Mercer's Theorem importance
$begingroup$
I understood that Mercer's Theorem extends the definition of kernels also for infinite input space.
In Machine Learning realm our training set is always finite and hence the input space is always finite. So why are we interested in infinite input spaces?
Where is the flaw in my reasoning?
machine-learning kernel
edited Jan 6 at 15:57
Goncalo Peres
111113
asked Jan 6 at 11:39
Tommaso BendinelliTommaso Bendinelli
407
$endgroup$
bumped to the homepage by Community♦ 6 hours ago
This question has answers that may be good or bad; the system has marked it active so that they can be reviewed.
add a comment |
$begingroup$
I understood that Mercer's Theorem extends the definition of kernels also for infinite input space.
In Machine Learning realm our training set is always finite and hence the input space is always finite. So why are we interested in infinite input spaces?
Where is the flaw in my reasoning?
machine-learning kernel
edited Jan 6 at 15:57
Goncalo Peres
111113
asked Jan 6 at 11:39
Tommaso BendinelliTommaso Bendinelli
407
$endgroup$
bumped to the homepage by Community♦ 6 hours ago
This question has answers that may be good or bad; the system has marked it active so that they can be reviewed.
add a comment |
$begingroup$
I understood that Mercer's Theorem extends the definition of kernels also for infinite input space.
In Machine Learning realm our training set is always finite and hence the input space is always finite. So why are we interested in infinite input spaces?
Where is the flaw in my reasoning?
machine-learning kernel
edited Jan 6 at 15:57
Goncalo Peres
111113
asked Jan 6 at 11:39
Tommaso BendinelliTommaso Bendinelli
407
$endgroup$
I understood that Mercer's Theorem extends the definition of kernels also for infinite input space.
In Machine Learning realm our training set is always finite and hence the input space is always finite. So why are we interested in infinite input spaces?
Where is the flaw in my reasoning?
machine-learning kernel
machine-learning kernel
edited Jan 6 at 15:57
Goncalo Peres
111113
asked Jan 6 at 11:39
Tommaso BendinelliTommaso Bendinelli
407
edited Jan 6 at 15:57
Goncalo Peres
111113
asked Jan 6 at 11:39
Tommaso BendinelliTommaso Bendinelli
407
edited Jan 6 at 15:57
Goncalo Peres
111113
edited Jan 6 at 15:57
Goncalo Peres
111113
edited Jan 6 at 15:57
Goncalo Peres
111113
111113
asked Jan 6 at 11:39
Tommaso BendinelliTommaso Bendinelli
407
asked Jan 6 at 11:39
Tommaso BendinelliTommaso Bendinelli
407
asked Jan 6 at 11:39
Tommaso BendinelliTommaso Bendinelli
407
407
bumped to the homepage by Community♦ 6 hours ago
This question has answers that may be good or bad; the system has marked it active so that they can be reviewed.
bumped to the homepage by Community♦ 6 hours ago
This question has answers that may be good or bad; the system has marked it active so that they can be reviewed.
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Mercer's Theorem and infinite-dimensional spaces aren't used directly. It justifies use of things like the Gaussian kernel in SVMs. Mercer's theorem says this kernel is just an inner product in some other space, but we need not figure out what that space is, or a mapping to it. The fact that it exists is essential to proving that the SVM still works.
answered Jan 6 at 16:48
Sean Owen♦Sean Owen
4,15141934
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "557"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: false,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: null,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fdatascience.stackexchange.com%2fquestions%2f43557%2fmercers-theorem-importance%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Mercer's Theorem and infinite-dimensional spaces aren't used directly. It justifies use of things like the Gaussian kernel in SVMs. Mercer's theorem says this kernel is just an inner product in some other space, but we need not figure out what that space is, or a mapping to it. The fact that it exists is essential to proving that the SVM still works.
answered Jan 6 at 16:48
Sean Owen♦Sean Owen
4,15141934
$endgroup$
add a comment |
$begingroup$
Mercer's Theorem and infinite-dimensional spaces aren't used directly. It justifies use of things like the Gaussian kernel in SVMs. Mercer's theorem says this kernel is just an inner product in some other space, but we need not figure out what that space is, or a mapping to it. The fact that it exists is essential to proving that the SVM still works.
answered Jan 6 at 16:48
Sean Owen♦Sean Owen
4,15141934
$endgroup$
add a comment |
$begingroup$
Mercer's Theorem and infinite-dimensional spaces aren't used directly. It justifies use of things like the Gaussian kernel in SVMs. Mercer's theorem says this kernel is just an inner product in some other space, but we need not figure out what that space is, or a mapping to it. The fact that it exists is essential to proving that the SVM still works.
answered Jan 6 at 16:48
Sean Owen♦Sean Owen
4,15141934
$endgroup$
Mercer's Theorem and infinite-dimensional spaces aren't used directly. It justifies use of things like the Gaussian kernel in SVMs. Mercer's theorem says this kernel is just an inner product in some other space, but we need not figure out what that space is, or a mapping to it. The fact that it exists is essential to proving that the SVM still works.
answered Jan 6 at 16:48
Sean Owen♦Sean Owen
4,15141934
answered Jan 6 at 16:48
Sean Owen♦Sean Owen
4,15141934
answered Jan 6 at 16:48
Sean Owen♦Sean Owen
4,15141934
answered Jan 6 at 16:48
Sean Owen♦Sean Owen
4,15141934
4,15141934
add a comment |
add a comment |
Thanks for contributing an answer to Data Science Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fdatascience.stackexchange.com%2fquestions%2f43557%2fmercers-theorem-importance%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
