Source
Rosen-Zvi, Chemudugunta, Griffiths, Smyth and Steyvers (2010), ACM TOIS 28(1), LIT-1546, read in full in NOTE-1180. The factorization is Appendix C, the special cases §6.1–6.2, and “the posterior probability of Φ and Θ is unaffected by permuting their columns” §3.2. The continuum, and the derivations below, are the record’s.
What was actually shown
The factorization. Under the generative process of §2, a word of document d is w with probability Σ_{a∈a_d} (1/A_d) Σ_t φ_wt θ_ta, since the author is uniform over a_d, the topic is drawn from θ_a, and the word from φ_t. Let A be the A × D matrix with entries 1/A_d where a ∈ a_d and 0 elsewhere. Then the word distribution of document d is column d of P = ΦΘA, and words are independent given (Φ, Θ). So the likelihood of the corpus is Π_d Π_i P_{w_di, d}, a function of P alone.
The special cases. If every document has its own single author, A is the identity (up to ordering) and P = ΦΘ with Θ a T × D matrix of per-document mixtures, which is LDA. If T = A and Θ is the identity, each author has one topic of their own and P = ΦA, the author model of §6.2.
The gauge. Let R be invertible with every column summing to one (1ᵀR = 1ᵀ). Then ΦR⁻¹ has columns summing to one, since 1ᵀΦR⁻¹ = 1ᵀR⁻¹ = 1ᵀ, and RΘ has columns summing to one. If both are nonnegative, (ΦR⁻¹, RΘ) is a valid parameter value and gives the same P = ΦR⁻¹RΘA. So the likelihood is constant on {(ΦR⁻¹, RΘ) : 1ᵀR = 1ᵀ, ΦR⁻¹ ≥ 0, RΘ ≥ 0}. Permutation matrices are in this set, which is §3.2’s statement.
The set contains a continuum when every φ_wt > 0. Take R = (1 − ε)I + (ε/T)J, with J the all-ones matrix. Its columns sum to one and it is nonnegative, so RΘ ≥ 0. Its inverse is (I − (ε/T)J)/(1 − ε), so ΦR⁻¹ = (Φ − ε φ̄1ᵀ)/(1 − ε), where φ̄ is the mean of Φ’s columns. That is nonnegative whenever φ_wt ≥ ε φ̄_w for all w and t, which holds for small enough ε > 0 if Φ is strictly positive. Every posterior mean of Φ is, since β > 0 in (6). The change mixes each author’s topic proportions toward uniform and sharpens the topics to compensate.
Checked numerically: with W = 8, T = 3, A = 4, D = 5, random Φ (Dirichlet(5)) and Θ, and ε = 0.05, the transformed Φ and Θ were column-stochastic and nonnegative, and P changed by at most 6 × 10⁻¹⁷.
What could have failed: a model with observed authorship might have pinned the topic basis, since the same θ_a must serve every document of author a. It does not. The authorship matrix A is to the right of Θ, and the gauge acts between Φ and Θ, where A does not reach.
What this does not say
- Not that the posterior is flat. The Dirichlet priors on Φ and Θ are not invariant under R, so the posterior prefers some points of the orbit. Which ones is set by α and β, not by the words.
- Not that the gauge is always a continuum. If some topic gives a word probability zero, as a point estimate might under β = 0, the continuum can shrink. Conditions like separability or anchor words, under which only permutations remain, are not examined here.
- Not a claim about the paper’s results. Its perplexities and retrievals depend on P, which the gauge leaves alone. Readings of individual topics or of an author’s profile are what it touches.
Connections
- CLAIM-148. This is that claim’s nonnegative-factorization gauge in three-factor form with one factor known: the set {R : WR ≥ 0, R⁻¹H ≥ 0}, with the column-sum constraint that probability matrices add.
- CLAIM-135. The AT model reaches the first achievement by perplexity. The second needs conditions the paper does not supply. Its §4.2 found about 80 of 100 topics aligned across two runs, the empirical face of the same gap.
- CASE-165. The additive latent-feature model of that case has the same kind of gauge between its features and their weights, and CLAIM-148’s note of 2026-10-26 lists it.
- THEORY-993. The mixed membership blockmodel has the bilinear analogue, between memberships and the block matrix.