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Our main result is that chaos in dimension $n+1$ is a one-dimensional geometrical object embedded in a geometrical object of dimension $n$ which corresponds to a $n$ dimensional object which is either singular or non-singular. Our main result is then that this chaos occurs in the first case as either on an isolated or non-isolated singularity. In the first case this chaos is either boundary chaos or spherical chaos which is what happens also in the non-singular case. In the case of an isolated singular geometry one has chaos which can either be boundary, spherical or tubular chaos. We furthermore prove that the prime numbers display quantum behaviour.

We discuss $\mathcal{D}$-modules and dynamical systems in the \'etale topology. We introduce the differential scheme associated to a morphism $f: X\to S$ of schemes of the same dimension. We introduce differential inertia group $I_{diff}^i$ which act trivially on the special fibers of these schemes. As an application we discuss the problem of counting points on elliptic curves which we show is connected to vanishing cycles of singularities and to closed orbits of dynamical systems.

We prove that to each real singularity $f: (\mathbb{R}^{n}, 0) \to (\mathbb{R}^k, 0)$ with $k\geq 2$ one can associate systems of differential equations $\mathfrak{g}^{k}_f$ which are pushforwards in the category of $\mathcal{D}$-modules over $\mathbb{R}^{k}$ of the sheaf of real analytic functions on the total space of the Milnor fibration. We then use this to study Hilbert's 16th problem on polynomial dynamical systems in the plane.

We propose two approaches to extend the notion of knowledge distillation to Gaussian Process Regression (GPR) and Gaussian Process Classification (GPC); data-centric and distribution-centric. The data-centric approach resembles most current distillation techniques for machine learning, and refits a model on deterministic predictions from the teacher, while the distribution-centric approach, re-uses the full probabilistic posterior for the next iteration. By analyzing the properties of these approaches, we show that the data-centric approach for GPR closely relates to known results for self-distillation of kernel ridge regression and that the distribution-centric approach for GPR corresponds to ordinary GPR with a very particular choice of hyperparameters. Furthermore, we demonstrate that the distribution-centric approach for GPC approximately corresponds to data duplication and a particular scaling of the covariance and that the data-centric approach for GPC requires redefining the model from a Binomial likelihood to a continuous Bernoulli likelihood to be well-specified. To the best of our knowledge, our proposed approaches are the first to formulate knowledge distillation specifically for Gaussian Process models., Comment: 10 pages; code at https://github.com/Kennethborup/gaussian_process_self_distillation

We prove that to each real singularity $f: (\mathbb{R}^{n+1}, 0) \to (\mathbb{R}, 0)$ one can associate two systems of differential equations $\mathfrak{g}^{k\pm}_f$ which are pushforwards in the category of $\mathcal{D}$-modules over $\mathbb{R}^{\pm}$, of the sheaf of real analytic functions on the total space of the positive, respectively negative, Milnor fibration. We prove that for $k=0$ if $f$ is an isolated singularity then $\mathfrak{g}^{\pm}$ determines the the $n$-th homology groups of the positive, respectively negative, Milnor fibre. We then calculate $\mathfrak{g}^{+}$ for ordinary quadratic singularities and prove that under certain conditions on the choice of morsification, one recovers the top homology groups of the Milnor fibers of any isolated singularity $f$. As an application we construct a public-key encryption scheme based on morsification of singularities.

Learning often involves sensitive data and as such, privacy preserving extensions to Stochastic Gradient Descent (SGD) and other machine learning algorithms have been developed using the definitions of Differential Privacy (DP). In differentially private SGD, the gradients computed at each training iteration are subject to two different types of noise. Firstly, inherent sampling noise arising from the use of minibatches. Secondly, additive Gaussian noise from the underlying mechanisms that introduce privacy. In this study, we show that these two types of noise are equivalent in their effect on the utility of private neural networks, however they are not accounted for equally in the privacy budget. Given this observation, we propose a training paradigm that shifts the proportions of noise towards less inherent and more additive noise, such that more of the overall noise can be accounted for in the privacy budget. With this paradigm, we are able to improve on the state-of-the-art in the privacy/utility tradeoff of private end-to-end CNNs., Comment: 2021 IEEE International Workshop on Machine Learning for Signal Processing (MLSP)

In this article we apply the results in the article "On Isolated Real Singularities I" to the study of real $ADE$-singularities. We show that said results enables us to find the homology groups of the Milnor fibres of real $ADE$-singularities of curves. As a corollary we show how to deduce the same for $ADE$-singularities in higher codimension, giving an almost complete answer in this case. This is followed by a discussion of the classification problem of simple real singularities., Comment: This article (as well as its predecessor) formed part of my thesis

In this article we prove two results concerning the motivic Milnor fibres $S^{\epsilon}(f)$ associated to a map germ $f: (\mathbb{R}^n,0)\to(\mathbb{R},0)$, defined by G. Comte and G. Fichou. Firstly, we prove that if $f,g:(\mathbb{R}^n,0)\to(\mathbb{R},0)$ are arc-analytically equivalent germs of Nash functions then the virtual Poincar\'e polynomial of the corresponding motivic Milnor fibres are equal. This extends (and provides a new proof) of a result of G. Fichou. Secondly, let $T\subset\mathbb{R}^m$ be a real algebraic set and $f: T\times(\mathbb{R}^n,0)\to\mathbb{R}$ a polynomial function of polynomial map germs such that $f(t,0)=0$ for any $t\in T$. Then we prove that there exists a locally finite real analytic stratification $\mathcal{S}$ of $T$ such that if $S\in\mathcal{S}$ is a stratum then $f_t\sim_a f_{t'}$ are arc-analytically equivalent, for any $t,t'\in S$. Furthermore, if $T$ is compact then the stratification $\mathcal{S}$ can be taken to be finite. These two results imply in particular that the virtual Poincar\'e polynomials of the respective zeta functions are equal i.e $\beta(S^{\epsilon}(f_t))=\beta(S^{\epsilon}(f_{t'}))$., Comment: This is part of my thesis completed in 2020 at LAMA

For analytic map germs $f: (\mathbb{R}^n, 0)\to (\mathbb{R}, 0)$ having an isolated critical value in the origin with $\dim V(f)>0$ and satisfying the transversality property of D.B. Massey we show that for $c>0$ a large enough constant, and $k$ a large enough natural number the local Milnor-L\^e fibers of $f$ and of the isolated singularity $g=f-c(\sum_{i=1}^n x_i^2)^k$ satisfies the following. There exists a homotopy equivalence between the negative Milnor-L\^e fiber of $f$, to which a cobordism between its boundary and the link of $f$ has been adjoined, and the negative Milnor-L\^e fiber of $g$.

Given a nonconstant polynomial map over the reals having an isolated critical point in the origin and with zero locus of positive dimension, we establish a formula for the singular homology groups of a Milnor fibre relative to its boundary.

The use of quantitative broadband echosounders for biological studies and surveys offers considerable advantages over narrowband echosounders. These include improved spectral-based target identification and significantly increased ability to resolve individual targets. Biological studies and surveys typically require accurate measures of backscatter strength and we present here a systematic and comprehensive explanation of how to derive quantitative estimates of target strength and volume backscattering, as a function of frequency from broadband echosounder signals., Comment: 20 pages, 5 figures

Knowledge distillation is classically a procedure where a neural network is trained on the output of another network along with the original targets in order to transfer knowledge between the architectures. The special case of self-distillation, where the network architectures are identical, has been observed to improve generalization accuracy. In this paper, we consider an iterative variant of self-distillation in a kernel regression setting, in which successive steps incorporate both model outputs and the ground-truth targets. This allows us to provide the first theoretical results on the importance of using the weighted ground-truth targets in self-distillation. Our focus is on fitting nonlinear functions to training data with a weighted mean square error objective function suitable for distillation, subject to $\ell_2$ regularization of the model parameters. We show that any such function obtained with self-distillation can be calculated directly as a function of the initial fit, and that infinite distillation steps yields the same optimization problem as the original with amplified regularization. Furthermore, we provide a closed form solution for the optimal choice of weighting parameter at each step, and show how to efficiently estimate this weighting parameter for deep learning and significantly reduce the computational requirements compared to a grid search., Comment: To be published at NeurIPS 2021; 21 pages, 14 figures

Linear functions of the site frequency spectrum (SFS) play a major role for understanding and investigating genetic diversity. Estimators of the mutation rate (e.g. based on the total number of segregating sites or average of the pairwise differences) and tests for neutrality (e.g. Tajima's D) are perhaps the most well-known examples. The distribution of linear functions of the SFS is important for constructing confidence intervals for the estimators, and to determine significance thresholds for neutrality tests. These distributions are often approximated using simulation procedures. In this paper we use multivariate phase-type theory to specify, characterize and calculate the distribution of linear functions of the site frequency spectrum. In particular, we show that many of the classical estimators of the mutation rate are distributed according to a discrete phase-type distribution. Neutrality tests, however, are generally not discrete phase-type distributed. For neutrality tests we derive the probability generating function using continuous multivariate phase-type theory, and numerically invert the function to obtain the distribution. A main result is an analytically tractable formula for the probability generating function of the SFS. Software implementation of the phase-type methodology is available in the R package phasty, and R code for the reproduction of our results is available as an accompanying vignette.