A Practical Algorithm for Topic Modeling with Provable Guarantees

Citation

Authors: Sanjeev Arora, Rong Ge, Yonatan Halpern, David Mimno, Ankur Moitra, David Sontag, Yichen Wu, Michael Zhu
Year: 2013
Venue: Proceedings of the 30th International Conference on Machine Learning (PMLR 28(2)), 280-288
arXiv: arXiv:1212.4777
URL: https://proceedings.mlr.press/v28/arora13.html

One-Line Takeaway

Arora et al. turn topic recovery into a tractable and provable problem by imposing anchor-word separability, showing how an additional geometric assumption can convert a latent-class model from merely plausible to stably recoverable.

Technical Summary

Setup: documents are mixtures over $K$ topics with word-topic matrix $A\in\mathbb{R}^{V\times K}$ , and $A$ is $p$ -separable in the sense that for each topic $k$ there is an anchor word $s_k$ with $A_{s_k,k}\ge p$ and $A_{s_k,k'}=0$ for $k'\neq k$ .
Observable object: from word-word co-occurrences, form $Q_{ij}=p(w_1=i,w_2=j)$ and row-normalize to $\bar Q_{ij}=p(w_2=j\mid w_1=i)$ . Under separability, $\bar Q_i = \sum_{k=1}^{K} C_{ik}\,\bar Q_{s_k}, \qquad C_{ik}=p(z_1=k\mid w_1=i),$ so every row lies in the convex hull of the anchor rows.
Method: first find anchor words combinatorially via farthest-point geometry on the rows of $\bar Q$ ; then recover $C$ by solving, for each word $i$ , a simplex-constrained optimization problem $\bar Q_i\approx \sum_k C_{ik}\bar Q_{s_k}$ using KL or $\ell_2$ loss instead of unstable matrix inversion; finally recover $A_{ik}\propto C_{ik}\,p(w=i)$ by Bayes' rule and estimate the topic-topic covariance matrix $R$ .
Main guarantee: under separability and a conditioning assumption on the topic-topic covariance $R$ , the algorithm recovers $A$ in polynomial time with polynomial sample complexity; empirically, it matches collapsed Gibbs quality on several corpora while running substantially faster.

Assumptions And Limits

The guarantees depend on anchor-word separability and are therefore not generic identifiability results for arbitrary topic models. The paper is about structured latent-class recovery, not unrestricted finite-mixture identifiability.

Relevance To This Project

This is a strong example of a theme that matters for x -> z -> y: latent decompositions often need an extra structural bias to become recoverable in practice. Anchor words are one such bias. In a latent-strategy setting, analogous "anchors" could be diagnostic outputs, privileged prompts, or views that are nearly exclusive to a strategy.

Comparison Notes

The paper sits downstream of Allman et al. (2009) in that it uses multi-view latent structure, but it adds a much stronger geometric assumption to obtain a practical recovery algorithm. It is also a useful foil to Yakowitz-Spragins (1968), where identifiability is purely algebraic and single-view.

Source Note

Retrieval: alphaXiv MCP get_paper_content on arXiv:1212.4777 plus alphaXiv MCP answer_pdf_queries on the PMLR PDF; no local cache or downloaded source artifacts were created.

Verified from source: title, author list, venue, year, page numbers, the separability definition, the convex-combination equation for $\bar Q_i$ , the Bayes-rule recovery step for $A$ , the switch from matrix inversion to simplex-constrained recovery, and the headline empirical claim of Gibbs-comparable quality with much faster runtime were checked against the alphaXiv-served arXiv/PMLR text.