Low-rank approximation

In mathematics, low-rank approximation is a minimization problem, in which the cost function measures the fit between a given matrix (the data) and an approximating matrix (the optimization variable), subject to a constraint that the approximating matrix has reduced rank. The problem is used for mathematical modeling and data compression. The rank constraint is related to a constraint on the complexity of a model that fits the data. In applications, often there are other constraints on the approximating matrix apart from the rank constraint, e.g., non-negativity and Hankel structure.

Low-rank approximation is closely related to:

Given

The unstructured problem with fit measured by the Frobenius norm, i.e.,

has analytic solution in terms of the singular value decomposition of the data matrix. The result is referred to as the matrix approximation lemma or Eckart–Young–Mirsky theorem. Let

be the singular value decomposition of ${\displaystyle D}$ and partition ${\displaystyle U}$, ${\displaystyle \Sigma =:\operatorname {diag} (\sigma _{1},\ldots ,\sigma _{m})}$, and ${\displaystyle V}$ as follows:

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