[Paper Review] Quantum Recommendation Systems

Iordanis Kerenidis, Anupam Prakash|arXiv (Cornell University)|Mar 29, 2016

Quantum Computing Algorithms and ArchitectureComputer Science14 references83 citations

TL;DR

This paper presents a quantum recommendation system that achieves polylogarithmic runtime in matrix dimensions by sampling from a low-rank approximation of the preference matrix, rather than reconstructing it. The key contribution is a quantum algorithm with $ O(\text{poly}(k)\text{polylog}(mn)) $ expected running time that provides high-quality recommendations with high probability, demonstrating a quantum advantage for real-world machine learning applications.

ABSTRACT

A recommendation system uses the past purchases or ratings of $n$ products by a group of $m$ users, in order to provide personalized recommendations to individual users. The information is modeled as an $m imes n$ preference matrix which is assumed to have a good rank-$k$ approximation, for a small constant $k$. In this work, we present a quantum algorithm for recommendation systems that has running time $O( ext{poly}(k) ext{polylog}(mn))$. All known classical algorithms for recommendation systems that work through reconstructing an approximation of the preference matrix run in time polynomial in the matrix dimension. Our algorithm provides good recommendations by sampling efficiently from an approximation of the preference matrix, without reconstructing the entire matrix. For this, we design an efficient quantum procedure to project a given vector onto the row space of a given matrix. This is the first algorithm for recommendation systems that runs in time polylogarithmic in the dimensions of the matrix and provides an example of a quantum machine learning algorithm for a real world application.

Motivation & Objective

• To develop a quantum algorithm for recommendation systems that operates in time polylogarithmic in the matrix dimensions, overcoming the polynomial-time bottleneck of classical methods.
• To enable efficient recommendation by sampling from a low-rank approximation of the preference matrix without full reconstruction.
• To provide a practical quantum machine learning algorithm for a real-world application with provable performance guarantees.
• To formalize and justify the low-rank assumption underlying user preference matrices in terms of user types and shared utility properties.
• To design a quantum procedure that projects a vector onto the row space of a matrix efficiently, forming the core of the recommendation engine.

Proposed method

• The algorithm uses quantum amplitude amplification and amplitude estimation to sample from the row space of the preference matrix, avoiding full matrix reconstruction.
• It constructs a quantum state representing the top $ k $ singular vectors of the preference matrix using quantum singular value transformation (SVT).
• A quantum projection procedure projects a user’s preference vector onto the row space of the matrix, enabling efficient sampling of high-utility products.
• The method relies on a truncated SVD approximation $ \widehat{T}_{\geq\sigma,\kappa} $, where $ \sigma $ is a threshold on singular values and $ \kappa $ controls the approximation error.
• The algorithm ensures high-probability correctness by setting parameters such that the Frobenius norm error $ \|T - \widehat{T}_{\geq\sigma,\kappa}\|_F \leq 9\epsilon\|T\|_F $.
• Running time is bounded by $ O(\text{poly}(k)\text{polylog}(mn)) $ for at least $ (1-\xi)(1-\delta-\zeta)m $ users, under appropriate parameter settings.

Experimental results

Research questions

• RQ1 Can a quantum algorithm achieve sub-polynomial runtime in the matrix dimensions for recommendation systems, given the low-rank structure of user preference matrices?
• RQ2 How can quantum sampling from a low-rank approximation outperform classical matrix reconstruction in terms of time complexity?
• RQ3 What is the role of quantum projection onto the row space in enabling efficient recommendation sampling?
• RQ4 Under what conditions does the quantum recommendation algorithm maintain high accuracy with high probability?
• RQ5 Can the algorithm be made robust to noise and partial data while preserving polylogarithmic runtime?

Key findings

• The quantum recommendation algorithm achieves an expected running time of $ O(\text{poly}(k)\text{polylog}(mn)) $, representing a significant speedup over classical methods.
• For at least $ (1-\xi)(1-\delta-\zeta)m $ users, the algorithm produces good recommendations with high probability, as quantified by a bounded error in the recommendation quality.
• The Frobenius norm error between the true preference matrix $ T $ and its approximation $ \widehat{T}_{\geq\sigma,\kappa} $ is bounded by $ 9\epsilon\|T\|_F $ with high probability.
• The quantum projection procedure successfully outputs a state proportional to the projected row of the matrix with probability at least $ 1 - 1/\text{poly}(n) $.
• The expected running time is constant in the matrix size for most users, provided the sampling fraction $ p $ is constant and the spectral properties of the matrix are well-behaved.
• The algorithm’s correctness is maintained even under sub-constant sampling rates, though a constant $ p $ is required for optimal runtime.