Efficiently compute pairwise squared Euclidean distance in Matlab

Question

Given two sets of `d`-dimensional points. How can I most efficiently compute the pairwise squared euclidean distance matrix in Matlab? Notation: Set one is given by a `(numA,d)`-matrix `A` and set two is given by a `(numB,d)`-matrix `B`. The resulting distance matrix shall be of the format `(numA,numB)`. Example points: ``` d = 4; % dimension numA = 100; % number of set 1 points numB = 200; % number of set 2 points A = rand(numA,d); % set 1 given as matrix A B = rand(numB,d); % set 2 given as matrix B ```

Accepted Answer

The usually given answer here is based on `bsxfun` (cf. e.g. [1]). My proposed approach is based on matrix multiplication and turns out to be much faster than any comparable algorithm I could find:

```
helpA = zeros(numA,3*d);
helpB = zeros(numB,3*d);
for idx = 1:d
    helpA(:,3*idx-2:3*idx) = [ones(numA,1), -2*A(:,idx), A(:,idx).^2 ];
    helpB(:,3*idx-2:3*idx) = [B(:,idx).^2 ,    B(:,idx), ones(numB,1)];
end
distMat = helpA * helpB';
```

Please note: For constant `d` one can replace the `for`-loop by hardcoded implementations, e.g.

```
helpA(:,3*idx-2:3*idx) = [ones(numA,1), -2*A(:,1), A(:,1).^2, ... % d == 2
                          ones(numA,1), -2*A(:,2), A(:,2).^2 ];   % etc.
```

Evaluation:

```
%% create some points
d = 2; % dimension
numA = 20000;
numB = 20000;
A = rand(numA,d);
B = rand(numB,d);

%% pairwise distance matrix
% proposed method:
tic;
helpA = zeros(numA,3*d);
helpB = zeros(numB,3*d);
for idx = 1:d
    helpA(:,3*idx-2:3*idx) = [ones(numA,1), -2*A(:,idx), A(:,idx).^2 ];
    helpB(:,3*idx-2:3*idx) = [B(:,idx).^2 ,    B(:,idx), ones(numB,1)];
end
distMat = helpA * helpB';
toc;

% compare to pdist2:
tic;
pdist2(A,B).^2;
toc;

% compare to [1]:
tic;
bsxfun(@plus,dot(A,A,2),dot(B,B,2)')-2*(A*B');
toc;

% Another method: added 07/2014
% compare to ndgrid method (cf. Dan's comment)
tic;
[idxA,idxB] = ndgrid(1:numA,1:numB);
distMat = zeros(numA,numB);
distMat(:) = sum((A(idxA,:) - B(idxB,:)).^2,2);
toc;
```

Result:

```
Elapsed time is 1.796201 seconds.
Elapsed time is 5.653246 seconds.
Elapsed time is 3.551636 seconds.
Elapsed time is 22.461185 seconds.
```

For a more detailed evaluation w.r.t. dimension and number of data points follow the discussion below (@comments). It turns out that different algos should be preferred in different settings. In non time critical situations just use the `pdist2` version.

Further development: One can think of replacing the squared euclidean by any other metric based on the same principle:

```
help = zeros(numA,numB,d);
for idx = 1:d
    help(:,:,idx) = [ones(numA,1), A(:,idx)     ] * ...
                    [B(:,idx)'   ; -ones(1,numB)];
end
distMat = sum(ANYFUNCTION(help),3);
```

Nevertheless, this is quite time consuming. It could be useful to replace for smaller `d` the 3-dimensional matrix `help` by `d` 2-dimensional matrices. Especially for `d = 1` it provides a method to compute the pairwise difference by a simple matrix multiplication:

```
pairDiffs = [ones(numA,1), A ] * [B'; -ones(1,numB)];
```

Do you have any further ideas?

Efficiently compute pairwise squared Euclidean distance in Matlab

Solution

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