Family Of Metrics Research Articles

Decisional value scores: A new family of metrics for ethical AI-ML

AbstractResearch in ethical AI has made strides in quantitative expression of ethical values such as fairness, transparency, and privacy. Here we contribute to this effort by proposing a new family of metrics called “decisional value scores” (DVS). DVSs are scores assigned to a system based on whether the decisions it makes meet or fail to meet a particular standard (either individually, in total, or as a ratio or average over decisions made). Advantages of DVS include greater discrimination capacity between types of ethically relevant decisions and facilitation of ethical comparisons between decisions and decision-making systems, including across different modalities (for instance: human, machine, or coupled human–machine systems). After clarifying ambiguities in the concept of “decision” itself, including the question of how to individuate the decisions made by a system, we discuss the role and meaning of “decision” in common AI and machine learning approaches such as decision trees, neural networks, SVMs, and unsupervised classifiers. We then show how DVSs may be defined for several ethical values of interest, with an extended discussion of transparency. Finally, we explore how such metrics can be applied to real decision-making systems through two case studies: evaluations of LLMs for transparency; and evaluations of criminal risk assessment tools for utility, rights violations, fairness, and transparency.

The nonlinear stability of (n + 1)-dimensional FLRW spacetimes

We prove nonlinear Lyapunov stability of a family of “([Formula: see text])”-dimensional cosmological models of general relativity locally isometric to the Friedman–Lemaître–Robertson–Walker (FLRW) spacetimes including a positive cosmological constant. In particular, we show that the perturbed solutions to the Einstein–Euler field equations around a class of spatially compact FLRW metrics (for which the spatial slices are compact negative Einstein spaces in general and hyperbolic for the physically relevant [Formula: see text] case) arising from regular Cauchy data remain uniformly bounded and decay to a family of metrics with constant negative spatial scalar curvature. To accomplish this result, we employ an energy method for the coupled Einstein–Euler field equations in constant mean extrinsic curvature spatial harmonic (CMCSH) gauge. In order to handle Euler’s equations, we construct energy from a current that is similar to the one derived by Christodoulou [The Formation of Shocks in 3-dimensional Fluids, EMS Monographs in Mathematics, Vol. 2 (European Mathematical Society, 2007)] (and which coincides with Christodoulou’s current on the Minkowski space) and show that this energy controls the desired norm of the fluid degrees of freedom. The use of a fluid energy current together with the CMCSH gauge condition casts the Einstein–Euler field equations into a coupled elliptic–hyperbolic system. Utilizing the estimates derived from the elliptic equations, we first show that the gravity-fluid energy functional remains uniformly bounded in the expanding direction. Using this uniform boundedness property, we later obtain sharp decay estimates if a positive cosmological constant [Formula: see text] is included, which confirms that the accelerated expansion of the physical universe that is induced by the positive cosmological constant is sufficient to control the nonlinearities in the case of small data. A few physical consequences of this stability result are discussed.

Families of Metrics with Positive Scalar Curvature on Spectral Sequence Cobordisms

Families of Metrics with Positive Scalar Curvature on Spectral Sequence Cobordisms

Beyond correlation: optimal transport metrics for characterizing representational stability and remapping in neurons encoding spatial memory.

Spatial representations in the entorhinal cortex (EC) and hippocampus (HPC) are fundamental to cognitive functions like navigation and memory. These representations, embodied in spatial field maps, dynamically remap in response to environmental changes. However, current methods, such as Pearson's correlation coefficient, struggle to capture the complexity of these remapping events, especially when fields do not overlap, or transformations are non-linear. This limitation hinders our understanding and quantification of remapping, a key aspect of spatial memory function. We propose a family of metrics based on the Earth Mover's Distance (EMD) as a versatile framework for characterizing remapping. The EMD provides a granular, noise-resistant, and rate-robust description of remapping. This approach enables the identification of specific cell types and the characterization of remapping in various scenarios, including disease models. Furthermore, the EMD's properties can be manipulated to identify spatially tuned cell types and to explore remapping as it relates to alternate information forms such as spatiotemporal coding. We present a feasible, lightweight approach that complements traditional methods. Our findings underscore the potential of the EMD as a powerful tool for enhancing our understanding of remapping in the brain and its implications for spatial navigation, memory studies and beyond.

An abelian way approach to study random extended intervals and their ARMA processes

<abstract><p>An extended interval is a range $ A = [\underline{A}, \overline{A}] $ where $ \underline{A} $ may be bigger than $ \overline{A} $. This is not really natural, but is what has been used as the definition of an extended interval so far. In the present work we introduce a new, natural, and very intuitive way to see an extended interval. From now on, an extended interval is a subset of the Cartesian product $ {\mathbb R}\times {\mathbb Z}_2 $, where $ {\mathbb Z}_2 = \{0, 1\} $ is the set of directions; the direction $ 0 $ is for increasing intervals, and the direction $ 1 $ for decreasing ones. For instance, $ [3, 6]\times\{1\} $ is the decreasing version of $ [6, 3] $. Thereafter, we introduce on the set of extended intervals a family of metrics $ d_\gamma $, depending on a function $ \gamma(t) $, and show that there exists a unique metric $ d_\gamma $ for which $ \gamma(t)dt $ is what we have called an "adapted measure". This unique metric has very good properties, is simple to compute, and has been implemented in the software $ R $. Furthermore, we use this metric to {define variability for random extended intervals. We further study extended interval-valued ARMA} time series and prove the Wold decomposition theorem for stationary extended interval-valued times series.</p></abstract>

Comparison results for Gromov–Wasserstein and Gromov–Monge distances

Inspired by the Kantorovich formulation of optimal transport distance between probability measures on a metric space, Gromov–Wasserstein (GW) distances comprise a family of metrics on the space of isomorphism classes of metric measure spaces. In previous work, the authors introduced a variant of this construction which was inspired by the original Monge formulation of optimal transport; elements of the resulting family are referred to Gromov–Monge (GM) distances. These GM distances, and related ideas, have since become a subject of interest from both theoretical and applications-oriented perspectives. In this note, we establish several theoretical properties of GM distances, focusing on comparisons between GM and GW distances. In particular, we show that GM and GW distances are equal for non-atomic metric measure spaces. We also consider variants of GM distance, such as a Monge version of Sturm’s Lp-transportion distance, and give precise comparisons to GW distance. Finally, we establish bi-Hölder equivalence between GM distance and an isometry-invariant Monge optimal transport distance between Euclidean metric measure spaces that has been utilized in shape and image analysis applications.

Prediction rigidities for data-driven chemistry.

The widespread application of machine learning (ML) to the chemical sciences is making it very important to understand how the ML models learn to correlate chemical structures with their properties, and what can be done to improve the training efficiency whilst guaranteeing interpretability and transferability. In this work, we demonstrate the wide utility of prediction rigidities, a family of metrics derived from the loss function, in understanding the robustness of ML model predictions. We show that the prediction rigidities allow the assessment of the model not only at the global level, but also on the local or the component-wise level at which the intermediate (e.g. atomic, body-ordered, or range-separated) predictions are made. We leverage these metrics to understand the learning behavior of different ML models, and to guide efficient dataset construction for model training. We finally implement the formalism for a ML model targeting a coarse-grained system to demonstrate the applicability of the prediction rigidities to an even broader class of atomistic modeling problems.

Families of degenerating Poincaré–Einstein metrics on R4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^4$$\\end{document}

We provide the first example of continuous families of Poincaré–Einstein metrics developing cusps on the trivial topology R4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^4$$\\end{document}. We also exhibit families of metrics with unexpected degenerations in their conformal infinity only. These are obtained from the Riemannian version of an ansatz of Debever and Plebański–Demiański. We additionally indicate how to construct similar examples on more complicated topologies.

Characterizing Families of Spectral Similarity Scores and Their Use Cases for Gas Chromatography-Mass Spectrometry Small Molecule Identification.

Metabolomics provides a unique snapshot into the world of small molecules and the complex biological processes that govern the human, animal, plant, and environmental ecosystems encapsulated by the One Health modeling framework. However, this "molecular snapshot" is only as informative as the number of metabolites confidently identified within it. The spectral similarity (SS) score is traditionally used to identify compound(s) in mass spectrometry approaches to metabolomics, where spectra are matched to reference libraries of candidate spectra. Unfortunately, there is little consensus on which of the dozens of available SS metrics should be used. This lack of standard SS score creates analytic uncertainty and potentially leads to issues in reproducibility, especially as these data are integrated across other domains. In this work, we use metabolomic spectral similarity as a case study to showcase the challenges in consistency within just one piece of the One Health framework that must be addressed to enable data science approaches for One Health problems. Here, using a large cohort of datasets comprising both standard and complex datasets with expert-verified truth annotations, we evaluated the effectiveness of 66 similarity metrics to delineate between correct matches (true positives) and incorrect matches (true negatives). We additionally characterize the families of these metrics to make informed recommendations for their use. Our results indicate that specific families of metrics (the Inner Product, Correlative, and Intersection families of scores) tend to perform better than others, with no single similarity metric performing optimally for all queried spectra. This work and its findings provide an empirically-based resource for researchers to use in their selection of similarity metrics for GC-MS identification, increasing scientific reproducibility through taking steps towards standardizing identification workflows.

Symmetries of Ricci flows

Abstract In the present work, we find the Lie point symmetries of the Ricci flow on an n-dimensional manifold, and we introduce a method in order to reutilize these symmetries to obtain the Lie point symmetries of particular metrics. We apply this method to retrieve the Lie point symmetries of the Einstein equations (seen as a “static” Ricci flow) and of some particular types of metrics of interest, such as, on warped products of manifolds. Finally, we use the symmetries found to obtain invariant solutions of the Ricci flow for the particular families of metrics considered.

Staticity of asymptotically hyperbolic minimal mass extensions

In this paper, we define the Bartnik mass of a domain whose boundary is connected and compact, has scalar curvature bounded below −n(n − 1), and whose extensions are asymptotically hyperbolic manifolds. With this definition, we show that asymptotically hyperbolic admissible extensions of a domain that achieve the Bartnik mass must admit a static potential. Given a non-static admissible extension of a domain, we are able to construct a one-parameter family of metrics that are close to the original metric, have smaller mass, share the same bound on the scalar curvature, and contain the domain isometrically.

Increasing Intensity of Extreme Heatwaves: The Crucial Role of Metrics

AbstractIn weather and climate applications, a wide range of commonly employed heatwave intensity indices relies either on cumulative or averaged values of temperature‐based variables. In this study, by comparing four different heatwave intensity indices applied to reanalysis data we show that metrics based on cumulative or averaged values lead to important differences in the detection of the most intense events of the period 1950–2021. This suggests that particular attention is needed when using the two families of metrics for assessing heatwave intensity. Indices based on cumulative values should be preferred over the ones relying on temporal averages, better allowing for the comparison of events of different length. Under these considerations, one of the considered cumulative indices is used for characterizing heatwaves of the period 1950–2021, showing that heatwaves that were unlikely before 1986 have become almost 10 times more frequent and up to three times more intense during recent times.

Closeness Centrality on Uncertain Graphs

Centrality is a family of metrics for characterizing the importance of a vertex in a graph. Although a large number of centrality metrics have been proposed, a majority of them ignores uncertainty in graph data. In this article, we formulate closeness centrality on uncertain graphs and define the batch closeness centrality evaluation problem that computes the closeness centrality of a subset of vertices in an uncertain graph. We develop three algorithms, MS-BCC , MG-BCC, and MGMS-BCC , based on sampling to approximate the closeness centrality of the specified vertices. All these algorithms require to perform breadth-first searches (BFS) starting from the specified vertices on a large number of sampled possible worlds of the uncertain graph. To improve the efficiency of the algorithms, we exploit operation-level parallelism of the BFS traversals and simultaneously execute the shared sequences of operations in the breadth-first searches. Parallelization is realized at different levels in these algorithms. The experimental results show that the proposed algorithms can efficiently and accurately approximate the closeness centrality of the given vertices. MGMS-BCC is faster than both MS-BCC and MG-BCC because it avoids more repeated executions of the shared operation sequences in the BFS traversals.

A variational approach to SKT and balanced metrics

We investigate compact complex manifolds endowed with SKT or balanced metrics. In each case, we define a new functional whose critical points are proved to be precisely the Kähler metrics, if any, on the manifold. As general manifolds of either type need not admit Kähler metrics, this provides an approach to new obstructions to Kählerianity within these two families of metrics.

Generic properties of nonexpansive mappings on unbounded domains

We investigate typical properties of nonexpansive mappings on unbounded complete hyperbolic metric spaces. For two families of metrics of uniform convergence on bounded sets, we show that the typical nonexpansive mapping is a Rakotch contraction on every bounded subset and that there is a bounded set which is mapped into itself by this mapping. In particular, we obtain that the typical nonexpansive mapping in this setting has a unique fixed point which can be reached by iterating the mapping. Nevertheless, it turns out that the typical mapping is not a Rakotch contraction on the whole space and that it has the maximal possible Lipschitz constant of one on a residual subset of its domain. By typical we mean that the complement of the set of mappings with this property is σ-ϕ-porous, that is, small in a metric sense. For a metric of pointwise convergence, we show that the set of Rakotch contractions is meagre.

On the complementarity of ordinal patterns-based entropy and time asymmetry metrics.

Entropy and time asymmetry are two intertwined aspects of a system's dynamics, with the production of entropy marking a clear direction in the temporal dimension. In the last few years, metrics to quantify both properties in time series have been designed around the same concept, i.e., the use of ordinal patterns. In spite of this, the relationship between these two families of metrics is yet not well understood. In this contribution, we study this problem by constructing an entropy-time asymmetry plane and evaluating it on a large set of synthetic and real-world time series. We show how the two metrics can at times behave independently, the main reason being the presence of patterns with turning points; due to this, they yield complementary information about the underlying systems, and they have different discriminating performance.

An attack volume metric

AbstractFor more than a decade, the notion of attack surface has been used to define the set of vulnerable assets that an adversary may exploit to penetrate a system, and various metrics have been developed to quantify the extent of a system's attack surface. However, most approaches to tackle this problem have failed to consider the complex interdependencies that exist between the many components of a distributed system, its vulnerabilities, and its configuration parameters. In our work, building upon previous research on vulnerability metrics and on graphical models to capture such interdependencies, we propose a novel approach to evaluate the potential risk associated with exposed vulnerabilities by studying how the effect of each vulnerability exploit propagates through chains of dependencies. Our analysis goes beyond the scope of traditional attack surface metrics, and considers the depth and implications of potential attacks, leading to the definition of a new family of metrics, which we refer to as attack volume metrics. We present experimental results illustrating how the proposed metric scales for graphs of realistic sizes, and illustrate its application to real‐world testbeds.

A multi-parameter family of metrics on stiefel manifolds and applications

<abstract><p>The real (compact) Stiefel manifold realized as set of orthonormal frames is considered as a pseudo-Riemannian submanifold of an open subset of a vector space equipped with a multi-parameter family of pseudo-Riemannian metrics. This family contains several well-known metrics from the literature. Explicit matrix-type formulas for various differential geometric quantities are derived. The orthogonal projections onto tangent spaces are determined. Moreover, by computing the metric spray, the geodesic equation as an explicit second order matrix valued ODE is obtained. In addition, for a multi-parameter subfamily, explicit matrix-type formulas for pseudo-Riemannian gradients and pseudo-Riemannian Hessians are derived. Furthermore, an explicit expression for the second fundamental form and an explicit formula for the Levi-Civita covariant derivative are obtained. Detailed proofs are included.</p></abstract>

A Teichmüller space for negatively curved surfaces

AbstractWe first describe the action of the fundamental group of a closed surface of variable negative curvature on the oriented geodesics in its universal covering in terms of a naturally defined flat connection whose holonomy lies in the group of Hamiltonian diffeomorphisms of . Consideration of the holonomy necessitates an extension from Riemannian to Finsler metrics. The second part of the paper follows the Higgs bundle approach to flat connections adapted to this infinite‐dimensional group and focuses on a family of metrics, relying on a construction of O. Biquard, which is parametrised by the infinite‐dimensional space of CR functions on the unit circle bundle of a hyperbolic surface. This generates an alternative approach to defining a connection and offers the possibility of this vector space representing a moduli space which generalizes and includes the classical Teichmüller space.

Net Reclassification Index Statistics Do Not Help Assess New Risk Models.

When evaluating a new risk factor for disease (eg, a measurement from imaging studies), many investigators examine its value above and beyond existing biomarkers and risk factors. They compare the performance of an "old" risk model using established predictors and a "new" risk model that adds the new factor. Net reclassification index (NRI) statistics are a family of metrics for comparing two risk models. NRI statistics became popular in some medical fields and have appeared in high-impact journals. This article reviews NRI statistics and describes several issues with them. Problems include unacceptable statistical behavior, incorrect statistical inferences, and lack of interpretability. NRI statistics are unhelpful (at best) and misleading (at worst).