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Longer titles found: Multilinear subspace learning (view)

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which it is radial points of the set. If M {\displaystyle M} is a linear subspace of X {\displaystyle X} and A ⊆ X {\displaystyle A\subseteq X} then

 is a linear subspace } {\displaystyle \operatorname {qri} (A):=\left\{x\in A:\operatorname {\overline {cone}} (A-x){\text{ is a linear subspace}}\right\}}

set or a translate of a linear subspace (i.e., a set of the form v + M, where v is a given vector and M is a linear subspace). Riemann series theorem

the projection error, controlling the error of approximation by a linear subspace based on a linear projection relative to the optimal error together

the orthogonal projection of a larger real symmetric matrix A onto a linear subspace spanned by the columns of B. The theorem is named after Henri Poincaré

manifolds. The sectional curvature K(σp) depends on a two-dimensional linear subspace σp of the tangent space at a point p of the manifold. It can be defined

generate a contraction semigroup. Let A be a linear operator defined on a linear subspace D(A) of the Banach space X. Then A generates a contraction semigroup

holomorphic functions in D. Then L2,h(D) is a Hilbert space: it is a closed linear subspace of L2(D), and therefore complete in its own right. This follows from

(S)-s_{0}=\operatorname {span} (S-s_{0})=\operatorname {span} (S-S)} is a linear subspace of X {\displaystyle X} . aff ⁡ ( S − S ) = span ⁡ ( S − S ) {\displaystyle

functions X → R, again with the supremum norm, since Cb(X) is a closed linear subspace of ℓ ∞(X). These embedding results are useful because Banach spaces

ideal of bounded linear operators on a separable Hilbert space is the linear subspace spanned by commutators of operators in the ideal with bounded operators

in the linear space of phase space functions and projects onto the linear subspace of "slow" phase space functions. It was introduced by Robert Zwanzig

of the zero-dimensional intersection of the variety with a generic linear subspace of complementary dimension. More generally, the intersection multiplicity

X 1 {\displaystyle X_{1}} (resp. Y 1 {\displaystyle Y_{1}} ) is a linear subspace of X 2 {\displaystyle X_{2}} (resp. Y 2 {\displaystyle Y_{2}} ) then

bounded linear operators B(H) on a separable Hilbert space H is a linear subspace such that AB and BA belong to J for all operators A from J and B from

lines, or osculating planes, and so on. axial axis A special line or linear subspace associated with some family of geometric objects. For example, a special

(k+1)-dimensional linear subspace of L that intersects A. Every point of the affine subspace A is the intersection of A with a one-dimensional linear subspace of L

only if the cone generated by S − x {\displaystyle S-x} is a barreled linear subspace of X {\displaystyle X} or equivalently, if and only if ∪ n ∈ N n (

{\displaystyle A\in B(X,Y)} the kernel of A {\displaystyle A} is a closed linear subspace of X {\displaystyle X} . If B ( X , Y ) {\displaystyle B(X,Y)} is Banach

measurable additive subgroup of the vector space X (i.e. a measurable linear subspace), then the inner measure of G under μ, μ ∗ ( G ) = sup { μ ( K ) ∣

two-dimensional linear subspace (including the origin) of the n+1-dimensional Minkowski space. If we take u and v to be basis vectors of that linear subspace with

) = 0 } {\displaystyle \ker p:=\left\{x\in X:p(x)=0\right\}} is a linear subspace of X {\displaystyle X} . If p {\displaystyle p} is continuous then

{\displaystyle Y,} is a linear operator that is defined on a dense linear subspace dom ⁡ ( T ) {\displaystyle \operatorname {dom} (T)} of X {\displaystyle

norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} . Let Y {\displaystyle Y} be a linear subspace of X {\displaystyle X} and B : Y → X {\displaystyle B:Y\to X} be a

closed sets in a Grassmannian defined by conditions of incidence of a linear subspace in projective space with a given flag. For further details see Schubert

continuously embedded in a Hausdorff topological vector space Z when X is a linear subspace of Z such that the inclusion map from X into Z is continuous. A compatible

, the set of all linear combinations of these vectors is called a linear subspace and is therefore a vector space. The equation says that vectors a _

spaces that they contain. (In the context of projective geometry, a linear subspace of P N {\displaystyle {\mathbf {P} }^{N}} is isomorphic to P a {\displaystyle

closed sets in a Grassmannian defined by conditions of incidence of a linear subspace in projective space with a given flag. For details see Schubert variety

{\displaystyle \Omega } . A {\displaystyle A} can be viewed as a closed linear subspace of C ( Ω ) {\displaystyle C(\Omega )} (this is Kadison's function representation)

algebra, a Jacobi rotation is a rotation, Qkℓ, of a 2-dimensional linear subspace of an n-dimensional inner product space, chosen to zero a symmetric

of the cj's in all the Ck with k < i. This means that si is in the linear subspace of Qm spanned by the set of the cj's. Folkman's theorem, the statement

of lines containing L and L′. This corresponds to a one-dimensional linear subspace belonging to the Klein quadric. If three distinct and non-parallel

also for groups) has analogs for Lie algebras. A Lie subalgebra is a linear subspace h ⊆ g {\displaystyle {\mathfrak {h}}\subseteq {\mathfrak {g}}} which

complete flag, the stabilizer gives a Borel subalgebra. For a single linear subspace F k ⊂ F n {\displaystyle \mathbb {F} ^{k}\subset \mathbb {F} ^{n}}

that is also a linear map, then we have to choose an n-dimensional linear subspace of L 2 {\displaystyle L^{2}} . This fact that the dimensions have to

_{Y}(p)):\Sigma _{n,m}\to P^{(n+1)(m+1)-1}\ } for a fixed point p is a linear subspace of the codomain. For example with m = n = 1 we get an embedding of

{\displaystyle \sum _{i\in S_{j}}x_{i}=0} for them. Since it is a non-trivial linear subspace of R n {\displaystyle \mathbf {R} ^{n}} with fewer constraints than

) {\displaystyle {\mathcal {J}}(X,Y)} of an operator ideal forms a linear subspace of L ( X , Y ) {\displaystyle {\mathcal {L}}(X,Y)} , although in general

u(0)=u_{0}{\text{ (1)}}} , where A is a linear operator mapping a dense linear subspace D(A) of X into X, with u ( t ) = S ( t ) u 0 {\displaystyle u(t)=S(t)u_{0}}

{W} } is a vector space and B {\displaystyle {\mathcal {B}}} is a linear subspace of W T {\displaystyle \mathbb {W} ^{\mathbb {T} }} , "time-invariant"

matrix theory, the Radon–Hurwitz number counts the maximum size of a linear subspace of the real n × n {\displaystyle n\times n} matrices, for which each

of higher dimension, except in the trivial way of lying in a proper linear subspace. Projective normality may similarly be translated, by using enough

length of the orthogonal projection of S {\displaystyle S} to a random linear subspace of R n {\displaystyle \mathbb {R} ^{n}} . When n = 2 {\displaystyle

Hilbert space Unbounded operator – Linear operator defined on a dense linear subspace Kreyszig, Erwin (1978), Introductory functional analysis with applications

set of solutions of α(x) = 0, the kernel of α, is a one-dimensional linear subspace (that is, a complex line through the origin of X). But if c is some

space H, connected by α–Hölder continuous arcs with α > 1/2, is a linear subspace. There are closed additive subgroups of H, not linear subspaces, connected

that row or column equal one. The row and column constraints define a linear subspace of dimension n2 − 2n + 1 in which the Birkhoff polytope lies, and the

diffeomorphic mapping which transports the data onto a lower-dimensional linear subspace. The methods solves for a smooth time indexed vector field such that

functions obtained by imposing a linear constraint on a set of inputs (a linear subspace) are known as subfunctions. The Boolean derivative of the function

problem corresponds to minimization of a quadratic functional over a linear subspace of functions, the free boundary problem corresponds to minimization

action of any T in M. Its closure cl(Mh) in the norm of H is a closed linear subspace, with corresponding orthogonal projection P : H → cl(Mh) in L(H). In

whitened STA is a consistent estimator, i.e., it converges to the true linear subspace, if The stimulus distribution P ( x ) {\displaystyle P(\mathbf {x}

{\displaystyle \operatorname {rec} A\cap \operatorname {rec} B} is a linear subspace. If A or B is locally compact then A − B is closed. The notion of convexity

{Pr} _{X}(A)} is ideally convex. Let X 0 {\displaystyle X_{0}} be a linear subspace of X . {\displaystyle X.} Let R : X ⇉ Y {\displaystyle {\mathcal

the ReFT family is Low-rank Linear Subspace ReFT (LoReFT), which intervenes on hidden representations in the linear subspace spanned by a low-rank projection

{C}}^{k}\Omega ({\mathcal {O}})} be its k {\displaystyle k} -th power, i.e. the linear subspace of C Ω {\displaystyle {\mathcal {C}}\Omega } generated by w 1 ∧ ⋯ ∧

rank k bundle whose fiber at any point [L] ∊ G(k, V) is simply the linear subspace L ⊂ V.) M is called a base space of the family U. We say that such

rank k bundle whose fiber at any point [L] ∊ G(k, V) is simply the linear subspace L ⊂ V.) M is called a base space of the family U. We say that such

{C}}^{k}\Omega ({\mathcal {O}})} be its k {\displaystyle k} -th power, i.e. the linear subspace of C Ω {\displaystyle {\mathcal {C}}\Omega } generated by w 1 ∧ ⋯ ∧

which is valued in the g F ( p ) {\displaystyle g_{F(p)}} -orthogonal linear subspace to d F p ( T p Σ ) ⊂ T F ( p ) M . {\displaystyle dF_{p}(T_{p}\Sigma

linear equations: x11 = -x21 = x41 x12 = -x31 = -x42 x22 = -x32 The linear subspace where these equations are satisfied is one of the constraint spaces

linear maps Unbounded operator – Linear operator defined on a dense linear subspace Narici & Beckenstein 2011, pp. 126–128. Narici & Beckenstein 2011,

to consist of all tangent vectors to S at p, is a two-dimensional linear subspace of ℝ3; it is often denoted by TpS. The normal space to S at p, which

combinations of points in the set. The linear hull is the smallest linear subspace of a vector space containing a given set, or the union of all linear

of a hyperplane CPn−1 in CPn. More generally, xj is the class of a linear subspace CPn−j in CPn. The cohomology ring of the closed oriented surface X

parametrization of the intersection of S3 ⊂ R4 with the two-dimensional linear subspace R × R × {0} × {0} a constant-speed parametrization of the intersection

of (say) a group G {\displaystyle G} , and W {\displaystyle W} is a linear subspace of V {\displaystyle V} that is preserved by the action of G {\displaystyle

H, thus eigenvalues of T, when they exist, are real. When a closed linear subspace L of H is invariant under T, then the restriction of T to L is a self-adjoint

{\displaystyle \mathbf {P} ^{n}} be n-dimensional projective space. Fix a linear subspace L of codimension d. There are several explicit ways to describe the

infinitesimal increments normally constrain them to a two-dimensional linear subspace space of possible infinitesimal state changes, that depends on the

be identified (i.e., their values can only be estimated within some linear subspace of the full parameter space Rp). See partial least squares regression

V^{*}} its dual. A (linear) Dirac structure on V {\displaystyle V} is a linear subspace D {\displaystyle D} of V × V ∗ {\displaystyle V\times V^{*}} satisfying

each t, the kernel of e t {\displaystyle e_{t}} is a one-dimensional linear subspace of V. Hence t ↦ ker ⁡ e t {\displaystyle t\mapsto \ker e_{t}} defines

_{a}||=1} ). The factor vectors define an k {\displaystyle k} -dimensional linear subspace (i.e. a hyperplane) in this space, upon which the data vectors are

}|\log(f(x))|^{p}dP(x)<\infty \}} . For p = 2 {\displaystyle p=2} , this is a linear subspace and isometrically isomorphic to the Hilbert space L 0 2 ( P ) = { g

p. 291. Oppenheim & Schafer 2010, p. 55. "Characterizations of a linear subspace associated with Fourier series". MathOverflow. 2010-11-19. Retrieved

_{mn}} Finally, the sequence {en}n ∈ N spans a dense linear subspace of L2((0, L)). This shows that in effect we have diagonalized the operator

Clifford algebra. Now define an action of so(3,1) on the σμν, and the linear subspace Vσ ⊂ Cl4(C) they span in Cl4(C) ≈ MnC, given by The last equality in

the orthogonal projection of a larger real symmetric matrix A onto a linear subspace spanned by the columns of B. Poincaré–Birkhoff theorem: every area-preserving

surfaces. Let E ⊂ P n {\displaystyle E\subset \mathbb {P} ^{n}} be a linear subspace; i.e., E = { s 0 = s 1 = ⋯ = s r = 0 } {\displaystyle E=\{s_{0}=s_{1}=\cdots

equivalent to D, called the complete linear system of D. A projective linear subspace of this projective space is called a linear system of divisors. One

are characterized by functionals: if X is a normed vector space with linear subspace M (not necessarily closed) and if z {\displaystyle z} is an element

representation or conformal space. A homogeneous subspace refers to a linear subspace of the algebraic space. The terms for objects: point, line, circle

B=0} . Therefore, the variety of all circles is a three-dimensional linear subspace of P5. After rescaling and completing the square, these equations also

{MMSE} }-x)g(y)\}=0} for all g ( y ) {\displaystyle g(y)} in closed, linear subspace V = { g ( y ) ∣ g : R m → R , E ⁡ { g ( y ) 2 } < + ∞ } {\displaystyle

{\displaystyle t} of the domain of X {\displaystyle X} , a one-dimensional linear subspace of X {\displaystyle X} . That is, the kernel defines a mapping from

this space; and random variables of the form g (X) are a (closed, linear) subspace. The orthogonal projection of this vector to this subspace is well-defined

{\displaystyle x^{k+1}} is obtained by first constraining the update to the linear subspace spanned by the columns of the random matrix B − 1 A T S {\displaystyle

is continuous but it is not necessarily a topological embedding. A linear subspace of D ′ ( U ) {\displaystyle {\mathcal {D}}'(U)} carrying a locally

polyhedra. The Dehn invariants of Euclidean polyhedra form a real linear subspace of R ⊗ Z R / 2 π Z {\displaystyle \mathbb {R} \otimes _{\mathbb {Z}

derivation. If E is a Euclidean Jordan algebra an ideal F in E is a linear subspace closed under multiplication by elements of E, i.e. F is invariant under

inherits from D ′ ( U ) . {\displaystyle {\mathcal {D}}^{\prime }(U).} A linear subspace of D ′ ( U ) {\displaystyle {\mathcal {D}}^{\prime }(U)} carrying a