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Measuring the Predictability of Recommender Systems using Structural Complexity Metrics

Alfonso Valderrama, Andrés Abeliuk

TL;DR

The paper tackles the problem of quantifying the inherent predictability of recommender systems by treating the user-item rating matrix $M$ as a structural object and measuring its complexity under perturbations. It introduces two data-driven metrics, Analytical Structural Consistency (ASC) and Empirical Structural Consistency (ESC), derived from SVD-based perturbations and matrix factorization, respectively, and evaluates them against the best-performing CF algorithms using RMSE on real and synthetic data. Real-data results show a strong link between the metrics and predictive performance, with ASC and ESC achieving high correlations (e.g., $r=0.968$, $r=0.924$ for Pearson) and ESC demonstrating robustness on synthetic data where ASC may fail. The work suggests that these metrics can guide algorithm selection and monitor system evolution, while noting computational costs and pointing to future work on scalability and broader validation.

Abstract

Recommender systems (RS) are central to the filtering and curation of online content. These algorithms predict user ratings for unseen items based on past preferences. Despite their importance, the innate predictability of RS has received limited attention. This study introduces data-driven metrics to measure the predictability of RS based on the structural complexity of the user-item rating matrix. A low predictability score indicates complex and unpredictable user-item interactions, while a high predictability score reveals less complex patterns with predictive potential. We propose two strategies that use singular value decomposition (SVD) and matrix factorization (MF) to measure structural complexity. By perturbing the data and evaluating the prediction of the perturbed version, we explore the structural consistency indicated by the SVD singular vectors. The assumption is that a random perturbation of highly structured data does not change its structure. Empirical results show a high correlation between our metrics and the accuracy of the best-performing prediction algorithms on real data sets.

Measuring the Predictability of Recommender Systems using Structural Complexity Metrics

TL;DR

The paper tackles the problem of quantifying the inherent predictability of recommender systems by treating the user-item rating matrix as a structural object and measuring its complexity under perturbations. It introduces two data-driven metrics, Analytical Structural Consistency (ASC) and Empirical Structural Consistency (ESC), derived from SVD-based perturbations and matrix factorization, respectively, and evaluates them against the best-performing CF algorithms using RMSE on real and synthetic data. Real-data results show a strong link between the metrics and predictive performance, with ASC and ESC achieving high correlations (e.g., , for Pearson) and ESC demonstrating robustness on synthetic data where ASC may fail. The work suggests that these metrics can guide algorithm selection and monitor system evolution, while noting computational costs and pointing to future work on scalability and broader validation.

Abstract

Recommender systems (RS) are central to the filtering and curation of online content. These algorithms predict user ratings for unseen items based on past preferences. Despite their importance, the innate predictability of RS has received limited attention. This study introduces data-driven metrics to measure the predictability of RS based on the structural complexity of the user-item rating matrix. A low predictability score indicates complex and unpredictable user-item interactions, while a high predictability score reveals less complex patterns with predictive potential. We propose two strategies that use singular value decomposition (SVD) and matrix factorization (MF) to measure structural complexity. By perturbing the data and evaluating the prediction of the perturbed version, we explore the structural consistency indicated by the SVD singular vectors. The assumption is that a random perturbation of highly structured data does not change its structure. Empirical results show a high correlation between our metrics and the accuracy of the best-performing prediction algorithms on real data sets.
Paper Structure (17 sections, 13 equations, 3 figures, 3 tables)

This paper contains 17 sections, 13 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Predictability in complex networks
  4. Collaborative Filtering Algorithms
  5. Correlation between prediction performance and dataset meta-features
  6. Metrics
  7. Analytical Structural Consistency (ASC)
  8. Adjacency matrix perturbation by diagonalization
  9. Structural perturbation by SVD
  10. Empirical Structural Consistency (ESC)
  11. Methodology
  12. Results
  13. Outcome for the real datasets
  14. The case of synthetic data
  15. Discussion
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Graphical examples of the generated cases studied for the predictability metrics. The five dataset types have the same numerical rating distribution, but they vary in the relative distribution among agents. Figure \ref{['fig:gen-lines']} has all users rating items with one value; figure \ref{['fig:gen-mirror']} has users that rate at most two values, and items can be set into two groups according to the ratings given to them; figure \ref{['fig:gen-consec']} has users rating using consecutive ratings (5 is taken as consecutive to 1); figure \ref{['fig:gen-consec-2d']} does something similar, but with a wider variety of users; and figure \ref{['fig:gen-random']} has all ratings randomly and independently assigned. Intuitively, a suitable measurement for structural complexity $c$ should satisfy $c(\ref{['fig:gen-lines']}) < c(\ref{['fig:gen-mirror']}) \approx c(\ref{['fig:gen-consec']}) < c(\ref{['fig:gen-consec-2d']}) < c(\ref{['fig:gen-random']})$. The matrices used to obtain the results in figure \ref{['fig:generated']} follow this basic structure, but are of size $500 \times 500$.
  • Figure 2: Graphical correlation for real datasets between predictability metrics and the lowest RMSE from the tested recommendation techniques. Each point is obtained by iterating 10 times over each metric and algorithm and taking its mean; the error bars are taken from the standard deviation of the aforementioned iterations on each axis. Both metrics show a strong linear correlation with algorithmic performance, although ESC presents problems with one dataset (Steam reviews).
  • Figure 3: Graphical correlation for generated data between predictability measurements and the lowest RMSE from recommendation techniques, taking into account 5% of their ratings as known. Each point is obtained by iterating 10 times over each metric and algorithm and taking its mean; the error bars are taken from the standard deviation of the aforementioned iterations on each axis. Only the ESC metric has an identifiable correlation with the prediction errors for these cases.