionally convergent series, denoted by E2below, as a function of summation over increasingly large rectangles with arbitrary fixed aspect ratio

Multiplexed gradient descent: Fast online training of modern datasets on hardware neural networks without backpropagation

We present multiplexed gradient descent (MGD), a gradient descent framework designed to easily train analog or digital neural networks in hardware. MGD utilizes zero-order optimization techniques for online training of hardware neural networks. We demonstrate its ability to train neural networks on modern machine learning datasets, including CIFAR-10 and Fashion-MNIST, and compare its performance to backpropagation. Assuming realistic timescales and hardware parameters, our results indicate that these optimization techniques can train a network on emerging hardware platforms orders of magnitude faster than the wall-clock time of training via backpropagation on a standard GPU, even in the presence of imperfect weight updates or device-to-device variations in the hardware. We additionally describe how it can be applied to existing hardware as part of chip-in-the-loop training, or integrated directly at the hardware level. Crucially, the MGD framework is highly flexible, and its gradient descent process can be optimized to compensate for specific hardware limitations such as slow parameter-update speeds or limited input bandwidth

Uncertainty Quantification in Scientific Computing: 10th IFIP WG2.5Working Conference, WoCoUQ 2011, Boulder, CO, USA, August 1-4, 2011

International audienceBook Front Matter of AICT 37