How to compute gaussian kernel matrix efficiently?

2012 10 28
2 个回答
更新时间：2022 6 20
17 次查看（30 天）

0 个投票

Hi,
I have a matrix X(10000, 800). I want to compute gramm matrix K(10000,10000), where K(i,j)= exp(-(X(i,:)-X(j,:))^2).
First i used double for loop, but then it just hangs forever. Then I tried this:
[N d] = size(X); aa = repmat(X',[1 N]); bb = repmat(reshape(X',1,[]),[N 1]); K = reshape((aa-bb).^2, [N*N d]); K = reshape(sum(D,2),[N N]); But then it uses a lot of extra space and I run out of memory very soon. Is there any efficient vectorized method for this. I am sure there must be something as this is quite a standard intermediate step for many kernel svms and also in image processing.

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Matt J
Matt J 2012-10-28
编辑：Matt J 2012-10-28
Your expression for K(i,j) does not evaluate to a scalar. Are you sure you don't want something like
exp(-norm( X(i,:) - X(j,:) ))^2);
Also, please format your code so it's more readable.
Shahid Mahmood
Shahid Mahmood 2019-11-21
can you explain the whole procedure in detail to compute a kernel matrix in matlab

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回答（2 个）

Matt J
Matt J 2012-10-28
编辑：Matt J 2012-10-28

2 个投票

Assuming you really want exp(-norm( X(i,:) - X(j,:) ))^2), then one way is
nsq=sum(X.^2,2);
K=bsxfun(@minus,nsq,(2*X)*X.');
K=bsxfun(@plus,nsq.',K);
K=exp(-K);

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Jeon
Jeon 2015-3-10
How I can modify the code when I want to involve 'sigma', that is, I want to calculate 'exp(-norm(X1(:,i)-X2(:,j))^2/(2*sigma^2));' instead? Thanks
Matt J
Matt J 2015-3-10
编辑：Matt J 2015-3-10
Just pre-normalize X1 and X2 by sigma.
Farzan Zaheer
Farzan Zaheer 2015-8-4
编辑：Farzan Zaheer 2015-8-4
I am working on Kernel LMS, and I am having issues with the implementation of Kernel. I want to know what exactly is "X2" here. I am implementing the Kernel using recursion.
I am using the following statement,
for n=2:K-1
Kernel(n)=exp(-0.5*(dist(x(:,2:n),x(:,n)')/ker_bw^2));
end
where ker_bw is the kernel bandwidth/sigma and x is input of (1000,1) and I have reshaped the input x as
x = [x(1:end-1),x(2:end)];
as mentioned in the research paper I am following. Any help will be highly appreciated.

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Image Analyst
Image Analyst 2012-10-28

0 个投票

If you have the Image Processing Toolbox, why not use fspecial()?
h = fspecial('gaussian', hsize, sigma) returns a rotationally symmetric Gaussian lowpass filter of size hsize with standard deviation sigma (positive). hsize can be a vector specifying the number of rows and columns in h, or it can be a scalar, in which case h is a square matrix. The default value for hsize is [3 3]; the default value for sigma is 0.5.

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Mohammad Alwardat
Mohammad Alwardat 2020-3-23
Hi Image Analyst,
How can I study the similarity between 2 vectors x and y using Gaussian kernel similarity algorithm? And how can I determine the parameter sigma?
Thanks♥
amel kaouane
amel kaouane 2022-6-20
i have the same problem, don't know to get the parameter sigma
Image Analyst
Image Analyst 2022-6-20
Ran in:
Ran in:
@amel kaouane it comes from your mind. You think up some sigma that might work, assign it like
sigma = 5;
hsize = 21;
h = fspecial('gaussian', hsize, sigma)
h = 21×21
0.0001 0.0002 0.0003 0.0003 0.0005 0.0006 0.0007 0.0008 0.0009 0.0009 0.0009 0.0009 0.0009 0.0008 0.0007 0.0006 0.0005 0.0003 0.0003 0.0002 0.0001 0.0002 0.0003 0.0004 0.0005 0.0007 0.0008 0.0010 0.0011 0.0012 0.0013 0.0014 0.0013 0.0012 0.0011 0.0010 0.0008 0.0007 0.0005 0.0004 0.0003 0.0002 0.0003 0.0004 0.0005 0.0007 0.0009 0.0012 0.0014 0.0016 0.0018 0.0019 0.0019 0.0019 0.0018 0.0016 0.0014 0.0012 0.0009 0.0007 0.0005 0.0004 0.0003 0.0003 0.0005 0.0007 0.0010 0.0012 0.0016 0.0019 0.0021 0.0024 0.0025 0.0026 0.0025 0.0024 0.0021 0.0019 0.0016 0.0012 0.0010 0.0007 0.0005 0.0003 0.0005 0.0007 0.0009 0.0012 0.0016 0.0020 0.0024 0.0028 0.0031 0.0033 0.0033 0.0033 0.0031 0.0028 0.0024 0.0020 0.0016 0.0012 0.0009 0.0007 0.0005 0.0006 0.0008 0.0012 0.0016 0.0020 0.0025 0.0030 0.0035 0.0038 0.0041 0.0042 0.0041 0.0038 0.0035 0.0030 0.0025 0.0020 0.0016 0.0012 0.0008 0.0006 0.0007 0.0010 0.0014 0.0019 0.0024 0.0030 0.0036 0.0042 0.0046 0.0049 0.0050 0.0049 0.0046 0.0042 0.0036 0.0030 0.0024 0.0019 0.0014 0.0010 0.0007 0.0008 0.0011 0.0016 0.0021 0.0028 0.0035 0.0042 0.0048 0.0053 0.0056 0.0057 0.0056 0.0053 0.0048 0.0042 0.0035 0.0028 0.0021 0.0016 0.0011 0.0008 0.0009 0.0012 0.0018 0.0024 0.0031 0.0038 0.0046 0.0053 0.0058 0.0062 0.0063 0.0062 0.0058 0.0053 0.0046 0.0038 0.0031 0.0024 0.0018 0.0012 0.0009 0.0009 0.0013 0.0019 0.0025 0.0033 0.0041 0.0049 0.0056 0.0062 0.0066 0.0067 0.0066 0.0062 0.0056 0.0049 0.0041 0.0033 0.0025 0.0019 0.0013 0.0009
imshow(h, []);
axis('on', 'image')
If you don't like 5 for sigma then just try others until you get one that you like. It's not like I can tell you the perfect value of sigma because it really depends on your situation and image.
amel kaouane
amel kaouane 2022-6-20
am looking to get similarity between two time series by using this gaussian kernel, i think it's not the same situation, right?!

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