5 Best Ways to Return the Norm of a Vector Over Axis 1 in Python

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💡 Problem Formulation: In Python’s linear algebra context, calculating the norm of a vector is a common task. Specifically, returning the norm of a matrix over axis 1 involves computing the norm of each row vector within the matrix. For a matrix with vectors as rows, we’re looking at transforming input such as [[1, 2], [3, 4]] to output [2.236, 5.0], where each element is the norm of the corresponding vector.

Method 1: Using NumPy’s linalg.norm Function

The NumPy library’s linalg.norm function is a versatile tool that allows for the computation of the norm of an array. When applied over axis 1, it computes the norm of each row vector in a two-dimensional array, by default returning the Euclidean norm (L2 norm).

Here’s an example:

import numpy as np

matrix = np.array([[1, 2], [3, 4]])
norms = np.linalg.norm(matrix, axis=1)
print(norms)

Output:

[ 2.23606798  5.        ]

This snippet first imports the NumPy library and defines a 2D array (matrix). Then, it utilizes np.linalg.norm to compute the norm of each row (over axis 1), resulting in the array of norms which is then printed out.

Method 2: Using NumPy’s einsum Function

NumPy’s einsum function provides a powerful way to compute the Euclidean norm without explicitly calling a norm function, using Einstein summation convention. It is a compact and expressive way to perform operations, including computing the norm over axis 1.

Here’s an example:

import numpy as np

matrix = np.array([[1, 2], [3, 4]])
norms = np.sqrt(np.einsum('ij,ij->i', matrix, matrix))
print(norms)

Output:

[ 2.23606798  5.        ]

In this code block, after defining the matrix, the einsum function computes the sum of squares of each row, after which the square root is applied to each sum, computing the norm of each row. The resulting norms are printed out.

Method 3: Using List Comprehension with Built-in Functions

List comprehension in Python combined with the built-in math.sqrt and sum functions can be used to calculate the norm of each row vector over axis 1. While less efficient for large matrices, this method is accessible without additional libraries.

Here’s an example:

import math

matrix = [[1, 2], [3, 4]]
norms = [math.sqrt(sum(i**2 for i in row)) for row in matrix]
print(norms)

Output:

[2.23606797749979, 5.0]

The code uses list comprehension to iterate over each row of a standard Python list, computing the sum of squares of each element, and then applies the square root. The norms of each row are printed, and they correspond to the norms of the vectors over axis 1.

Method 4: Using SciPy’s norm Function

SciPy, an extension of NumPy, has its own norm function within the scipy.linalg module. This method uses more specialized routines and can be more suitable for certain types of norm calculations.

Here’s an example:

from scipy.linalg import norm

matrix = [[1, 2], [3, 4]]
norms = [norm(row) for row in matrix]
print(norms)

Output:

[2.23606797749979, 5.0]

SciPy’s norm function is applied to each row of the matrix separately within a list comprehension. The norms array resulting from this computation is then printed and illustrates the expected norms over axis 1 of the matrix.

Bonus One-Liner Method 5: Using np.sqrt and Summation Over Axis

A one-liner that utilizes NumPy’s np.sqrt function and direct summation over an axis can also achieve this computation. This method combines simplicity and efficiency.

Here’s an example:

import numpy as np

matrix = np.array([[1, 2], [3, 4]])
norms = np.sqrt((matrix ** 2).sum(axis=1))
print(norms)

Output:

[2.23606798 5.        ]

With a single line, after defining the matrix, this example computes the squares of each element, sums these squares over axis 1 (per row), and takes the square root, directly yielding the Euclidean norms for each row vector.

Summary/Discussion

  • Method 1: Using np.linalg.norm. This is straightforward and considered standard for such tasks. However, it requires the NumPy library, which may not always be available.
  • Method 2: Using np.einsum. Provides a powerful, versatile approach and may improve performance for complex operations. It can be a bit intimidating for beginners not familiar with Einstein summation.
  • Method 3: List Comprehension with Built-in Functions. It’s Pythonic and doesn’t need third-party libraries, making it suitable for small-scale operations. It is less efficient for large computations, though.
  • Method 4: Using SciPy’s norm function. Offers specialized and advanced norm calculation options, but requires the SciPy library, which is heavier than NumPy.
  • Bonus Method 5: Using np.sqrt and Summation Over Axis. Combines the performance of NumPy with the simplicity of a one-liner, potentially the best choice for readability and speed in many scenarios.