np.einsum() for efficient computation in NumPy

Introduction

np.einsum() helps you do math with arrays in a fast and clear way. It lets you write complex calculations simply and runs them efficiently.

When you want to multiply and sum elements of arrays in custom ways.

When you need to do matrix multiplication or dot products with less code.

When you want to avoid slow loops and speed up calculations.

When you want to combine multiple array operations into one step.

When you want to write clear code for complex math with arrays.

Syntax

NumPy

np.einsum(subscripts, *operands, optimize=False)

subscripts is a string that tells how to multiply and sum array elements.

operands are the arrays you want to calculate with.

Examples

Calculates the dot product of two 1D arrays a and b.

NumPy

np.einsum('i,i->', a, b)

Performs matrix multiplication of 2D arrays A and B.

NumPy

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

Extracts the diagonal elements of a square matrix M.

NumPy

np.einsum('ii->i', M)

Sums over the second axis of a 3D array T.

NumPy

np.einsum('ijk->ik', T)

Sample Program

This program shows how to use np.einsum() to do a dot product and matrix multiplication. It prints the results clearly.

NumPy

import numpy as np

# Define two 1D arrays
a = np.array([1, 2, 3])
b = np.array([4, 5, 6])

# Dot product using np.einsum
result_dot = np.einsum('i,i->', a, b)

# Define two 2D arrays for matrix multiplication
A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])

# Matrix multiplication using np.einsum
result_matmul = np.einsum('ij,jk->ik', A, B)

# Print results
print('Dot product:', result_dot)
print('Matrix multiplication:\n', result_matmul)

Important Notes

Use simple letters like i, j, k to label array axes in the subscripts.

np.einsum() can replace many common numpy functions like dot, sum, and transpose.

It often runs faster because it avoids creating temporary arrays.

Summary

np.einsum() lets you write complex array math clearly and efficiently.

It uses a string to describe how to multiply and sum array elements.

Great for dot products, matrix multiplication, and custom sums without slow loops.

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