Triple point in theO(2)ghost model with higher-order gradient term

Accelerating Natural Gradient with Higher-Order Invariance

An appealing property of the natural gradient is that it is invariant to arbitrary differentiable reparameterizations of the model. However, this invariance property requires infinitesimal steps and is lost in practical implementations with small but finite step sizes. In this paper, we study invariance properties from a combined perspective of Riemannian geometry and numerical differential equ...

Tarski's Fixed-Point Theorem and Higher-Order Term Rewrite Systems

Higher order multi-point fractional boundary value problems with integral boundary conditions

In this paper, we concerned with positive solutions for higher order m-point nonlinear fractional boundary value problems with integral boundary conditions. We establish the criteria for the existence of at least one, two and three positive solutions for higher order m-point nonlinear fractional boundary value problems with integral boundary conditions by using a result from the theory of fixed...

Gradient-based learning of higher-order features

We describe an auto-encoder with multiplicative connections whose hidden variables encode products of pixel intensities. The model allows for efficient learning of image transformations and of higher-order structure within an image-patch. Modelling higher-order structure can be an effective way to improve recognition performance, as shown recently with a similar, probabilistic model of image co...

Two - point higher order BVPs with singularities in phase variables ∗

The existence of solutions for singular higher order differential equations with the Lidstone or the (n, p) boundary conditions is proved. The righthand sides of differential equations can have singularities in the zero value of their phase variables and so higher derivatives of solutions changing their signs can pass through these singularities. Proofs are based on the method of a priori estim...