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"Pushed forward" from one measurable space to another
In measure theory, a pushforward measure (also known as push forward, push-forward or image measure) is obtained by transferring ("pushing forward") a
Pushforward_measure
Topics referred to by the same term
notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things. Pushforward (differential)
Pushforward
Linear approximation of smooth maps on tangent spaces
In differential geometry, pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces. Suppose that φ : M → N {\displaystyle
Pushforward_(differential)
In algebraic topology, the pushforward of a continuous function f {\displaystyle f} : X → Y {\displaystyle X\rightarrow Y} between two topological spaces
Pushforward_(homology)
Mathematical function
In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the
Outer_measure
In mathematics, a mapping between categories
can define a new sheaf f∗F on Y, called the direct image sheaf or the pushforward sheaf of F along f, such that the global sections of f∗F is given by
Direct_image_functor
The pushforward is harder to define formally. For simplicity, fix Lebesgue measures on U {\displaystyle U} and V . {\displaystyle V.} The pushforward can
Valuation_(geometry)
Degree of differentiability of a function or map
level of the tangent bundle, the pushforward is a vector bundle homomorphism: F∗:TM→TN. The dual to the pushforward is the pullback, which "pulls" covectors
Smoothness
Mathematical operation
{\displaystyle \phi } is a diffeomorphism, then the pullback, together with the pushforward, can be used to transform any tensor field from N {\displaystyle N} to
Pullback (differential geometry)
Pullback_(differential_geometry)
Mathematical description of spacetime used in relativity
under inclusion, pullback of covariant tensors under general maps and pushforward of vectors under general maps Behavior of tensors under inclusion: For
Minkowski_spacetime
Process in mathematics
precomposition is a special case of the general fiber-product. Its dual is a pushforward. Precomposition with a function probably provides the most elementary
Pullback
Concept from algebraic geometry
codimension as Y′. Conversely, if f is proper, for Y a subvariety of X the pushforward is defined to be f ∗ ( [ Y ] ) = n [ f ( Y ) ] {\displaystyle f_{*}([Y])=n[f(Y)]\
Algebraic_cycle
sheaves come from intersection cohomology sheaves or from the derived pushforward of a local system on a family of topological spaces parameterized by
Constructible_sheaf
Differential map between manifolds whose differential is everywhere surjective
differentiable map between differentiable manifolds whose differential pushforward is everywhere surjective. It is a basic concept in differential topology
Submersion_(mathematics)
Differentiable function whose derivative is everywhere injective
differentiable function between differentiable manifolds whose differential pushforward is everywhere injective. Explicitly, f : M → N is an immersion if D p
Immersion_(mathematics)
Mathematical Concept
generated by tautological classes. These are classes obtained from 1 by pushforward along various morphisms described below. The tautological cohomology
Tautological_ring
Average value of a random variable
valued in A {\displaystyle A} is also equal to zero; in other words, the pushforward measure P X {\displaystyle P_{X}} given by P X ( A ) = P ( X − 1 ( A
Expected_value
Mathematical notion of infinitesimal difference
generally, the differential or pushforward refers to the derivative of a map between smooth manifolds and the pushforward operations it defines. The differential
Differential_(mathematics)
Typographical symbol (*)
vector spaces ∗:Ak→An−k. as a unary operator, written as a subscript The pushforward (differential) of a smooth map f between two smooth manifolds, denoted
Asterisk
Mathematical technique for simplification
)}gd\mu =\int _{\Omega }g\circ TdT^{*}\mu } . Pushforward measure and transformation formula The pushforward measure in terms of a transformation T {\displaystyle
Change_of_variables
Discrete-variable probability distribution
{\displaystyle X\colon A\to B} is discrete provided its image is countable. The pushforward measure X ∗ ( P ) {\displaystyle X_{*}(P)} —called the distribution of
Probability_mass_function
Tool to track locally defined data attached to the open sets of a topological space
topological spaces, pushforward and pullback relate sheaves on X {\displaystyle X} to those on Y {\displaystyle Y} and vice versa. The pushforward (also known
Sheaf_(mathematics)
Integral expressing the amount of overlap of one function as it is shifted over another
are Radon measures on G, then their convolution μ∗ν is defined as the pushforward measure of the group action and can be written as (μ∗ν)(E)=∬1E(xy)dμ(x)dν(y)
Convolution
Variable representing a random phenomenon
sample space) to a measurable space. This allows consideration of the pushforward measure, which is called the distribution of the random variable; the
Random_variable
Type of category in mathematics
{\displaystyle u} is a continuous map between them, then the pullback and pushforward operations on sheaves yield a geometric morphism between the associated
Topos
Duistermaat–Heckman formula, due to Duistermaat and Heckman (1982), states that the pushforward of the canonical (Liouville) measure on a symplectic manifold under the
Duistermaat–Heckman_formula
Generalizations of codimension-1 subvarieties of algebraic varieties
the pushforward. (If X is not quasi-compact, then the pushforward may fail to be a locally finite sum.) This is a special case of the pushforward on Chow
Divisor_(algebraic_geometry)
Unsolved problem in geometry
that a cohomology class on X is of co-level c (coniveau c) if it is the pushforward of a cohomology class on a c-codimensional subvariety of X. The cohomology
Hodge_conjecture
Concept in differential geometry
Y, the pushforward diffeology on Y is the diffeology generated by the compositions f∘p, for plots p:U→X of X. In other words, the pushforward diffeology
Diffeology
Measure that changes under a transformation but keeps the same null sets
expressible as multiplication by the Jacobian determinant of the derivative (pushforward) of T. To express this idea more formally in measure theory terms, the
Quasi-invariant_measure
Group of rotations in 3 dimensions
3-area measure in 4 dimensions, the Haar measure on SO(3) is just the pushforward of the 3-area measure. Consequently, generating a uniformly random rotation
3D_rotation_group
Consistent set of finite-dimensional distributions will define a stochastic process
π F G ) ∗ μ G {\displaystyle (\pi _{F}^{G})_{*}\mu _{G}} denotes the pushforward measure of μ G {\displaystyle \mu _{G}} induced by the canonical projection
Kolmogorov_extension_theorem
Mathematical concept
of the tangent space TgG at each g ∈ G into TeG. It is given as the pushforward of a vector in TgG along the left-translation in the group: ω(v)=(Lg−1)∗v
Maurer–Cartan_form
Mathematical function for the probability a given outcome occurs in an experiment
variable. Then the probability distribution of X {\displaystyle X} is the pushforward measure of the probability measure P {\displaystyle P} onto ( E , E )
Probability_distribution
Vector space equipped with a bilinear product
pullbacks, while upper indices are contravariant, transforming under pushforwards. Thus, the structure coefficients are often written ci,jk, and their
Algebra_over_a_field
Vector field on a pseudo-Riemannian manifold that preserves the metric tensor
of isometries. In this broader sense, a Killing vector field is the pushforward of a right invariant vector field on G {\displaystyle G} by the group
Killing_vector_field
Theorem in probability and statistics
theorem of mathematical analysis on Lebesgue integration relative to a pushforward measure. This proposition is (sometimes) known as the law of the unconscious
Law of the unconscious statistician
Law_of_the_unconscious_statistician
Instantaneous rate of change (mathematics)
ℝm. If v is a vector starting at a, then f′(𝐚)𝐯 is called the pushforward of 𝐯 by f. If the total derivative exists at 𝐚, then all the partial
Derivative
Fiber bundle induced by a map of its base space
so that although the notion of a 'pushforward of a bundle' is defined in some contexts (for example, the pushforward by a diffeomorphism), in general it
Pullback_bundle
other, then they determine the same pullback: f* = g*. In contrast, a pushforward for de Rham cohomology for example is given by integration-along-fibers
Pullback_(cohomology)
Identity concerning harmonic maps between Riemannian manifolds
manifolds and let u : M → N be a harmonic map. Let du denote the derivative (pushforward) of u, ∇ the gradient, Δ the Laplace–Beltrami operator, RiemN the Riemann
Bochner_identity
a hyperplane section, f ∗ {\displaystyle f_{*}} is the direct image (pushforward) and R n f ∗ {\displaystyle R^{n}f_{*}} is the n-th derived functor of
Decomposition theorem of Beilinson, Bernstein and Deligne
Decomposition_theorem_of_Beilinson,_Bernstein_and_Deligne
Tangent spaces of a manifold
M {\displaystyle \pi :TM\rightarrow M} is the canonical projection. Pushforward (differential) Unit tangent bundle Cotangent bundle Frame bundle Musical
Tangent_bundle
a true measure on the second space. It acquired its name because the pushforward measure on the second space was historically thought of as a Radon measure
Radonifying_operator
Technique from algebraic geometry
morphism of schemes and f ∗ , f ∗ {\displaystyle f_{*},f^{*}} denote the pushforward as well the pullback for quasi-coherent sheaves (here, for simplicity
Faithfully_flat_descent
Matrix of partial derivatives of a vector-valued function
sensitivity and statistical diagnostics. Center manifold Hessian matrix Pushforward (differential) Differentiability at x implies, but is not implied by
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Mathematical result in differential geometry
above). If X is a compact submanifold of a manifold Y then there is a pushforward (or "shriek") map from K(TX) to K(TY). The topological index of an element
Atiyah–Singer_index_theorem
Mathematical structure
{\displaystyle F} to F u {\displaystyle Fu} . These functors are called pushforwards. If C ~ {\displaystyle {\tilde {\mathcal {C}}}} and D ~ {\displaystyle
Grothendieck_topology
Expression that may be integrated over a region
denoted f∗ and called the pushforward. For any point p ∈ M and any tangent vector v ∈ TpM, there is a well-defined pushforward vector f∗(v) in Tf(p)N. However
Differential_form
Assignment of vector fields to manifolds
called variously the derivative, total derivative, differential, or pushforward of φ {\displaystyle \varphi } at x {\displaystyle x} . It is frequently
Tangent_space
Result in algebraic geometry
{\displaystyle X.} The Grothendieck–Riemann–Roch theorem relates the pushforward map f ! = ∑ ( − 1 ) i R i f ∗ : K 0 ( X ) → K 0 ( Y ) {\displaystyle
Grothendieck–Riemann–Roch theorem
Grothendieck–Riemann–Roch_theorem
Description of continuous random distribution
Borel sets as measurable subsets) has as probability distribution the pushforward measure X∗P on ( X , A ) {\displaystyle ({\mathcal {X}},{\mathcal {A}})}
Probability_density_function
Branch of mathematics
map of topological spaces gives rise to a homomorphism (also called pushforward) between their nth homology groups for all n. This basic fact of algebraic
Homological_algebra
Theorem
how a large deviation principle on one space "pushes forward" (via the pushforward of a probability measure) to a large deviation principle on another space
Contraction principle (large deviations theory)
Contraction_principle_(large_deviations_theory)
Random process of binary (boolean) random variables
probability measure P:ℬ→ℝ, consider instead some arbitrary function f:ℬ→ℝ. The pushforward f∘T−1 defined by (f∘T−1)(σ)=f(T−1(σ)) is again some function ℬ→ℝ. Thus
Bernoulli_process
Lie group of complex numbers of unit modulus; topologically a circle
normalized counting measure μN(A)=|A|/N, whose associated Loeb measure (the pushforward measure under the standard part) gives the usual Haar measure on the
Circle_group
Dual space to the tangent space in differential geometry
{\displaystyle f:M\to N} between manifolds induces a linear map (called the pushforward or derivative) between the tangent spaces f ∗ : T x M → T f ( x ) N {\displaystyle
Cotangent_space
Generalization of vector bundles
quasi-coherent sheaf on X {\displaystyle X} , then the direct image sheaf (or pushforward) f ∗ F {\displaystyle f_{*}{\mathcal {F}}} is quasi-coherent on Y {\displaystyle
Coherent_sheaf
Exterior derivative Lie derivative pullback (differential geometry) pushforward (differential) jet (mathematics) Contact (mathematics) jet bundle Frobenius
List of differential geometry topics
List_of_differential_geometry_topics
the dual vector space to the cotangent space. Analytic mappings induce pushforward maps on tangent spaces and pullback maps on cotangent spaces. The dimension
Analytic_space
Intuitively, a perfect measure μ is one for which, if we consider the pushforward measure on the real line R, then every measurable set is "μ-approximately
Perfect_measure
Defines a notion of parallel transport on a bundle
smooth manifold E . {\displaystyle E.} As such, one may consider the pushforward d X ( v ) , {\displaystyle dX(v),} which is an element of the tangent
Connection_(vector_bundle)
Algebra in algebraic topology
H^{0}(Y)} represent the fundamental class of Y {\displaystyle Y} then the pushforward map f ∗ ( 1 ) = α {\displaystyle f_{*}(1)=\alpha } gives a representation
Steenrod_algebra
Concept in probability theory
, Xn) is called comonotonic, if its multivariate distribution (the pushforward measure) is comonotonic, this means Pr ( X 1 ≤ x 1 , … , X n ≤ x n )
Comonotonicity
Generalization of mass, length, area and volume
Measurable function Minkowski content Outer measure Product measure Pushforward measure Random measure Regular measure Vector measure Valuation (measure
Measure_(mathematics)
Collection of random variables
the law of stochastic process X {\displaystyle X} is defined as the pushforward measure: μ = X ∗ P = P ∘ X − 1 , {\displaystyle \mu =X_{*}P=P\circ X^{-1}
Stochastic_process
Mathematics concept
The finite-dimensional distributions of μ {\displaystyle \mu } are the pushforward measures f ∗ ( μ ) {\displaystyle f_{*}(\mu )} , where f : X → R k {\displaystyle
Finite-dimensional distribution
Finite-dimensional_distribution
Distance-preserving mathematical transformation
{\displaystyle g'} by f {\displaystyle f} . Equivalently, in terms of the pushforward f ∗ , {\displaystyle f_{*},} we have that for any two vector fields v
Isometry
Smooth manifold with an inner product on each tangent space
di_{p}(w){\big )},} where d i p ( v ) {\displaystyle di_{p}(v)} is the pushforward of v {\displaystyle v} by i . {\displaystyle i.} Examples: The n {\displaystyle
Riemannian_manifold
Technique for the generative modeling of a continuous probability distribution
solution ϕt of the above ODE define a probability path pt=[ϕt]#π0 by the pushforward measure operator. In particular, [ϕ1]#π0=π1. The probability path and
Diffusion_model
Expected value of a random variable given that certain conditions are known to occur
\mathbb {R} ^{n}} . The measure used is the pushforward measure induced by Y. In the first example, the pushforward measure is a Dirac distribution at 1. In
Conditional_expectation
Long exact sequence
H^{*}(E).\,} In the case of a fiber bundle, one can also define a pushforward map π ∗ {\displaystyle \pi _{\ast }} π ∗ : H ∗ ( E ) ⟶ H ∗ − k ( M )
Gysin_homomorphism
object K ∈ D(X×Y). Most natural functors, including basic ones like pushforwards and pullbacks, are of this type. These kinds of functors were introduced
Fourier–Mukai_transform
Theorem in measure theory
that π−1(a)=a×[0,1], a slice we want to capture. Let ν∈𝒫(X) be the pushforward measure ν=π∗(μ)=μ∘π−1. This measure provides the distribution of x (which
Disintegration_theorem
Straight path on a curved surface or a Riemannian manifold
here d π : T T M → T M {\displaystyle d\pi :TTM\to TM} denotes the pushforward (differential) along the projection π : T M → M {\displaystyle \pi :TM\to
Geodesic
Transformation that preserves area measure of regions
} denotes the Euclidean wedge product of vectors and df denotes the pushforward along f. Every Euclidean isometry of the Euclidean plane is equiareal
Equiareal_map
{\displaystyle \ P_{*}f(q)} where P ∗ f {\displaystyle \ P_{*}f} denotes the pushforward of f {\displaystyle \ f} by P {\displaystyle \ P} , f {\displaystyle
Orbit_(control_theory)
Fundamental construction of differential calculus
vector will agree with the directional derivative. The differential or pushforward of a map between manifolds is the induced map between tangent spaces
Generalizations of the derivative
Generalizations_of_the_derivative
Nonlinear differential operator used to study conformal mappings
orientation-preserving diffeomorphisms of S1, Diff(S1), acts on Fλ(S1) via pushforwards. If f ∈ D i f f ( S 1 ) {\displaystyle f\in \mathrm {Diff} (S^{1})} then
Schwarzian_derivative
Specialized notation for multivariable calculus
components represent a space is known as the pushforward (or differential), or the Jacobian matrix. The pushforward along a vector function f with respect to
Matrix_calculus
Subject of study in ergodic theory
systems are defined in terms of the pushforward. For example, the transfer operator is defined in terms of the pushforward of the transformation map T {\displaystyle
Measure-preserving dynamical system
Measure-preserving_dynamical_system
Function in mathematics
is a point of U0 ⊂ S, then a vector field may be represented by the pushforward of a vector field v0 on R2 by φ 0 {\displaystyle \varphi _{0}} : where
Connection_(mathematics)
Differential geometry construct on fiber bundles
s : T M → T E {\displaystyle {\rm {d}}s\colon TM\to TE} is then the pushforward of tangent vectors. The horizontal spaces together form a vector subbundle
Ehresmann_connection
Type of probability space
Ω → R {\displaystyle \textstyle f:\Omega \to \mathbb {R} } induces a pushforward measure f ∗ P {\displaystyle f_{*}P} , – the probability measure μ {\displaystyle
Standard_probability_space
Technique in integral evaluation
Weierstrass substitution Euler substitution Glasser's master theorem Pushforward measure Swokowski 1983, p. 257 Swokowski 1983, p. 258 Briggs & Cochran
Integration_by_substitution
Mathematical theory
Riemann–Roch theorem then describes how the Chern class behaves under pushforward of vector bundles under a proper map of arithmetic varieties. A complete
Arakelov_theory
Stochastic process generalizing Brownian motion
surely. The image of the Lebesgue measure on [0, t] under the map w (the pushforward measure) has a density Lt. Thus, ∫ 0 t f ( w ( s ) ) d s = ∫ − ∞ + ∞
Wiener_process
respective dimensions m≥n. Let F:M⟶N be a smooth surjection such that the pushforward (differential) of F is surjective almost everywhere. Let φ:M⟶[0,∞) a
Smooth_coarea_formula
Deep learning generative model to encode data representation
be a Multivariate normal distribution. We will denote Eϕ♯ℙreal this pushforward measure which in practice is just the empirical distribution obtained
Variational_autoencoder
Array of numbers describing a metric connection
properties of a tensor are given by pullbacks for covariant indices, and pushforwards for contravariant indices. The article on covariant derivatives provides
Christoffel_symbols
Algebraic structure used in topology
homology: A continuous map f : X → Y {\displaystyle f:X\to Y} determines a pushforward homomorphism f ∗ : H i ( X ) → H i ( Y ) {\displaystyle f_{*}:H_{i}(X)\to
Cohomology
Topics referred to by the same term
relating derivatives of a function Differential topology Differential (pushforward) The total derivative of a map between manifolds. Differential exponent
Differential
Algebraic tool for computing topological spaces' invariants
f_{*}:H_{k}(X_{1})\to H_{k}(X_{2})} such that the composition of pushforwards is the pushforward of a composition: that is, ( g ∘ h ) ∗ = g ∗ ∘ h ∗ . {\displaystyle
Mayer–Vietoris_sequence
Locally constant sheaf of abelian groups on topological space
system. These are typically found by taking the cohomology of the derived pushforward for some continuous map f : X → Y {\displaystyle f:X\to Y} . For example
Local_system
Machine learning framework
probability distribution, in practice, it is usually implemented as a pushforward: μG=μZ∘G−1. That is, start with a random variable z∼μZ, where μZ is a
Generative adversarial network
Generative_adversarial_network
Concept in topology
the winding number): Let f:Sn→Sn be a continuous map. Then f induces a pushforward homomorphism f∗:Hn(Sn)→Hn(Sn), where Hn(⋅) is the nth homology group
Degree of a continuous mapping
Degree_of_a_continuous_mapping
Module over a sheaf of differential operators
D-modules on different algebraic varieties are connected by pullback and pushforward functors comparable to the ones for coherent sheaves. For a map f: X
D-module
Tool in homological algebra
groups of the pushforward of the sheaf. The relation involved an infinite process. Leray found that the cohomology groups of the pushforward formed a natural
Spectral_sequence
Notion in calculus
one treats the differential in this manner, then it is known as the pushforward since it "pushes" velocities from a source space into velocities in a
Differential_of_a_function
Cohomology with real coefficients computed using differential forms
Hodge theory Integration along fibers (for de Rham cohomology, the pushforward is given by integration) Sheaf theory ∂ ∂ ¯ {\displaystyle \partial {\bar
De_Rham_cohomology
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