Question 1

What is SVD (Singular Value Decomposition)?

Accepted Answer

SVD is a matrix factorization technique that decomposes any matrix A into three matrices: A = U × Σ × V^T, where U and V are orthogonal matrices and Σ is a diagonal matrix containing singular values. It's a fundamental tool in linear algebra used for data compression, dimensionality reduction, and solving least squares problems.

Question 2

How do you calculate SVD step by step?

Accepted Answer

The SVD calculation process involves: 1) Compute A^T × A, 2) Find eigenvalues of A^T × A, 3) Calculate singular values as square roots of eigenvalues, 4) Find V matrix (eigenvectors of A^T × A), 5) Calculate U matrix using U = A × V × Σ⁻¹. The result is A = U × Σ × V^T where U and V are orthogonal and Σ contains singular values.

Question 3

What are the properties of U, Σ, and V matrices?

Accepted Answer

U matrix: Contains left singular vectors, is orthogonal (U^T × U = I). Σ matrix: Diagonal matrix with singular values in descending order, non-negative values. V matrix: Contains right singular vectors, is orthogonal (V^T × V = I). The singular values represent the 'importance' of each dimension in the original matrix.

Question 4

What are the applications of SVD?

Accepted Answer

SVD has numerous applications including: 1) Data compression (image compression, PCA), 2) Dimensionality reduction, 3) Solving least squares problems, 4) Matrix approximation, 5) Signal processing, 6) Machine learning (collaborative filtering, latent semantic analysis), 7) Numerical analysis, 8) Image processing and computer vision.

Question 5

How do you use SVD for dimensionality reduction?

Accepted Answer

For dimensionality reduction: 1) Compute SVD of your data matrix, 2) Select the k largest singular values, 3) Keep only the first k columns of U and V, 4) Form the reduced matrix A_k = U_k × Σ_k × V_k^T. This gives the best rank-k approximation of the original matrix, preserving the most important information while reducing dimensions.

Question 6

What's the difference between SVD and eigenvalue decomposition?

Accepted Answer

SVD works on any matrix (rectangular or square), while eigenvalue decomposition only works on square matrices. SVD decomposes A into U × Σ × V^T, while eigenvalue decomposition decomposes A into P × D × P⁻¹. SVD always exists for any matrix, while eigenvalue decomposition only exists for diagonalizable matrices. SVD is more numerically stable.

Question 7

How do you interpret singular values?

Accepted Answer

Singular values represent the 'energy' or 'importance' of each dimension in the original matrix. Larger singular values correspond to more important directions in the data. The ratio of singular values indicates the relative importance of different components. In data analysis, you can truncate SVD by keeping only singular values above a certain threshold.

Question 8

What are the computational complexity considerations for SVD?

Accepted Answer

SVD computation is O(mn²) for an m×n matrix, which can be expensive for large matrices. For very large matrices, approximate SVD methods like randomized SVD are used. The full SVD is rarely needed in practice; often only the first k singular values and vectors are computed. Modern libraries use optimized algorithms like LAPACK for efficient computation.

Question 9

How do you use SVD for matrix approximation?

Accepted Answer

SVD provides the best low-rank approximation of a matrix. For a matrix A with SVD A = U × Σ × V^T, the best rank-k approximation is A_k = U_k × Σ_k × V_k^T where U_k, Σ_k, V_k contain only the first k columns/rows. This minimizes the Frobenius norm error ||A - A_k||_F among all rank-k matrices.

Question 10

What are common numerical issues with SVD?

Accepted Answer

Common issues include: 1) Ill-conditioned matrices (very small singular values), 2) Rounding errors in floating-point arithmetic, 3) Convergence problems for iterative methods, 4) Memory requirements for large matrices. Solutions include using regularized SVD, setting minimum singular value thresholds, and using specialized algorithms for sparse matrices.

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