np.einsum() for efficient computation in NumPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of np.einsum for matrix multiplication

What is the output of the following code using np.einsum to multiply two matrices?

NumPy

import numpy as np
A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])
result = np.einsum('ik,kj->ij', A, B)
print(result)

A

[[17 20]
 [39 46]]

B

[[12 14]
 [30 32]]

C

[[19 22]
 [43 50]]

D

[[5 12]
 [21 32]]

💡 Hint

Think about how matrix multiplication sums over the shared dimension.

The einsum string 'ik,kj->ij' performs standard matrix multiplication by summing over the k index. The result matches the dot product of A and B.

❓ data_output

intermediate

Sum of diagonal elements using np.einsum

What is the output of this code that sums the diagonal elements of a matrix using np.einsum?

NumPy

import numpy as np
M = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
diag_sum = np.einsum('ii', M)
print(diag_sum)

A 45

B 15

C 9

D 6

💡 Hint

The diagonal elements are where the row and column indices are equal.

The einsum string 'ii' sums elements where the two indices are the same, which are the diagonal elements 1, 5, and 9. Their sum is 15.

❓ visualization

advanced

Visualizing outer product with np.einsum

Which option shows the correct 2D array output of the outer product computed by np.einsum?

NumPy

import numpy as np
x = np.array([1, 2, 3])
y = np.array([4, 5])
out = np.einsum('i,j->ij', x, y)
print(out)

A

[[4 5]
 [8 10]
 [12 15]]

B

[[4 8 12]
 [5 10 15]]

C

[[5 4]
 [10 8]
 [15 12]]

D

[[1 2 3]
 [4 5 6]]

💡 Hint

Outer product multiplies each element of x by each element of y.

The einsum string 'i,j->ij' computes the outer product, resulting in a matrix where each row is x[i] times all elements of y.

🧠 Conceptual

advanced

Understanding einsum subscripts for tensor contraction

Given two 3D arrays A and B with shapes (2,3,4) and (3,4,5), which einsum subscript string correctly contracts over the last two axes of A and the first two axes of B to produce a (2,5) array?

A 'ijk,ljk->il'

B 'ijk,klj->il'

C 'ijk,jlk->il'

D 'ijk,jkl->il'

💡 Hint

Look for matching indices in the middle dimensions to sum over.

The subscript 'ijk,jkl->il' sums over j and k, contracting the last two axes of A and first two of B, resulting in shape (2,5).

🔧 Debug

expert

Identify the error in this einsum usage

What error does the following code raise when executed?

NumPy

import numpy as np
A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])
result = np.einsum('ij,jkl->ik', A, B)
print(result)

A ValueError: einstein sum subscripts string contains too many subscripts for operand 1

B TypeError: unsupported operand type(s) for einsum

C No error, prints [[19 22]\n [43 50]]

D SyntaxError: invalid syntax in einsum string

💡 Hint

Check the number of indices in the subscript string vs array dimensions.

The subscript 'ij,jkl->ik' expects B to have 3 dimensions (indices j,k,l), but B has shape (2,2) with only 2 dimensions, so einsum raises a ValueError.

Practice

1. What is the main purpose of np.einsum() in numpy?

easy

A. To perform efficient and readable array operations using Einstein summation notation

B. To create random arrays with specific distributions

C. To sort arrays in ascending order

D. To reshape arrays into different dimensions

Solution

1. Step 1: Understand the function's goal

   np.einsum() uses Einstein summation notation to express complex array operations clearly and efficiently.

2. Step 2: Compare with other options

   Options B, C, and D describe unrelated numpy functions like random generation, sorting, and reshaping.

3. Final Answer:

   To perform efficient and readable array operations using Einstein summation notation -> Option A

4. Quick Check:

   Purpose of np.einsum() = To perform efficient and readable array operations using Einstein summation notation [OK]

Hint: Remember: einsum is for smart, fast array math [OK]

Common Mistakes:

• Confusing einsum with random array creation
• Thinking einsum sorts or reshapes arrays
• Assuming einsum is only for dot products

2. Which of the following is the correct syntax to compute the matrix product of two 2D arrays A and B using np.einsum()?

easy

A. np.einsum('ij,jk->ik', A, B)

B. np.einsum('ik,jk->ij', A, B)

C. np.einsum('ij,ik->jk', A, B)

D. np.einsum('ji,kj->ik', A, B)

Solution

1. Step 1: Recall matrix multiplication indices

   Matrix multiplication of A (shape ixj) and B (shape jxk) results in shape ixk, summing over j.

2. Step 2: Match einsum subscripts

   np.einsum('ij,jk->ik', A, B) uses 'ij,jk->ik', which correctly sums over j and produces ixk output.

3. Final Answer:

   np.einsum('ij,jk->ik', A, B) -> Option A

4. Quick Check:

   Matrix product einsum = np.einsum('ij,jk->ik', A, B) [OK]

Hint: Use 'ij,jk->ik' for matrix multiplication [OK]

Common Mistakes:

• Mixing up indices order in the subscript string
• Using wrong output indices after ->
• Confusing which axis to sum over

3. Given the code:

import numpy as np
A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])
result = np.einsum('ij,ij->i', A, B)
print(result)

What is the output?

medium

A. [19 43]

B. [17 43]

C. [23 53]

D. [17 53]

Solution

1. Step 1: Understand the einsum subscripts 'ij,ij->i'

   This sums element-wise multiplication over the second index j, keeping the first index i.

2. Step 2: Calculate for each i

   For i=0: (1*5)+(2*6)=5+12=17; For i=1: (3*7)+(4*8)=21+32=53.

3. Final Answer:

   [17 53] -> Option D

4. Quick Check:

   Element-wise sum over j = [17 53] [OK]

Hint: Sum over repeated indices, keep others [OK]

Common Mistakes:

• Summing over wrong axis
• Confusing output indices in einsum
• Calculating element-wise product incorrectly

4. Identify the error in this code snippet:

import numpy as np
A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])
result = np.einsum('ij,jk->ik', A, B)
print(result)

and choose the correct fix.

medium

A. Ensure B has shape (2, 2) and use 'ij,jk->ik' as is

B. No error; code runs correctly

C. Swap arrays to np.einsum('jk,ij->ik', B, A)

D. Change subscript to 'ij,ik->jk' to match shapes

Solution

1. Step 1: Check array shapes and einsum subscripts

   A and B both have shape (2,2). The subscript 'ij,jk->ik' expects B's first dimension (j) to match A's second dimension (j), which it does.

2. Step 2: Verify if code runs

   Shapes match perfectly for matrix multiplication. No error; the code runs correctly.

3. Final Answer:

   No error; code runs correctly -> Option B

4. Quick Check:

   No error; code runs correctly [OK]

Hint: Check array shapes match einsum indices [OK]

Common Mistakes:

• Assuming code errors without checking shapes
• Changing subscripts incorrectly
• Swapping arrays unnecessarily

5. You have a 3D array data with shape (10, 5, 4) representing 10 samples, 5 features, and 4 time points. Using np.einsum(), how do you compute the sum of products of features and time points for each sample, resulting in a 1D array of length 10?

hard

A. np.einsum('ijk,ijk->k', data, data)

B. np.einsum('ijk,ijk->j', data, data)

C. np.einsum('ijk,ijk->i', data, data)

D. np.einsum('ijk,ijk->ij', data, data)

Solution

1. Step 1: Understand the dimensions and desired output

   Data shape is (samples=10, features=5, time=4). We want to sum over features and time, keeping samples dimension.

2. Step 2: Use einsum to sum over j and k, keep i

   Subscript 'ijk,ijk->i' multiplies element-wise and sums over j and k, resulting in length 10 array.

3. Final Answer:

   np.einsum('ijk,ijk->i', data, data) -> Option C

4. Quick Check:

   Sum over features and time = np.einsum('ijk,ijk->i', data, data) [OK]

Hint: Sum over repeated indices, keep sample index [OK]

Common Mistakes:

• Summing over wrong axes
• Confusing output indices in einsum
• Expecting 2D output instead of 1D