Matrix factorization and prediction for high dimensional co-occurrence count data via shared parameter alternating zero inflated Gamma model

TL;DR

The paper tackles high-dimensional, sparse co-occurrence data by modeling entries as zero-inflated Gamma variables with shared latent parameters, enabling dense vector representations through cosine similarity. It introduces the SA-ZIG framework, deriving two-link (canonical and log) formulations and an alternating Fisher-scoring algorithm with inner-outer loops and learning-rate adjustments to ensure stable convergence. The approach is analyzed for convergence, including conditions for MLE existence and non-identifiability concerns, and validated through simulations and a word-embedding application that demonstrates meaningful latent word relationships. This method offers a likelihood-based alternative to conventional ALS/weighted MSE approaches for skewed, zero-heavy co-occurrence matrices, with practical implications for recommendations and NLP embeddings.

Abstract

High-dimensional sparse matrix data frequently arise in various applications. A notable example is the weighted word-word co-occurrence count data, which summarizes the weighted frequency of word pairs appearing within the same context window. This type of data typically contains highly skewed non-negative values with an abundance of zeros. Another example is the co-occurrence of item-item or user-item pairs in e-commerce, which also generates high-dimensional data. The objective is to utilize this data to predict the relevance between items or users. In this paper, we assume that items or users can be represented by unknown dense vectors. The model treats the co-occurrence counts as arising from zero-inflated Gamma random variables and employs cosine similarity between the unknown vectors to summarize item-item relevance. The unknown values are estimated using the shared parameter alternating zero-inflated Gamma regression models (SA-ZIG). Both canonical link and log link models are considered. Two parameter updating schemes are proposed, along with an algorithm to estimate the unknown parameters. Convergence analysis is presented analytically. Numerical studies demonstrate that the SA-ZIG using Fisher scoring without learning rate adjustment may fail to fi nd the maximum likelihood estimate. However, the SA-ZIG with learning rate adjustment performs satisfactorily in our simulation studies.

Figures (5)

• Figure 1: $L_2$ norm of the score functions ${\bf U}_{\hbox{\boldmath $\theta$}}$ and ${\bf U}_{\hbox{\boldmath ${\widetilde{\theta}}$}}$ (Uthetas norm), and overall loss of the ZIG model in log$_{10}$ scale. Top two panels: without learning rate adjustment; bottom two panels: with learning rate adjustment. All parameters were initialized with true parameter values except for $\hbox{\boldmath ${\widetilde{w}}$}$. In the top panels, the norms of the score vectors reduce drastically in early iterations but increase over later iterations even though the overall loss is consistently reduced. In the bottom panels, the norms of the score vectors first show similar pattern as the top panel but consistently reduce in later iterations. The overall loss shows the desired reducing trend throughout all iterations.
• Figure 2: Comparing performance of the alternating ZIG regression using log link with or without learning rate adjustment over 8 simulated datasets. Each row is for one dataset. First column: $\log_{10}(L_2$ norm of $U_{\hbox{\boldmath $\theta$}})$; Second column: $\log_{10}(L_2$ norm of $U_{\hbox{\boldmath ${\widetilde{\theta}}$}})$; Third column: $\log_{10}($overall loss$)$; Fourth column: Overall loss of the algorithm with learning rate adjustment.
• Figure 3: Histogram of weighted count from each of the first 12 rows of the data matrix
• Figure 4: Left: The absolute value of change in the log$_{10}$ norms of ${\bf U}_{\hbox{\boldmath $\theta$}}$ and ${\bf U}_{\hbox{\boldmath ${\widetilde{\theta}}$}}$ from successive iterations and overall loss curve. Right: Cosine similarity of word vector representations shown as heatmap.
• Figure 5: Plot of word vector representations using t-SNE on first 10 principal components.