Notion Of Distance Research Articles

Machine Learning Monte Carlo Approaches and Statistical Physics Notions to Characterize Bacterial Species in Human Microbiota

Recent studies have shown correlations between the microbiota’s composition and various health conditions. Machine learning (ML) techniques are essential for analyzing complex biological data, particularly in microbiome research. ML methods help analyze large datasets to uncover microbiota patterns and understand how these patterns affect human health. This study introduces a novel approach combining statistical physics with the Monte Carlo (MC) methods to characterize bacterial species in the human microbiota. We assess the significance of bacterial species in different age groups by using notions of statistical distances to evaluate species prevalence and abundance across age groups and employing MC simulations based on statistical mechanics principles. Our findings show that the microbiota composition experiences a significant transition from early childhood to adulthood. Species such as Bifidobacterium breve and Veillonella parvula decrease with age, while others like Agathobaculum butyriciproducens and Eubacterium rectale increase. Additionally, low-prevalence species may hold significant importance in characterizing age groups. Finally, we propose an overall species ranking by integrating the methods proposed here in a multicriteria classification strategy. Our research provides a comprehensive tool for microbiota analysis using statistical notions, ML techniques, and MC simulations.

Exploring The Didactic Transposition of The Metric Space

The didactic transposition is used to adapt scientific knowledge into taught knowledge for students through two steps, namely external and internal transposition. This process is crucial when applied to mathematics content as it enhances the understanding of complex concepts and makes knowledge more accessible to learners. To create effective lessons and scholarly knowledge objects, teachers must establish connections between their learned knowledge and the lesson structure. Metric and probabilistic metric spaces are two complex structures of knowledge that require understanding and teaching. This article explores the didactic transposition approach for teaching the concept of metric space, including probabilistic metric space, through external and internal steps of transposition, it provides two examples of transposing the Metric Distance notion in the Euclidean space, as well as two examples of the probabilistic metric space, aiming to transpose function distribution distance. It also outlines strategies for introducing metric spaces to students, referencing examples while acknowledging the teaching obstacles posed by this concept.

The Gromov–Wasserstein Distance Between Spheres

AbstractThe Gromov–Wasserstein distance—a generalization of the usual Wasserstein distance—permits comparing probability measures defined on possibly different metric spaces. Recently, this notion of distance has found several applications in Data Science and in Machine Learning. With the goal of aiding both the interpretability of dissimilarity measures computed through the Gromov–Wasserstein distance and the assessment of the approximation quality of computational techniques designed to estimate the Gromov–Wasserstein distance, we determine the precise value of a certain variant of the Gromov–Wasserstein distance between unit spheres of different dimensions. Indeed, we consider a two-parameter family $$\{d_{{{\text {GW}}}p,q}\}_{p,q=1}^{\infty }$$ { d GW p , q } p , q = 1 ∞ of Gromov–Wasserstein distances between metric measure spaces. By exploiting a suitable interaction between specific values of the parameters p and q and the metric of the underlying spaces, we are able to determine the exact value of the distance $$d_{{{\text {GW}}}4,2}$$ d GW 4 , 2 between all pairs of unit spheres of different dimensions endowed with their Euclidean distance and their uniform measure.

Approximate Quantum Codes From Long Wormholes

We discuss families of approximate quantum error correcting codes which arise as the nearly-degenerate ground states of certain quantum many-body Hamiltonians composed of non-commuting terms. For exact codes, the conditions for error correction can be formulated in terms of the vanishing of a two-sided mutual information in a low-temperature thermofield double state. We consider a notion of distance for approximate codes obtained by demanding that this mutual information instead be small, and we evaluate this mutual information for the SYK model and for a family of low-rank SYK models. After an extrapolation to nearly zero temperature, we find that both kinds of models produce fermionic codes with constant rate as the number, N, of fermions goes to infinity. For SYK, the distance scales as N1/2, and for low-rank SYK, the distance can be arbitrarily close to linear scaling, e.g. N.99, while maintaining a constant rate. We also consider an analog of the no low-energy trivial states property which we dub the no low-energy adiabatically accessible states property and show that these models do have low-energy states that can be prepared adiabatically in a time that does not scale with system size N. We discuss a holographic model of these codes in which the large code distance is a consequence of the emergence of a long wormhole geometry in a simple model of quantum gravity.

Quantifying Quantum Chaos through Microcanonical Distributions of Entanglement

A characteristic feature of “quantum chaotic” systems is that their eigenspectra and eigenstates display universal statistical properties described by random matrix theory (RMT). However, eigenstates of local systems also encode structure beyond RMT. To capture this feature, we introduce a framework that allows us to compare the properties of eigenstates in local systems with those of pure random states. In particular, our framework defines a notion of distance between quantum state ensembles that utilizes the Kullback-Leibler divergence to compare the microcanonical distribution of entanglement entropy (EE) of eigenstates with a reference RMT distribution generated by pure random states (with appropriate constraints). This notion gives rise to a quantitative metric for quantum chaos that not only accounts for averages of the distributions but also higher moments. The differences in moments are compared on a highly resolved scale set by the standard deviation of the RMT distribution, which is exponentially small in system size. As a result, the metric can distinguish between chaotic and integrable behaviors and, in addition, quantify and compare the of chaos (in terms of proximity to RMT behavior) between two systems that are assumed to be chaotic. We implement our framework in local, minimally structured, Floquet random circuits, as well as a canonical family of many-body Hamiltonians, the mixed-field Ising model (MFIM). Importantly, for Hamiltonian systems, we find that the reference random distribution must be appropriately constrained to incorporate the effect of energy conservation in order to describe the ensemble properties of midspectrum eigenstates. The metric captures deviations from RMT across all models and parameters, including those that have been previously identified as strongly chaotic, and for which other diagnostics of chaos such as level spacing statistics look strongly thermal. In Floquet circuits, the dominant source of deviations is the second moment of the distribution, and this persists for all system sizes. For the MFIM, we find significant variation of the KL divergence in parameter space. Notably, we find a small region where deviations from RMT are minimized, suggesting that “maximally chaotic” Hamiltonians may exist in fine-tuned pockets of parameter space. Published by the American Physical Society 2024

A Heat Method for Generalized Signed Distance

We introduce a method for approximating the signed distance function (SDF) of geometry corrupted by holes, noise, or self-intersections. The method implicitly defines a completed version of the shape, rather than explicitly repairing the given input. Our starting point is a modified version of the heat method for geodesic distance, which diffuses normal vectors rather than a scalar distribution. This formulation provides robustness akin to generalized winding numbers (GWN) , but provides distance function rather than just an inside/outside classification. Our formulation also offers several features not common to classic distance algorithms, such as the ability to simultaneously fit multiple level sets, a notion of distance for geometry that does not topologically bound any region, and the ability to mix and match signed and unsigned distance. The method can be applied in any dimension and to any spatial discretization, including triangle meshes, tet meshes, point clouds, polygonal meshes, voxelized surfaces, and regular grids. We evaluate the method on several challenging examples, implementing normal offsets and other morphological operations directly on imperfect curve and surface data. In many cases we also obtain an inside/outside classification dramatically more robust than the one obtained provided by GWN.

Amplitudes in persistence theory

The use of persistent homology in applications is justified by the validity of certain stability results. At the core of such results is a notion of distance between the invariants that one associates with data sets. Here we introduce a general framework to compare distances and invariants in multiparameter persistence, where there is no natural choice of invariants and distances between them. We define amplitudes, monotone, and subadditive invariants that arise from assigning a non-negative real number to objects of an abelian category. We then present different ways to associate distances to such invariants, and we provide a classification of classes of amplitudes relevant to topological data analysis. In addition, we study the relationships as well as the discriminative power of such amplitude distances arising in topological data analysis scenarios.

Exploring a modification of [formula omitted] convergence

In the work by M. C. Lee, A. Naber, and R. Neumayer (Lee et al., 2023) a beautiful ɛ-regularity theorem is proved under small negative scalar curvature and entropy bounds. In that paper, the dp distance for Riemannian manifolds is introduced and the quantitative stability results are given in terms of this notion of distance, with important examples showing why other existing notions of convergence are not adequate in their setting. Due to the presence of an entropy bound, the possibility of long, thin splines forming along a sequence of Riemannian manifolds whose scalar curvature is becoming almost positive is ruled out. It is important to rule out such examples since the dp distance is not well behaved in the presences of splines that persist in the limit. Since there are many geometric stability conjectures (Gromov, 2023; Sormani, 2023) where we want to allow for the presence of splines that persist in the limit, it is crucial to be able to modify the dp distance to retain its positive qualities and prevent it from being sensitive to splines. In this paper we explore one such modification of the dp distance and give a theorem which allows one to estimate the modified dp distance, which we expect to be useful in practice.

Linear-time logics -- a coalgebraic perspective

We describe a general approach to deriving linear-time logics for a wide variety of state-based, quantitative systems, by modelling the latter as coalgebras whose type incorporates both branching and linear behaviour. Concretely, we define logics whose syntax is determined by the type of linear behaviour, and whose domain of truth values is determined by the type of branching behaviour, and we provide two semantics for them: a step-wise semantics akin to that of standard coalgebraic logics, and a path-based semantics akin to that of standard linear-time logics. The former semantics is useful for model checking, whereas the latter is the more natural semantics, as it measures the extent with which qualitative properties hold along computation paths from a given state. Our main result is the equivalence of the two semantics. We also provide a semantic characterisation of a notion of logical distance induced by these logics. Instances of our logics support reasoning about the possibility, likelihood or minimal cost of exhibiting a given linear-time property.

Schatten class Hankel operators on doubling Fock spaces and the Berger-Coburn phenomenon

Using the notion of integral distance to analytic functions, we give a characterization of Schatten class Hankel operators acting on doubling Fock spaces on the complex plane and use it to show that for f∈L∞, if Hf is Hilbert-Schmidt, then so is Hf¯. This property is known as the Berger-Coburn phenomenon. When 0<p≤1, we show that the Berger-Coburn phenomenon fails for a large class of doubling Fock spaces. Along the way, we illustrate our results for the canonical weights |z|m when m>0.

Simultaneous consideration of time and cost impacts of machine failures on cellular manufacturing systems

Cellular manufacturing systems (CMS) are essential for achieving efficient production processes, and their success relies on effective cell formation and layout design. This research paper presents a novel nonlinear mathematical programming model that employs the rectilinear distance notion to define the layout within a continuous space. The proposed model incorporates the stochastic nature of machine failures, considering the unreliability with the inclusion of a stochastic time between failures. With a bi-objective approach, the model aims to minimize inter and intra-cell movements of parts, as well as the associated costs related to exceptional elements (EEs), cell reconfigurations, and machine failures. To optimize the proposed model, several multi-objective meta-heuristic algorithms, including the multi-objective particle swarm optimization (MOPSO), multi-objective taboo search (MOTS), and non-dominated sorting genetic algorithm (NSGA-II), are introduced. The effectiveness of these algorithms is validated through numerical instances and a real case study. The results indicate that the MOPSO algorithm exhibits superior performance in optimizing various multi-objective criteria. The presented mathematical model and algorithms provide valuable tools for manufacturers to optimize their CMS, resulting in reduced costs, enhanced production efficiency, and increased competitiveness. By considering the simultaneous effects of time and cost associated with machine failures, this research offers practical insights and solutions for improving the performance of cellular manufacturing systems.

Predicting Progression From Mild Cognitive Impairment to Alzheimer's Dementia With Adversarial Attacks.

Early diagnosis of Alzheimer's disease plays a crucial role in treatment planning that might slow down the disease's progression. This problem is commonly posed as a classification task performed by machine learning and deep learning techniques. Although data-driven techniques set the state-of-the-art in many domains, the scale of the available datasets in Alzheimer's research is not sufficient to learn complex models from patient data. This study proposes a simple yet promising framework to predict the conversion from Mild Cognitive Impairment (MCI) to Alzheimer's Disease (AD). The proposed framework comprises a shallow neural network for binary classification and a single-step gradient-based adversarial attack to find an adversarial progression direction in the input space. The step size required for the adversarial attack to change a patient's diagnosis from MCI to AD indicates the distance to the decision boundary. The patient's diagnosis at the next visit is predicted by employing this notion of distance to the decision boundary. We also present a potential application of the proposed framework to patient subtyping. Experiments with two publicly available datasets for Alzheimer's disease research imply that the proposed framework can predict MCI-to-AD conversions and assist in subtyping by only training a shallow neural network.

International students’ cultural engagement through constructing distance or proximity

International students’ contact and engagement with various cultures has received increased scholarly attention. This scholarship tends to either celebrate students’ cosmopolitanism or highlight their difficulties ‘adapting’ in their receiving countries. In this paper, we examine students’ own perceptions of and engagement with their sending and receiving countries’ cultures through the dialectic of distance and proximity, gleaned from mobilities studies. Based on 20 in-depth online interviews with Vietnamese nationals studying in Vancouver and Paris, our analysis highlights how these students construct or deconstruct notions of distance and proximity between Vietnam and their receiving countries (i.e. France and Canada), as well as between themselves and each of these countries. First, we examine how, before their departure, students cultivate a sense of cultural proximity to their geographically distant countries of destination, through studying and consuming media in French or English. Second, we address students’ rapport with French and Canadian societies as well as their sense of proximity to or distance from Vietnamese culture while studying in France and Canada. We examine how these (de-)constructions of distance can be related to students’ cosmopolitanism. We argue that notions of distance and proximity help foster a nuanced understanding of international students’ mobilities and cosmopolitanism.

Regions, extensions, distances, diameters

AbstractExtended simple regions have been the focus of recent developments in philosophical logic, metaphysics, and philosophy of physics. However, only a handful of works provides a rigorous characterization of an extended simple region. In particular, a recent paper in this journal defends a definition based on an extrinsic notion of least distance. Call it the Least Distance proposal. This paper provides the first assessment of it. It argues that Least Distance faces difficulties and drawbacks. The paper then goes on to suggest a different proposal, the Diameter proposal that is able to handle such drawbacks and difficulties.

Signature-based validation of real-world economic scenarios

AbstractMotivated by insurance applications, we propose a new approach for the validation of real-world economic scenarios. This approach is based on the statistical test developed by Chevyrev and Oberhauser ((2022) Journal of Machine Learning Research, 23(176), 1–42.) and relies on the notions of signature and maximum mean distance. This test allows to check whether two samples of stochastic processes paths come from the same distribution. Our contribution is to apply this test to a variety of stochastic processes exhibiting different pathwise properties (Hölder regularity, autocorrelation, and regime switches) and which are relevant for the modelling of stock prices and stock volatility as well as of inflation in view of actuarial applications.

Persistent homotopy groups of metric spaces

In this paper, we study notions of persistent homotopy groups of compact metric spaces. We pay particular attention to the case of fundamental groups, for which we obtain a more precise description via a persistent version of the notion of discrete fundamental groups due to Berestovskii–Plaut and Barcelo et al. Under fairly mild assumptions on the spaces, we prove that the persistent fundamental group admits a tree structure which encodes more information than its persistent homology counterpart. We also consider the rationalization of the persistent homotopy groups and by invoking results of Adamaszek–Adams and Serre, we completely characterize them in the case of the circle. Finally, we establish that persistent homotopy groups enjoy stability in the Gromov–Hausdorff sense. We then discuss several implications of this result including that the critical spectrum of Plaut et al. is also stable under this notion of distance.

Domain walls and distances in discrete landscapes

We explore a notion of distance between vacua of a discrete landscape that takes into account scalar potentials and fluxes via transitions mediated by domain walls. Such settings commonly arise in supergravity and string compactifications with stabilized moduli. We derive general bounds and simple estimates in supergravity which constrain deviations from the ordinary swampland distance conjecture based on moduli space geodesics, and we connect this picture to renormalization group flows via holography.

Low-Shot Unsupervised Visual Anomaly Detection via Sparse Feature Representation.

Visual anomaly detection is an essential component in modern industrial manufacturing. Existing studies using notions of pairwise similarity distance between a test feature and nominal features have achieved great breakthroughs. However, the absolute similarity distance lacks certain generalizations, making it challenging to extend the comparison beyond the available samples. This limitation could potentially hamper anomaly detection performance in scenarios with limited samples. This article presents a novel sparse feature representation anomaly detection (SFRAD) framework, which formulates the anomaly detection as a sparse feature representation problem; and notably proposes an anomaly score by orthogonal matching pursuit (ASOMP) as a novel detection metric. Specifically, SFRAD calculates the Gaussian kernel distance between the test feature and its sparse representation in the nominal feature space for anomaly detection. Here, the orthogonal matching pursuit (OMP) algorithm is adopted to achieve the sparse feature representation. Moreover, to construct a low-redundancy memory bank storing the basis features for sparse representation, a novel basis feature sampling (BFS) algorithm is proposed by considering both the maximum coverage and the optimum feature representation simultaneously. As a result, SFRAD incorporates both the advantages of absolute similarity and linear representation; and this enhances the generalization in low-shot scenarios. Extensive experiments on the MVTec anomaly detection (MVTec AD), Kolektor surface-defect dataset (KolektorSDD), Kolektor surface-defect dataset 2 (KolektorSDD2), MVTec logical constraints anomaly detection (MVTec LOCO AD), Visual anomaly (VISA), Modified national institute of standards and technology (MNIST), and CIFAR-10 datasets demonstrate that our proposed SFRAD outperforms the previous methods and achieves state-of-the-art unsupervised anomaly detection performance. Notably, significantly improved outcomes and results have also been achieved on low-shot anomaly detection. Code is available at https://github.com/fanghuisky/SFRAD.

Exploring elliptic geometry in neural networks : The concept of elliptic neural networks (ENN)

The exploration of non-Euclidean geometries in the context of neural network architectures presents a novel avenue for enhancing the processing of complex data structures. This paper introduces the concept of Elliptic Neural Networks (ENN), a framework that integrates the intrinsic properties of elliptic geometry into the fabric of neural computation. Characterized by constant positive curvature and closed geodesic paths, elliptic geometry provides a unique perspective for representing data, especially advantageous for configurations exhibiting cyclic and dense hierarchical interconnections. We begin by delineating the fundamental aspects of elliptic geometry, focusing on its impact on conventional notions of distance and global structural interpretation within data sets. This foundational understanding paves the way for our proposed mathematical schema for ENNs. Within this framework, we articulate the methodologies for embedding data in elliptic spaces and adapt neural network operations by encompassing neuron activation, signal propagation, and learning paradigms to conform to the topological idiosyncrasies of elliptic geometry. In this paper addressing the implementation of ENNs, we discuss the computational challenges and the development of novel algorithms requisite for navigating the elliptic geometric landscape. The prospective applications of ENNs are extensive and diverse, ranging from the enhancement of cyclic pattern recognition in time-series analysis to more sophisticated representations in graph-based data and natural language processing tasks. This theoretical exposition aims to set the groundwork for subsequent empirical research to validate the proposed model and assess its practical performance relative to conventional neural network architectures. By harnessing the distinctive characteristics of elliptic geometry, ENNs mark a significant stride towards enriching the arsenal of machine learning methodologies, potentially leading to more versatile and efficacious computational tools in data analysis and artificial intelligence.

Machine Learning based on Probabilistic Models Applied to Medical Data: The Case of Prostate Cancer

The growth in the amount of data in companies puts analysts in difficulties when extracting hidden knowledge from data. Several models have emerged that focus on the notion of distances while ignoring the notion of conditional probability density. This research study focuses on segmentation using mixture models and Bayesian networks for medical data mining. As enterprise data becomes large, there is a way to apply data mining methods to make sense of it using classification methods. We designed different models with different architectures and then applied these models to the medical database. The algorithms were implemented for the real data. The objective is to classify individuals according to the conditional probability density of random variables, in addition to identifying causalities between traits from tests of conditional independence and a correlation measure, both based on χ2. After a quick illustration of several models (decision tree, SVM, K-means, Bayes), we applied our method to data from an epidemiological study (done at the University of Kinshasa University clinics) of case-control of prostate cancer. Thus, we found after interpretation of the results followed by discussion that our model allows us to classify a new individual with an accuracy of 96%.