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Independent Poisson counts have the stated NMF log-likelihood

Statement

For an admissible count matrix $X = (x_{i j})$ and a Poisson NMF parameter with rates $λ_{i j} = (W H)_{i j}$ , the joint log-likelihood is

$\log L (W, H) = \sum_{i = 1}^{n} \sum_{j = 1}^{p} [x_{i j} \log λ_{i j} - λ_{i j} - \log (x_{i j}!)] .$

After dropping the data-only term $- \sum_{i, j} \log (x_{i j}!)$ , the objective is

$ℓ (W, H) = \sum_{i = 1}^{n} \sum_{j = 1}^{p} [x_{i j} \log λ_{i j} - λ_{i j}] .$

Use $0 \log 0 = 0$ ; if $x_{i j} > 0$ and $λ_{i j} = 0$ , the log-likelihood is $- \infty$ .

Statement references

Hastie, Tibshirani, and Friedman, §14.6

Lee and Seung, §3

Facts & Assumptions

Given: An admissible observed count matrix, nonnegative factors $W, H$ , and rates $λ_{i j} = (W H)_{i j}$ .

[F1]

Poisson nonnegative matrix-factorization model: the cell counts are mutually independent; a positive-rate cell has Poisson mass $e^{- λ} λ^{m} / m!$ , and a zero-rate cell is concentrated at zero.

Proof

1.1F1givenalgebra

When all rates are positive, [F1] and independence give joint likelihood $L (W, H) = \prod_{i, j} e^{- λ_{i j}} λ_{i j}^{x_{i j}} / x_{i j}!$ . Taking logarithms and using $\log \prod a_{i j} = \sum \log a_{i j}$ gives the displayed formula.

2.1F1step 1.1algebra∎

If $λ_{i j} = 0$ , [F1] gives cell likelihood $1$ when $x_{i j} = 0$ and $0$ when $x_{i j} > 0$ ; their log-likelihoods are respectively $0$ and $- \infty$ , exactly the stated conventions. The factorial term depends only on the fixed data, so removing it leaves the same maximizing objective in $W, H$ .

Depends on

Poisson nonnegative matrix-factorization model

Used by

Finite Poisson NMF multiplicative-update sweeps Algorithm

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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)