🧠 Principal Component Analysis (PCA) – ML/DL Interview Series
Welcome back to our Machine Learning/Deep Learning interview series.
After diving into K-Means Clustering and other supervised learning techniques, we now explore Principal Component Analysis (PCA)—a fundamental unsupervised learning technique used for dimensionality reduction and data visualization.
📘1.Conceptual Understanding
What is PCA?
Principal Component Analysis is an unsupervised linear transformation technique that converts a dataset with possibly correlated variables into a set of linearly uncorrelated variables called principal components.
These components are ordered such that the first few retain most of the variation present in the original dataset.
Why PCA?
Real-world data often has high dimensions, making it hard to visualize and model.
PCA helps reduce noise and redundancy, speeding up ML models.
It improves generalization by removing irrelevant variance.
🔍 2. Applied Perspective
How does PCA work?
Standardize the dataset
Subtract the mean and divide by standard deviation so all features have zero mean and unit variance.Compute the covariance matrix
\(\Sigma = \frac{1}{n - 1} X^\top X\)Compute eigenvectors and eigenvalues
Solve the eigenvalue equation:\(\Sigma v = \lambda v\)where:
v = eigenvector (principal component)
λ = eigenvalue (variance explained by that component)
Sort eigenvectors by descending eigenvalues
The eigenvector with the highest eigenvalue captures the most variance.Select top k components
Choose the first k eigenvectors (based on cumulative variance explained).Project data onto new axes
\(Z = XW_k\)where Wkis the matrix of selected k eigenvectors and Z is the reduced representation.
🛠️ 3.System Design Perspective
When do we use PCA in a system?
Preprocessing before training ML models, especially with high-dimensional input.
Visualizing high-dimensional data in 2D or 3D.
Noise reduction for sensor or image data.
Compressing features to reduce storage and improve performance.
Key Considerations:
PCA assumes linearity and Gaussian distributions.
PCA does not handle categorical variables unless one-hot encoded.
Interpretability of principal components can be low.
Interview Questions
What is PCA and what problem does it solve in machine learning?
How are principal components computed from the original dataset?
What is the relationship between eigenvalues and the variance in PCA?
How do you choose the number of principal components (k)?
What are the limitations of PCA?
Solutions
Q1: What is PCA and what problem does it solve in machine learning?
Answer: PCA is a dimensionality reduction technique that transforms data into a new coordinate system such that the greatest variance lies along the first axis (principal component), the second greatest variance on the second axis, and so on. It reduces data complexity, removes redundancy, and helps with visualization and model efficiency.
Q2: How are principal components computed from the original dataset?
Answer:
Standardize the data (zero mean and unit variance).
Compute the covariance matrix:
\(\Sigma = \frac{1}{n - 1} X^\top X\)Compute the eigenvectors (principal components) and their corresponding eigenvalues (amount of variance).
Sort the eigenvectors by eigenvalues in descending order.
Select top-k eigenvectors to form the transformation matrix.
Project the data onto the new k-dimensional space using:
\(Z = XW_k\)
Q3: What is the relationship between eigenvalues and the variance in PCA?
Answer:Each eigenvalue in PCA represents the amount of variance captured by its corresponding principal component. A higher eigenvalue means that component captures more variance. The total variance is the sum of all eigenvalues, and the proportion of variance explained by a component is:
Q4: How do you choose the number of principal components (k)?
A4:
You can choose k by:
Explained Variance Threshold: Select k such that cumulative explained variance exceeds a threshold (e.g., 95%).
Scree Plot: Plot eigenvalues and find the "elbow" point.
Cross-validation: Evaluate performance of downstream models for different values of k.
Q5: What are the limitations of PCA?
A5:
Assumes linear relationships between variables.
Does not perform well if features are not normalized.
Not suitable for categorical data.
Principal components are not easily interpretable.
Sensitive to outliers which can distort the direction of components.
📚 Further Reading
📌 Conclusion
PCA is a powerful tool in the data scientist’s toolbox—capable of simplifying high-dimensional data without losing essential information. It’s a must-know concept for ML interviews and real-world applications alike.
In our next article, we’ll explore advanced clustering methods like DBSCAN, Hierarchical Clustering, and Gaussian Mixture Models—clustering methods that overcome K-Means’ limitations.
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