How can matrix decomposition techniques be utilized to enhance clustering algorithms in optimization problems?

Matrix decomposition techniques, such as singular value decomposition, can break a large data matrix into simpler, low‑rank components that capture the main patterns while discarding noise. By projecting the data onto these few components, a clustering algorithm works in a reduced, more informative space, which speeds up the optimization and reduces the chance of getting stuck in poor local minima. This focused representation lets the algorithm emphasize the most relevant features, improving the objective function of the clustering problem. For instance, applying SVD to a customer purchase matrix produces a handful of principal components; clustering these components with k‑means yields clearer customer segments. The overall effect is faster, more accurate clustering that better satisfies the optimization goals.