Fdtxjr

Infinite past with a beginning?

Do airline pilots ever risk not hearing communication directed to them specifically, from traffic controllers?

How is it possible for user's password to be changed after storage was encrypted? (on OS X, Android)

A function which translates a sentence to title-case

How do I create uniquely male characters?

Pronouncing Dictionary.com's W.O.D "vade mecum" in English

A Journey Through Space and Time

What would the Romans have called "sorcery"?

What are these boxed doors outside store fronts in New York?

Why has Russell's definition of numbers using equivalence classes been finally abandoned? ( If it has actually been abandoned).

The use of multiple foreign keys on same column in SQL Server

Prevent a directory in /tmp from being deleted

Is there a familial term for apples and pears?

Why is this code 6.5x slower with optimizations enabled?

Are tax years 2016 & 2017 back taxes deductible for tax year 2018?

How is it possible to have an ability score that is less than 3?

I probably found a bug with the sudo apt install function

Draw simple lines in Inkscape

Can Medicine checks be used, with decent rolls, to completely mitigate the risk of death from ongoing damage?

TGV timetables / schedules?

Copycat chess is back

Why doesn't Newton's third law mean a person bounces back to where they started when they hit the ground?

Why are 150k or 200k jobs considered good when there are 300k+ births a month?

Accidentally leaked the solution to an assignment, what to do now? (I'm the prof)

Kernel function with a feature space equipped with an inner product that is not the dot product

Inner product space computationThe complex inner product spacedot product vs inner product?Implicit feature space of Power KernelWhy is $mathbbR^3$ not the Reproducing Kernel Hilbert Space defined by kernel $k(x,y)=(x_1y_1+x_2y_2)^2$Hilbert space with an additional semi-inner product structureComplexification of real inner product spaces and how the inner product extends to a complex spaceLinear regression with feature representation confusion - is design matrix column space the feature space?Semi-inner product structure in complex Hilbert spacesSimple (?) Quesiton on Inner Product in Reproducing Kernel Hilbert Space

0

$begingroup$

Premise:

A function $K: mathbb R^d times mathbb R^d to mathbb R$ is called
a kernel function on $mathbbR^d$ if there exists a Hilbert space
$mathcalH$ and a map $phi: mathbb R^d to mathcalH$ such that
for any $mathbf x, mathbf yin mathbbR^d$: beginequation
labeleq:kerdef K(mathbf x,mathbf y) = langle phi(mathbf x),
phi(mathbf y) rangle_mathcalH, endequation where $
langlecdot,cdotrangle$ is an inner product.

I have recently noticed that under this definition $ langlecdot,cdotrangle$ is not necessary the standard inner product (dot product).

I have thought of the following example.
Consider the degree-two polynomial kernel on $mathbb R^2$:
beginequation
K(mathbf x,mathbf y) = (mathbf xcdot mathbf y)^2.
endequation
Then, the following is a valid feature map for this kernel:
beginequation
phi(mathbf x) = (2mathbf x_1 mathbf x_1,~ 2mathbf x_1 mathbf x_2,~ 2mathbf x_2 mathbf x_1,~ 2mathbf x_2 mathbf x_2 ),
endequation
considering the feature space $mathcal H=mathbb R^4$, with the custom inner product:
beginequation
langlemathbf x, mathbf y rangle = fracmathbf xcdot mathbf y4.
endequation

Proof:
beginalign
langle phi(mathbf x), phi(mathbf y) rangle_mathcalH &= (2mathbf x_1 mathbf x_1,~ 2mathbf x_1 mathbf x_2,~ 2mathbf x_2 mathbf x_1,~ 2mathbf x_2 mathbf x_2 ) cdot (2mathbf y_1 mathbf y_1,~ 2mathbf y_1 mathbf y_2,~ 2mathbf y_2 mathbf y_1,~ 2mathbf y_2 mathbf y_2 )/4\
&= (mathbf x cdot mathbf y)^2 \&= K(mathbf x, mathbf y)
endalign

Question:

Can you provide a less trivial example of kernel function, feature map and feature space, with an inner product that is not the dot product?

hilbert-spaces inner-product-space machine-learning reproducing-kernel-hilbert-spaces

share|cite|improve this question

asked Mar 22 at 9:06

Daniel LópezDaniel López

1185

$endgroup$

add a comment | 

0

$begingroup$

Premise:

A function $K: mathbb R^d times mathbb R^d to mathbb R$ is called
a kernel function on $mathbbR^d$ if there exists a Hilbert space
$mathcalH$ and a map $phi: mathbb R^d to mathcalH$ such that
for any $mathbf x, mathbf yin mathbbR^d$: beginequation
labeleq:kerdef K(mathbf x,mathbf y) = langle phi(mathbf x),
phi(mathbf y) rangle_mathcalH, endequation where $
langlecdot,cdotrangle$ is an inner product.

I have recently noticed that under this definition $ langlecdot,cdotrangle$ is not necessary the standard inner product (dot product).

I have thought of the following example.
Consider the degree-two polynomial kernel on $mathbb R^2$:
beginequation
K(mathbf x,mathbf y) = (mathbf xcdot mathbf y)^2.
endequation
Then, the following is a valid feature map for this kernel:
beginequation
phi(mathbf x) = (2mathbf x_1 mathbf x_1,~ 2mathbf x_1 mathbf x_2,~ 2mathbf x_2 mathbf x_1,~ 2mathbf x_2 mathbf x_2 ),
endequation
considering the feature space $mathcal H=mathbb R^4$, with the custom inner product:
beginequation
langlemathbf x, mathbf y rangle = fracmathbf xcdot mathbf y4.
endequation

Proof:
beginalign
langle phi(mathbf x), phi(mathbf y) rangle_mathcalH &= (2mathbf x_1 mathbf x_1,~ 2mathbf x_1 mathbf x_2,~ 2mathbf x_2 mathbf x_1,~ 2mathbf x_2 mathbf x_2 ) cdot (2mathbf y_1 mathbf y_1,~ 2mathbf y_1 mathbf y_2,~ 2mathbf y_2 mathbf y_1,~ 2mathbf y_2 mathbf y_2 )/4\
&= (mathbf x cdot mathbf y)^2 \&= K(mathbf x, mathbf y)
endalign

Question:

Can you provide a less trivial example of kernel function, feature map and feature space, with an inner product that is not the dot product?

hilbert-spaces inner-product-space machine-learning reproducing-kernel-hilbert-spaces

share|cite|improve this question

asked Mar 22 at 9:06

Daniel LópezDaniel López

1185

$endgroup$

add a comment | 

$begingroup$

Premise:

A function $K: mathbb R^d times mathbb R^d to mathbb R$ is called
a kernel function on $mathbbR^d$ if there exists a Hilbert space
$mathcalH$ and a map $phi: mathbb R^d to mathcalH$ such that
for any $mathbf x, mathbf yin mathbbR^d$: beginequation
labeleq:kerdef K(mathbf x,mathbf y) = langle phi(mathbf x),
phi(mathbf y) rangle_mathcalH, endequation where $
langlecdot,cdotrangle$ is an inner product.

I have recently noticed that under this definition $ langlecdot,cdotrangle$ is not necessary the standard inner product (dot product).

I have thought of the following example.
Consider the degree-two polynomial kernel on $mathbb R^2$:
beginequation
K(mathbf x,mathbf y) = (mathbf xcdot mathbf y)^2.
endequation
Then, the following is a valid feature map for this kernel:
beginequation
phi(mathbf x) = (2mathbf x_1 mathbf x_1,~ 2mathbf x_1 mathbf x_2,~ 2mathbf x_2 mathbf x_1,~ 2mathbf x_2 mathbf x_2 ),
endequation
considering the feature space $mathcal H=mathbb R^4$, with the custom inner product:
beginequation
langlemathbf x, mathbf y rangle = fracmathbf xcdot mathbf y4.
endequation

Proof:
beginalign
langle phi(mathbf x), phi(mathbf y) rangle_mathcalH &= (2mathbf x_1 mathbf x_1,~ 2mathbf x_1 mathbf x_2,~ 2mathbf x_2 mathbf x_1,~ 2mathbf x_2 mathbf x_2 ) cdot (2mathbf y_1 mathbf y_1,~ 2mathbf y_1 mathbf y_2,~ 2mathbf y_2 mathbf y_1,~ 2mathbf y_2 mathbf y_2 )/4\
&= (mathbf x cdot mathbf y)^2 \&= K(mathbf x, mathbf y)
endalign

Question:

Can you provide a less trivial example of kernel function, feature map and feature space, with an inner product that is not the dot product?

hilbert-spaces inner-product-space machine-learning reproducing-kernel-hilbert-spaces

share|cite|improve this question

asked Mar 22 at 9:06

Daniel LópezDaniel López

1185

$endgroup$

share|cite|improve this question

asked Mar 22 at 9:06

Daniel LópezDaniel López

1185

share|cite|improve this question

asked Mar 22 at 9:06

Daniel LópezDaniel López

1185

asked Mar 22 at 9:06

Daniel LópezDaniel López

1185

asked Mar 22 at 9:06

Daniel LópezDaniel López

1185