I just gave a talk about this at SCaLE 18x. Here are the video of the talk and the “slides”.
numpysane: more-reasonable core functionality for numpy
>>> import numpy as np
>>> import numpysane as nps
>>> a = np.arange(6).reshape(2,3)
>>> b = a + 100
>>> row = a[0,:] + 1000
>>> a
array([[0, 1, 2],
[3, 4, 5]])
>>> b
array([[100, 101, 102],
[103, 104, 105]])
>>> row
array([1000, 1001, 1002])
>>> nps.glue(a,b, axis=-1)
array([[ 0, 1, 2, 100, 101, 102],
[ 3, 4, 5, 103, 104, 105]])
>>> nps.glue(a,b,row, axis=-2)
array([[ 0, 1, 2],
[ 3, 4, 5],
[ 100, 101, 102],
[ 103, 104, 105],
[1000, 1001, 1002]])
>>> nps.cat(a,b)
array([[[ 0, 1, 2],
[ 3, 4, 5]],
[[100, 101, 102],
[103, 104, 105]]])
>>> @nps.broadcast_define( (('n',), ('n',)) )
... def inner_product(a, b):
... return a.dot(b)
>>> inner_product(a,b)
array([ 305, 1250])
Numpy is a very widely used toolkit for numerical computation in Python. Despite its popularity, some of its core functionality is mysterious and/or incomplete. The numpysane library seeks to fill those gaps by providing its own replacement routines. Many of the replacement functions are direct translations from PDL (http://pdl.perl.org), a numerical computation library for perl. The functions provided by this module fall into three broad categories:
- Broadcasting support
- Nicer array manipulation
- Basic linear algebra
Numpy has a limited support for broadcasting (http://docs.scipy.org/doc/numpy/user/basics.broadcasting.html), a generic way to vectorize functions. A broadcasting-aware function knows the dimensionality of its inputs, and any extra dimensions in the input are automatically used for vectorization.
A basic example is an inner product: a function that takes in two identically-sized 1-dimensional arrays (input prototype ((‘n’,), (‘n’,)) ) and returns a scalar (output prototype () ). If one calls a broadcasting-aware inner product with two arrays of shape (2,3,4) as input, it would compute 6 inner products of length-4 each, and report the output in an array of shape (2,3).
In short:
- The most significant dimension in a numpy array is the LAST one, so the prototype of an input argument must exactly match a given input’s trailing shape. So a prototype shape of (a,b,c) accepts an argument shape of (......, a,b,c), with as many or as few leading dimensions as desired.
- The extra leading dimensions must be compatible across all the inputs. This
means that each leading dimension must either
- equal 1
- be missing (thus assumed to equal 1)
- equal to some positive integer >1, consistent across all arguments
- The output is collected into an array that’s sized as a superset of the above-prototype shape of each argument
More involved example: A function with input prototype ( (3,), (‘n’,3), (‘n’,), (‘m’,) ) given inputs of shape
(1,5, 3) (2,1, 8,3) ( 8) ( 5, 9)
will return an output array of shape (2,5, …), where … is the shape of each output slice. Note again that the prototype dictates the TRAILING shape of the inputs.
The numpy documentation dedicates a whole page explaining the broadcasting rules, but only a small number of numpy functions provide any broadcasting support. It’s fairly inconsistent, and most functions have no broadcasting support and no mention of it in the documentation. And as a result, this is not a prominent part of the numpy ecosystem and there’s little user awareness that it exists.
This module contains functionality to make any arbitrary function broadcastable, in either C or Python. In both cases, the input and output prototypes are declared, and these are used for shape-checking and vectorization each time the function is called.
The functions can have either
- A single output, returned as a numpy array. The output specification in the prototype is a single shape tuple
- Multiple outputs, returned as a tuple of numpy arrays. The output specification in the prototype is a tuple of shape tuples
This is invoked as a decorator, applied to any function. An example:
>>> import numpysane as nps
>>> @nps.broadcast_define( (('n',), ('n',)) )
... def inner_product(a, b):
... return a.dot(b)
Here we have a simple inner product function to compute ONE inner product. The ‘broadcast_define’ decorator adds broadcasting-awareness: ‘inner_product()’ expects two 1D vectors of length ‘n’ each (same ‘n’ for the two inputs), vectorizing extra dimensions, as needed. The inputs are shape-checked, and incompatible dimensions will trigger an exception. Example:
>>> import numpy as np
>>> a = np.arange(6).reshape(2,3)
>>> b = a + 100
>>> a
array([[0, 1, 2],
[3, 4, 5]])
>>> b
array([[100, 101, 102],
[103, 104, 105]])
>>> inner_product(a,b)
array([ 305, 1250])
Another related function in this module broadcast_generate(). It’s similar to broadcast_define(), but instead of adding broadcasting-awareness to an existing function, it returns a generator that produces tuples from a set of arguments according to a given prototype. Similarly, broadcast_extra_dims() is available to report the outer shape of a potential broadcasting operation.
Stock numpy has some rudimentary support for all this with its vectorize() function, but it assumes only scalar inputs and outputs, which severely limits its usefulness. See the docstrings for ‘broadcast_define’ and ‘broadcast_generate’ in the INTERFACE section below for usage details.
The python broadcasting is useful, but it is a python loop, so the loop itself is computationally expensive if we have many iterations. If the function being wrapped is available in C, we can apply broadcasting awareness in C, which makes a much faster loop.
The “numpysane_pywrap” module generates code to wrap arbitrary C code in a broadcasting-aware wrapper callable from python. This is an analogue of PDL::PP (http://pdl.perl.org/PDLdocs/PP.html). This generated code is compiled and linked into a python extension module, as usual. This functionality documented separately: https://github.com/dkogan/numpysane/blob/master/README-pywrap.org
After I wrote this, I realized there is some support for this in stock numpy:
https://docs.scipy.org/doc/numpy-1.13.0/reference/c-api.ufunc.html
Note: I have not tried using these APIs.
Numpy functions that move dimensions around and concatenate matrices are unintuitive. For instance, a simple concatenation of a row-vector or a column-vector to a matrix requires arcane knowledge to accomplish reliably. This module provides new functions that can be used for these basic operations. These new functions do have well-defined and sensible behavior, and they largely come from the interfaces in PDL (http://pdl.perl.org). These all respect the core rules of numpy broadcasting:
- LEADING length-1 dimensions don’t affect the meaning of an array, so the routines handle missing or extra length-1 dimensions at the front
- The inner-most dimensions of an array are the TRAILING ones, so whenever an axis specification is used, it is strongly recommended (sometimes required) to count the axes from the back by passing in axis<0
A high level description of the functionality is given here, and each function is described in detail in the INTERFACE section below. In the following examples, I use a function “arr” that returns a numpy array with given dimensions:
>>> def arr(*shape):
... product = reduce( lambda x,y: x*y, shape)
... return numpy.arange(product).reshape(*shape)
>>> arr(1,2,3)
array([[[0, 1, 2],
[3, 4, 5]]])
>>> arr(1,2,3).shape
(1, 2, 3)
This module provides two functions to do this
Concatenates some number of arrays along a given axis (‘axis’ must be given in a kwarg). Implicit length-1 dimensions are added at the start as needed. Dimensions other than the glueing axis must match exactly. Basic usage:
>>> row_vector = arr( 3,) >>> col_vector = arr(5,1,) >>> matrix = arr(5,3,) >>> numpysane.glue(matrix, row_vector, axis = -2).shape (6,3) >>> numpysane.glue(matrix, col_vector, axis = -1).shape (5,4)
Concatenate some number of arrays along a new leading axis. Implicit length-1 dimensions are added, and the logical shapes of the inputs must match. This function is a logical inverse of numpy array iteration: iteration splits an array over its leading dimension, while cat joins a number of arrays via a new leading dimension. Basic usage:
>>> numpysane.cat(arr(5,), arr(5,)).shape (2,5) >>> numpysane.cat(arr(5,), arr(1,1,5,)).shape (2,1,1,5)
Several functions are available, all being fairly direct ports of their PDL (http://pdl.perl.org) equivalents
Reshapes the array by grouping together ‘n’ dimensions, where ‘n’ is given in a kwarg. If ‘n’ > 0, then n leading dimensions are clumped; if ‘n’ < 0, then -n trailing dimensions are clumped. Basic usage:
>>> numpysane.clump( arr(2,3,4), n = -2).shape (2, 12) >>> numpysane.clump( arr(2,3,4), n = 2).shape (6, 4)
Adds length-1 dimensions at the front of an array so that all the given dimensions are in-bounds. Any axis<0 may expand the shape. Adding new leading dimensions (axis>=0) is never useful, since numpy broadcasts from the end, so clump() treats axis>0 as a check only: the requested axis MUST already be in-bounds, or an exception is thrown. This function always preserves the meaning of all the axes in the array: axis=-1 is the same axis before and after the call. Basic usage:
>>> numpysane.atleast_dims(arr(2,3), -1).shape (2, 3) >>> numpysane.atleast_dims(arr(2,3), -2).shape (2, 3) >>> numpysane.atleast_dims(arr(2,3), -3).shape (1, 2, 3) >>> numpysane.atleast_dims(arr(2,3), 0).shape (2, 3) >>> numpysane.atleast_dims(arr(2,3), 1).shape (2, 3) >>> numpysane.atleast_dims(arr(2,3), 2).shape [exception]
Moves a dimension from one position to another. Basic usage to move the last dimension (-1) to the front (0)
>>> numpysane.mv(arr(2,3,4), -1, 0).shape (4, 2, 3)
Or to move a dimension -5 (added implicitly) to the end
>>> numpysane.mv(arr(2,3,4), -5, -1).shape (1, 2, 3, 4, 1)
Exchanges the positions of two dimensions. Basic usage to move the last dimension (-1) to the front (0), and the front to the back.
>>> numpysane.xchg(arr(2,3,4), -1, 0).shape (4, 3, 2)
Or to swap a dimension -5 (added implicitly) with dimension -2
>>> numpysane.xchg(arr(2,3,4), -5, -2).shape (3, 1, 2, 1, 4)
Reverses the order of the two trailing dimensions in an array. The whole array is seen as being an array of 2D matrices, each matrix living in the 2 most significant dimensions, which implies this definition. Basic usage:
>>> numpysane.transpose( arr(2,3) ).shape (3,2) >>> numpysane.transpose( arr(5,2,3) ).shape (5,3,2) >>> numpysane.transpose( arr(3,) ).shape (3,1)
Note that in the second example we had 5 matrices, and we transposed each one. And in the last example we turned a row vector into a column vector by adding an implicit leading length-1 dimension before transposing.
Adds a single length-1 dimension at the given position. Basic usage:
>>> numpysane.dummy(arr(2,3,4), -1).shape (2, 3, 4, 1)
Reorders the dimensions in an array using the given order. Basic usage:
>>> numpysane.reorder( arr(2,3,4), -1, -2, -3 ).shape (4, 3, 2) >>> numpysane.reorder( arr(2,3,4), 0, -1, 1 ).shape (2, 4, 3) >>> numpysane.reorder( arr(2,3,4), -2 , -1, 0 ).shape (3, 4, 2) >>> numpysane.reorder( arr(2,3,4), -4 , -2, -5, -1, 0 ).shape (1, 3, 1, 4, 2)
Broadcast-aware inner product. Identical to numpysane.dot(). Basic usage to compute 4 inner products of length 3 each:
>>> numpysane.inner(arr( 3,),
arr(4,3,)).shape
(4,)
>>> numpysane.inner(arr( 3,),
arr(4,3,))
array([5, 14, 23, 32])
Broadcast-aware non-conjugating dot product. Identical to numpysane.inner().
Broadcast-aware conjugating dot product. Same as numpysane.dot(), except this one conjugates complex input, which numpysane.dot() does not
Broadcast-aware outer product. Basic usage to compute 4 outer products of length 3 each:
>>> numpysane.outer(arr( 3,),
arr(4,3,)).shape
array(4, 3, 3)
Broadcast-aware 2-norm. numpysane.norm2(x) is identical to numpysane.inner(x,x):
>>> numpysane.norm2(arr(4,3)) array([5, 50, 149, 302])
Broadcast-aware vector magnitude. mag(x) is functionally identical to sqrt(numpysane.norm2(x)) and sqrt(numpysane.inner(x,x))
>>> numpysane.mag(arr(4,3)) array([ 2.23606798, 7.07106781, 12.20655562, 17.3781472 ])
Broadcast-aware matrix trace.
>>> numpysane.trace(arr(4,3,3)) array([12., 39., 66., 93.])
Broadcast-aware matrix multiplication. This accepts an arbitrary number of inputs, and adds leading length-1 dimensions as needed. Multiplying a row-vector by a matrix
>>> numpysane.matmult( arr(3,), arr(3,2) ).shape (2,)
Multiplying a row-vector by 5 different matrices:
>>> numpysane.matmult( arr(3,), arr(5,3,2) ).shape (5, 2)
Multiplying a matrix by a col-vector:
>>> numpysane.matmult( arr(3,2), arr(2,1) ).shape (3, 1)
Multiplying a row-vector by a matrix by a col-vector:
>>> numpysane.matmult( arr(3,), arr(3,2), arr(2,1) ).shape (1,)
Why did I go through all the trouble to reimplement all this? Doesn’t numpy already do all these things? Yes, it does. But in a very nonintuitive way.
hstack() performs a “horizontal” concatenation. When numpy prints an array, this is the last dimension (the most significant dimensions in numpy are at the end). So one would expect that this function concatenates arrays along this last dimension. In the special case of 1D and 2D arrays, one would be right:
>>> numpy.hstack( (arr(3), arr(3))).shape (6,) >>> numpy.hstack( (arr(2,3), arr(2,3))).shape (2, 6)
but in any other case, one would be wrong:
>>> numpy.hstack( (arr(1,2,3), arr(1,2,3))).shape (1, 4, 3) <------ I expect (1, 2, 6) >>> numpy.hstack( (arr(1,2,3), arr(1,2,4))).shape [exception] <------ I expect (1, 2, 7) >>> numpy.hstack( (arr(3), arr(1,3))).shape [exception] <------ I expect (1, 6) >>> numpy.hstack( (arr(1,3), arr(3))).shape [exception] <------ I expect (1, 6)
The above should all succeed, and should produce the shapes as indicated. Cases such as “numpy.hstack( (arr(3), arr(1,3)))” are maybe up for debate, but broadcasting rules allow adding as many extra length-1 dimensions as we want without changing the meaning of the object, so I claim this should work. Either way, if you print out the operands for any of the above, you too would expect a “horizontal” stack() to work as stated above.
It turns out that normally hstack() concatenates along axis=1, unless the first argument only has one dimension, in which case axis=0 is used. In a system where the most significant dimension is the last one, this is only correct if everyone has only 2D arrays. The correct way to do this is to concatenate along axis=-1. It works for n-dimensionsal objects, and doesn’t require the special case logic for 1-dimensional objects.
Similarly, vstack() performs a “vertical” concatenation. When numpy prints an array, this is the second-to-last dimension (remember, the most significant dimensions in numpy are at the end). So one would expect that this function concatenates arrays along this second-to-last dimension. Again, in the special case of 1D and 2D arrays, one would be right:
>>> numpy.vstack( (arr(2,3), arr(2,3))).shape (4, 3) >>> numpy.vstack( (arr(3), arr(3))).shape (2, 3) >>> numpy.vstack( (arr(1,3), arr(3))).shape (2, 3) >>> numpy.vstack( (arr(3), arr(1,3))).shape (2, 3) >>> numpy.vstack( (arr(2,3), arr(3))).shape (3, 3)
Note that this function appears to tolerate some amount of shape mismatches. It does it in a form one would expect, but given the state of the rest of this system, I found it surprising. For instance “numpy.hstack( (arr(1,3), arr(3)))” fails, so one would think that “numpy.vstack( (arr(1,3), arr(3)))” would fail too.
And once again, adding more dimensions make it confused, for the same reason:
>>> numpy.vstack( (arr(1,2,3), arr(2,3))).shape [exception] <------ I expect (1, 4, 3) >>> numpy.vstack( (arr(1,2,3), arr(1,2,3))).shape (2, 2, 3) <------ I expect (1, 4, 3)
Similarly to hstack(), vstack() concatenates along axis=0, which is “vertical” only for 2D arrays, but not for any others. And similarly to hstack(), the 1D case has special-cased logic to make it work properly.
The correct way to do this is to concatenate along axis=-2. It works for n-dimensionsal objects, and doesn’t require the special case for 1-dimensional objects.
I’ll skip the detailed description, since this is similar to hstack() and vstack(). The intent was to concatenate across axis=-3, but the implementation takes axis=2 instead. This is wrong, as before. And I find it strange that these 3 functions even exist, since they are all special-cases: the concatenation axis should be an argument, and at most, the edge special case (hstack()) should exist. This brings us to the next function
This is a more general function, and unlike hstack(), vstack() and dstack(), it takes as input a list of arrays AND the concatenation dimension. It accepts negative concatenation dimensions to allow us to count from the end, so things should work better. And in many cases that failed previously, they do:
>>> numpy.concatenate( (arr(1,2,3), arr(1,2,3)), axis=-1).shape (1, 2, 6) >>> numpy.concatenate( (arr(1,2,3), arr(1,2,4)), axis=-1).shape (1, 2, 7) >>> numpy.concatenate( (arr(1,2,3), arr(1,2,3)), axis=-2).shape (1, 4, 3)
But many things still don’t work as I would expect:
>>> numpy.concatenate( (arr(1,3), arr(3)), axis=-1).shape [exception] <------ I expect (1, 6) >>> numpy.concatenate( (arr(3), arr(1,3)), axis=-1).shape [exception] <------ I expect (1, 6) >>> numpy.concatenate( (arr(1,3), arr(3)), axis=-2).shape [exception] <------ I expect (3, 3) >>> numpy.concatenate( (arr(3), arr(1,3)), axis=-2).shape [exception] <------ I expect (2, 3) >>> numpy.concatenate( (arr(2,3), arr(2,3)), axis=-3).shape [exception] <------ I expect (2, 2, 3)
This function works as expected only if
- All inputs have the same number of dimensions
- All inputs have a matching shape, except for the dimension along which we’re concatenating
- All inputs HAVE the dimension along which we’re concatenating
A common use case that violates these conditions: I have an object that contains N 3D vectors, and I want to add another 3D vector to it. This is essentially the first failing example above.
The name makes it sound exactly like concatenate(), and it takes the same arguments, but it is very different. stack() requires that all inputs have EXACTLY the same shape. It then concatenates all the inputs along a new dimension, and places that dimension in the location given by the ‘axis’ input. If this is the exact type of concatenation you want, this function works fine. But it’s one of many things a user may want to do.
This module introduces numpysane.glue() and numpysane.cat() to replace all the above functions. These do not refer to anything being “horizontal” or “vertical”, nor do they talk about “rows” or “columns”: these concepts simply don’t apply in a generic N-dimensional system. These functions are very explicit about the dimensionality of the inputs/outputs, and fit well into a broadcasting-aware system.
Since these functions assume that broadcasting is an important concept in the system, the given axis indices should be counted from the most significant dimension: the last dimension in numpy. This means that where an axis index is specified, negative indices are encouraged. glue() forbids axis>=0 outright.
An array containing N 3D vectors would have shape (N,3). Another array containing a single 3D vector would have shape (3,). Counting the dimensions from the end, each vector is indexed in dimension -1. However, counting from the front, the vector is indexed in dimension 0 or 1, depending on which of the two arrays we’re looking at. If we want to add the single vector to the array containing the N vectors, and we mistakenly try to concatenate along the first dimension, it would fail if N != 3. But if we’re unlucky, and N=3, then we’d get a nonsensical output array of shape (3,4). Why would an array of N 3D vectors have shape (N,3) and not (3,N)? Because if we apply python iteration to it, we’d expect to get N iterates of arrays with shape (3,) each, and numpy iterates from the first dimension:
>>> a = numpy.arange(2*3).reshape(2,3)
>>> a
array([[0, 1, 2],
[3, 4, 5]])
>>> [x for x in a]
[array([0, 1, 2]), array([3, 4, 5])]
Numpy has 3 special-case functions atleast_1d(), atleast_2d() and atleast_3d(). For 4d and higher, you need to do something else. These do surprising things:
>>> numpy.atleast_3d(arr(3)).shape (1, 3, 1)
Given a matrix (a 2D array), numpy.transpose() swaps the two dimensions, as expected. Given anything else, it does not do what is expected:
>>> numpy.transpose(arr(3, )).shape (3,) >>> numpy.transpose(arr(3,4, )).shape (4, 3) >>> numpy.transpose(arr(3,4,5,6,)).shape (6, 5, 4, 3)
I.e. numpy.transpose() reverses the order of ALL dimensions in the array. So if we have N 2D matrices in a single array, numpy.transpose() doesn’t transpose each matrix.
numpy.inner() and numpy.dot() are strange. In a real-valued n-dimensional Euclidean space, a “dot product” is just another name for an “inner product”. Numpy disagrees.
It looks like numpy.dot() is matrix multiplication, with some wonky behaviors when given higher-dimension objects, and with some special-case behaviors for 1-dimensional and 0-dimensional objects:
>>> numpy.dot( arr(4,5,2,3), arr(3,5)).shape
(4, 5, 2, 5) <--- expected result for a broadcasted matrix multiplication
>>> numpy.dot( arr(3,5), arr(4,5,2,3)).shape
[exception] <--- numpy.dot() is not commutative.
Expected for matrix multiplication, but not for a dot
product
>>> numpy.dot( arr(4,5,2,3), arr(1,3,5)).shape
(4, 5, 2, 1, 5) <--- don't know where this came from at all
>>> numpy.dot( arr(4,5,2,3), arr(3)).shape
(4, 5, 2) <--- 1D special case. This is a dot product.
>>> numpy.dot( arr(4,5,2,3), 3).shape
(4, 5, 2, 3) <--- 0D special case. This is a scaling.
It looks like numpy.inner() is some sort of quasi-broadcastable inner product, also with some funny dimensioning rules. In many cases it looks like numpy.dot(a,b) is the same as numpy.inner(a, transpose(b)) where transpose() swaps the last two dimensions:
>>> numpy.inner( arr(4,5,2,3), arr(5,3)).shape
(4, 5, 2, 5) <--- All the length-3 inner products collected into a shape
with not-quite-broadcasting rules
>>> numpy.inner( arr(5,3), arr(4,5,2,3)).shape
(5, 4, 5, 2) <--- numpy.inner() is not commutative. Unexpected
for an inner product
>>> numpy.inner( arr(4,5,2,3), arr(1,5,3)).shape
(4, 5, 2, 1, 5) <--- No idea
>>> numpy.inner( arr(4,5,2,3), arr(3)).shape
(4, 5, 2) <--- 1D special case. This is a dot product.
>>> numpy.inner( arr(4,5,2,3), 3).shape
(4, 5, 2, 3) <--- 0D special case. This is a scaling.
Vectorizes an arbitrary function, expecting input as in the given prototype.
SYNOPSIS
>>> import numpy as np
>>> import numpysane as nps
>>> @nps.broadcast_define( (('n',), ('n',)) )
... def inner_product(a, b):
... return a.dot(b)
>>> a = np.arange(6).reshape(2,3)
>>> b = a + 100
>>> a
array([[0, 1, 2],
[3, 4, 5]])
>>> b
array([[100, 101, 102],
[103, 104, 105]])
>>> inner_product(a,b)
array([ 305, 1250])
The prototype defines the dimensionality of the inputs. In the inner product example above, the input is two 1D n-dimensional vectors. In particular, the ‘n’ is the same for the two inputs. This function is intended to be used as a decorator, applied to a function defining the operation to be vectorized. Each element in the prototype list refers to each input, in order. In turn, each such element is a list that describes the shape of that input. Each of these shape descriptors can be any of
- a positive integer, indicating an input dimension of exactly that length
- a string, indicating an arbitrary, but internally consistent dimension
The normal numpy broadcasting rules (as described elsewhere) apply. In summary:
- Dimensions are aligned at the end of the shape list, and must match the prototype
- Extra dimensions left over at the front must be consistent for all the
input arguments, meaning:
- All dimensions of length != 1 must match
- Dimensions of length 1 match corresponding dimensions of any length in other arrays
- Missing leading dimensions are implicitly set to length 1
- The output(s) have a shape where
- The trailing dimensions are whatever the function being broadcasted returns
- The leading dimensions come from the extra dimensions in the inputs
Calling a function wrapped with broadcast_define() with extra arguments (either positional or keyword), passes these verbatim to the inner function. Only the arguments declared in the prototype are broadcast.
Scalars are represented as 0-dimensional numpy arrays: arrays with shape (), and these broadcast as one would expect:
>>> @nps.broadcast_define( (('n',), ('n',), ()))
... def scaled_inner_product(a, b, scale):
... return a.dot(b)*scale
>>> a = np.arange(6).reshape(2,3)
>>> b = a + 100
>>> scale = np.array((10,100))
>>> a
array([[0, 1, 2],
[3, 4, 5]])
>>> b
array([[100, 101, 102],
[103, 104, 105]])
>>> scale
array([ 10, 100])
>>> scaled_inner_product(a,b,scale)
array([[ 3050],
[125000]])
Let’s look at a more involved example. Let’s say we have a function that takes a set of points in R^2 and a single center point in R^2, and finds a best-fit least-squares line that passes through the given center point. Let it return a 3D vector containing the slope, y-intercept and the RMS residual of the fit. This broadcasting-enabled function can be defined like this:
import numpy as np
import numpysane as nps
@nps.broadcast_define( (('n',2), (2,)) )
def fit(xy, c):
# line-through-origin-model: y = m*x
# E = sum( (m*x - y)**2 )
# dE/dm = 2*sum( (m*x-y)*x ) = 0
# ----> m = sum(x*y)/sum(x*x)
x,y = (xy - c).transpose()
m = np.sum(x*y) / np.sum(x*x)
err = m*x - y
err **= 2
rms = np.sqrt(err.mean())
# I return m,b because I need to translate the line back
b = c[1] - m*c[0]
return np.array((m,b,rms))
And I can use broadcasting to compute a number of these fits at once. Let’s say I want to compute 4 different fits of 5 points each. I can do this:
n = 5
m = 4
c = np.array((20,300))
xy = np.arange(m*n*2, dtype=np.float64).reshape(m,n,2) + c
xy += np.random.rand(*xy.shape)*5
res = fit( xy, c )
mb = res[..., 0:2]
rms = res[..., 2]
print "RMS residuals: {}".format(rms)
Here I had 4 different sets of points, but a single center point c. If I wanted 4 different center points, I could pass c as an array of shape (4,2). I can use broadcasting to plot all the results (the points and the fitted lines):
import gnuplotlib as gp
gp.plot( *nps.mv(xy,-1,0), _with='linespoints',
equation=['{}*x + {}'.format(mb_single[0],
mb_single[1]) for mb_single in mb],
unset='grid', square=1)
The examples above all create a separate output array for each broadcasted slice, and copy the contents from each such slice into the larger output array that contains all the results. This is inefficient, and it is possible to pre-allocate an array to forgo these extra allocation and copy operations. There are several settings to control this. If the function being broadcasted can write its output to a given array instead of creating a new one, most of the inefficiency goes away. broadcast_define() supports the case where this function takes this array in a kwarg: the name of this kwarg can be given to broadcast_define() like so:
@nps.broadcast_define( ....., out_kwarg = "out" )
def func( ....., *, out):
.....
out[:] = result
When used this way, the return value of the broadcasted function is ignored. In order for broadcast_define() to pass such an output array to the inner function, this output array must be available, which means that it must be given to us somehow, or we must create it.
The most efficient way to make a broadcasted call is to create the full output array beforehand, and to pass that to the broadcasted function. In this case, nothing extra will be allocated, and no unnecessary copies will be made. This can be done like this:
@nps.broadcast_define( (('n',), ('n',)), ....., out_kwarg = "out" )
def inner_product(a, b, *, out):
.....
out.setfield(a.dot(b), out.dtype)
out = np.empty((2,4), np.float)
inner_product( np.arange(3), np.arange(2*4*3).reshape(2,4,3), out=out)
In this example, the caller knows that it’s calling an inner_product function, and that the shape of each output slice would be (). The caller also knows the input dimensions and that we have an extra broadcasting dimension (2,4), so the output array will have shape (2,4) + () = (2,4). With this knowledge, the caller preallocates the array, and passes it to the broadcasted function call. Furthermore, in this case the inner function will be called with an output array EVERY time, and this is the only mode the inner function needs to support.
If the caller doesn’t know (or doesn’t want to pre-compute) the shape of the output, it can let the broadcasting machinery create this array for them. In order for this to be possible, the shape of the output should be pre-declared, and the dtype of the output should be known:
@nps.broadcast_define( (('n',), ('n',)),
(),
out_kwarg = "out" )
def inner_product(a, b, *, out, dtype):
.....
out.setfield(a.dot(b), out.dtype)
out = inner_product( np.arange(3), np.arange(2*4*3).reshape(2,4,3), dtype=int)
Note that the caller didn’t need to specify the prototype of the output or the extra broadcasting dimensions (output prototype is in the broadcast_define() call, but not the inner_product() call). Specifying the dtype here is optional: it defaults to float if omitted. If the dtype IS given, the inner function must take a “dtype” argument; to use in cases where out_kwarg isn’t given, and the output array must be created by the inner function.
If we want the output array to be pre-allocated, the output prototype (it is () in this example) is required: we must know the shape of the output array in order to create it.
Without a declared output prototype, we can still make mostly- efficient calls: the broadcasting mechanism can call the inner function for the first slice as we showed earlier, by creating a new array for the slice. This new array required an extra allocation and copy, but it contains the required shape information. This infomation will be used to allocate the output, and the subsequent calls to the inner function will be efficient:
@nps.broadcast_define( (('n',), ('n',)),
out_kwarg = "out" )
def inner_product(a, b, *, out=None):
.....
if out is None:
return a.dot(b)
out.setfield(a.dot(b), out.dtype)
return out
out = inner_product( np.arange(3), np.arange(2*4*3).reshape(2,4,3))
Here we were slighly inefficient, but the ONLY required extra specification was out_kwarg: that’s all you need. Also it is important to note that in this case the inner function is called both with passing it an output array to fill in, and with asking it to create a new one (by passing out=None to the inner function). This inner function then must support both modes of operation. If the inner function does not support filling in an output array, none of these efficiency improvements are possible.
It is possible for a function to return more than one output, and this is supported by broadcast_define(). This case works exactly like the one-output case, except the output prototype is REQUIRED, and this output prototype contains multiple tuples, one for each output. The inner function must return the outputs in a tuple, and each individual output will be broadcasted as expected.
broadcast_define() is analogous to thread_define() in PDL.
A generator that produces broadcasted slices
SYNOPSIS
>>> import numpy as np
>>> import numpysane as nps
>>> a = np.arange(6).reshape(2,3)
>>> b = a + 100
>>> a
array([[0, 1, 2],
[3, 4, 5]])
>>> b
array([[100, 101, 102],
[103, 104, 105]])
>>> for s in nps.broadcast_generate( (('n',), ('n',)), (a,b)):
... print "slice: {}".format(s)
slice: (array([0, 1, 2]), array([100, 101, 102]))
slice: (array([3, 4, 5]), array([103, 104, 105]))
The broadcasting operation of numpysane is described in detail in the numpysane.broadcast_define() docstring and in the main README of numpysane. This function can be used as a Python generator to produce each broadcasted slice one by one
Since Python generators are inherently 1-dimensional, this function effectively flattens the broadcasted results. If the correct output shape needs to be reconstituted, the leading shape is available by calling numpysane.broadcast_extra_dims() with the same arguments as this function.
Report the extra leading dimensions a broadcasted call would produce
SYNOPSIS
>>> import numpy as np
>>> import numpysane as nps
>>> a = np.arange(6). reshape( 2,3)
>>> b = np.arange(15).reshape(5,1,3)
>>> print(nps.broadcast_extra_dims((('n',), ('n',)),
(a,b)))
[5,2]
The broadcasting operation of numpysane is described in detail in the numpysane.broadcast_define() docstring and in the main README of numpysane. This function applies the broadcasting rules to report the leading dimensions of a broadcasted result if a broadcasted function was called with the given arguments. This is most useful to reconstitute the desired shape from flattened output produced by numpysane.broadcast_generate()
Concatenates a given list of arrays along the given ‘axis’ keyword argument.
SYNOPSIS
>>> import numpy as np
>>> import numpysane as nps
>>> a = np.arange(6).reshape(2,3)
>>> b = a + 100
>>> row = a[0,:] + 1000
>>> a
array([[0, 1, 2],
[3, 4, 5]])
>>> b
array([[100, 101, 102],
[103, 104, 105]])
>>> row
array([1000, 1001, 1002])
>>> nps.glue(a,b, axis=-1)
array([[ 0, 1, 2, 100, 101, 102],
[ 3, 4, 5, 103, 104, 105]])
# empty arrays ignored when glueing. Useful for initializing an accumulation
>>> nps.glue(a,b, np.array(()), axis=-1)
array([[ 0, 1, 2, 100, 101, 102],
[ 3, 4, 5, 103, 104, 105]])
>>> nps.glue(a,b,row, axis=-2)
array([[ 0, 1, 2],
[ 3, 4, 5],
[ 100, 101, 102],
[ 103, 104, 105],
[1000, 1001, 1002]])
>>> nps.glue(a,b, axis=-3)
array([[[ 0, 1, 2],
[ 3, 4, 5]],
[[100, 101, 102],
[103, 104, 105]]])
The ‘axis’ must be given in a keyword argument.
In order to count dimensions from the inner-most outwards, this function accepts only negative axis arguments. This is because numpy broadcasts from the last dimension, and the last dimension is the inner-most in the (usual) internal storage scheme. Allowing glue() to look at dimensions at the start would allow it to unalign the broadcasting dimensions, which is never what you want.
To glue along the last dimension, pass axis=-1; to glue along the second-to-last dimension, pass axis=-2, and so on.
Unlike in PDL, this function refuses to create duplicated data to make the shapes fit. In my experience, this isn’t what you want, and can create bugs. For instance, PDL does this:
pdl> p sequence(3,2) [ [0 1 2] [3 4 5] ] pdl> p sequence(3) [0 1 2] pdl> p PDL::glue( 0, sequence(3,2), sequence(3) ) [ [0 1 2 0 1 2] <--- Note the duplicated "0,1,2" [3 4 5 0 1 2] ]
while numpysane.glue() does this:
>>> import numpy as np
>>> import numpysane as nps
>>> a = np.arange(6).reshape(2,3)
>>> b = a[0:1,:]
>>> a
array([[0, 1, 2],
[3, 4, 5]])
>>> b
array([[0, 1, 2]])
>>> nps.glue(a,b,axis=-1)
[exception]
Finally, this function adds as many length-1 dimensions at the front as required. Note that this does not create new data, just new degenerate dimensions. Example:
>>> import numpy as np
>>> import numpysane as nps
>>> a = np.arange(6).reshape(2,3)
>>> b = a + 100
>>> a
array([[0, 1, 2],
[3, 4, 5]])
>>> b
array([[100, 101, 102],
[103, 104, 105]])
>>> res = nps.glue(a,b, axis=-5)
>>> res
array([[[[[ 0, 1, 2],
[ 3, 4, 5]]]],
[[[[100, 101, 102],
[103, 104, 105]]]]])
>>> res.shape
(2, 1, 1, 2, 3)
In numpysane older than 0.10 the semantics were slightly different: the axis kwarg was optional, and glue(*args) would glue along a new leading dimension, and thus would be equivalent to cat(*args). This resulted in very confusing error messages if the user accidentally omitted the kwarg. To request the legacy behavior, do
nps.glue.legacy_version = '0.9'
Concatenates a given list of arrays along a new first (outer) dimension.
SYNOPSIS
>>> import numpy as np
>>> import numpysane as nps
>>> a = np.arange(6).reshape(2,3)
>>> b = a + 100
>>> c = a - 100
>>> a
array([[0, 1, 2],
[3, 4, 5]])
>>> b
array([[100, 101, 102],
[103, 104, 105]])
>>> c
array([[-100, -99, -98],
[ -97, -96, -95]])
>>> res = nps.cat(a,b,c)
>>> res
array([[[ 0, 1, 2],
[ 3, 4, 5]],
[[ 100, 101, 102],
[ 103, 104, 105]],
[[-100, -99, -98],
[ -97, -96, -95]]])
>>> res.shape
(3, 2, 3)
>>> [x for x in res]
[array([[0, 1, 2],
[3, 4, 5]]),
array([[100, 101, 102],
[103, 104, 105]]),
array([[-100, -99, -98],
[ -97, -96, -95]])]
### Note that this is the same as [a,b,c]: cat is the reverse of
### iterating on an array
This function concatenates the input arrays into an array shaped like the highest-dimensioned input, but with a new outer (at the start) dimension. The concatenation axis is this new dimension.
As usual, the dimensions are aligned along the last one, so broadcasting will continue to work as expected. Note that this is the opposite operation from iterating a numpy array: iteration splits an array over its leading dimension, while cat joins a number of arrays via a new leading dimension.
Groups the given n dimensions together.
SYNOPSIS
>>> import numpysane as nps >>> nps.clump( arr(2,3,4), n = -2).shape (2, 12)
Reshapes the array by grouping together ‘n’ dimensions, where ‘n’ is given in a kwarg. If ‘n’ > 0, then n leading dimensions are clumped; if ‘n’ < 0, then -n trailing dimensions are clumped
So for instance, if x.shape is (2,3,4) then nps.clump(x, n = -2).shape is (2,12) and nps.clump(x, n = 2).shape is (6, 4)
In numpysane older than 0.10 the semantics were different: n > 0 was required, and we ALWAYS clumped the trailing dimensions. Thus the new clump(-n) is equivalent to the old clump(n). To request the legacy behavior, do
nps.clump.legacy_version = '0.9'
Returns an array with extra length-1 dimensions to contain all given axes.
SYNOPSIS
>>> import numpy as np
>>> import numpysane as nps
>>> a = np.arange(6).reshape(2,3)
>>> a
array([[0, 1, 2],
[3, 4, 5]])
>>> nps.atleast_dims(a, -1).shape
(2, 3)
>>> nps.atleast_dims(a, -2).shape
(2, 3)
>>> nps.atleast_dims(a, -3).shape
(1, 2, 3)
>>> nps.atleast_dims(a, 0).shape
(2, 3)
>>> nps.atleast_dims(a, 1).shape
(2, 3)
>>> nps.atleast_dims(a, 2).shape
[exception]
>>> l = [-3,-2,-1,0,1]
>>> nps.atleast_dims(a, l).shape
(1, 2, 3)
>>> l
[-3, -2, -1, 1, 2]
If the given axes already exist in the given array, the given array itself is returned. Otherwise length-1 dimensions are added to the front until all the requested dimensions exist. The given axis>=0 dimensions MUST all be in-bounds from the start, otherwise the most-significant axis becomes unaligned; an exception is thrown if this is violated. The given axis<0 dimensions that are out-of-bounds result in new dimensions added at the front.
If new dimensions need to be added at the front, then any axis>=0 indices become offset. For instance:
>>> x.shape (2, 3, 4) >>> [x.shape[i] for i in (0,-1)] [2, 4] >>> x = nps.atleast_dims(x, 0, -1, -5) >>> x.shape (1, 1, 2, 3, 4) >>> [x.shape[i] for i in (0,-1)] [1, 4]
Before the call, axis=0 refers to the length-2 dimension and axis=-1 refers to the length=4 dimension. After the call, axis=-1 refers to the same dimension as before, but axis=0 now refers to a new length=1 dimension. If it is desired to compensate for this offset, then instead of passing the axes as separate arguments, pass in a single list of the axes indices. This list will be modified to offset the axis>=0 appropriately. Ideally, you only pass in axes<0, and this does not apply. Doing this in the above example:
>>> l [0, -1, -5] >>> x.shape (2, 3, 4) >>> [x.shape[i] for i in (l[0],l[1])] [2, 4] >>> x=nps.atleast_dims(x, l) >>> x.shape (1, 1, 2, 3, 4) >>> l [2, -1, -5] >>> [x.shape[i] for i in (l[0],l[1])] [2, 4]
We passed the axis indices in a list, and this list was modified to reflect the new indices: The original axis=0 becomes known as axis=2. Again, if you pass in only axis<0, then you don’t need to care about this.
Moves a given axis to a new position. Similar to numpy.moveaxis().
SYNOPSIS
>>> import numpy as np >>> import numpysane as nps >>> a = np.arange(24).reshape(2,3,4) >>> a.shape (2, 3, 4) >>> nps.mv( a, -1, 0).shape (4, 2, 3) >>> nps.mv( a, -1, -5).shape (4, 1, 1, 2, 3) >>> nps.mv( a, 0, -5).shape (2, 1, 1, 3, 4)
New length-1 dimensions are added at the front, as required, and any axis>=0 that are passed in refer to the array BEFORE these new dimensions are added.
Exchanges the positions of the two given axes. Similar to numpy.swapaxes()
SYNOPSIS
>>> import numpy as np >>> import numpysane as nps >>> a = np.arange(24).reshape(2,3,4) >>> a.shape (2, 3, 4) >>> nps.xchg( a, -1, 0).shape (4, 3, 2) >>> nps.xchg( a, -1, -5).shape (4, 1, 2, 3, 1) >>> nps.xchg( a, 0, -5).shape (2, 1, 1, 3, 4)
New length-1 dimensions are added at the front, as required, and any axis>=0 that are passed in refer to the array BEFORE these new dimensions are added.
Reverses the order of the last two dimensions.
SYNOPSIS
>>> import numpy as np >>> import numpysane as nps >>> a = np.arange(24).reshape(2,3,4) >>> a.shape (2, 3, 4) >>> nps.transpose(a).shape (2, 4, 3) >>> nps.transpose( np.arange(3) ).shape (3, 1)
A “matrix” is generally seen as a 2D array that we can transpose by looking at the 2 dimensions in the opposite order. Here we treat an n-dimensional array as an n-2 dimensional object containing 2D matrices. As usual, the last two dimensions contain the matrix.
New length-1 dimensions are added at the front, as required, meaning that 1D input of shape (n,) is interpreted as a 2D input of shape (1,n), and the transpose is 2 of shape (n,1).
Adds length-1 dimensions at the given positions.
SYNOPSIS
>>> import numpy as np >>> import numpysane as nps >>> a = np.arange(24).reshape(2,3,4) >>> a.shape (2, 3, 4) >>> nps.dummy(a, 0).shape (1, 2, 3, 4) >>> nps.dummy(a, 1).shape (2, 1, 3, 4) >>> nps.dummy(a, -1).shape (2, 3, 4, 1) >>> nps.dummy(a, -2).shape (2, 3, 1, 4) >>> nps.dummy(a, -2, -2).shape (2, 3, 1, 1, 4) >>> nps.dummy(a, -5).shape (1, 1, 2, 3, 4)
This is similar to numpy.expand_dims(), but handles out-of-bounds dimensions better. New length-1 dimensions are added at the front, as required, and any axis>=0 that are passed in refer to the array BEFORE these new dimensions are added.
Reorders the dimensions of an array.
SYNOPSIS
>>> import numpy as np >>> import numpysane as nps >>> a = np.arange(24).reshape(2,3,4) >>> a.shape (2, 3, 4) >>> nps.reorder( a, 0, -1, 1 ).shape (2, 4, 3) >>> nps.reorder( a, -2 , -1, 0 ).shape (3, 4, 2) >>> nps.reorder( a, -4 , -2, -5, -1, 0 ).shape (1, 3, 1, 4, 2)
This is very similar to numpy.transpose(), but handles out-of-bounds dimensions much better.
New length-1 dimensions are added at the front, as required, and any axis>=0 that are passed in refer to the array BEFORE these new dimensions are added.
Non-conjugating dot product of two 1-dimensional n-long vectors.
SYNOPSIS
>>> import numpy as np >>> import numpysane as nps >>> a = np.arange(3) >>> b = a+5 >>> a array([0, 1, 2]) >>> b array([5, 6, 7]) >>> nps.dot(a,b) 20
This is identical to numpysane.inner(). for a conjugating version of this function, use nps.vdot(). Note that the stock numpy dot() has some special handling when its dot() is given more than 1-dimensional input. THIS function has no special handling: normal broadcasting rules are applied, as expected.
In-place operation is available with the “out” kwarg. The output dtype can be selected with the “dtype” kwarg. If omitted, the dtype of the input is used
Conjugating dot product of two 1-dimensional n-long vectors.
vdot(a,b) is equivalent to dot(np.conj(a), b)
SYNOPSIS
>>> import numpy as np >>> import numpysane as nps >>> a = np.array(( 1 + 2j, 3 + 4j, 5 + 6j)) >>> b = a+5 >>> a array([ 1.+2.j, 3.+4.j, 5.+6.j]) >>> b array([ 6.+2.j, 8.+4.j, 10.+6.j]) >>> nps.vdot(a,b) array((136-60j)) >>> nps.dot(a,b) array((24+148j))
For a non-conjugating version of this function, use nps.dot(). Note that the numpy vdot() has some special handling when its vdot() is given more than 1-dimensional input. THIS function has no special handling: normal broadcasting rules are applied.
Outer product of two 1-dimensional n-long vectors.
SYNOPSIS
>>> import numpy as np
>>> import numpysane as nps
>>> a = np.arange(3)
>>> b = a+5
>>> a
array([0, 1, 2])
>>> b
array([5, 6, 7])
>>> nps.outer(a,b)
array([[ 0, 0, 0],
[ 5, 6, 7],
[10, 12, 14]])
This function is broadcast-aware through numpysane.broadcast_define(). The expected inputs have input prototype:
(('n',), ('m',))
and output prototype
('n', 'm')
The first 2 positional arguments will broadcast. The trailing shape of those arguments must match the input prototype; the leading shape must follow the standard broadcasting rules. Positional arguments past the first 2 and all the keyword arguments are passed through untouched.
Broadcast-aware 2-norm. norm2(x) is identical to inner(x,x)
SYNOPSIS
>>> import numpy as np >>> import numpysane as nps >>> a = np.arange(3) >>> a array([0, 1, 2]) >>> nps.norm2(a) 5
This is a convenience function to compute a 2-norm
Magnitude of a vector. mag(x) is functionally identical to sqrt(inner(x,x))
SYNOPSIS
>>> import numpy as np >>> import numpysane as nps >>> a = np.arange(3) >>> a array([0, 1, 2]) >>> nps.mag(a) 2.23606797749979
This is a convenience function to compute a magnitude of a vector, with full broadcasting support.
In-place operation is available with the “out” kwarg. The output dtype can be selected with the “dtype” kwarg. If omitted, dtype=float is selected.
Broadcast-aware trace
SYNOPSIS
>>> import numpy as np
>>> import numpysane as nps
>>> a = np.arange(3*4*4).reshape(3,4,4)
>>> a
array([[[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11],
[12, 13, 14, 15]],
[[16, 17, 18, 19],
[20, 21, 22, 23],
[24, 25, 26, 27],
[28, 29, 30, 31]],
[[32, 33, 34, 35],
[36, 37, 38, 39],
[40, 41, 42, 43],
[44, 45, 46, 47]]])
>>> nps.trace(a)
array([ 30, 94, 158])
This function is broadcast-aware through numpysane.broadcast_define(). The expected inputs have input prototype:
(('n', 'n'),)
and output prototype
()
The first 1 positional arguments will broadcast. The trailing shape of those arguments must match the input prototype; the leading shape must follow the standard broadcasting rules. Positional arguments past the first 1 and all the keyword arguments are passed through untouched.
Multiplication of two matrices
SYNOPSIS
>>> import numpy as np
>>> import numpysane as nps
>>> a = np.arange(6) .reshape(2,3)
>>> b = np.arange(12).reshape(3,4)
>>> a
array([[0, 1, 2],
[3, 4, 5]])
>>> b
array([[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11]])
>>> nps.matmult2(a,b)
array([[20, 23, 26, 29],
[56, 68, 80, 92]])
This function is exposed publically mostly for legacy compatibility. Use numpysane.matmult() instead
Multiplication of N matrices
SYNOPSIS
>>> import numpy as np
>>> import numpysane as nps
>>> a = np.arange(6) .reshape(2,3)
>>> b = np.arange(12).reshape(3,4)
>>> c = np.arange(4) .reshape(4,1)
>>> a
array([[0, 1, 2],
[3, 4, 5]])
>>> b
array([[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11]])
>>> c
array([[0],
[1],
[2],
[3]])
>>> nps.matmult(a,b,c)
array([[162],
[504]])
>>> abc = np.zeros((2,1), dtype=float)
>>> nps.matmult(a,b,c, out=abc)
>>> abc
array([[162],
[504]])
This multiplies N matrices together by repeatedly calling matmult2() for each adjacent pair. In-place output is supported with the ‘out’ keyword argument
This function supports broadcasting fully, in C internally
Python 2 and Python 3 should both be supported. Please report a bug if either one doesn’t work.
https://github.com/dkogan/numpysane
Dima Kogan <[email protected]>
Copyright 2016-2020 Dima Kogan.
This program is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License (any version) as published by the Free Software Foundation