I have two 2-D tensors of shape say m X d and n X d. What is the optimized(i.e. without for loops) or the tensorflow way of evaluating the pairwise euclidean distance between these two tensors so that I get an output tensor of shape m X n. I need it for creating the squared term of a Gaussian kernel for ultimately having a covariance matrix of size m x n.
The equivalent unoptimized numpy code would look like this
difference_squared = np.zeros((x.shape[0], x_.shape[0]))
for row_iterator in range(difference_squared.shape[0]):
for column_iterator in range(difference_squared.shape[1]):
difference_squared[row_iterator, column_iterator] = np.sum(np.power(x[row_iterator]-x_[column_iterator], 2))
I found the answer by taking help from here. Assuming the two tensors are x1 and x2, and their dimensions are m X d and n X d, their pair-wise Euclidean distance is given by
tile_1 = tf.tile(tf.expand_dims(x1, 0), [n, 1, 1])
tile_2 = tf.tile(tf.expand_dims(x2, 1), [1, m, 1])
pairwise_euclidean_distance = tf.reduce_sum(tf.square(tf.subtract(tile_1, tile_2)), 2))
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I have an idea for a tensor operation that would not be difficult to implement via iteration, with batch size one. However I would like to parallelize it as much as possible.
I have two tensors with shape (n, 5) called X and Y. X is actually supposed to represent 5 one-dimensional tensors with shape (n, 1): (x_1, ..., x_n). Ditto for Y.
I would like to compute a tensor with shape (n, 25) where each column represents the output of the tensor operation f(x_i, y_j), where f is fixed for all 1 <= i, j <= 5. The operation f has output shape (n, 1), just like x_i and y_i.
I feel it is important to clarify that f is essentially a fully-connected layer from the concatenated [...x_i, ...y_i] tensor with shape (1, 10), to an output layer with shape (1,5).
Again, it is easy to see how to do this manually with iteration and slicing. However this is probably very slow. Performing this operation in batches, where the tensors X, Y now have shape (n, 5, batch_size) is also desirable, particularly for mini-batch gradient descent.
It is difficult to really articulate here why I desire to create this network; I feel it is suited for my domain of 'itemized tabular data' and cuts down significantly on the number of weights per operation, compared to a fully connected network.
Is this possible using tensorflow? Certainly not using just keras.
Below is an example in numpy per AloneTogether's request
import numpy as np
features = 16
batch_size = 256
X_batch = np.random.random((features, 5, batch_size))
Y_batch = np.random.random((features, 5, batch_size))
# one tensor operation to reduce weights in this custom 'layer'
f = np.random.random((features, 2 * features))
for b in range(batch_size):
X = X_batch[:, :, b]
Y = Y_batch[:, :, b]
for i in range(5):
x_i = X[:, i:i+1]
for j in range(5):
y_j = Y[:, j:j+1]
x_i_y_j = np.concatenate([x_i, y_j], axis=0)
# f(x_i, y_j)
# implemented by a fully-connected layer
f_i_j = np.matmul(f, x_i_y_j)
All operations you need (concatenation and matrix multiplication) can be batched.
Difficult part here is, that you want to concatenate features of all items in X with features of all items in Y (all combinations).
My recommended solution is to expand the dimensions of X to [batch, features, 5, 1], expand dimensions of Y to [batch, features, 1, 5]
Than tf.repeat() both tensors so their shapes become [batch, features, 5, 5].
Now you can concatenate X and Y. You will have a tensor of shape [batch, 2*features, 5, 5]. Observe that this way all combinations are built.
Next step is matrix multiplication. tf.matmul() can also do batch matrix multiplication, but I use here tf.einsum() because I want more control over which dimensions are considered as batch.
Full code:
import tensorflow as tf
import numpy as np
batch_size=3
features=6
items=5
x = np.random.uniform(size=[batch_size,features,items])
y = np.random.uniform(size=[batch_size,features,items])
f = np.random.uniform(size=[2*features,features])
x_reps= tf.repeat(x[:,:,:,tf.newaxis], items, axis=3)
y_reps= tf.repeat(y[:,:,tf.newaxis,:], items, axis=2)
xy_conc = tf.concat([x_reps,y_reps], axis=1)
f_i_j = tf.einsum("bfij, fg->bgij", xy_conc,f)
f_i_j = tf.reshape(f_i_j , [batch_size,features,items*items])
I have a tensor phi = np.random.rand(n, n, 3) and a matrix D = np.random.rand(3, 3). I want to multiply the matrix D along the last axis of phi so that the output has shape (n, n, 3). I have tried this
np.einsum("klj,ij->kli", phi, D)
But I am not confident in this notation at all. Basically I want to do
res = np.zeros_like(phi)
for i in range(n):
for j in range(n):
res[i, j, :] = D.dot(phi[i, j, :])
You are treating phi as an n, n array of vectors, each of which is to be left-multiplied by D. So you want to keep the n, n portion of the shape exactly as-is. The last (only) dimension of the vectors should be multiplied and summed with the last dimension of the matrix (the vectors are implicitly 3x1):
np.einsum('ijk,lk->ijl', phi, D)
OR
np.einsum('ij,klj->kli', D, phi)
It's likely much simpler to use broadcasting with np.matmul (the # operator):
np.squeeze(D # phi[..., None])
You can omit the squeeze if you don't mind the extra unit dimension at the end.
I have N Gaussian distributions (multivariate) with N different means (covariance is the same for all of them) in D dimensions.
I also have N evaluation points, where I want to evaluate each of these (log) PDFs.
This means I need to get a NxN matrix, call it "kernels". That is, the (i,j)-th entry is the j-th Gaussian evaluated at the i-th point. A naive approach is:
from torch.distributions.multivariate_normal import MultivariateNormal
import numpy as np
# means contains all N means as rows and is thus N x D
# same for eval_points
# cov is not a problem , just a DxD matrix that is equal for all N Gaussians
kernels = np.empty((N,N))
for i in range(N):
for j in range(N):
kernels[i][j] = MultivariateNormal(means[j], cov).log_prob(eval_points[i])
Now one for loop we can get rid of easily, since for example if we wanted all the evaluations of the first Gaussian , we simply do:
MultivariateNormal(means[0], cov).log_prob(eval_points).squeeze()
and this gives us a N x 1 list of values, that is the first Gaussian evaluated at all N points.
My problem is that , in order to get the full N x N matrix , this doesn't work:
kernels = MultivariateNormal(means, cov).log_prob(eval_points).squeeze()
It doesn't figure out that it should evaluate each mean with all evaluation points in eval_points, and it doesn't return a NxN matrix with these which would be what I want. Therefore, I am not able to get rid of the second for loop, over all N Gaussians.
You are passing wrong shaped tensors to MultivariateNormal's constructor. You should pass a collection of mean vectors of shape (N, D) and a collection of precision matrix cov of shape (N, D, D) for N D-dimensional gaussian.
You are passing mu of shape (N, D) but your precision matrix is not well-shaped. You will need to repeat the precision matrix N number of times before passing it to the MultivariateNormal constructor. Here's one way to do it.
N = 10
D = 3
# means contains all N means as rows and is thus N x D
# same for eval_points
# cov is not a problem , just a DxD matrix that is equal for all N Gaussians
mu = torch.from_numpy(np.random.randn(N, D))
cov = torch.from_numpy(make_spd_matrix(D, D))
cov_n = cov[None, ...].repeat_interleave(N, 0)
assert cov_n.shape == (N, D, D)
kernels = MultivariateNormal(mu, cov_n)
Whenever I'm finding covariance of 2 arrays, I've always seen it done like
(np.cov(X,Y)[0,1])
What purpose does the [0,1] serve?
For two 1d arrays x and y, np.cov(x, y) returns:
np.array([[variance(x), covariance(x, y)],
[covariance(y, x), variance(y) ]])
Thus for the covariance, you need the [0,1] term.
When formulated as cov(x ,y), numpy creates np.cov(X) where X = np.stack(x, y, axis = 1)
The confusion occurs because for np.cov(X) is really optimized for many vectors at once, with X.shape = (m, n), and np.cov(X)[i,j], i, j < n to be the covariance between rows i and j. And the covariance of rows i and i is just the variance of row i.
I know tensorflow can calculate expressions like [ [a,b,c] ] x [ [x],[y],[z] ] when the elements are primitive data type (integer or float).
Is it possible to perform a similar computation when each of a, b and c is a 1x3 matrix and x, y and z are 3x1 matrices?
Can TensorFlow calculate and optimize this formula?
The tf.batch_matmul() operator can perform matrix multiplications on batches of matrices. In this case, you would have a tensor abc of shape (3, 1, 3) (where abc[0, :, :] = a, abc[1, :, :] = b, etc.) and a tensor xyz of shape (3, 3, 1) (where xyz[0, :, :] = x, etc.).
abc = ...
xyz = ...
result = tf.batch_matmul(abc, xyz)
print result.get_shape() # ==> "(3, 1, 1)"
result is a 3-D tensor with contents equivalent to tf.pack([tf.matmul(a, x), tf.matmul(b, y), tf.matmul(c, z)]).