I have tried searching for an answer on this, but can't find one that specifically addresses my issue. Although vectorizing in MATLAB must draw many questions in, the problem I am having is less general than typical examples I have found on the web. My background is more in C++ than MATLAB, so this is an odd concept for me to get my head around.
I am trying to evolve a Hamiltonian matrix from it's initial state (being a column vector where all elements but the last is a 0, and the last is a 1) to a final state as time increases. This is achieved by sequentially applying a time evolution operator U to the state. I also want to use the new state at each time interval to calculate an observable property.
I have achieved this, as can be seen in the code below. However, I need to make this code as efficient as possible, and so I was hoping to vectorize, rather than rely on for loops. However, I am unsure of how to vectorize this code. The problem I have is that on each iteration of the for loop, the column vector psi should change its values. Each new psi is then used to calculate my observable M for each interval of time. I am unsure of how to track the evolution of psi such that I can end up with a row vector for M, giving the outcome of the application of each new psi.
time = tmin:dt:tmax;
H = magic(2^N)
X = [0,1;1,0]
%%% INITIALISE COLUMN VECTOR
init = sparse(2^N,1);
init(2^N) = 1;
%%% UNITARY TIME EVOLUTION OPERATOR
U = expm(-1i*H*dt);
%%% TIME EVOLVUTION
for num = 1:length(time)
psi = U*init;
init = psi;
%%% CALCULATE OBSERVABLE
M(num) = psi' * kron(X,speye(2^(N-1))) * psi
end
Any help would be greatly appreciated.
I have quickly come up with the following partially vectorized code:
time = tmin:dt:tmax;
H = magic(2^N);
X = [0,1;1,0];
%%% INITIALISE COLUMN VECTOR
init = sparse(2^N,1);
init(2^N) = 1;
%%% UNITARY TIME EVOLUTION OPERATOR
U = expm(-1i*H*dt);
%%% TIME EVOLVUTION
% preallocate psi
psi = complex(zeros(2^N, length(time)));
% compute psi for all timesteps
psi(:,1) = U*init;
for num = 2:length(time)
psi(:,num) = U*psi(:, num-1);
end
% precompute kronecker product (if X is constant through time)
F = kron(X,speye(2^(N-1)));
%%% CALCULATE OBSERVABLE
M = sum((psi' * F) .* psi.', 2);
However, it seems that the most computationally intensive part of your problem is computation of the psi. For that I can't see any obvious way to vectorize as it depends on the value computed in the previous step.
This line:
M = sum((psi' * F) .* psi.', 2);
is a little Matlab trick to compute psi(:,i)'*F*psi(:,i) in a vectorized way.
Related
Summary: This question deals with the improvement of an algorithm for the computation of linear regression.
I have a 3D (dlMAT) array representing monochrome photographs of the same scene taken at different exposure times (the vector IT) . Mathematically, every vector along the 3rd dimension of dlMAT represents a separate linear regression problem that needs to be solved. The equation whose coefficients need to be estimated is of the form:
DL = R*IT^P, where DL and IT are obtained experimentally and R and P must be estimated.
The above equation can be transformed into a simple linear model after applying a logarithm:
log(DL) = log(R) + P*log(IT) => y = a + b*x
Presented below is the most "naive" way to solve this system of equations, which essentially involves iterating over all "3rd dimension vectors" and fitting a polynomial of order 1 to (IT,DL(ind1,ind2,:):
%// Define some nominal values:
R = 0.3;
IT = 600:600:3000;
P = 0.97;
%// Impose some believable spatial variations:
pMAT = 0.01*randn(3)+P;
rMAT = 0.1*randn(3)+R;
%// Generate "fake" observation data:
dlMAT = bsxfun(#times,rMAT,bsxfun(#power,permute(IT,[3,1,2]),pMAT));
%// Regression:
sol = cell(size(rMAT)); %// preallocation
for ind1 = 1:size(dlMAT,1)
for ind2 = 1:size(dlMAT,2)
sol{ind1,ind2} = polyfit(log(IT(:)),log(squeeze(dlMAT(ind1,ind2,:))),1);
end
end
fittedP = cellfun(#(x)x(1),sol); %// Estimate of pMAT
fittedR = cellfun(#(x)exp(x(2)),sol); %// Estimate of rMAT
The above approach seems like a good candidate for vectorization, since it does not utilize MATLAB's main strength that is MATrix operations. For this reason, it does not scale very well and takes much longer to execute than I think it should.
There exist alternative ways to perform this computation based on matrix division, as demonstrated here and here, which involve something like this:
sol = [ones(size(x)),log(x)]\log(y);
That is, appending a vector of 1s to the observations, followed by mldivide to solve the equation system.
The main challenge I'm facing is how to adapt my data to the algorithm (or vice versa).
Question #1: How can the matrix-division-based solution be extended to solve the problem presented above (and potentially replace the loops I am using)?
Question #2 (bonus): What is the principle behind this matrix-division-based solution?
The secret ingredient behind the solution that includes matrix division is the Vandermonde matrix. The question discusses a linear problem (linear regression), and those can always be formulated as a matrix problem, which \ (mldivide) can solve in a mean-square error sense‡. Such an algorithm, solving a similar problem, is demonstrated and explained in this answer.
Below is benchmarking code that compares the original solution with two alternatives suggested in chat1, 2 :
function regressionBenchmark(numEl)
clc
if nargin<1, numEl=10; end
%// Define some nominal values:
R = 5;
IT = 600:600:3000;
P = 0.97;
%// Impose some believable spatial variations:
pMAT = 0.01*randn(numEl)+P;
rMAT = 0.1*randn(numEl)+R;
%// Generate "fake" measurement data using the relation "DL = R*IT.^P"
dlMAT = bsxfun(#times,rMAT,bsxfun(#power,permute(IT,[3,1,2]),pMAT));
%% // Method1: loops + polyval
disp('-------------------------------Method 1: loops + polyval')
tic; [fR,fP] = method1(IT,dlMAT); toc;
fprintf(1,'Regression performance:\nR: %d\nP: %d\n',norm(fR-rMAT,1),norm(fP-pMAT,1));
%% // Method2: loops + Vandermonde
disp('-------------------------------Method 2: loops + Vandermonde')
tic; [fR,fP] = method2(IT,dlMAT); toc;
fprintf(1,'Regression performance:\nR: %d\nP: %d\n',norm(fR-rMAT,1),norm(fP-pMAT,1));
%% // Method3: vectorized Vandermonde
disp('-------------------------------Method 3: vectorized Vandermonde')
tic; [fR,fP] = method3(IT,dlMAT); toc;
fprintf(1,'Regression performance:\nR: %d\nP: %d\n',norm(fR-rMAT,1),norm(fP-pMAT,1));
function [fittedR,fittedP] = method1(IT,dlMAT)
sol = cell(size(dlMAT,1),size(dlMAT,2));
for ind1 = 1:size(dlMAT,1)
for ind2 = 1:size(dlMAT,2)
sol{ind1,ind2} = polyfit(log(IT(:)),log(squeeze(dlMAT(ind1,ind2,:))),1);
end
end
fittedR = cellfun(#(x)exp(x(2)),sol);
fittedP = cellfun(#(x)x(1),sol);
function [fittedR,fittedP] = method2(IT,dlMAT)
sol = cell(size(dlMAT,1),size(dlMAT,2));
for ind1 = 1:size(dlMAT,1)
for ind2 = 1:size(dlMAT,2)
sol{ind1,ind2} = flipud([ones(numel(IT),1) log(IT(:))]\log(squeeze(dlMAT(ind1,ind2,:)))).'; %'
end
end
fittedR = cellfun(#(x)exp(x(2)),sol);
fittedP = cellfun(#(x)x(1),sol);
function [fittedR,fittedP] = method3(IT,dlMAT)
N = 1; %// Degree of polynomial
VM = bsxfun(#power, log(IT(:)), 0:N); %// Vandermonde matrix
result = fliplr((VM\log(reshape(dlMAT,[],size(dlMAT,3)).')).');
%// Compressed version:
%// result = fliplr(([ones(numel(IT),1) log(IT(:))]\log(reshape(dlMAT,[],size(dlMAT,3)).')).');
fittedR = exp(real(reshape(result(:,2),size(dlMAT,1),size(dlMAT,2))));
fittedP = real(reshape(result(:,1),size(dlMAT,1),size(dlMAT,2)));
The reason why method 2 can be vectorized into method 3 is essentially that matrix multiplication can be separated by the columns of the second matrix. If A*B produces matrix X, then by definition A*B(:,n) gives X(:,n) for any n. Moving A to the right-hand side with mldivide, this means that the divisions A\X(:,n) can be done in one go for all n with A\X. The same holds for an overdetermined system (linear regression problem), in which there is no exact solution in general, and mldivide finds the matrix that minimizes the mean-square error. In this case too, the operations A\X(:,n) (method 2) can be done in one go for all n with A\X (method 3).
The implications of improving the algorithm when increasing the size of dlMAT can be seen below:
For the case of 500*500 (or 2.5E5) elements, the speedup from Method 1 to Method 3 is about x3500!
It is also interesting to observe the output of profile (here, for the case of 500*500):
Method 1
Method 2
Method 3
From the above it is seen that rearranging the elements via squeeze and flipud takes up about half (!) of the runtime of Method 2. It is also seen that some time is lost on the conversion of the solution from cells to matrices.
Since the 3rd solution avoids all of these pitfalls, as well as the loops altogether (which mostly means re-evaluation of the script on every iteration) - it unsurprisingly results in a considerable speedup.
Notes:
There was very little difference between the "compressed" and the "explicit" versions of Method 3 in favor of the "explicit" version. For this reason it was not included in the comparison.
A solution was attempted where the inputs to Method 3 were gpuArray-ed. This did not provide improved performance (and even somewhat degradaed them), possibly due to wrong implementation, or the overhead associated with copying matrices back and forth between RAM and VRAM.
I have the for loop (outlined below) in my code which takes a while to run. CALC is a function I have defined; Dis a matrix; Y is a matrix; k is a vector. Is there a way I can vectorize this code such that I do away with the for loop? Any contribution will be highly appreciated.
for column = 1:n
q(:,column) = CALC(D,Y(:,column), k(column));
end
The CALC function is outlined below:
function [x] = CALC(A, y, s)
[m, n] = size(A);
% y is an m x 1 vector
% s is an integer
r = y;
index_cols = [];
atoms = [];
for i = 1 : s
[max_r, lambda_t] = max(abs(r'*A));
index_cols = [index_cols, lambda_t];
atoms = [atoms, A(:,lambda_t)];
x_t = pinv(atoms)*y;
r = y - atoms*x_t;
end
x = zeros(n,1);
x(index_cols) = x_t;
end
I will expand on rayryeng's comment. Vectorization means grouping some elementary operations together in such a way that they can be jointly handled by a low-level routine. But the bulk of execution time of your code is the computation of pinv(atoms); everything else is not nearly as expensive.
If your task is to saw several pieces of wood, you can clamp them together and saw them all at once. That's vectorization.
But that does not work when you're a mechanic whose task is to repair several cars. The bulk of your time will have to be spent working on an individual car.
Things you can consider:
Caching. Your code computes pseudoinverses of matrices that are always made of the columns of the same matrix D. So it may happen to call pinv with the same atoms input multiple times. Investigate whether this happens often enough to warrant caching the pseudoinverses. Here's an example of caching Matlab results
Parallelizing, if you have the hardware and software for this.
Rethink the algorithm...
I have a code which works perfectly, and I'm looking to make it more efficient.
t = -1:.001:1;
t_for_y = -50:.01:50;
x = zeros(size(t));
x(1001:end) = exp(-3 * t(1001:end));
h = zeros(size(t));
h(1001:end) = exp(-2 * t(1001:end)); % FIXED TYPO
for k = 1:length(t_for_y)
X(k)=trapz(t,x.*exp(-1i*t*t_for_y(k)));
H(k)=trapz(t,h.*exp(-1i*t*t_for_y(k)));
end
Y = X.*H;
for k = 1:length(t)
y(k) = (1/(2*pi))*trapz(t_for_y,Y.*exp(1i*t(k)*t_for_y));
end
plot(t,real(y));grid on;
I'd like to only use one for-loop or no for loops is this possible?
Is there a way of using doing this faster?
The trapz function can take a matrix as the second input (see help trapz for more info). This means that your first column can be replaced by the following:
t_i = 1i*t';
exp_t = bsxfun(#times,t_i,t_for_y); % Precompute for speed
xexp = bsxfun(#times,x',exp_t);
hexp = bsxfun(#times,h',exp_t);
% NOTE: As you've got it, X and H are identical - I assume this is a typo
X = trapz(t,xexp,1);
H = trapz(t,xexp,1);
Be aware that this will generate some fairly large matrices (~2000 X 10000), which can eat up your memory if you're not careful.
The second loop can be linearised in a similar manner:
% Using exp_t from the previous loop
yexp = bsxfun(#times,Y,exp_t);
% NOTE: As you've got it, X and H are identical - I assume this is a typo
y = trapz(t_for_y,xexp,2);
Again, this will use a lot of memory. You may find that you will save memory by using sparse matrices.
If memory is at a premium for you, then your original code is better (though you should preallocate X, H and y for a slight speed boost), as the time saved by linearising it is not really enough to justify the extra memory. If you've got memory aplenty, then this method is slightly faster.
I am trying to get the angle between every vector in a large array (1896378x4 -EDIT: this means I need 1.7981e+12 angles... TOO LARGE, but if there's room to optimize the code, let me know anyways). It's too slow - I haven't seen it finish yet. Here's the steps towards optimizing I've taken:
First, logically what I (think I) want (just use Bt=rand(N,4) for testing):
[ro,col]=size(Bt);
angbtwn = zeros(ro-1); %too long to compute!! total non-zero = ro*(ro-1)/2
count=1;
for ii=1:ro-1
for jj=ii+1:ro
angbtwn(count) = atan2(norm(cross(Bt(ii,1:3),Bt(jj,1:3))), dot(Bt(ii,1:3),Bt(jj,1:3))).*180/pi;
count=count+1;
end
end
So, I though I'd try and vectorize it, and get rid of the non-built-in functions:
[ro,col]=size(Bt);
% angbtwn = zeros(ro-1); %TOO LONG!
for ii=1:ro-1
allAxes=Bt(ii:ro,1:3);
repeachAxis = allAxes(ones(ro-ii+1,1),1:3);
c = [repeachAxis(:,2).*allAxes(:,3)-repeachAxis(:,3).*allAxes(:,2)
repeachAxis(:,3).*allAxes(:,1)-repeachAxis(:,1).*allAxes(:,3)
repeachAxis(:,1).*allAxes(:,2)-repeachAxis(:,2).*allAxes(:,1)];
crossedAxis = reshape(c,size(repeachAxis));
normedAxis = sqrt(sum(crossedAxis.^2,2));
dottedAxis = sum(repeachAxis.*allAxes,2);
angbtwn(1:ro-ii+1,ii) = atan2(normedAxis,dottedAxis)*180/pi;
end
angbtwn(1,:)=[]; %angle btwn vec and itself
%only upper left triangle are values...
Still too long, even to pre-allocate... So I try to do sparse, but not implemented right:
[ro,col]=size(Bt);
%spalloc:
angbtwn = sparse([],[],[],ro,ro,ro*(ro-1)/2);%zeros(ro-1); %cell(ro,1)
for ii=1:ro-1
...same
angbtwn(1:ro-ii+1,ii) = atan2(normedAxis,dottedAxis)*180/pi; %WARNED: indexing = >overhead
% WHAT? Can't index sparse?? what's the point of spalloc then?
end
So if my logic can be improved, or if sparse is really the way to go, and I just can't implement it right, let me know where to improve. THANKS for your help.
Are you trying to get the angle between every pair of vectors in Bt? If Bt has 2 million vectors that's a trillion pairs each (apparently) requiring an inner product to get the angle between. I don't know that any kind of optimization is going to help have this operation finish in a reasonable amount of time in MATLAB on a single machine.
In any case, you can turn this problem into a matrix multiplication between matrices of unit vectors:
N=1000;
Bt=rand(N,4); % for testing. A matrix of N (row) vectors of length 4.
[ro,col]=size(Bt);
magnitude = zeros(N,1); % the magnitude of each row vector.
units = zeros(size(Bt)); % the unit vectors
% Compute the unit vectors for the row vectors
for ii=1:ro
magnitude(ii) = norm(Bt(ii,:));
units(ii,:) = Bt(ii,:) / magnitude(ii);
end
angbtwn = acos(units * units') * 360 / (2*pi);
But you'll run out of memory during the matrix multiplication for largish N.
You might want to use pdist with 'cosine' distance to compute the 1-cos(angbtwn).
Another perk for this approach that it does not compute n^2 values but exaxtly .5*(n-1)*n unique values :)
Suppose I have a vector J of jump sizes and an initial starting point X_0. Also I have boundaries 0, B (assume 0 < X_0 < B). I want to do a random walk where X_i = [min(X_{i-1} + J_i,B)]^+. (positive part). Basically if it goes over a boundary, it is made equal to the boundary. Anyone know a vectorized way to do this? The current way I am doing it consists of doing cumsums and then finding places where it violates a condition, and then starting from there and repeating the cumsum calculation, etc until I find that I stop violating the boundaries. It works when the boundaries are rarely hit, but if they are hit all the time, it basically becomes a for loop.
In the code below, I am doing this across many samples. To 'fix' the ones that go out of the boundary, I have to loop through the samples to check...(don't think there is a vectorized 'find')
% X_init is a row vector describing initial resource values to use for
% each sample
% J is matrix where each col is a sequence of Jumps (columns = sample #)
% In this code the jumps are subtracted, but same thing
X_intvl = repmat(X_init,NumJumps,1) - cumsum(J);
X = [X_init; X_intvl];
for sample = 1:NumSamples
k = find(or(X_intvl(:,sample) > B, X_intvl(:,sample) < 0),1);
while(~isempty(k))
change = X_intvl(k-1,sample) - X_intvl(k,sample);
X_intvl(k:end,sample) = X_intvl(k:end,sample)+change;
k = find(or(X_intvl(:,sample) > B, X_intvl(:,sample) < 0),1);
end
end
Interesting question (+1).
I faced a similar problem a while back, although slightly more complex as my lower and upper bound depended on t. I never did work out a fully-vectorized solution. In the end, the fastest solution I found was a single loop which incorporates the constraints at each step. Adapting the code to your situation yields the following:
%# Set the parameters
LB = 0; %# Lower bound
UB = 5; %# Upper bound
T = 100; %# Number of observations
N = 3; %# Number of samples
X0 = (1/2) * (LB + UB); %# Arbitrary start point halfway between LB and UB
%# Generate the jumps
Jump = randn(N, T-1);
%# Build the constrained random walk
X = X0 * ones(N, T);
for t = 2:T
X(:, t) = max(min(X(:, t-1) + Jump(:, t-1), UB), 0);
end
X = X';
I would be interested in hearing if this method proves faster than what you are currently doing. I suspect it will be for cases where the constraint is binding in more than one or two places. I can't test it myself as the code you provided is not a "working" example, ie I can't just copy and paste it into Matlab and run it, as it depends on several variables for which example (or simulated) values are not provided. I tried adapting it myself, but couldn't get it to work properly?
UPDATE: I just switched the code around so that observations are indexed on columns and samples are indexed on rows, and then I transpose X in the last step. This will make the routine more efficient, since Matlab allocates memory for numeric arrays column-wise - hence it is faster when performing operations down the columns of an array (as opposed to across the rows). Note, you will only notice the speed-up for large N.
FINAL THOUGHT: These days, the JIT accelerator is very good at making single loops in Matlab efficient (double loops are still pretty slow). Therefore personally I'm of the opinion that every time you try and obtain a fully-vectorized solution in Matlab, ie no loops, you should weigh up whether the effort involved in finding a clever solution is worth the slight gains in efficiency to be made over an easier-to-obtain method that utilizes a single loop. And it is important to remember that fully-vectorized solutions are sometimes slower than solutions involving single loops when T and N are small!
I'd like to propose another vectorized solution.
So, first we should set the parameters and generate random Jumpls. I used the same set of parameters as Colin T Bowers:
% Set the parameters
LB = 0; % Lower bound
UB = 20; % Upper bound
T = 1000; % Number of observations
N = 3; % Number of samples
X0 = (1/2) * (UB + LB); % Arbitrary start point halfway between LB and UB
% Generate the jumps
Jump = randn(N, T-1);
But I changed generation code:
% Generate initial data without bounds
X = cumsum(Jump, 2);
% Apply bounds
Amplitude = UB - LB;
nsteps = ceil( max(abs(X(:))) / Amplitude - 0.5 );
for ii = 1:nsteps
ind = abs(X) > (1/2) * Amplitude;
X(ind) = Amplitude * sign(X(ind)) - X(ind);
end
% Shifting X
X = X0 + X;
So, instead of for loop I'm using cumsum function with smart post-processing.
N.B. This solution works significantly slower than Colin T Bowers's one for tight bounds (Amplitude < 5), but for loose bounds (Amplitude > 20) it works much faster.