如何将元素值元素分配给theano矩阵？ Numpy和Theano的区别？

Question

I'm new to theano. I would like to replace the numpy functions in my scripts with theano functions in order to speed up the calculation process. I'm not sure how to do it.

My final goal is to apply affine transformation to 3D rigid body, assign a score to the conformation after each transformation, and do some optimization on the parameters determining the scores.

Here's an example of what I'm trying to do.

import numpy as numpy 
import theano 
import theano.tensor as T 

pi = 3.141592653
deg2rad = lambda angle: (angle/180.)*pi 

# generate 3D transformation matrix for rotation around x axis by angle 

def rotate_x_axis_numpy(angle):  # my old numpy function 
    a    = deg2rad(angle)
    cosa = np.cos(a)
    sina = np.sin(a)
    R    = np.identity(4)
    R[1][1] = cosa; R[1][2] = -sina
    R[2][1] = sina; R[2][2] =  cosa
    return R    

angle_var = T.dscalar()

def rotate_x_axis_expr(angle): # new theano function expression I expected to work  
    a    = T.deg2rad(angle)
    cosa = T.cos(a)
    sina = T.sin(a)   
    R    = theano.shared(np.identity(4))
    R[1][1] = cosa; R[1][2] = -sina
    R[2][1] = sina; R[2][2] =  cosa
    return R

rotate_x_axis_theano = theano.function([angle_var], rotate_x_axis_expr(angle_var))

The above theano function didn't pass compilation. I've got the following error message.

---------------------------------------------------------------------------
TypeError     Traceback (most recent call last)<ipython-input-85-8d98ae1d1c9b> in <module>()
      17     return R
      18 
 ---> 19 rotate_x_axis_theano = theano.function([angle_var],rotate_x_axis_expr(angle_var))

<ipython-input-85-8d98ae1d1c9b> in rotate_x_axis_expr(angle)
      12   
      13 
 ---> 14     R[1][1] = cosa; R[1][2] = -sina
      15     R[2][1] = sina; R[2][2] =  cosa
      16 

TypeError: 'TensorVariable' object does not support item assignment

In general, my questions are

(1) is there a way to assign or update or initialize a theano matrix with a specific shape elementwise,

(2) as theano is closely related with numpy, what's the difference between theano and numpy in defining, optimizing, and evaluating mathematical expressions,

and (3) can theano replace numpy in the sense that we can use theano functions solely in defining, optimizing, and evaluating mathematical expressions without calling numpy functions.

Answer 1

I can't answer your questions 1, 2, 3, since I haven't used theano before ten minutes ago. But, to define the function in theano, you don't seem to use the def construction; you want to do something more like this:

angle_var = T.dscalar('angle_var')
a    = T.deg2rad(angle_var)
cosa = T.cos(a)
sina = T.sin(a)   

R = theano.shared(np.identity(4))
R = T.set_subtensor(R[1,1],  cosa)
R = T.set_subtensor(R[1,2], -sina)
R = T.set_subtensor(R[2,1],  sina)
R = T.set_subtensor(R[2,2],  cosa)

rotate_x_axis_theano = theano.function([angle_var], R)

Doesn't help much with speed though, for a scalar angle at least:

In [368]: timeit rotate_x_axis_theano(10)
10000 loops, best of 3: 67.7 µs per loop

In [369]: timeit rotate_x_axis_numpy(10)
The slowest run took 4.23 times longer than the fastest. This could mean that an intermediate result is being cached
10000 loops, best of 3: 22.7 µs per loop

In [370]: np.allclose(rotate_x_axis_theano(10), rotate_x_axis_numpy(10))
Out[370]: True

Answer 2

Just for getting the theano function posted above work, my version is:

angle_var = T.dscalar()

def rotate_x_axis_expr(angle):
    a    = T.deg2rad(angle)
    cosa = T.cos(a)
    sina = T.sin(a)   

    R = theano.shared(np.identity(4))
    R = T.set_subtensor(R[1,1],  cosa)
    R = T.set_subtensor(R[1,2], -sina)
    R = T.set_subtensor(R[2,1],  sina)
    R = T.set_subtensor(R[2,2],  cosa)

    return R

rotate_x_axis = theano.function([angle_var],rotate_x_axis_expr(angle_var))