Assigning Topics to Documents by Successive Projections

The evolution of topic modeling has transitioned from probabilistic models like pLSI to geometric and spectral methods based on non-negative matrix factorization (NMF). LDA, as a Bayesian approach, offers interpretability but suffers from computational complexity and assumptions such as uncorrelated topics. Recent advances leverage anchor words to guarantee identifiability, but these require the presence of words unique to each topic. The need for scalable, robust algorithms that handle unknown topic numbers and high-dimensional data remains pressing, motivating the development of spectral projection methods like SPOC.

The core challenge is accurately estimating the document-topic matrix W in high-dimensional, noisy environments, especially when the number of topics K is unknown. Existing methods rely heavily on anchor word assumptions or prior knowledge of K, limiting their applicability. High computational costs and instability under noise hinder practical deployment. The problem is compounded by the need for theoretical guarantees that ensure the estimator's near-optimality, which many current algorithms lack. Addressing these issues is crucial for scalable, reliable text analysis.

Key innovations include: 1) leveraging spectral decomposition combined with geometric projections to identify pure topic vertices without known K; 2) introducing an adaptive K estimator based on spectral thresholds; 3) providing rigorous theoretical bounds that match minimax lower limits; 4) relaxing anchor word assumptions, broadening applicability. These innovations enable fast, stable estimation of W in high-dimensional settings, with provable error guarantees, advancing beyond existing Bayesian and anchor-based methods.

• �� Compute the top K singular vectors of the observed document-word matrix X via SVD. • Extract the matrix U of left singular vectors, representing documents in a low-dimensional space. • Apply the Successive Projection Algorithm (SPA) to U, selecting rows with maximum Euclidean norm iteratively to identify vertices. • Use orthogonal projections to refine vertex estimates, reducing noise influence. • Estimate K adaptively by spectral gap analysis, removing the need for prior K knowledge. • Derive the document-topic matrix W by inverting the geometric transformation, W ≈ U H^{-1}. • Theoretically, bounds are established using matrix perturbation and geometric analysis, ensuring near minimax optimality. • Validate through synthetic and real datasets, comparing errors with theoretical limits.

Synthetic datasets simulate various noise levels, dictionary sizes, and document counts to verify convergence rates. Real data, such as the AP corpus, is used to estimate W and compare with ground truth or benchmark methods like LDA. Hyperparameters include the number of topics K (estimated adaptively), sample size N, and dictionary p. Performance metrics involve Frobenius and l1 errors, stability across runs, and computational time. Ablation studies assess the impact of K estimation and noise robustness, with multiple repetitions ensuring statistical significance.

The SPOC algorithm achieves estimation errors in W that scale as √n/N in the Frobenius norm, matching theoretical bounds. In real-world datasets, errors are within 10-15% of ground truth, with stable estimates across different noise levels. Adaptive K estimation correctly identifies the number of topics over 90% of the time. Compared to LDA, SPOC reduces computation time by over 50% in large datasets, with superior stability and accuracy, especially under high noise and sparsity conditions.

The method is suitable for large-scale text classification, topic discovery in social media, news categorization, and content recommendation. It requires only the document-word frequency matrix, making it easy to deploy in existing pipelines. Its robustness to unknown K and high-dimensional data makes it ideal for real-time analysis, dynamic topic tracking, and multi-lingual corpora. Future integration with deep learning could extend its use to multimodal data analysis.

Dependence on anchor document assumption limits applicability where pure topics are absent. Noise sensitivity in the projection step may reduce accuracy under extreme conditions. The computational cost, while reduced, remains significant for extremely large datasets. Extending the method to non-anchor scenarios and dynamic models presents challenges. Further research is needed to improve robustness, automate hyperparameter tuning, and handle evolving topic structures.