Welcome to our explanation of the Karhunen-Loève Transform, also known as Principal Component Analysis or PCA. The KLT is a data-dependent linear transformation that provides optimal decorrelation and energy compaction. It works by finding the eigenvectors of the data's covariance matrix. In this visualization, we have a set of correlated data points in blue. The red arrow represents the first principal component, which points in the direction of maximum variance. The green arrow shows the second principal component, which is orthogonal to the first. These principal components form a new coordinate system that better represents the data's inherent structure.
Now let's look at the mathematical formulation of the KLT. The process begins with a data matrix X containing our observations. We then compute the covariance matrix C, which measures how variables change together. Next, we find the eigenvectors and eigenvalues of this covariance matrix. The eigenvectors represent the directions of maximum variance, while the eigenvalues indicate the amount of variance in those directions. We sort these eigenvectors by their corresponding eigenvalues in descending order. Finally, we transform our original data by projecting it onto these eigenvectors. In the visualization, you can see our original correlated data on top, with principal components shown as arrows. Below is the transformed data, where the axes now align with the principal components. Notice how the transformed data is decorrelated, with the maximum variance along the first principal component.
One of the most powerful applications of the Karhunen-Loève Transform is dimensionality reduction. In this process, we retain only the top k principal components with the largest eigenvalues, and discard components with small eigenvalues. This allows us to represent high-dimensional data in a lower-dimensional space while preserving most of the information. In our visualization, we start with 3D data shown at the top. By applying KLT, we can reduce it to 2D data shown at the bottom, by projecting onto the two principal components with the highest variance. The bar chart shows the percentage of variance explained by each principal component. Notice how the first two components capture most of the variance, while the third component contributes very little. This makes KLT optimal in terms of mean square error for dimensionality reduction, as it preserves the maximum possible variance in the reduced representation.
The Karhunen-Loève Transform has numerous practical applications across various fields. One of the most common applications is image compression. In this example, we start with an original grayscale image on the left. By applying KLT and keeping only the top three principal components, we can create a compressed version shown on the right. The bar chart below shows the eigenvalues, which represent the amount of variance captured by each component. Notice how the first few components (highlighted in red) capture most of the variance, allowing us to achieve good compression with minimal information loss. Beyond compression, KLT is also used for noise reduction by discarding components with small eigenvalues that often represent noise. In feature extraction and pattern recognition, KLT helps identify the most important features in the data. And for data visualization, it allows us to project high-dimensional data onto lower dimensions for easier interpretation.