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Generalization of a positive-definite matrix
branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced
Positive-definite_kernel
Mathematical theorem
in the reproducing kernel Hilbert space theory where it characterizes a symmetric positive-definite kernel as a reproducing kernel. To explain Mercer's
Mercer's_theorem
Index of articles associated with the same name
Positive-definite kernel Positive-definite matrix Positive-definite operator Positive-definite quadratic form Fasshauer, Gregory E. (2011), "Positive
Positive_definiteness
and algebraic groups. It can be viewed as a particular type of positive-definite kernel where the underlying set has the additional group structure. Let
Positive-definite function on a group
Positive-definite_function_on_a_group
Property of a mathematical matrix
with real entries is positive-definite if the real number x T M x {\displaystyle \mathbf {x} ^{\mathsf {T}}M\mathbf {x} } is positive for every nonzero real
Definite_matrix
Class of algorithms for pattern analysis
coefficients ( c 1 , … , c n ) {\displaystyle (c_{1},\dots ,c_{n})} (cf. positive definite kernel), then the function k {\displaystyle k} satisfies Mercer's condition
Kernel_method
In functional analysis, a Hilbert space
in neural network settings.[citation needed] Positive definite kernel Mercer's theorem Kernel trick Kernel embedding of distributions Representer theorem
Reproducing kernel Hilbert space
Reproducing_kernel_Hilbert_space
Bimodal function
In mathematics, a positive-definite function is, depending on the context, either of two types of function. Let R {\displaystyle \mathbb {R} } be the
Positive-definite_function
Concept in statistics
Kernel density estimation Kernel smoother Stochastic kernel Positive-definite kernel Density estimation Multivariate kernel density estimation Kernel
Kernel_(statistics)
In machine learning, tree kernels are the application of the more general concept of positive-definite kernel to tree structures. They find applications
Tree_kernel
Topics referred to by the same term
mathematical finance Positive-definite kernel, a generalization of a positive-definite matrix Kernel trick, in statistics Reproducing kernel Hilbert space Seed
Kernel
Class of nonparametric methods
\Omega } and distribution P {\displaystyle P} . Given a symmetric, positive-definite kernel k : Ω × Ω → R {\displaystyle k:\Omega \times \Omega \rightarrow
Kernel embedding of distributions
Kernel_embedding_of_distributions
Concept in regression analysis mathematics
(f)+\lambda R(f),\lambda >0} A RKHS can be defined by a symmetric positive-definite kernel function K ( x , z ) {\displaystyle K(x,z)} with the reproducing
Regularized_least_squares
states that positive-definite kernels can be expressed as a dot product in a high-dimensional space. This theorem is the basis of the kernel trick (applied
James_Mercer_(mathematician)
Mathematical function
TW(x-x')}{\pi (x-x')}}.} The trace of the positive definite kernel is the sum of the infinite number of real and positive eigenvalues, N = ∑ α = 1 ∞ λ α = ∫
Slepian_function
Statistical model
{\displaystyle {\mathcal {H}}(R)} be a reproducing kernel Hilbert space with positive definite kernel R {\displaystyle R} . Driscoll's zero-one law is a
Gaussian_process
Theorem of Fourier transforms of Borel measures
{\displaystyle h(g)=0} for all but finitely many g {\displaystyle g} . The positive-definite kernel K ( g 1 , g 2 ) = f ( g 1 − g 2 ) {\displaystyle K(g_{1},g_{2})=f(g_{1}-g_{2})}
Bochner's_theorem
Type of kernel induced by artificial neural networks
allows ANNs to be studied using theoretical tools from kernel methods. In general, a kernel is a positive-semidefinite symmetric function of two inputs which
Neural_tangent_kernel
Matrix decomposition method
(pronounced /ʃəˈlɛski/ shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate
Cholesky_decomposition
Mathematical concept
speaking, the inner product is required only to be positive semi-definite rather than positive definite, so that it gives rise to a seminorm rather than
Semi-Hilbert_space
Approximations used in machine learning
of the Nyström method from integral equation theory. Consider a positive-definite kernel function k : X × X → R {\displaystyle k:X\times X\to \mathbb {R}
Low-rank matrix approximations
Low-rank_matrix_approximations
Method of data analysis
generalization is kernel PCA, which corresponds to PCA performed in a reproducing kernel Hilbert space associated with a positive definite kernel. In multilinear
Principal_component_analysis
Mathematical function
b b c ] {\displaystyle {\begin{bmatrix}a&b\\b&c\end{bmatrix}}} is positive-definite. Using this formulation, the figure on the right can be created using
Gaussian_function
Statistical technique
belong to the Reproducing Kernel Hilbert Space associated with any arbitrary (possibly non-linear), symmetric positive-definite kernel. The linear regression
Principal component regression
Principal_component_regression
Function in probability theory
matrix Covariance operator – Operator in probability theory Kriging Positive-definite kernel Random field Stochastic process Variogram Wackernagel, Hans (2003)
Covariance_function
Relationships between music and mathematics
Tatlow (In Our Time, May 25, 2006) Measuring note similarity with positive definite kernels, Measuring note similarity with positive definite kernels
Music_and_mathematics
Mathematical function
in functional analysis, a seminorm is like a norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm
Seminorm
Statistical learning theory
due to Schölkopf, Herbrich, and Smola: Theorem: Consider a positive-definite real-valued kernel k : X × X → R {\displaystyle k:{\mathcal {X}}\times {\mathcal
Representer_theorem
Mathematical concept of energy in physics
h {\displaystyle u_{h}} , see Céa's lemma. Inner product space Positive-definite kernel Zeidler, Eberhard (1995). Applied functional analysis: applications
Energetic_space
Concept in statistics mathematics
bandwidth (or smoothing) d×d matrix which is symmetric and positive definite; K is the kernel function which is a symmetric multivariate density; K H (
Multivariate kernel density estimation
Multivariate_kernel_density_estimation
Machine learning technique
the above construction can be generalized to arbitrary positive definite shift-invariant kernel k ( x , y ) = k ( x − y ) {\displaystyle k(x,y)=k(x-y)}
Random_feature
Compact operator for which a finite trace can be defined
That is, suppose K {\displaystyle K} is a continuous symmetric positive-definite kernel on L 2 ( [ a , b ] ) {\displaystyle L^{2}([a,b])} , defined as
Trace_class
matrices with the positive elements being the positive-definite matrices. The trace function defined on this C*-algebra is a positive functional, as the
Positive_linear_functional
Vector space with generalized dot product
{\displaystyle x} , then one obtains the definition of positive semi-definite Hermitian form. A positive semi-definite Hermitian form ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle
Inner_product_space
Statistical tool
weighting scheme also ensures that the resulting covariance matrix is positive semi-definite. L = 0 reduces the Newey–West estimator to Huber–White standard
Newey–West_estimator
Kernel methods are a well-established tool to analyze the relationship between input data and the corresponding output of a function. Kernels encapsulate
Kernel methods for vector output
Kernel_methods_for_vector_output
Integral of sin(x)/x from 0 to infinity
integrals. It is a good illustration of special techniques for evaluating definite integrals, particularly when it is not useful to directly apply the fundamental
Dirichlet_integral
Matrix B such that B² equals a given matrix A
semidefinite matrix is real. The principal square root of a positive definite matrix is positive definite; more generally, the rank of the principal square root
Square_root_of_a_matrix
Austrian mathematician
harmonic analysis, operator algebras and probability theory. Positive definite kernels, continuous tensor products, and central limit theorems in probability
Klaus_Schmidt_(mathematician)
Indian statistician (1936–2023)
Mathematical Society, 1967. Parthasarathy, K. R.; Schmidt, K. (1972). Positive Definite Kernels, Continuous Tensor Products, and Central Limit Theorems of Probability
K. R. Parthasarathy (probabilist)
K._R._Parthasarathy_(probabilist)
Idempotent linear transformation from a vector space to itself
{\displaystyle P} is always a positive semi-definite matrix. In general, the corresponding eigenspaces are (respectively) the kernel and range of the projection
Projection_(linear_algebra)
positive-definite function k : X × X → R {\displaystyle k:{\mathcal {X}}\times {\mathcal {X}}\rightarrow \mathbb {R} } called the reproducing kernel such
Bayesian interpretation of kernel regularization
Bayesian_interpretation_of_kernel_regularization
Matrix of second derivatives
a positive-definite or negative-definite Hessian cannot apply here since a bordered Hessian can neither be negative-definite nor positive-definite, as
Hessian_matrix
Matrix of inner products of vectors
the last from the positive definiteness of the inner product. Note that this also shows that the Gramian matrix is positive definite if and only if the
Gram_matrix
Type of array in math
(and positive eigenvalues). A symmetric totally positive matrix is therefore also positive-definite. A totally non-negative matrix is defined similarly
Totally_positive_matrix
German computer scientist
positive definite. Both insights together led to the foundation of the field of kernel methods, encompassing SVMs and many other algorithms. Kernel methods
Bernhard_Schölkopf
Generalized function whose value is zero everywhere except at zero
represented by integration against a kernel K z ( ζ ) {\displaystyle K_{z}(\zeta )} , the Bergman kernel. This kernel is the analog of the delta function
Dirac_delta_function
Problem in complex analysis
positive semi-definite matrix, for all τ {\displaystyle \tau } in the n-torus. Here, the K τ {\displaystyle K_{\tau }} s are the reproducing kernels corresponding
Nevanlinna–Pick_interpolation
Polynomial with all terms of degree two
positive definite (all +1) or negative definite (all −1). If none of the terms are 0, then the form is called nondegenerate; this includes positive definite
Quadratic_form
Irish mathematician (1948–2006)
operators, J. Integr. Equ. Oper. Theory, 27 (1997), 439–445. Positive definite kernels and Hilbert C*-modules, Proc. Edinburgh Math. Soc., 40 (1997)
Gerard_Murphy_(mathematician)
Mathematical concept
other characterizations of the topology on the spectrum in terms of positive definite functions are desirable. In fact, the topology on  is intimately
Spectrum_of_a_C*-algebra
Space of complex matrices with positive definite imaginary part
symmetric matrices over the complex numbers whose imaginary part is positive definite. It was introduced by Siegel (1939). The space H g {\displaystyle
Siegel_upper_half-space
with positive real part. Such functions had already been characterized in 1907 by Constantin Carathéodory in terms of the positive definiteness of their
Positive_harmonic_function
Generalization of gamma distribution to multiple dimensions
is a family of probability distributions defined over symmetric, positive-definite random matrices (i.e. matrix-valued random variables). These distributions
Wishart_distribution
Bounded linear operator
_{x}^{1}f(y)dy=\int _{0}^{1}1_{y\geq x}f(y)dy} The positive-definite integral operator K := V ∗ V {\displaystyle K:=V^{*}V} has kernel form K f ( x ) = ∫ 0 1 min ( 1 −
Volterra_operator
A heat kernel signature (HKS) is a feature descriptor for use in deformable shape analysis and belongs to the group of spectral shape analysis methods
Heat_kernel_signature
Equation from probability theory
{\displaystyle B_{ij}(x,t)} is the diffusion matrix (symmetric and positive semi-definite), W ( x | x ′ , t ) {\displaystyle W(x|x',t)} is the transition
Chapman–Kolmogorov_equation
Mathematical representation
chosen to be a transcendental unit complex number near 1, it is a positive-definite Hermitian pairing. Thus the Burau representation of the braid group
Burau_representation
Mathematical function, in linear algebra
{im} (f).} This is the dual notion to the kernel: just as the kernel is a subspace of the domain, the co-kernel is a quotient space of the target. Formally
Linear_map
Probability problem
the family of complex valued sequences with finitary support. The positive Hankel kernel A induces a (possibly degenerate) sesquilinear product on the family
Hamburger_moment_problem
Type of mathematical function
kernel): Infinitely Smooth RBFs These radial basis functions are from C ∞ ( R ) {\displaystyle C^{\infty }(\mathbb {R} )} and are strictly positive-definite
Radial_basis_function
Equation that describes density changes of a material that is diffusing in a medium
isotropic; in the case of anisotropic diffusion, D is a symmetric positive definite matrix, and the equation is written (for three dimensional diffusion)
Diffusion_equation
inversion theorem Plancherel's theorem Convolution Convolution theorem Positive-definite function Poisson summation formula Paley–Wiener theorem Sobolev space
List of harmonic analysis topics
List_of_harmonic_analysis_topics
Subgroup of the Clifford algebra associated to a quadratic space
but if the quadratic form is definite (and dimension is greater than 2), it is both. The non-trivial element of the kernel is denoted − 1 , {\displaystyle
Pin_group
Representation of a matrix as a product
Uniqueness: for positive definite matrices Cholesky decomposition is unique. However, it is not unique in the positive semi-definite case. Comment: if
Matrix_decomposition
Matrix decomposition
A {\displaystyle \mathbf {A} } . Positive definite matrices are matrices for which all eigenvalues are positive. They can be decomposed as A = L L T
Eigendecomposition of a matrix
Eigendecomposition_of_a_matrix
dG is called the Bergman distance. The Bergman metric is in fact a positive definite matrix at each point if G is a bounded domain. More importantly, the
Bergman_metric
Supervised learning of a similarity function
W} is a symmetric positive definite matrix, D W {\displaystyle D_{W}} is a metric. Moreover, as any symmetric positive semi-definite matrix W ∈ S + d {\displaystyle
Similarity_learning
Geometric inequality or concentration inequality in mathematics and probability theory
i = 1 m c i B i ∗ A i B i ) ∏ i = 1 m ( det A i ) c i | A i is a positive-definite n i × n i matrix } . {\displaystyle D=\inf \left\{\left.{\frac {\det
Brascamp–Lieb_inequality
Generalization of finite-dimensional Euclidean spaces different from Hilbert spaces
{\displaystyle Y_{B^{\prime }}^{\prime },} respectively. Any continuous positive-definite functional C {\displaystyle C} on a nuclear space A {\displaystyle
Nuclear_space
Algebraic operation on coordinate vectors
the dot product is a bilinear form. Moreover, this bilinear form is positive definite, which means that a ⋅ a {\displaystyle \mathbf {a} \cdot \mathbf {a}
Dot_product
Inequality for Harmonic Functions
u}{\partial x_{i}}}+c(t,x)u} with smooth and bounded coefficients and a positive definite matrix ( a i j ) {\displaystyle (a_{ij})} . Suppose that u ( t , x
Harnack's_inequality
Software library for numerical linear algebra
of numerical libraries List of open-source mathematical libraries Math Kernel Library (MKL) NAG Numerical Library SLATEC, a FORTRAN 77 library of mathematical
LAPACK
analysis topics. See also: Glossary of functional analysis. Bra–ket notation Definite bilinear form Direct integral Euclidean space Fundamental theorem of Hilbert
List of functional analysis topics
List_of_functional_analysis_topics
Type of group in mathematics
of the two maximal subspaces where the quadratic form is positive definite or negative definite. The component of the identity, whose elements preserve
Orthogonal_group
Bounded operators with sub-unit norm
every operator-valued positive-definite function arises in this way. Recall that every (continuous) scalar-valued positive-definite function on a topological
Contraction_(operator_theory)
with the ordinary orthogonal group, the main emphasis is on the real positive definite projective orthogonal group; other fields are elaborated in generalizations
Projective_orthogonal_group
German computer scientist
which can be applied to non-square kernel matrices and can be used with kernels that are not positive definite. Hochreiter and his collaborators have
Sepp_Hochreiter
Length in a vector space
Although this article defined "positive" to be a synonym of "positive definite", some authors instead define "positive" to be a synonym of "non-negative";
Norm_(mathematics)
Solving multiple machine learning tasks at the same time
x_{j})=k(x_{i},x_{j})A} , where k is a scalar reproducing kernel, and A is a symmetric positive semi-definite T × T {\displaystyle T\times T} matrix. Henceforth
Multi-task_learning
Group of matrices with determinant 1
unitary matrix (or special orthogonal matrix in the real case) and a positive definite Hermitian matrix (or symmetric matrix in the real case) having determinant
Special_linear_group
Statistical formula
{X}}\rightarrow \mathbb {R} } be a reproducing kernel. For a probability distribution P {\displaystyle P} with positive and differentiable density function p {\displaystyle
Stein_discrepancy
Matrix decomposition
along the diagonal. When M {\displaystyle \mathbf {M} } is positive semi-definite, the σ i {\displaystyle \sigma _{i}} will be non-negative real
Singular_value_decomposition
Operator theorem
commonly referred to as the Szegő kernel. The tricky part of the proof is showing that the condition of positive semi-definiteness implies the existence of said
Commutant_lifting_theorem
Verbs of the Hungarian language
information on the definiteness of their direct objects. This results in two types of conjugations: definite (used if there is a definite object) and indefinite
Hungarian_verbs
Concept in linear algebra
x , v ) ) {\displaystyle v\mapsto (x\mapsto f(x,v))} has a non-trivial kernel, i.e. there exist some non-zero x {\displaystyle x} in V {\displaystyle
Degenerate_bilinear_form
Non-Euclidean geometry
the hyperboloid given by q ( x ) = − 1 {\displaystyle q(x)=-1} are definite positive, hence they endow it with a Riemannian metric that turns out to be
Hyperbolic_space
Technique in numerical linear algebra
categorical. Distance matrix completion, in which case there is a positive definiteness constraint. Natural language processing, in which case the approximation
Low-rank_approximation
Type of vector space in math
b(x2 ⋅ y) for any scalars a, b, and vectors x1, x2, and y. It is positive definite: for all vectors x, x ⋅ x ≥ 0 , with equality if and only if x = 0
Hilbert_space
Fundamental space of geometry
product of a real vector space is a positive definite bilinear form, and so characterized by a positive definite quadratic form. A pseudo-Euclidean space
Euclidean_space
Types of mappings in mathematics
is a vector subspace of X , {\displaystyle X,} called the null space or kernel of the functional, or the orthogonal complement of x → , {\displaystyle
Functional_(mathematics)
Array of numbers
neither positive-semidefinite nor negative-semidefinite. A symmetric matrix is positive-definite if and only if all its eigenvalues are positive, that is
Matrix_(mathematics)
First-order differential linear operator on spinor bundle, whose square is the Laplacian
\varphi (x)} which is not identically zero, and the "geometric", positive-definite Laplacian defined by Δ = − ∇ 2 = − ∑ j = 1 n ( ∂ ∂ x j ) 2 . {\displaystyle
Dirac_operator
Structure defining distance on a manifold
tensor g is positive-definite if g ( v , v ) > 0 {\displaystyle g(v,v)>0} for every nonzero vector v. A manifold equipped with a positive-definite metric tensor
Metric_tensor
Branch of mathematical analysis
{\displaystyle f(x)} be a function defined for x > 0 {\displaystyle x>0} . Form the definite integral from 0 to x {\displaystyle x} . Call this ( J f ) ( x ) = ∫ 0
Fractional_calculus
Theorem in abstract algebra
Furthermore, since a2 ≤ 0, we have: B(a, a) > 0 for a ≠ 0. Thus B is a positive-definite symmetric bilinear form, in other words, an inner product on V. Let
Frobenius theorem (real division algebras)
Frobenius_theorem_(real_division_algebras)
Smooth and compactly supported function
prescribed set and vanish outside a larger set, and as standard examples of kernels used to construct mollifiers. Some authors use the term more broadly for
Bump_function
Matrix representation of a graph
{\displaystyle L} , the signless Laplacian Q {\displaystyle Q} also is positive semi-definite as it can be factored as Q = R R T {\displaystyle Q=RR^{\textsf
Laplacian_matrix
Optimization algorithm
system matrix A {\displaystyle \mathbf {A} } is real symmetric and positive-definite, an objective function is defined as the quadratic function, with
Gradient_descent
On surjectivity of linear map to anti-dual
sesquilinear form B {\displaystyle B} on H {\displaystyle H} is called positive definite if B ( x , x ) > 0 {\displaystyle B(x,x)>0} for all non-0 x ∈ H {\displaystyle
Fundamental theorem of Hilbert spaces
Fundamental_theorem_of_Hilbert_spaces
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POSITIVE DEFINITE-KERNEL
POSITIVE DEFINITE-KERNEL
Boy/Male
Indian, Tamil
Positive; The Lord Ganesh
Boy/Male
Tamil
Positive energy, Horseless
Boy/Male
Hindu, Indian, Tamil
Positive Energy
Girl/Female
Hindu
Positive energy, Horseless
Girl/Female
Tamil
Positive energy, Horseless
Boy/Male
Tamil
Positive, Suitable
Boy/Male
Indian
Positive Power
Boy/Male
Hindu
Positive, Suitable
Boy/Male
Hindu, Indian
Positive
Boy/Male
Hindu
Positive, Suitable
Girl/Female
Arabic, Australian, Muslim
Inspiring; Positive Attitude
Boy/Male
Hindu
Positive energy, Horseless
Boy/Male
Hindu, Indian, Japanese
Yonit; Good; Positive
Boy/Male
Tamil
Positive, Suitable
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Mythological, Sanskrit, Tamil, Telugu, Traditional
Powerful; Positive Thinker; Self Confidence; Positive; Frank; Powerful Character of Mahabharat
Boy/Male
Tamil
Anirved | அநீரà¯à®µà¯‡à®¤
Positive, Courageous, Resilient, Independent
Anirved | அநீரà¯à®µà¯‡à®¤
Boy/Male
Hindu
Positive, Courageous, Resilient, Independent
Boy/Male
Arabic, Australian, Chinese, German, Muslim
Definite; Decisive
Boy/Male
Hindu, Indian
Positive Thinking
Boy/Male
Hindu, Indian
Positive Thinking Person
POSITIVE DEFINITE-KERNEL
POSITIVE DEFINITE-KERNEL
POSITIVE DEFINITE-KERNEL
POSITIVE DEFINITE-KERNEL
POSITIVE DEFINITE-KERNEL
POSITIVE DEFINITE-KERNEL
POSITIVE DEFINITE-KERNEL
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