Low-rank approximation: compressing a matrix with the SVD

Any matrix has an SVD A = Σ σ_i u_i v_i^⊤. Keeping only the k largest singular values gives A_k = Σ_i<k σ_i u_i v_i^⊤, the best rank-k approximation of A (Eckart–Young). It stores U[:, :k], σ[:k], V^⊤[:k, :]: only k(m+n) numbers instead of mn. Sweep k and watch the image rebuild from a few components; the singular-value spectrum shows why some matrices compress and noise does not.