These are notes for the course with the same title given by the first named author during the workshop ”Resonances and periodic orbits: spectrum and zeta functions in quantum and classical chaos” at IHP, Paris, June 27-July 5, 2005. We refer to our joint paper ([4], arxiv.org, 19 pages) for a complete self-contained proof in a more general setting. Our goal here is to give a readerfriendly pres...

One way to define the “randomness” of a fixed individual sequence is to ask how hard it is to predict. When prediction error is measured via squared loss, it has been established that memoryless sequences (which are, in a precise sense, hard to predict) have some of the stochastic attributes of truly random sequences. In this paper, we ask how changing the loss function used changes the set of ...

Can we incorporate discrete optimization algorithms within modern machine learning models? For example, is it possible to incorporate in deep architectures a layer whose output is the minimal cut of a parametrized graph? Given that these models are trained end-to-end by leveraging gradient information, the introduction of such layers seems very challenging due to their non-continuous output. In...

The circle group G is said to act pseudofreely if it is not free and if every orbit is one-dimensional and if the isotropy group is the identity except on isolated exceptional orbits where the isotropy group is the finite cyclic group Z(k), k > 1. We consider differentiable actions of this kind on homotopy seven spheres and classify all such actions. As a corollary, it follows that there can be...

Introduction. It is shown in [4] that there is an infinite family of semifree Zm actions on odd dimensional homotopy spheres. There is also an infinite family of semifree S actions on odd dimensional homotopy spheres (see [2], [5]). On the other hand, it is announced in [2] that there are only finitely many inequivalent semifree S actions on even dimensional homotopy spheres. Hence it is intere...

We present some features of the smooth structure and of the canonical stratification on the orbit space of a proper Lie groupoid. One of the main features is that of Morita invariance of these structures-it allows us to talk about the canonical structure of differentiable stratified space on the orbispace (an object analogous to a separated stack in algebraic geometry) presented by the proper L...

Exponential Linear Units (ELUs) are a useful rectifier for constructing deep learning architectures, as they may speed up and otherwise improve learning by virtue of not have vanishing gradients and by having mean activations near zero [1]. However, the ELU activation as parametrized in [1] is not continuously differentiable with respect to its input when the shape parameter α is not equal to 1...

If A(t) is a C1,α-curve of unbounded self-adjoint operators with compact resolvents and common domain of definition, then the eigenvalues can be parameterized C1 in t. If A is C∞ then the eigenvalues can be parameterized twice differentiable. Theorem. Let t 7→ A(t) for t ∈ R be a curve of unbounded self-adjoint operators in a Hilbert space with common domain of definition and with compact resol...

Inverse graphics attempts to take sensor data and infer 3D geometry, illumination, materials, and motions such that a graphics renderer could realistically reproduce the observed scene. Renderers, however, are designed to solve the forward process of image synthesis. To go in the other direction, we propose an approximate differentiable renderer (DR) that explicitly models the relationship betw...

Recently, several exact cone-beam reconstruction algorithms, such as the generalized filtered-backprojection (FBP) and backprojection-filtration (BPF) methods, have been developed to solve the long object problem. Although the well-known 3D Shepp-Logan phantom (SLP) is often used to validate these algorithms, it is deficient due to the discontinuity of the SLP. In this paper, we first construct...