680 research outputs found

    Generative Learning of Continuous Data by Tensor Networks

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    Beyond their origin in modeling many-body quantum systems, tensor networks have emerged as a promising class of models for solving machine learning problems, notably in unsupervised generative learning. While possessing many desirable features arising from their quantum-inspired nature, tensor network generative models have previously been largely restricted to binary or categorical data, limiting their utility in real-world modeling problems. We overcome this by introducing a new family of tensor network generative models for continuous data, which are capable of learning from distributions containing continuous random variables. We develop our method in the setting of matrix product states, first deriving a universal expressivity theorem proving the ability of this model family to approximate any reasonably smooth probability density function with arbitrary precision. We then benchmark the performance of this model on several synthetic and real-world datasets, finding that the model learns and generalizes well on distributions of continuous and discrete variables. We develop methods for modeling different data domains, and introduce a trainable compression layer which is found to increase model performance given limited memory or computational resources. Overall, our methods give important theoretical and empirical evidence of the efficacy of quantum-inspired methods for the rapidly growing field of generative learning.Comment: 21 pages, 15 figure

    On the Cauchy problem of defocusing mKdV equation with finite density initial data: long time asymptotics in soliton-less regions

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    We investigate the long-time asymptotics for the solutions to the Cauchy problem of defocusing modified Kortweg-de Vries (mKdV) equation with finite density initial data. The present paper is the subsequent work of our previous paper [arXiv:2108.03650], which gives the soliton resolution for the defocusing mKdV equation in the central asymptotic sector {(x,t):∣ξ∣<6}\{(x,t): \vert \xi \vert<6\} with ξ:=x/t\xi:=x/t. In the present paper, via the Riemann-Hilbert (RH) problem associated to the Cauchy problem, the long-time asymptotics in the soliton-less regions {(x,t):∣ξ∣>6,∣ξ∣=O(1)}\{(x,t): \vert \xi \vert>6, |\xi|=\mathcal{O}(1)\} for the defocusing mKdV equation are further obtained. It is shown that the leading term of the asymptotics are in compatible with the ``background solution'' and the error terms are derived via rigorous analysis.Comment: 51 page

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Contributions in functional data analysis and functional-analytic statistics

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    Functional data analysis is the study of statistical algorithms which are applied in the scenario when the observed data is a collection of functions. Since this type of data is becoming cheaper and easier to collect, there is an increased need to develop statistical tools to handle such data. The first part of this thesis focuses on deriving distances between distributions over function spaces and applying these to two-sample testing, goodness-of-fit testing and sample quality assessment. This presents a wide range of contributions since currently there exists either very few or no methods at all to tackle these problems for functional data. The second part of this thesis adopts the functional-analytic perspective to two statistical algorithms. This is a perspective where functions are viewed as living in specific function spaces and the tool box of functional analysis is applied to identify and prove properties of the algorithms. The two algorithms are variational Gaussian processes, used widely throughout machine learning for function modelling with large observation data sets, and functional statistical depth, used widely as a means to evaluate outliers and perform testing for functional data sets. The results presented contribute a taxonomy of the variational Gaussian process methodology and multiple new results in the theory of functional depth including the open problem of providing a depth which characterises distributions on function spaces.Open Acces

    Data analysis with merge trees

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    Today’s data are increasingly complex and classical statistical techniques need growingly more refined mathematical tools to be able to model and investigate them. Paradigmatic situations are represented by data which need to be considered up to some kind of trans- formation and all those circumstances in which the analyst finds himself in the need of defining a general concept of shape. Topological Data Analysis (TDA) is a field which is fundamentally contributing to such challenges by extracting topological information from data with a plethora of interpretable and computationally accessible pipelines. We con- tribute to this field by developing a series of novel tools, techniques and applications to work with a particular topological summary called merge tree. To analyze sets of merge trees we introduce a novel metric structure along with an algorithm to compute it, define a framework to compare different functions defined on merge trees and investigate the metric space obtained with the aforementioned metric. Different geometric and topolog- ical properties of the space of merge trees are established, with the aim of obtaining a deeper understanding of such trees. To showcase the effectiveness of the proposed metric, we develop an application in the field of Functional Data Analysis, working with functions up to homeomorphic reparametrization, and in the field of radiomics, where each patient is represented via a clustering dendrogram

    Electromagnetic scattering from thin tubular objects and an application in electromagnetic chirality

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    Asymptotic perturbation formulas characterize the effective behavior of waves as the volume of the scattering object tends to zero. In this work, wave propagation is described by time-harmonic Maxwell\u27s equations in free space and the corresponding scattering objects are thin tubular objects that feature a different electric permittivity and a different magnetic permeability than their surrounding medium. For this setting, we derive an asymptotic representation of the scattered electric field away from the thin tubular object and use the corresponding leading order term in a shape identification problem and for designing highly electromagnetically chiral objects. In inverse problems, the leading order term may be used to find the center curve of a thin wire that is supposed to emit a scattered field, which is reasonably close to a given measured field. For the optimal design of electromagnetically chiral structures, the representation formula provides an explicit formula for the leading order term of an asymptotic far field operator expansion. A chirality measure, usually requiring the far field operator, will now map aforementioned leading order term to a value between 00 and 11 dependent on the level of electromagnetic chirality of the thin tubular scatterer. This approximation greatly simplifies the challenge to maximize the chirality measure with respect to thin tubular objects. The fact that neither the evaluation of the leading order term nor the calculation of corresponding derivatives require a Maxwell system to be solved implies that the shape optimization scheme is highly efficient compared to shape optimization algorithms that use e.g. domain derivatives. In the visible range, the metallic nanowires obtained by our optimization scheme attain high values of electromagnetic chirality and even exceed those attained by traditional metallic helices

    Long-time asymptotics of a complex cubic Camassa-Holm equation

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    n this paper, we study the Cauchy problem of the following complex cubic Camassa-Holm (ccCH) equation mt=bux+12[m(|u|2−|ux|2)]x−12m(uu¯x−uxu¯),m=u−uxx, where b is an arbitrary real constant. %By applying ∂¯-steepest descent method, l Long-time asymptotics of the equation is obtained through the ∂¯-steepest descent method. Firstly, based on the spectral analysis of the Lax pair and scattering matrix, the solution of the equation is able to be constructed %can be expressed by %the solution of via solving the corresponding Riemann-Hilbert problem (RHP). Then, we present %obtain different long time asymptotic expansions of the solution u(y,t) in different space-time solitonic regions of ξ=y/t. The half-plane (y,t):−∞0 is divided into four asymptotic regions: ξ∈(−∞,−1), ξ∈(−1,0), ξ∈(0,18) and ξ∈(18,+∞). When ξ falls in (−∞,−1)∪(18,+∞), no stationary phase point of the phase function θ(z) exists on the jump profile in the space-time region. In this case, corresponding asymptotic approximations can be characterized with an N(Λ)-solitons with diverse residual error order O(t−1+2ε). %While There are four stationary phase points and eight stationary phase points on the jump curve as ξ∈(−1,0) and ξ∈(0,18), respectively. The corresponding asymptotic form is accompanied by a residual error order O(t−34)

    Long time and Painlev\'{e}-type asymptotics for the defocusing Hirota equation with finite density initial data

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    In this work, we consider the Cauchy problem for the defocusing Hirota equation with a nonzero background \begin{align} \begin{cases} iq_{t}+\alpha\left[q_{xx}-2\left(\left\vert q\right\vert^{2}-1\right)q\right]+i\beta\left(q_{xxx}-6\left\vert q\right\vert^{2}q_{x}\right)=0,\quad (x,t)\in \mathbb{R}\times(0,+\infty),\\ q(x,0)=q_{0}(x),\qquad \underset{x\rightarrow\pm\infty 1}{\lim} q_{0}(x)=\pm 1, \qquad q_{0}\mp 1\in H^{4,4}(\mathbb{R}). \end{cases} \nonumber \end{align} According to the Riemann-Hilbert problem representation of the Cauchy problem and the ∂ˉ\bar{\partial} generalization of the nonlinear steepest descent method, we find different long time asymptotics types for the defocusing Hirota equation in oscillating region and transition region, respectively. For the oscillating region ξ<−8\xi<-8, four phase points appear on the jump contour R\mathbb{R}, which arrives at an asymptotic expansion,given by \begin{align} q(x,t)=-1+t^{-1/2}h+O(t^{-3/4}).\nonumber \end{align} It consists of three terms. The first term −1-1 is leading term representing a nonzero background, the second term t−1/2ht^{-1/2}h originates from the continuous spectrum and the third term O(t−3/4)O(t^{-3/4}) is the error term due to pure ∂ˉ\bar{\partial}-RH problem. For the transition region ∣ξ+8∣t2/3<C\vert\xi+8\vert t^{2/3}<C, three phase points raise on the jump contour R\mathbb{R}. Painlev\'{e} asymptotics expansion is obtained \begin{align} q(x,t)=-1-(\frac{15}{4}t)^{-1/3}\varrho+O(t^{-1/2}),\nonumber \end{align} in which the leading term is a solution to the Painlev\'{e} II equation, the last term is a residual error being from pure ∂ˉ\bar{\partial}-RH problem and parabolic cylinder model

    Adaptive Discontinuous Galerkin Methods for Variational Inequalities with Applications to Phase Field Models

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    Solutions of variational inequalities often have limited regularity. In particular, the nonsmooth parts are local, while other parts of the solution have higher regularity. To overcome this limitation, we apply hp-adaptivity, which uses a combination of locally finer meshes and varying polynomial degrees to separate the different features of the the solution. For this, we employ Discontinuous Galerkin (DG) methods and show some novel error estimates for the obstacle problem which emphasize the use in hp-adaptive algorithms. Besides this analysis, we present how to efficiently compute numerical solutions using error estimators, fast algebraic solvers which can also be employed in a parallel setup, and discuss implementation details. Finally, some numerical examples and applications to phase field models are presented
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