All MCQs

Matrix rank measures the dimensionality of the space spanned by its rows or columns. In exact SVD, it equals the number of nonzero singular values.

V contains orthonormal directions defined over the columns or features of X. Vᵀ expresses those directions as rows in the factorization.

Singular values quantify how much matrix structure is associated with each paired singular-vector direction. They are conventionally ordered from largest to smallest.

U contains orthonormal directions on the left side of the matrix transformation. Together with singular values, they describe how observations load onto latent dimensions.

SVD factors a matrix into left singular vectors, singular values, and right singular vectors. The factorization exists for any real or complex matrix under the standard formulation.

High-dimensional spaces are sparse, and local relationships can become difficult to estimate. Reducing the space to informative directions can restore meaningful structure.

Fewer dimensions can make algorithms faster and patterns easier to inspect. A compact representation may also suppress low-value variation.

Dimensionality reduction compresses the feature space into a smaller representation. The aim is to preserve information relevant to analysis, visualization, or prediction.

Redundant features add complexity without contributing much independent information. Removing one can simplify the model and reduce instability in some algorithms.

Many weak or irrelevant variables give a model more opportunities to fit noise. A smaller representation can be faster and may generalize better.

Test data must represent unseen cases and cannot influence which variables the model receives. Selection based on all data can make the final evaluation optimistically biased.

Backward elimination begins with all candidate variables and assesses which one can be removed with least harm. It requires that the initial model can be fitted using the full set.