ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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A zero-count cell remains well-defined in a positive NMF update

Example

Let

$X = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}), W^{(0)} = H^{(0)} = (\begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array}), r = 2.$

After one complete Poisson NMF sweep, the algorithm gives $W^{(1)} = \frac{1}{4} (\begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array})$ and $H^{(1)} = H^{(0)}$ . The zero-count cells contribute zero to the update numerators, and every rate used in the sweep is positive.

Facts & Assumptions

Given: The count matrix, positive initialization, and one sweep of the finite Poisson NMF updates.

[F1]

Poisson nonnegative matrix-factorization model: every row and column has a positive count, and cell rates are the entries of $W H$ .

[F2]

Finite Poisson NMF multiplicative-update sweeps: a sweep updates $W$ first, recomputes the rates, then updates $H$ using the new $W$ .

[F3]

Strictly positive initialization keeps the Poisson NMF updates well-defined: positive initialization and the nonzero row/column condition keep all rates and denominators positive.

Verification

1.1F1F2givenalgebra

Initially every rate is $2$ . For each $i, k$ , the $W$ numerator is $0 / 2 + 1 / 2 = 1 / 2$ and the denominator is $1 + 1 = 2$ , so $W_{i k}^{(1)} = 1 / 4$ . The recomputed intermediate rates are all $1 / 2$ .

2.1F2F3step 1.1algebra∎

For each $k, j$ , the $H$ numerator is $\frac{1}{4} (0 / (1 / 2)) + \frac{1}{4} (1 / (1 / 2)) = 1 / 2$ and the denominator is $1 / 4 + 1 / 4 = 1 / 2$ , hence $H_{k j}^{(1)} = 1$ . Every division is by a positive rate as guaranteed by [F3], and a zero-count cell contributes $0 / (1 / 2) = 0$ , not an indeterminate quotient.

Depends on

Used by

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