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Positive component rescaling and permutation preserve an NMF rate matrix

Statement

Let $W \in [0, \infty)^{n \times r}$ and $H \in [0, \infty)^{r \times p}$ . For any positive scalars $c_{1}, \dots, c_{r}$ , replacing column $k$ of $W$ by $c_{k} W_{\cdot k}$ and row $k$ of $H$ by $c_{k}^{- 1} H_{k \cdot}$ leaves $W H$ unchanged. For any permutation $π$ of ${1, \dots, r}$ , replacing $W_{\cdot k}$ by $W_{\cdot π (k)}$ and $H_{k \cdot}$ by $H_{π (k) \cdot}$ also leaves $W H$ unchanged. Therefore both transformations preserve every Poisson rate and the corresponding likelihood.

Statement references

Hastie, Tibshirani, and Friedman, §14.6

Lee and Seung, §2

Facts & Assumptions

Given: Nonnegative factor matrices $W, H$ and positive component scales $c_{1}, \dots, c_{r}$ or a permutation $π$ .

[F1]

Poisson nonnegative matrix-factorization model: the rate matrix is the matrix product $W H$ .

Proof

1.1F1givenalgebra

For the rescaled factors, each entry is $(W^{'} H^{'})_{i j} = \sum_{k = 1}^{r} (c_{k} W_{i k}) (c_{k}^{- 1} H_{k j}) = \sum_{k = 1}^{r} W_{i k} H_{k j} = (W H)_{i j}$ . Thus every rate is unchanged.

2.1F1step 1.1algebra∎

For the permuted factors, $(W^{''} H^{''})_{i j} = \sum_{k = 1}^{r} W_{i, π (k)} H_{π (k), j} = (W H)_{i j}$ because $π$ reorders the finite sum. Equal rate matrices give the same independent Poisson law and likelihood by [F1].

Depends on

Poisson nonnegative matrix-factorization model

Used by

A positive rate matrix has NMFs beyond scale and permutation Example

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Sources

Trevor Hastie, Robert Tibshirani, and Jerome Friedman, The Elements of Statistical Learning, second edition, corrected 12th printing (standard reference, not scraped)
Daniel D. Lee and H. Sebastian Seung, Algorithms for Non-negative Matrix Factorization, NIPS 2000 (standard reference, not scraped)