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arXiv:1910.01262

Quantum tensor singular value decomposition with applications to recommendation systems

Xiaoqiang Wang, Lejia Gu, Joseph H. W. Lee, Guofeng Zhang

Published Oct 3, 2019• 2 citations
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Abstract

In this paper, we present a quantum singular value decomposition algorithm for third-order tensors inspired by the classical algorithm of tensor singular value decomposition (t-svd) and then extend it to order-$p$ tensors. It can be proved that the quantum version of the t-svd for a third-order tensor $\mathcal{A} \in \mathbb{R}^{N\times N \times N}$ achieves the complexity of $\mathcal{O}(N{\rm polylog}(N))$, an exponential speedup compared with its classical counterpart. As an application, we propose a quantum algorithm for recommendation systems which incorporates the contextual situation of users to the personalized recommendation. We provide recommendations varying with contexts by measuring the output quantum state corresponding to an approximation of this user's preferences. This algorithm runs in expected time $\mathcal{O}(N{\rm polylog}(N){\rm poly}(k)),$ if every frontal slice of the preference tensor has a good rank-$k$ approximation. At last, we provide a quantum algorithm for tensor completion based on a different truncation method which is tested to have a good performance in dynamic video completion.

DOI
10.48550/arxiv.1910.01262

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Abstract Tensors are increasingly ubiquitous in various areas of applied mathematics and computing, and tensor decompositions are of practical significance and benefit many applications in data completion, image processing, computer vision, collaborative filtering, etc. Recently, Kilmer and Martin propose a new tensor factorization strategy, tensor singular value decomposition (t-svd), which extends the matrix singular value decomposition to tensors. However, computing t-svd for high dimensional tensors costs much computation and thus cannot efficiently handle large scale datasets. Motivated by advantage of quantum computation, in this paper, we present a quantum algorithm of t-svd for third-order tensors and then extend it to order-p tensors. We prove that our quantum t-svd algorithm for a third-order N dimensional tensor runs in time  N polylog ( N ) if we do not recover classical information from the quantum output state. Moreover, we apply our quantum t-svd algorithm to context-aware multidimensional recommendation systems, where we just need to extract partial classical information from the quantum output state, thus achieving low time complexity.
A recommendation system uses the past purchases or ratings of nnn products by a group of mmm users, in order to provide personalized recommendations to individual users. The information is modeled as an m×nm \times nm×n preference matrix which is assumed to have a good rank-kkk approximation, for a small constant kkk. In this work, we present a quantum algorithm for recommendation systems that has running time O(poly(k)polylog(mn))O(\text{poly}(k)\text{polylog}(mn))O(poly(k)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.
A recent breakthrough by Tang (STOC 2019) showed how to "dequantize" the quantum algorithm for recommendation systems by Kerenidis and Prakash (ITCS 2017). The resulting algorithm, classical but "quantum-inspired", efficiently computes a low-rank approximation of the users' preference matrix. Subsequent works have shown how to construct efficient quantum-inspired algorithms for approximating the pseudo-inverse of a low-rank matrix as well, which can be used to (approximately) solve low-rank linear systems of equations. In the present paper, we pursue this line of research and develop quantum-inspired algorithms for a large class of matrix transformations that are defined via the singular value decomposition of the matrix. In particular, we obtain classical algorithms with complexity polynomially related (in most parameters) to the complexity of the best quantum algorithms for singular value transformation recently developed by Chakraborty, Gilyén and Jeffery (ICALP 2019) and Gilyén, Su, Low and Wiebe (STOC 2019).
Abstract Faster than classical algorithms, quantum algorithms benefit from the superposition property of quantum information to offer significant speedup to complex algorithms. Therefore, quantum computing can be used to help machine learning algorithms by boosting their performance and accelerate the processing of time‐consuming ones. Clustering algorithms are very complex unsupervised learning algorithms. Indeed, the similarity calculation (distance) between input vectors is a resource‐consuming step, especially when working with large datasets. In this article, we propose a new better‐performing recommender system that operates as a combination of an adapted quantum K‐means algorithm and the singular value decomposition (SVT) algorithm. We integrate the developed quantum clustering algorithm to a prediction process of the proposed recommender system using matrix completion. To the best of our knowledge, no system with such details was proposed in the literature. The system was applied on the MovieLens dataset without a dimensionality reduction step and evaluated according to measures of information retrieval systems. The results of the quantum K‐means algorithm show that the quantum version leads to a logarithmic reduction of the time complexity compared to the classical algorithm. The proposed system has proved to be better than the previous tested ones in terms of precision and recall.
Abstract Researchers have put a lot of effort into reducing the gap between current quantum processing units (QPU) capabilities and their potential supremacy.One approach is to keep supplementary computations in the CPU, and use QPU only for the core of the problem. In this work, we address the complexity of quantum algorithm of arbitrary quantum state initialization, an important building block of quantum data analysis and machine learning.QPUs do not outperform classical machines with existing precise initialization algorithms.Hence, many studies propose an approximate but robust quantum state initialization.Cutting a quantum state into a product of (almost) independent partitions with the help of CPU reduces the number of two-qubit gates, and correspondingly minimizes the loss of state fidelity in the quantum part of the algorithm.To find the least entangled qubits, current methods compute the singular value decomposition (SVD) for each qubit separately with CPU.In this paper, we optimize CPU usage and memory resource bottlenecks.We consider Tucker tensor decomposition as an alternative to the CPU-based SVD in a single low-entangled qubit detection task without the loss of solution quality.Both proposed methods outperform the SVD in time and memory for systems of at least ten qubits. We achieve an order faster implementation and two orders less memory usage for a system of 15 qubits.