Suppose I have a very basic function in Python:
def f(x, y):
return x + y
Then I can call this with scalars, f(1, 5.4) == 6.4 or with numpy vectors of arbitrary (but the same) shape. E.g. this works:
x = np.arange(3)
y = np.array([1,4,2.3])
f(x, y)
which gives an array with entries 1, 5, 4.3.
But what if f is more complicated? For example, xx and yy are 1D numpy arrays here.
def g(x, y):
return np.sum((xx - x)**2 + (yy - y)**2)
(I hasten to add that I'm not interested in this specific g, but in general strategies...) Then g(5, 6) works fine, but if I want to pass numpy arrays, I seem to have to write a very different function with explict broadcasting etc. For example:
def gg(x, y):
xfull = np.stack([x]*len(xx),axis=-1)
yfull = np.stack([y]*len(xx),axis=-1)
return np.sum((xfull - xx)**2 + (yfull - yy)**2, axis=-1)
This does now work with scalars and arrays. But it seems like a mess, and is hard to read.
Is there a better way?
Given:
def g(x, y):
return np.sum((xx - x)**2 + (yy - y)**2)
my first questions are:
this is written with scalar x and y in mind?
what are xx and yy? You say 1d arrays. Same length?
why aren't they parameters? Because in this context they are fixed?
in words, this offsets xx and yy by constant amounts and takes the sum of their squares, returning a single value?
My next step is to explore the 'broadcasting' limits of this expression. For example it runs for any x that can be used in xx-x. That could be a 0d array, a one element 1d array, an array with the same shape as xx, or anything else that can 'broadcast' with `xx. That's where a thorough understanding of 'broadcasting' is essential.
g(1,2)
g(xx,xx)
g(xx[:,None],yy[None,:])
xx-xx[:,None] though produces a 2d array. np.sum as written takes the sum over all values, i.e. a flattened. Your gg suggests you want to sum on the last axis. If so go ahead and put that in g
def g(x, y):
return np.sum((xx - x)**2 + (yy - y)**2, axis=-1)
Your use of stack in gg produces:
In [101]: xx
Out[101]: array([0, 1, 2, 3, 4])
In [103]: np.stack([np.arange(3)]*len(xx), axis=-1)
Out[103]:
array([[0, 0, 0, 0, 0],
[1, 1, 1, 1, 1],
[2, 2, 2, 2, 2]])
I would have written that as x[:,None]
In [104]: xx-_
Out[104]:
array([[ 0, 1, 2, 3, 4],
[-1, 0, 1, 2, 3],
[-2, -1, 0, 1, 2]])
In [105]: xx-np.arange(3)[:,None]
Out[105]:
array([[ 0, 1, 2, 3, 4],
[-1, 0, 1, 2, 3],
[-2, -1, 0, 1, 2]])
That does not work with scalar x; but this does
xx-np.asarray(x)[...,None]
np.array or np.asarray is commonly used as the start of numpy functions to accommodate scalar or list inputs. ... is handy when dealing with a variable number of dimensions. reshape(...,-1) and [...,None] are widely used to expand or generalize dimensions.
I've learned a lot by looking the Python code of numpy functions. I've also learned from years of work with MATLAB to be pedantic about dimensions. Keep track of intended and actual array shapes. It helps to use test shapes that will highlight errors. Test with a (2,3) array instead of an ambiguous (3,3) one.
Related
I could not for the life of me get array indexing to work properly with higher dimensional lexsort.
I have an ndarray lines of shape (N, 2, 3). You can think of it as N pairs (start and end of a line) of three-dimensional coordinates. These pairs of vectors can contain duplicates, which should be removed.
points = np.array([[1,1,0],[-1,1,0],[-1,-1,0],[1,-1,0]])
lines = np.dstack([points, np.roll(points, shift=1, axis=0)]) # create point pairs / lines
lines = np.vstack([lines, lines[..., ::-1]]) # add duplicates w/reversed direction
lines = lines.transpose(0,2,1) # change shape from N,3,2 to N,2,3
Since the pair (v1, v2) is not equal to (v2, v1), I am sorting the vectors with lexsort as follows
idx = np.lexsort((lines[..., 0], lines[..., 1], lines[..., 2]))
which gives me an array idx of shape (N, 2) indicating the order along axis 1:
array([[0, 1],
[0, 1],
[1, 0],
[1, 0],
[1, 0],
[1, 0],
[0, 1],
[0, 1]])
However, lines[idx] results in something with shape (N, 2, 2, 3). I had tried all manner of newaxis padding, axis reordering etc. to get broadcasting to work, but everything results in the output having even more dimensions, not less. I also tried lines[:, idx], but this gives (N, N, 2, 3).
Based on https://numpy.org/doc/stable/user/basics.indexing.html#integer-array-indexing
for my concrete problem I eventually figured out I need to add an additional
idx_n = np.arange(len(lines))[:, np.newaxis]
lines[idx_n, idx]
due to mixing "advanced" and "simple" indexing lines[:, idx] did not work as I expected.
but is this really the most succinct it can be?
Eventually I found out I wanted
np.take_along_axis(lines, idx[..., np.newaxis] , axis=1)
I've got a 3D tensor x (e.g 4x4x100). I want to obtain a subset of this by explicitly choosing elements across the last dimension. This would have been easy if I was choosing the same elements across last dimension (e.g. x[:,:,30:50] but I want to target different elements across that dimension using the 2D tensor indices which specifies the idx across third dimension. Is there an easy way to do this in numpy?
A simpler 2D example:
x = [[1,2,3,4,5,6],[10,20,30,40,50,60]]
indices = [1,3]
Let's say I want to grab two elements across third dimension of x starting from points specified by indices. So my desired output is:
[[2,3],[40,50]]
Update: I think I could use a combination of take() and ravel_multi_index() but some of the platforms that are inspired by numpy (like PyTorch) don't seem to have ravel_multi_index so I'm looking for alternative solutions
Iterating over the idx, and collecting the slices is not a bad option if the number of 'rows' isn't too large (and the size of the sizes is relatively big).
In [55]: x = np.array([[1,2,3,4,5,6],[10,20,30,40,50,60]])
In [56]: idx = [1,3]
In [57]: np.array([x[j,i:i+2] for j,i in enumerate(idx)])
Out[57]:
array([[ 2, 3],
[40, 50]])
Joining the slices like this only works if they all are the same size.
An alternative is to collect the indices into an array, and do one indexing.
For example with a similar iteration:
idxs = np.array([np.arange(i,i+2) for i in idx])
But broadcasted addition may be better:
In [58]: idxs = np.array(idx)[:,None]+np.arange(2)
In [59]: idxs
Out[59]:
array([[1, 2],
[3, 4]])
In [60]: x[np.arange(2)[:,None], idxs]
Out[60]:
array([[ 2, 3],
[40, 50]])
ravel_multi_index is not hard to replicate (if you don't need clipping etc):
In [65]: np.ravel_multi_index((np.arange(2)[:,None],idxs),x.shape)
Out[65]:
array([[ 1, 2],
[ 9, 10]])
In [66]: x.flat[_]
Out[66]:
array([[ 2, 3],
[40, 50]])
In [67]: np.arange(2)[:,None]*x.shape[1]+idxs
Out[67]:
array([[ 1, 2],
[ 9, 10]])
along the 3D axis:
x = [x[:,i].narrow(2,index,2) for i,index in enumerate(indices)]
x = torch.stack(x,dim=1)
by enumerating you get the index of the axis and index from where you want to start slicing in one.
narrow gives you a zero-copy length long slice from a starting index start along a certain axis
you said you wanted:
dim = 2
start = index
length = 2
then you simply have to stack these tensors back to a single 3D.
This is the least work intensive thing i can think of for pytorch.
EDIT
if you just want different indices along different axis and indices is a 2D tensor you can do:
x = [x[:,i,index] for i,index in enumerate(indices)]
x = torch.stack(x,dim=1)
You really should have given a proper working example, making it unnecessarily confusing.
Here is how to do it in numpy, now clue about torch, though.
The following picks a slice of length n along the third dimension starting from points idx depending on the other two dimensions:
# example
a = np.arange(60).reshape(2, 3, 10)
idx = [(1,2,3),(4,3,2)]
n = 4
# build auxiliary 4D array where the last two dimensions represent
# a sliding n-window of the original last dimension
j,k,l = a.shape
s,t,u = a.strides
aux = np.lib.stride_tricks.as_strided(a, (j,k,l-n+1,n), (s,t,u,u))
# pick desired offsets from sliding windows
aux[(*np.ogrid[:j, :k], idx)]
# array([[[ 1, 2, 3, 4],
# [12, 13, 14, 15],
# [23, 24, 25, 26]],
# [[34, 35, 36, 37],
# [43, 44, 45, 46],
# [52, 53, 54, 55]]])
I came up with below using broadcasting:
x = np.array([[1,2,3,4,5,6,7,8,9,10],[10,20,30,40,50,60,70,80,90,100]])
i = np.array([1,5])
N = 2 # number of elements I want to extract along each dimension. Starting points specified in i
r = np.arange(x.shape[-1])
r = np.broadcast_to(r, x.shape)
ii = i[:, np.newaxis]
ii = np.broadcast_to(ii, x.shape)
mask = np.logical_and(r-ii>=0, r-ii<=N)
output = x[mask].reshape(2,3)
Does this look alright?
In a previous question (fastest way to use numpy.interp on a 2-D array) someone asked for the fastest way to implement the following:
np.array([np.interp(X[i], x, Y[i]) for i in range(len(X))])
assume X and Y are matrices with many rows so the for loop is costly. There is a nice solution in this case that avoids the for loop (see linked answer above).
I am faced with a very similar problem, but I am unclear on whether the for loop can be avoided in this case:
np.array([np.interp(x, X[i], Y[i]) for i in range(len(X))])
In other words, I want to use linear interpolation to upsample a large number of signals stored in the rows of two matrices X and Y.
I was hoping to find a function in numpy or scipy (scipy.interpolate.interp1d) that supported this operation via broadcasting semantics but I so far can't seem to find one.
Other points:
If it helps, the rows X[i] and x are pre-sorted in my application. Also, in my case len(x) is quite a bit larger than len(X[i]).
The function scipy.signal.resample almost does what I want, but it doesn't use linear interpolation...
This is a vectorized approach that directly implements linear interpolation. First, for each x value and each i, j compute the weight w expressing how much of the interval (X[i, j], X[i, j+1]) is to the left of x.
If the entire interval is to the left of x, the weight of that interval is 1.
If none of the subinterval is to the left, the weight is 0
Otherwise, the weight is a number between 0 and 1, expressing the proportion of that interval to the left of x.
Then the value of PL interpolant is computed as Y[i, 0] + sum of differences dY[i, j] multiplied by the corresponding weight. The logic is to follow by how much the interpolant changes from interval to interval. The differences dY = np.diff(Y, axis=1) show how much it changes over the entire interval. Multiplication by the weight prorates that change accordingly.
Setup, with some small data arrays
import numpy as np
X = np.array([[0, 2, 5, 6, 9], [1, 3, 4, 7, 8]])
Y = np.array([[3, 5, 2, 4, 1], [8, 6, 9, 5, 4]])
x = np.linspace(1, 8, 20)
The computation
dX = np.diff(X, axis=1)
dY = np.diff(Y, axis=1)
w = np.clip((x - X[:, :-1, None])/dX[:, :, None], 0, 1)
y = Y[:, [0]] + np.sum(w*dY[:, :, None], axis=1)
Demonstration
This is only to show that the interpolation is correct. Blue points: original data, red ones are computed.
import matplotlib.pyplot as plt
plt.plot(x, y[0], 'ro')
plt.plot(X[0], Y[0], 'bo')
plt.plot(x, y[1], 'rd')
plt.plot(X[1], Y[1], 'bd')
plt.show()
For a given NumPy array, it is easy to perform a "normal" sum along one dimension. For example:
X = np.array([[1, 0, 0], [0, 2, 2], [0, 0, 3]])
X.sum(0)
=array([1, 2, 5])
X.sum(1)
=array([1, 4, 3])
Instead, is there an "efficient" way of computing the bitwise OR along one dimension of an array similarly? Something like the following, except without requiring for-loops or nested function calls.
Example: bitwise OR along zeroeth dimension as I currently am doing it:
np.bitwise_or(np.bitwise_or(X[:,0],X[:,1]),X[:,2])
=array([1, 2, 3])
What I would like:
X.bitwise_sum(0)
=array([1, 2, 3])
numpy.bitwise_or.reduce(X, axis=whichever_one_you_wanted)
Use the reduce method of the numpy.bitwise_or ufunc.
I have a tensor of shape (30, 116, 10), and I want to swap the first two dimensions, so that I have a tensor of shape (116, 30, 10)
I saw that numpy as such a function implemented (np.swapaxes) and I searched for something similar in tensorflow but I found nothing.
Do you have any idea?
tf.transpose provides the same functionality as np.swapaxes, although in a more generalized form. In your case, you can do tf.transpose(orig_tensor, [1, 0, 2]) which would be equivalent to np.swapaxes(orig_np_array, 0, 1).
It is possible to use tf.einsum to swap axes if the number of input dimensions is unknown. For example:
tf.einsum("ij...->ji...", input) will swap the first two dimensions of input;
tf.einsum("...ij->...ji", input) will swap the last two dimensions;
tf.einsum("aij...->aji...", input) will swap the second and the third
dimension;
tf.einsum("ijk...->kij...", input) will permute the first three dimensions;
and so on.
You can transpose just the last two axes with tf.linalg.matrix_transpose, or more generally, you can swap any number of trailing axes by working out what the leading indices are dynamically, and using relative indices for the axes you want to transpose
x = tf.ones([5, 3, 7, 11])
trailing_axes = [-1, -2]
leading = tf.range(tf.rank(x) - len(trailing_axes)) # [0, 1]
trailing = trailing_axes + tf.rank(x) # [3, 2]
new_order = tf.concat([leading, trailing], axis=0) # [0, 1, 3, 2]
res = tf.transpose(x, new_order)
res.shape # [5, 3, 11, 7]