The self-similar gravitational collapse solutions to the Einstein-axion-dilaton system have already been found out. Those become invariants after combining spacetime dilation with transformations of internal SL(2, R). We apply nonlinear statistical models estimate functions that appear in physics Black Holes axion-dilaton four dimensions. These include polynomial regression, nonparametric kerne...

∆u = ∂2u ∂x1 + ∂2u ∂x2 where x = (x1, x2) ∈ IR 2 are Cartesian coordinates. Even though the Poisson equation looks very special it is an important model case representing several problems from physics based on energy minimisation. Variations of the techniques we will study apply to a wide class of second order so-called elliptic problems. It is known that there are cases where no classical (i.e...

Categorical variables are a natural choice for representing discrete structure in the world. However, stochastic neural networks rarely use categorical latent variables due to the inability to backpropagate through samples. In this work, we present an efficient gradient estimator that replaces the non-differentiable sample from a categorical distribution with a differentiable sample from a nove...

The minimum sum-of-squares clustering problem is considered. The mathematical modeling of this problem leads to a min − sum −min formulation which, in addition to its intrinsic bi-level nature, has the significant characteristic of being strongly nondifferentiable. To overcome these difficulties, the proposed resolution method, called Hyperbolic Smoothing, adopts a smoothing strategy using a sp...

The theory of scale relativity provides a new insight into the origin of fundamental laws in physics. Its application to microphysics allows us to recover quantum mechanics as mechanics on a non-differentiable (fractal) spacetime. The Schrödinger and Klein–Gordon equations are demonstrated as geodesic equations in this framework. A development of the intrinsic properties of this theory, using t...

This paper is twofold. In a first part, we extend the classical differential calculus to continuous non differentiable functions by developping the notion of scale calculus. The scale calculus is based on a new approach of continuous non differentiable functions by constructing a one parameter family of differentiable functions f(t, ǫ) such that f(t, ǫ) → f(t) when ǫ goes to zero. This lead to ...

In natural language processing tasks performance of the models is often measured with some non-differentiable metric, such as BLEU score. To use efficient gradientbased methods for optimization, it is a common workaround to optimize some surrogate loss function. This approach is effective if optimization of such loss also results in improving target metric. The corresponding problem is referred...

Categorical variables are a natural choice for representing discrete structure in the world. However, stochastic neural networks rarely use categorical latent variables due to the inability to backpropagate through samples. In this work, we present an efficient gradient estimator that replaces the non-differentiable sample from a categorical distribution with a differentiable sample from a nove...