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Popularity Bias Alignment Estimates

Anton Lyubinin

TL;DR

This work addresses how popularity-driven exposure in recommender systems aligns with spectral directions of the user-item interaction matrix without assuming a power-law tail. It (i) generalizes the Popularity Bias Memorization framework to distribution-agnostic item popularities, establishing Π₁-memorization and Πₖ-memorization results, and (ii) develops a suite of bounds—combinatorial, Ky Fan, interlacing, and LP-based—for the alignment with the top-$k$ singular subspace, tying these bounds to tail mass, effective ranks, and graph topology via Kumar's bounds. The results quantify when recommendation scores align with the principal or top-$k$ singular directions and reveal regimes (e.g., log-normal tails, truncation) where alignment weakens, providing interpretable quantities that can be data-driven estimates. Overall, the paper offers distribution-agnostic guarantees on spectral alignment in embedding-based recommender systems, informing debiasing and evaluation strategies grounded in spectral graph theory.

Abstract

We are extending Popularity Bias Memorization theorem from arXiv:archive/2404.12008 in several directions. We extend it to arbitrary degree distributions and also prove both upper and lower estimates for the alignment with top-k singular hyperspace.

Popularity Bias Alignment Estimates

TL;DR

This work addresses how popularity-driven exposure in recommender systems aligns with spectral directions of the user-item interaction matrix without assuming a power-law tail. It (i) generalizes the Popularity Bias Memorization framework to distribution-agnostic item popularities, establishing Π₁-memorization and Πₖ-memorization results, and (ii) develops a suite of bounds—combinatorial, Ky Fan, interlacing, and LP-based—for the alignment with the top- singular subspace, tying these bounds to tail mass, effective ranks, and graph topology via Kumar's bounds. The results quantify when recommendation scores align with the principal or top- singular directions and reveal regimes (e.g., log-normal tails, truncation) where alignment weakens, providing interpretable quantities that can be data-driven estimates. Overall, the paper offers distribution-agnostic guarantees on spectral alignment in embedding-based recommender systems, informing debiasing and evaluation strategies grounded in spectral graph theory.

Abstract

We are extending Popularity Bias Memorization theorem from arXiv:archive/2404.12008 in several directions. We extend it to arbitrary degree distributions and also prove both upper and lower estimates for the alignment with top-k singular hyperspace.

Paper Structure

This paper contains 9 sections, 5 theorems, 91 equations.

Table of Contents

  1. Preliminaries
  2. Popularity bias memorization.
  3. $\Pi_{1}$-memorization.
  4. $\Pi_{k}$-memorization.
  5. Linear programming estimate.
  6. Appendix.
  7. A. Kumar's bounds.
  8. Theorem 1C
  9. LP theorem.

Key Result

Theorem 1

$\boldsymbol{1A.}$ In the above notations,

Theorems & Definitions (13)

  • Remark
  • Theorem
  • proof
  • Remark 1
  • Lemma
  • Lemma
  • Theorem
  • Remark 2
  • Remark 3
  • Theorem
  • ...and 3 more