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Research draft

metric function

vr.tr.metric-function · XCT.QLT

Let an agent explain metric functions, relay axioms, examples and uses from mathematics sources, describe the named examples and distinguish the metric tensor sense, flag the misfiled alias, and distinguish metrics from norms, pseudometrics, similarity measures and performance metrics in business or software.

Thing Registry Cross-cutting context

Research draft, second pass

A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.

written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify

Researched by: Claude

Purpose and description

Let an agent explain metric functions, relay axioms, examples and uses from mathematics sources, describe the named examples and distinguish the metric tensor sense, flag the misfiled alias, and distinguish metrics from norms, pseudometrics, similarity measures and performance metrics in business or software.

In mathematics, a function that defines a distance between each pair of elements of a set, satisfying non-negativity with zero only for identical points, symmetry and the triangle inequality, making the set a metric space; examples the registry aliases name include Minkowski distance, the Lp family that includes Euclidean and Manhattan distance, and graph distance, the length of the shortest path between vertices, while the Kerr metric, spacetime metrics and the Fubini-Study metric are metric tensors in differential geometry and general relativity, a related but different sense in which a metric measures lengths infinitesimally and may not be positive definite, and Wikidata:Statistics is a project page and appears misfiled. Metrics are used in analysis, topology, machine learning and data science.

What it is for: Measuring distance in mathematical spaces.

It can be explain axioms; relay examples; distinguish metric tensors; flag misfiled aliases.

Distinguishing features

Triangle inequality

Symmetry

Defines topology

Many concrete examples

What it looks like

Not a physical object; defined by formulas and axioms.

Physical character

metric space concept: 1906 year - Maurice Frechet

axioms: identity, symmetry, triangle inequality list - plus non-negativity

Kerr metric: 1963 year - Roy Kerr

How it is recognised

Distance function satisfying metric axioms

Kerr metric, spacetime metric, Wikidata:Statistics, Minkowski distance, graph distance, Fubini-Study metric

Norms measure vector size; pseudometrics allow zero distance between distinct points; similarity measures may not satisfy axioms; performance metrics are indicators

Related models

is a kind of - in registry terms

function

defines -

metric space

is contrasted with - related sense

metric tensor

is contrasted with -

norm (mathematics)

In practice

Families and kinds

Lp and Minkowski distances

graph and path metrics

edit and string metrics

Riemannian and pseudo-Riemannian metric tensors

Standards and regulation

Not applicable

Failure modes and hazards

Confusing metric functions and metric tensors

Using non-metric similarities where axioms are assumed

Misfiled project page alias

Also called

Kerr metricspacetime metricWikidata:StatisticsMinkowski distancegraph distanceFubini–Study metricLévy metricproduct metricWasserstein metricVariation of informationstring metricphylogenetic distancegraph diametergraph radiusdegree of kinshipgraph eccentricityEarth mover's distanceinclusion measure

Where this came from

wikidata · CC0 1.0

Source repository Registry entry point

Drafted structure

Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.

Understand What a metric is.

Science.

Definition

Definition.

Definition

Definition.

  1. What is a metric function, and how does it differ from norms, pseudometrics, similarity measures and performance metrics? definition
  2. Is the question about distance functions or metric tensors in geometry and physics? boundary

Examples

Named examples.

Examples

Examples.

  1. What are Minkowski distance and graph distance, and how are the Kerr, spacetime and Fubini-Study metrics different? definition
  2. Which entry fits the specific metric? action
Axioms Axioms.

Science.

Triangle inequality

Triangle inequality.

Triangle inequality

Triangle inequality.

  1. Why is the triangle inequality essential? provenance
  2. Which references are standard? provenance

Metric spaces

Metric spaces.

Metric spaces

Metric spaces.

  1. How do metrics define open sets and convergence? provenance
  2. Which sources are cited? provenance
Geometry Metric tensors.

Science.

Tensors

Metric tensors.

Tensors

Tensors.

  1. How does a metric tensor define lengths and angles on manifolds? provenance
  2. Which entry fits metric tensor? action

Relativity

Spacetime metrics.

Relativity

Relativity.

  1. What does the Kerr metric describe for rotating black holes? provenance
  2. Is the information accurate? boundary
Context Applications.

Context.

Machine learning

Distance-based methods.

Machine learning

Machine learning.

  1. How are metrics used in nearest-neighbour methods and clustering? provenance
  2. Which entry fits k-nearest neighbors algorithm? action

Graphs

Graph distance.

Graphs

Graphs.

  1. How is shortest-path distance computed in graphs? measurement
  2. Is the information current? boundary

What the second pass must settle

  • The registry alias Wikidata:Statistics should be removed
  • The registry metric tensor aliases should be moved to differential geometry entries
  • Should graph distance be a separate entry?
Draft as JSON