Star (game theory)

Figure illustrates related but varying approaches to viewing RKHS

In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space is a Hilbert space of functions in which pointwise evaluation is a continuous linear functional. Equivalently, they are spaces that can be defined by reproducing kernels. The subject was originally and simultaneously developed by Nachman Aronszajn (1907–1980) and Stefan Bergman (1895–1977) in 1950.

In this article we assume that Hilbert spaces are complex. The main reason for this is that many of the examples of reproducing kernel Hilbert spaces are spaces of analytic functions, although some real Hilbert spaces also have reproducing kernels.

An important subset of the reproducing kernel Hilbert spaces are the reproducing kernel Hilbert spaces associated to a continuous kernel. These spaces have wide applications, including complex analysis, harmonic analysis, quantum mechanics, statistics and machine learning.

Definition

Let X be an arbitrary set and H a Hilbert space of complex-valued functions on X. We say that H is a reproducing kernel Hilbert space if the linear map

$L_x:f\mapsto f(x)$

from H to the complex numbers is continuous for any x in X; equivalently, if $L_x$ is a Bounded operator for any x in X. By the Riesz representation theorem, this implies that for every x in X there exists a unique element K_x of H with the property that:

$f(x)=L_x(f)=⟨f,K_x⟩\mathrm{\forall }f\in H(*).$

The function K_x is called the point-evaluation function at the point x.

Since H is a space of functions, the element K_x is itself a function with domain X, and may be written as K_x(y). The family of functions K_x(y) may be encoded into a single function K : X × X → C by defining

$K(x,y)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\overline{K_x(y)}.$

This function is called the reproducing kernel for the Hilbert space H and it is determined entirely by H because the Riesz representation theorem guarantees, for every x in X, that the element K_x satisfying (*) is unique.

Properties

The Reproducing Property

It is clear from the discussion above that

$K(x,y)=\overline{K(y,x)}=⟨K_y,K_x⟩.$

In particular,

$K(x,x)=⟨K_x,K_x⟩\ge 0,\mathrm{\forall }x\in X.$

Note that

$K_x=0\text{ if and only if }f(x)=0\mathrm{\forall }f\in H.$

Orthonormal Bases

If {φ₁, φ₂, ...} is an orthonormal basis of H, then

$K(x,y)=\sum _{k=1}^{\mathrm{\infty }}{\phi }_k(x)\overline{{\phi }_k(y)}.$

Reproducing Kernel

Definition

Let X be some set. A function $K:X\times X\to \mathbb{ℝ}$ is a reproducing kernel if it is symmetric and positive definite:

$\sum _{i,j=1}^n{c}_i{c}_{j}K(t_i,t_j)\ge 0$

We have now defined what a reproducing kernel is in its own right. And we can check that K, the reproducing kernel for the Hilbert space H above, is in fact a reproducing kernel.

-Symmetry follows from the symmetry property of inner products: $<K_t,K_x{>}_H=<K_x,K_t{>}_H$.

-Positive-definiteness follows because $\sum _{i,j=1}^n{c}_i{c}_{j}K(t_i,t_j)=\sum _{i,j=1}^n{c}_i{c}_j<K_{t_i},K_{t_j}{>}_H=\Vert \sum _{j=1}^n{c}_j{K}_{t_j}{\Vert }_H^2\ge 0$

Moore-Aronszajn Theorem

We have seen how a reproducing kernel Hilbert space defines reproducing kernel function. It follows from the definition of an inner product that the kernel defined is (conjugate) symmetric and positive definite. The Moore-Aronszajn theorem goes in the other direction; it says that every symmetric, positive definite kernel defines a unique reproducing kernel Hilbert space. The theorem first appeared in Aronszajn's Theory of Reproducing Kernels, although he attributes it to E. H. Moore.

Theorem. Suppose K is a symmetric, positive definite kernel on a set E. Then there is a unique Hilbert space of functions on E for which K is a reproducing kernel.

Proof. For all x in E, define K_x = K(x, ⋅ ). Let H₀ be the linear span of {K_x : x ∈ E}. Define an inner product on H₀ by

$⟨\sum _{j=1}^n{b}_j{K}_{y_j},\sum _{i=1}^m{a}_i{K}_{x_i}⟩=\sum _{i=1}^m\sum _{j=1}^n\overline{a_i}{b}_{j}K(y_j,x_i).$

The symmetry of this inner product follows from the symmetry of K and the non-degeneracy follows from the fact that K is positive definite.

Let H be the completion of H₀ with respect to this inner product. Then H consists of functions of the form

$f(x)=\sum _{i=1}^{\mathrm{\infty }}{a}_i{K}_{x_i}(x)$

where $\sum _{i=1}^{\mathrm{\infty }}{a}_i^{2}K(x_i,x_i)<\mathrm{\infty }$. The fact that the above sum converges for every x follows from the Cauchy-Schwarz inequality.

Now we can check the RKHS property, (*):

$⟨f,K_x⟩=⟨\sum _{i=1}^{\mathrm{\infty }}{a}_i{K}_{x_i},K_x⟩=\sum _{i=1}^{\mathrm{\infty }}{a}_{i}K(x_i,x)=f(x).$

To prove uniqueness, let G be another Hilbert space of functions for which K is a reproducing kernel. For any x and y in E, (*) implies that

$⟨K_x,K_y{⟩}_H=K(x,y)=⟨K_x,K_y{⟩}_G.$

By linearity, $⟨\cdot ,\cdot {⟩}_H=⟨\cdot ,\cdot {⟩}_G$ on the span of {K_x : x ∈ E}. Then G = H by the uniqueness of the completion.

Integral Operators and Mercer Theorem

We may characterize a symmetric positive semi-definite kernel K via the integral operator, using the Mercer's theorem.

Suppose K is a continuous positive semi-definite kernel on a compact set $X\subset {\mathbb{ℝ}}^n,\mu$ a Borel measure on $X$, and the integral operator $T_K:L_2(X)\to L_2(X)$ defined by $[T_{K}f](\cdot )=\underset{X}{\int }K(\cdot ,t)f(t)d\mu (t)$ is positive semi-definite.

By the Mercer Theorem, there is an orthonormal basis $\{{\varphi }_i{\}}_i$ of $L_2(X)$ consisting of eigenfunctions of $T_K$ such that the corresponding sequence of eigenvalues $\{{\sigma }_i{\}}_i$ is nonnegative. The eigenfunctions corresponding to non-zero eigenvalues are continuous on X and K has the representation

$K(s,t)=\sum _{j=1}^{\mathrm{\infty }}{\sigma }_j{\varphi }_j(s){\varphi }_j(t)$

where the convergence is absolute and uniform, that is, $\underset{n\to \mathrm{\infty }}{\lim }{\sup }_{u,v}|K(u,v)-\sum _{j=1}^n{\sigma }_j{\varphi }_j(u){\varphi }_j(v)|=0$

If X is a compact set, then $T_K$ is a compact, positive, and self-adjoint operator, which yields a decreasing sequence $({\sigma }_i{)}_i\ge 0$ such that $\underset{i\to \mathrm{\infty }}{\lim }{\sigma }_i=0$ and $T_K\varphi (x)={\sigma }_i{\varphi }_i(x)$ , where the $\{{\varphi }_i\}$ form an orthonormal basis of $L_2(X)$.

This means that $\mathrm{\forall }f\in L_2(X)f=\sum _{i=1}^{\mathrm{\infty }}<f,{\varphi }_i>{\varphi }_i$, so that the action of $T_K$ can be written as $[T_{K}f]=\sum _{i=1}^{\mathrm{\infty }}{\sigma }_i<f,{\varphi }_i>{\varphi }_i$

Furthermore, a Reproducing Kernel Hilbert Space $H$ can now be constructed, with $K$ as the reproducing kernel.

It can be shown to be $H=\{f\in L_2(X)|\sum _{i=1}^{\mathrm{\infty }}\frac{<f,{\varphi }_i{>}^2}{{\sigma }_i}<\mathrm{\infty }\}$,

with the inner product of $H$ given by $<f,g{>}_H=\sum _{i=1}^{\mathrm{\infty }}\frac{<f,{\varphi }_i{>}_{L_2}<g,{\varphi }_i{>}_{L_2}}{{\sigma }_i}$

Feature Map

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. A feature map is a map $\phi :X\to F$, where $F$ is a Hilbert space called the Feature Space.

Every feature map defines a kernel via $K(x,s)=<\phi (x),\phi (s)>$

For example, for the reproducing kernel in the previous section, we can define $\phi (x)=(\sqrt{(}{\sigma }_i){\varphi }_i(x){)}_i$. We can always associate a feature map to every kernel; in fact, there exist infinitely many feature maps associated with $K$.

With feature maps, we can now view functions as hyperplanes in the Feature Space. For example, for any $x\in X$, $\{w\in F|f(x)=<w,\phi (x)>\}$ is a hyperplane.

In machine learning, kernel-based procedures allow us to map the data from the input space X into the Feature Space where linear methods may then be used.

Examples

Linear Kernel: $K(x,y)=<x,y>$

Radial Basis Function Kernels:

These are kernels which satisfy $K(x,y)=K(\Vert x-y\Vert )$

-Gaussian Kernel:

Sometimes referred to as the Radial basis function kernel,

$K(x,y)=e^{-\frac{\Vert x-y{\Vert }^2}{2{\sigma }^2}},\sigma >0$

-Polynomial Kernel:

$K(x,y)=(\alpha ⟨x,y⟩+1{)}^d,\alpha \in \mathbb{ℝ},d\in \mathbb{\mathbb{N}}$

-Exponential Kernel:

$K(x,y)=e^{-\frac{\Vert x-y\Vert }{2{\sigma }^2}},\sigma >0$

-Laplacian Kernel:

$K(x,y)=e^{-\frac{\Vert x-y\Vert }{\sigma }},\sigma >0$

In another example, when X is finite and H consists of all complex-valued functions on X, then an element of H can be represented as an array of complex numbers. If the usual inner product is used, then K_x is the function whose value is 1 at x and 0 everywhere else, and K(x,y) can be thought of as an identity matrix since K(x,y)=1 when x=y and K(x,y)=0 otherwise. In this case, H is isomorphic to Cⁿ.

The case of X = D is more sophisticated, here the Bergman space H²(D) is the space of square-integrable holomorphic functions on D. It can be shown that the reproducing kernel for H²(D) is

$K(x,y)=\frac{1}{\pi }\frac{1}{(1-x\bar{y}{)}^2}.$

This kernel is an example of a Bergman kernel, named for Stefan Bergman.

See also

• Positive definite kernel
• Mercer's theorem
• Kernel trick
• Kernel embedding of distributions

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

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  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• Alain Berlinet and Christine Thomas, Reproducing kernel Hilbert spaces in Probability and Statistics, Kluwer Academic Publishers, 2004.
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• Grace Wahba, Spline Models for Observational Data, SIAM, 1990.
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang