Math Part 2



In [1]:

    
from __future__ import print_function
import tensorflow as tf
import numpy as np



In [2]:

    
from datetime import date
date.today()









    Out[2]:





datetime.date(2017, 2, 23)



In [3]:

    
author = "kyubyong. https://github.com/Kyubyong/tensorflow-exercises"



In [4]:

    
tf.__version__









    Out[4]:





'1.0.0'



In [5]:

    
np.__version__









    Out[5]:





'1.12.0'



In [2]:

    
sess = tf.InteractiveSession()

NOTE on notation

_x, _y, _z, ...: NumPy 0-d or 1-d arrays
_X, _Y, _Z, ...: NumPy 2-d or higer dimensional arrays
x, y, z, ...: 0-d or 1-d tensors
X, Y, Z, ...: 2-d or higher dimensional tensors

Matrix Math Functions

Q1. Create a diagonal tensor with the diagonal values of x.



In [7]:

    
_x = np.array([1, 2, 3, 4])
x = tf.convert_to_tensor(_x)

Q2. Extract the diagonal of X.



In [8]:

    
_X = np.array(
[[1, 0, 0, 0],
 [0, 2, 0, 0],
 [0, 0, 3, 0],
 [0, 0, 0, 4]])
X = tf.convert_to_tensor(_X)

Q3. Permutate the dimensions of x such that the new tensor has shape (3, 4, 2).



In [9]:

    
_X = np.random.rand(2,3,4)
X = tf.convert_to_tensor(_X)









    



(3, 4, 2)

Q4. Construct a 3 by 3 identity matrix.



In [10]:









    



[[ 1.  0.  0.]
 [ 0.  1.  0.]
 [ 0.  0.  1.]]

Q5. Predict the result of this.



In [3]:

    
_X = np.array([[1, 2, 3, 4], [5, 6, 7, 8]])
X = tf.convert_to_tensor(_X)

diagonal_tensor = tf.matrix_diag(X)
diagonal_part = tf.matrix_diag_part(diagonal_tensor)

print("diagonal_tensor =\n", diagonal_tensor.eval())
print("diagonal_part =\n", diagonal_part.eval())

Q6. Transpose the last two dimensions of X.



In [4]:

    
_X= np.random.rand(1, 2, 3, 4)
X = tf.convert_to_tensor(_X)

Q7. Multiply X by Y.



In [13]:

    
_X = np.array([[1, 2, 3], [4, 5, 6]])
_Y = np.array([[1, 1], [2, 2], [3, 3]])
X = tf.convert_to_tensor(_X)
Y = tf.convert_to_tensor(_Y)









    



[[14 14]
 [32 32]]

Q8. Multiply X and Y. The first axis represents batches.



In [14]:

    
_X = np.arange(1, 13, dtype=np.int32).reshape((2, 2, 3))
_Y = np.arange(13, 25, dtype=np.int32).reshape((2, 3, 2))
X = tf.convert_to_tensor(_X)
Y = tf.convert_to_tensor(_Y)









    



[[[ 94 100]
  [229 244]]

 [[508 532]
  [697 730]]]

Q9. Compute the determinant of X.



In [15]:

    
_X = np.arange(1, 5, dtype=np.float32).reshape((2, 2))
X = tf.convert_to_tensor(_X)

Q10. Compute the inverse of X.



In [16]:

    
_X = np.arange(1, 5, dtype=np.float64).reshape((2, 2))
X = tf.convert_to_tensor(_X)









    



[[-2.   1. ]
 [ 1.5 -0.5]]

Q11. Get the lower-trianglular in the Cholesky decomposition of X.



In [17]:

    
_X = np.array([[4, 12, -16], [12, 37, -43], [-16, -43, 98]], np.float32)
X = tf.convert_to_tensor(_X)









    



[[ 2.  0.  0.]
 [ 6.  1.  0.]
 [-8.  5.  3.]]

Q12. Compute the eigenvalues and eigenvectors of X.



In [18]:

    
_X = np.diag((1, 2, 3))
X = tf.convert_to_tensor(_X, tf.float32)









    



eigentvalues =
 [ 1.  2.  3.]
eigenvectors =
 [[ 1.  0.  0.]
 [ 0.  1.  0.]
 [ 0.  0.  1.]]

Q13. Compute the singular values of X.



In [19]:

    
_X = np.array(
[[1, 0, 0, 0, 2], 
 [0, 0, 3, 0, 0], 
 [0, 0, 0, 0, 0], 
 [0, 2, 0, 0, 0]], dtype=np.float32)
X = tf.convert_to_tensor(_X)









    



[ 3.          2.23606801  2.          0.        ]

Reduction

Q14. Predict the results of these.



In [5]:

    
_X = np.array(
    [[1, 2, 3, 4],
     [5, 6, 7, 8]])
X = tf.convert_to_tensor(_X)

outs = [tf.reduce_sum(X),
        tf.reduce_sum(X, axis=0),
        tf.reduce_sum(X, axis=1, keep_dims=True),
        "",
        tf.reduce_prod(X),
        tf.reduce_prod(X, axis=0),
        tf.reduce_prod(X, axis=1, keep_dims=True),
        "",
        tf.reduce_min(X),
        tf.reduce_min(X, axis=0),
        tf.reduce_min(X, axis=1, keep_dims=True),
        "",
        tf.reduce_max(X),
        tf.reduce_max(X, axis=0),
        tf.reduce_max(X, axis=1, keep_dims=True),
        "",
        tf.reduce_mean(X),
        tf.reduce_mean(X, axis=0),
        tf.reduce_mean(X, axis=1, keep_dims=True)
           ]
           
for out in outs:
    if out == "":
        print()
    else:
        print("->", out.eval())
# If you remove the common suffix "reduce_", you will get the same 
# result in numpy.

Q15. Predict the results of these.



In [6]:

    
_X = np.array([[True, True],
              [False, False]], np.bool)
X = tf.convert_to_tensor(_X)

outs = [tf.reduce_all(X),
        tf.reduce_all(X, axis=0),
        tf.reduce_all(X, axis=1, keep_dims=True),
        "",
        tf.reduce_any(X),
        tf.reduce_any(X, axis=0),
        tf.reduce_any(X, axis=1, keep_dims=True),
        ]

# for out in outs:
    if out == "":
        print()
    else:
        print("->", out.eval())

# If you remove the common suffix "reduce_", you will get the same 
# result in numpy.

Q16. Predict the results of these.



In [7]:

    
_X = np.array([[0, 1, 0],
              [1, 1, 0]])
X = tf.convert_to_tensor(_X)

outs = [tf.count_nonzero(X),
        tf.count_nonzero(X, axis=0),
        tf.count_nonzero(X, axis=1, keep_dims=True),
        ]

for out in outs:
    print("->", out.eval())

# tf.count_nonzero == np.count_nonzero

Q17. Complete the einsum function that would yield the same result as the given function.



In [23]:

    
_X = np.arange(1, 7).reshape((2, 3))
_Y = np.arange(1, 7).reshape((3, 2))

X = tf.convert_to_tensor(_X)
Y = tf.convert_to_tensor(_Y)

# Matrix multiplication
out1 = tf.matmul(X, Y)
out1_ = tf.einsum(..., X, Y)
assert np.allclose(out1.eval(), out1_.eval())

# Dot product
flattened = tf.reshape(X, [-1])
out2 = tf.reduce_sum(flattened * flattened)
out2_ = tf.einsum(..., flattened, flattened)
assert np.allclose(out2.eval(), out2_.eval())

# Outer product
expanded_a = tf.expand_dims(flattened, 1) # shape: (6, 1)
expanded_b = tf.expand_dims(flattened, 0) # shape: (1, 6)
out3 = tf.matmul(expanded_a, expanded_b)
out3_ = tf.einsum(..., flattened, flattened)
assert np.allclose(out3.eval(), out3_.eval())

# Transpose
out4 = tf.transpose(X) # shape: (3, 2)
out4_ = tf.einsum(..., X)
assert np.allclose(out4.eval(), out4_.eval())



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