Viva Questions and Answers on Singular Value Decomposition
(SVD) and PCA
1 Basic Questions
1. What is Singular Value Decomposition (SVD)? SVD is a matrix factorization technique
that decomposes a matrix X into three matrices:
X = U SV T
where:
• U is an orthogonal matrix of left singular vectors.
• S is a diagonal matrix containing singular values.
• V T is an orthogonal matrix of right singular vectors.
2. How is SVD different from PCA?
• PCA (Principal Component Analysis) finds the eigenvectors of the covariance matrix of
data.
• SVD directly decomposes the data matrix into singular vectors and singular values.
• PCA is dependent on mean-centering, while SVD can be applied to any matrix.
3. What are singular values, and what do they represent? Singular values in the diagonal
matrix S represent the importance or energy of each singular vector. Higher singular values
indicate more significant components in data.
4. What are left and right singular vectors in SVD?
• Left singular vectors (U ): Represent the directions in the original space.
• Right singular vectors (V T ): Represent principal directions in feature space.
• These vectors are orthonormal and help in dimensionality reduction.
5. Why do we use SVD in image processing?
• Compression: We can store only the top singular values and vectors.
• Noise Reduction: Removing small singular values reduces noise.
• Feature Extraction: Singular vectors capture important patterns in images.
2 Conceptual Questions
6. What is the mathematical formula for SVD? For any matrix X of size m × n, SVD is defined
as:
X = U SV T
where:
• U is m × m (left singular vectors),
• S is m × n (singular values on diagonal),
• V T is n × n (right singular vectors).
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7. What is the significance of the singular values in SVD?
• They represent the energy or variance in the data.
• Larger singular values indicate more important patterns.
• Smaller singular values correspond to noise or less important features.
8. Why do we take only the top k singular values instead of using all of them?
• The top k singular values capture most of the data variation.
• Lower singular values contribute to noise, so we ignore them.
• This reduces dimensionality without significant information loss.
9. How does SVD help in dimensionality reduction?
• We keep only the top k singular vectors corresponding to the largest singular values.
• This results in a low-rank approximation of the data matrix, reducing its size.
3 Practical & Implementation-Based Questions
11. Why do we reshape singular vectors into images?
• Singular vectors are stored as 1D arrays, but they correspond to 2D patterns in images.
• Reshaping them back into 64 × 64 helps visualize the important features.
12. Why do we scale singular vectors before displaying them?
• Singular vectors have values that may not be in a visible range (e.g., negative values).
• Scaling makes them easier to interpret.
13. What happens if we take more singular vectors?
• More details from the original image are preserved.
• Less compression but higher accuracy.
4 Advanced Questions
16. Can SVD be applied to color images? How? Yes!
• Apply SVD separately to each color channel (Red, Green, Blue).
• Compress or modify each channel independently and then recombine.
17. What is the relationship between SVD and eigen decomposition?
• Eigen decomposition applies to square symmetric matrices.
• SVD applies to any matrix (even non-square) and is more general.
18. How does SVD help in image compression?
• We keep only the top k singular values and vectors.
• This reduces the data size while preserving the most important details.
19. How can noise be reduced using SVD?
• Noise is typically represented by small singular values.
• Ignoring the smallest singular values removes noise while keeping important features.
