Understanding the Curse of Dimensionality

Building models is often the easy part. The real work lies in preparing the data—cleaning it, transforming it, and deciding which features actually matter.
In most real-world projects, data preprocessing and feature engineering consume the majority of effort, often determining whether a model succeeds or fails.
One of the most powerful techniques used during this stage is dimensionality reduction, and among all such techniques, Principal Component Analysis (PCA) remains one of the most widely used and best understood.
This article walks you through:
Why dimensionality reduction is necessary
The intuition behind PCA
The mathematics that power it
A step-by-step PCA implementation in R
How PCA impacts model performance

Why Feature Selection Alone Is Not Enough
Feature engineering is not just about removing columns that “look unimportant.”
Sometimes, important information is spread across multiple correlated features. Removing any one of them individually may destroy valuable signal.
This is where dimensionality reduction comes in.
Instead of selecting features directly, dimensionality reduction:
Transforms existing features
Combines correlated variables
Creates new, compact representations of the data
PCA is one of the most effective techniques to achieve this.

Understanding the Curse of Dimensionality
A common misconception in analytics is:
“More features always lead to better models.”
In reality, the opposite is often true.
As the number of features increases:
Model complexity grows exponentially
Data becomes sparse
Distance-based measures lose meaning
Models overfit easily
This phenomenon is known as the curse of dimensionality.
In Simple Terms
When you increase the number of features without increasing the amount of data proportionally, models struggle to generalize. Accuracy often decreases, not improves.

How Do We Escape the Curse?
There are two possible strategies:
Collect more data
Reduce the number of features
In practice, collecting more data is often expensive, slow, or impossible.
Reducing dimensions intelligently is usually the better option—and this is where PCA shines.

The Core Idea Behind Principal Component Analysis
PCA transforms your original feature space into a new coordinate system such that:
The new variables (principal components) are orthogonal
Each component captures the maximum possible variance
Components are ordered by importance
The first principal component explains the most variance.
The second explains the next most, and so on.
By keeping only the top components, you:
Retain most of the information
Remove noise and redundancy
Reduce computational cost

A Helpful Intuition: Shlens’ PCA Example
In his well-known paper, Jonathon Shlens explains PCA using a simple example.
Imagine observing the motion of a pendulum:
The pendulum moves in one direction
But you don’t know that direction beforehand
You place multiple cameras at different angles
Each camera captures a different projection of the same motion.
PCA helps:
Identify the true direction of motion
Combine redundant views
Represent the system using fewer dimensions
In real datasets, features behave like those cameras—multiple noisy views of the same underlying structure.

PCA Conceptual Foundation
Assume a dataset with:
m observations
n features
We represent it as a matrix A (m × n).
PCA transforms A into A′ (m × k), where k < n, such that:
The new features are uncorrelated
Variance is maximized
Information loss is minimized
Why Normalization Matters
PCA is sensitive to scale.
Without normalization, features with larger numerical ranges dominate the transformation.
That’s why standardization is critical before applying PCA—a step often guided by experienced analytics or AI consulting teams to avoid incorrect results.

Mathematical Steps in PCA
Standardize the data
Compute the covariance matrix
Compute eigenvalues and eigenvectors
Sort eigenvectors by decreasing eigenvalues
Project data onto top components
Eigenvalues indicate how much variance each component explains.

Implementing PCA in R
Let’s apply PCA using the classic Iris dataset.
Step 1: Load Numeric Features
data_iris <- iris[, 1:4]

The dataset contains:
150 observations
4 numerical features

Step 2: Compute Covariance Matrix
Cov_data <- cov(data_iris)

Step 3: Compute Eigenvalues and Eigenvectors
Eigen_data <- eigen(Cov_data)

Step 4: Perform PCA Using princomp()
PCA_data <- princomp(data_iris, cor = FALSE)

Step 5: Compare Variances
Eigen_data$values
PCA_data$sdev^2

The results closely match, confirming the correctness of PCA.

Step 6: Examine Loadings (Eigenvectors)
PCA_data$loadings[,1:4]
Eigen_data$vectors

These loadings show how original features contribute to each principal component.

Understanding Component Importance
summary(PCA_data)

Interpretation:
PC1 explains ~92.5% variance
PC2 increases cumulative variance to ~97.7%
Remaining components contribute very little
This means:
One component captures most information
Two components are often sufficient

Visualizing PCA Results
Biplot
biplot(PCA_data)

This visualization shows:
Feature contributions
Direction and magnitude of variance
Relationships between variables
Petal Length and Petal Width dominate PC1, explaining why the first component is so powerful.

Scree Plot
screeplot(PCA_data, type = "lines")

The “elbow” at the second component confirms that retaining two components is ideal.

PCA and Model Performance: A Practical Test
Let’s compare two Naive Bayes models:
Using all four features
Using only the first principal component
Model Using All Features
library(e1071)
mod1 <- naiveBayes(iris[,1:4], iris[,5])
table(predict(mod1, iris[,1:4]), iris[,5])

Model Using First Principal Component
model2 <- PCA_data$loadings[,1]
model2_scores <- as.matrix(data_iris) %*% model2

mod2 <- naiveBayes(model2_scores, iris[,5])
table(predict(mod2, model2_scores), iris[,5])

Result:
Only 3 misclassifications difference
75% reduction in feature count
This trade-off is often worth it in real-world systems.

Key Takeaways
PCA is a powerful and intuitive dimensionality reduction technique
It reduces noise and redundancy while preserving information
PCA improves model efficiency and scalability
Interpretation can be challenging, as transformed features lose business meaning
PCA is best suited for large, high-dimensional datasets
When Not to Use PCA
When features are already uncorrelated
When interpretability is critical
When higher-order statistics (skewness, kurtosis) matter

Final Thoughts
Principal Component Analysis remains a foundational technique in data science, from image compression to genomics to predictive modeling.
When used thoughtfully, PCA allows you to sharpen your axe before chopping the tree—making downstream modeling faster, simpler, and more effective.
At Perceptive Analytics, our mission is “to enable businesses to unlock value in data.” For over 20 years, we’ve partnered with more than 100 clients—from Fortune 500 companies to mid-sized firms—to solve complex data analytics challenges. Our services include working as a reliable Power BI development company and helping organizations hire Power BI consultants, turning data into strategic insight. We would love to talk to you. Do reach out to us.