How to Compute the Euclidean Distance Between Two Complex Matrices with Vectorization in MATLAB

Learn how to efficiently compute the Euclidean distance between complex matrices in MATLAB using vectorization techniques to optimize performance and execution speed.
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How to Compute the Euclidean Distance Between Two Complex Matrices with Vectorization in MATLAB

Working with complex matrices can be a challenging task in MATLAB, especially when it comes to calculating distances between different vectors. If you find yourself dealing with two complex matrices and need to compute their Euclidean distance efficiently, this guide is for you!

Understanding the Problem

Let's say you have a complex matrix X of size [S, N], where:

S is the number of rows (or features) in each vector.

N is the number of complex vectors you are dealing with.

For example, you might have:

[[See Video to Reveal this Text or Code Snippet]]

with S=3 and x_1 = [1+ 2j, 2+ 3j, 3+ 4j]'.

You want to create a distance matrix D, such that:

D(i, j) represents the Euclidean distance between x_i and x_j.

Initial Code Implementation

Initially, you might employ nested loops to compute the distances, like this:

[[See Video to Reveal this Text or Code Snippet]]

However, as N increases, this loop approach can become computationally expensive, especially since you cannot utilize pdist on complex matrices.

Vectorization: The Solution

To enhance your solution using vectorization—a critical technique for speeding up computations in MATLAB—you can follow these steps:

Building a 3D Matrix

By restructuring your code, you can create an intermediate 3D matrix that is essential for vectorization:

[[See Video to Reveal this Text or Code Snippet]]

This method may consume more memory; however, it generally speeds up the computation significantly. Since our example has S=3, the additional space is manageable and should not hinder performance.

Importance of Modern MATLAB

It is essential to note that previous versions of MATLAB and Octave had a consistent performance boost from avoiding loops. However, in modern MATLAB, that is not always the case. Always profile your code to identify what offers you the fastest run times.

Alternative Approach: Improving the Original Code

If you prefer not to eliminate loops yet want to optimize your existing method, consider the following modifications:

[[See Video to Reveal this Text or Code Snippet]]

Key Improvements Made:

Loop Structure: Ensure that when looping through matrices, the inner loop should always iterate over the first index. This optimizes cache usage.

Using vecnorm: This function is newer and executes faster than norm as it performs a single task effectively.

Skipping the Diagonal: Avoid computation of distances for elements where i = j, as known distances will always be zero.

Conclusion

Calculating the Euclidean distance between complex matrices is a common requirement in various mathematical and engineering applications. By implementing vectorization techniques or optimizing your loop methods, you can significantly reduce computation time and improve the efficiency of your MATLAB programs.

This post should empower you to tackle complex matrices with confidence, providing you with tools and methods to optimize them effectively. Happy coding!