Measuring distance preservation in dimensionality reduction
$begingroup$
I am looking to compare the distance preserved during dimension reductions for several techniques. I have read some papers on similar topics here and here.
For example, I would like to use the Euclidean Distance to measure the distance preserved during PCA's dimension reduction. However, my point of confusion what are $X$ and $Y$ in
$$d(X, Y) = sqrt{sum^n_{i=1}left(x_i - y_iright)^2}$$
I understand how to calculate $dleft(X, Yright)$ given two vectors/matrices, but I don't understand with context to PCA. Let me try to explain.
Let $W_{k times n}$ be the matrix of $k$ eigenvectors, $X_{dtimes n}$ be the original data, and $Z_{ktimes n}$ be the projection of $X$ onto the reduced subspace.
$Z = W^{T}X$
Back to calculating $dleft(X, Yright)$. My guess is that the PCA's $X$ correspond to $X$ and $Y$ can correspond to $Z$. But how does this work since $X$ and $Y$ have different dimensions? I have to be oblivious to something here.
Also, I am not concerned if a Euclidean Distance measure is not a good choice for measuring PCA's distance preservation (unless they are incompatible). This is simply exploration.
pca dimensionality-reduction distance
asked 5 mins ago
qq3254qq3254
1
New contributor
qq3254 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
I am looking to compare the distance preserved during dimension reductions for several techniques. I have read some papers on similar topics here and here.
For example, I would like to use the Euclidean Distance to measure the distance preserved during PCA's dimension reduction. However, my point of confusion what are $X$ and $Y$ in
$$d(X, Y) = sqrt{sum^n_{i=1}left(x_i - y_iright)^2}$$
I understand how to calculate $dleft(X, Yright)$ given two vectors/matrices, but I don't understand with context to PCA. Let me try to explain.
Let $W_{k times n}$ be the matrix of $k$ eigenvectors, $X_{dtimes n}$ be the original data, and $Z_{ktimes n}$ be the projection of $X$ onto the reduced subspace.
$Z = W^{T}X$
Back to calculating $dleft(X, Yright)$. My guess is that the PCA's $X$ correspond to $X$ and $Y$ can correspond to $Z$. But how does this work since $X$ and $Y$ have different dimensions? I have to be oblivious to something here.
Also, I am not concerned if a Euclidean Distance measure is not a good choice for measuring PCA's distance preservation (unless they are incompatible). This is simply exploration.
pca dimensionality-reduction distance
asked 5 mins ago
qq3254qq3254
1
New contributor
qq3254 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
I am looking to compare the distance preserved during dimension reductions for several techniques. I have read some papers on similar topics here and here.
For example, I would like to use the Euclidean Distance to measure the distance preserved during PCA's dimension reduction. However, my point of confusion what are $X$ and $Y$ in
$$d(X, Y) = sqrt{sum^n_{i=1}left(x_i - y_iright)^2}$$
I understand how to calculate $dleft(X, Yright)$ given two vectors/matrices, but I don't understand with context to PCA. Let me try to explain.
Let $W_{k times n}$ be the matrix of $k$ eigenvectors, $X_{dtimes n}$ be the original data, and $Z_{ktimes n}$ be the projection of $X$ onto the reduced subspace.
$Z = W^{T}X$
Back to calculating $dleft(X, Yright)$. My guess is that the PCA's $X$ correspond to $X$ and $Y$ can correspond to $Z$. But how does this work since $X$ and $Y$ have different dimensions? I have to be oblivious to something here.
Also, I am not concerned if a Euclidean Distance measure is not a good choice for measuring PCA's distance preservation (unless they are incompatible). This is simply exploration.
pca dimensionality-reduction distance
asked 5 mins ago
qq3254qq3254
1
New contributor
qq3254 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
I am looking to compare the distance preserved during dimension reductions for several techniques. I have read some papers on similar topics here and here.
For example, I would like to use the Euclidean Distance to measure the distance preserved during PCA's dimension reduction. However, my point of confusion what are $X$ and $Y$ in
$$d(X, Y) = sqrt{sum^n_{i=1}left(x_i - y_iright)^2}$$
I understand how to calculate $dleft(X, Yright)$ given two vectors/matrices, but I don't understand with context to PCA. Let me try to explain.
Let $W_{k times n}$ be the matrix of $k$ eigenvectors, $X_{dtimes n}$ be the original data, and $Z_{ktimes n}$ be the projection of $X$ onto the reduced subspace.
$Z = W^{T}X$
Back to calculating $dleft(X, Yright)$. My guess is that the PCA's $X$ correspond to $X$ and $Y$ can correspond to $Z$. But how does this work since $X$ and $Y$ have different dimensions? I have to be oblivious to something here.
Also, I am not concerned if a Euclidean Distance measure is not a good choice for measuring PCA's distance preservation (unless they are incompatible). This is simply exploration.
pca dimensionality-reduction distance
pca dimensionality-reduction distance
asked 5 mins ago
qq3254qq3254
1
New contributor
qq3254 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 5 mins ago
qq3254qq3254
1
New contributor
qq3254 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 5 mins ago
qq3254qq3254
1
New contributor
qq3254 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 5 mins ago
qq3254qq3254
1
asked 5 mins ago
qq3254qq3254
1
1
New contributor
qq3254 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
qq3254 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
qq3254 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
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
});
}
});
qq3254 is a new contributor. Be nice, and check out our Code of Conduct.
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%2f49661%2fmeasuring-distance-preservation-in-dimensionality-reduction%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
qq3254 is a new contributor. Be nice, and check out our Code of Conduct.
qq3254 is a new contributor. Be nice, and check out our Code of Conduct.
qq3254 is a new contributor. Be nice, and check out our Code of Conduct.
qq3254 is a new contributor. Be nice, and check out our Code of Conduct.
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%2f49661%2fmeasuring-distance-preservation-in-dimensionality-reduction%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
