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# Isometry

In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. 

59 relations: Affine transformation, Banach space, Bijection, Cauchy sequence, Closed set, Complete metric space, Conference on Neural Information Processing Systems, Congruence (geometry), Continuous function, Differentiable manifold, Distance, Embedding, Equivalence class, Euclidean plane isometry, Euclidean space, Felix Hausdorff, Flat (geometry), Function (mathematics), Function composition, Geometric transformation, Group (mathematics), Hilbert space, Homeomorphism group, Injective function, Inverse function, Involution (mathematics), Isometry group, Isomorphism, John Wiley & Sons, Journal of Machine Learning Research, Linear map, Local tangent space alignment, Mathematics, Mazur–Ulam theorem, Metric space, Motion (geometry), Nonlinear dimensionality reduction, Normed vector space, Order embedding, Partial isometry, Partially ordered set, Polarization identity, Proceedings of the American Mathematical Society, Pseudo-Euclidean space, Quasi-isometry, Reflection (mathematics), Restricted isometry property, Riesz representation theorem, Rotation, Science (journal), ... Expand index (9 more) »

## Affine transformation

In geometry, an affine transformation, affine mapBerger, Marcel (1987), p. 38.

## Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.

## Bijection

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.

## Cauchy sequence

In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses.

## Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.

## Complete metric space

In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary).

## Conference on Neural Information Processing Systems

The Conference and Workshop on Neural Information Processing Systems (NIPS) is a machine learning and computational neuroscience conference held every December.

## Congruence (geometry)

In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.

## Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

## Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.

## Distance

Distance is a numerical measurement of how far apart objects are.

## Embedding

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

## Equivalence class

In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes.

## Euclidean plane isometry

In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length.

## Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

## Felix Hausdorff

Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis.

## Flat (geometry)

In geometry, a flat is a subset of n-dimensional space that is congruent to a Euclidean space of lower dimension.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## Function composition

In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.

## Geometric transformation

A geometric transformation is any bijection of a set having some geometric structure to itself or another such set.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

## Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

## Homeomorphism group

In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation.

## Injective function

In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.

## Inverse function

In mathematics, an inverse function (or anti-function) is a function that "reverses" another function: if the function applied to an input gives a result of, then applying its inverse function to gives the result, and vice versa.

## Involution (mathematics)

In mathematics, an involution, or an involutory function, is a function that is its own inverse, for all in the domain of.

## Isometry group

In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation.

## Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

## John Wiley & Sons

John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.

## Journal of Machine Learning Research

The Journal of Machine Learning Research is a peer-reviewed open access scientific journal covering machine learning.

## Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

## Local tangent space alignment

Local tangent space alignment (LTSA) is a method for manifold learning, which can efficiently learn a nonlinear embedding into low-dimensional coordinates from high-dimensional data, and can also reconstruct high-dimensional coordinates from embedding coordinates.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Mazur–Ulam theorem

In mathematics, the Mazur–Ulam theorem states that if V and W are normed spaces over R and the mapping is a surjective isometry, then f is affine.

## Metric space

In mathematics, a metric space is a set for which distances between all members of the set are defined.

## Motion (geometry)

In geometry, a motion is an isometry of a metric space.

## Nonlinear dimensionality reduction

High-dimensional data, meaning data that requires more than two or three dimensions to represent, can be difficult to interpret.

## Normed vector space

In mathematics, a normed vector space is a vector space over the real or complex numbers, on which a norm is defined.

## Order embedding

In mathematical order theory, an order embedding is a special kind of monotone function, which provides a way to include one partially ordered set into another.

## Partial isometry

In functional analysis a partial isometry is a linear map between Hilbert spaces such that it is an isometry on the orthogonal complement of its kernel.

## Partially ordered set

In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.

## Polarization identity

In mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.

## Proceedings of the American Mathematical Society

Proceedings of the American Mathematical Society is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society.

## Pseudo-Euclidean space

In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional ''n''-space together with a non-degenerate quadratic form.

## Quasi-isometry

In mathematics, quasi-isometry is an equivalence relation on metric spaces that ignores their small-scale details in favor of their coarse structure.

## Reflection (mathematics)

In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection.

## Restricted isometry property

In linear algebra, the restricted isometry property (RIP) characterizes matrices which are nearly orthonormal, at least when operating on sparse vectors.

## Riesz representation theorem

There are several well-known theorems in functional analysis known as the Riesz representation theorem.

## Rotation

A rotation is a circular movement of an object around a center (or point) of rotation.

## Science (journal)

Science, also widely referred to as Science Magazine, is the peer-reviewed academic journal of the American Association for the Advancement of Science (AAAS) and one of the world's top academic journals.

## Semidefinite embedding

Semidefinite embedding (SDE) or maximum variance unfolding (MVU) is an algorithm in computer science that uses semidefinite programming to perform non-linear dimensionality reduction of high-dimensional vectorial input data.

## Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

## Space group

In mathematics, physics and chemistry, a space group is the symmetry group of a configuration in space, usually in three dimensions.

## Surjective function

In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x).

## Symmetry in mathematics

Symmetry occurs not only in geometry, but also in other branches of mathematics.

## Translation (geometry)

In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction.

## Unitary matrix

In mathematics, a complex square matrix is unitary if its conjugate transpose is also its inverse—that is, if where is the identity matrix.

## Unitary operator

In functional analysis, a branch of mathematics, a unitary operator is a surjective bounded operator on a Hilbert space preserving the inner product.

## Unsupervised learning

Unsupervised machine learning is the machine learning task of inferring a function that describes the structure of "unlabeled" data (i.e. data that has not been classified or categorized).

## References

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