Quantum divide-and-conquer anchoring for separable non-negative matrix factorization

© 2018 International Joint Conferences on Artificial Intelligence. All right reserved. It is NP-complete to find non-negative factors W and H with fixed rank r from a non-negative matrix X by minimizing ||X − WHτ||2F. Although the separability assumption (all data points are in the conical hull of the extreme rows) enables polynomial-time algorithms, the computational cost is not affordable for big data. This paper investigates how the power of quantum computation can be capitalized to solve the non-negative matrix factorization with the separability assumption (SNMF) by devising a quantum algorithm based on the divide-and-conquer anchoring (DCA) scheme [Zhou et al., 2013]. The design of quantum DCA (QDCA) is challenging. In the divide step, the random projections in DCA is completed by a quantum algorithm for linear operations, which achieves the exponential speedup. We then devise a heuristic post-selection procedure which extracts the information of anchors stored in the quantum states efficiently. Under a plausible assumption, QDCA performs efficiently, achieves the quantum speedup, and is beneficial for high dimensional problems

A quantum-inspired classical algorithm for separable Non-negative Matrix Factorization

Non-negative Matrix Factorization (NMF) asks to decompose a (entry-wise) non-negative matrix into the product of two smaller-sized nonnegative matrices, which has been shown intractable in general. In order to overcome this issue, separability assumption is introduced which assumes all data points are in a conical hull. This assumption makes NMF tractable and is widely used in text analysis and image processing, but still impractical for huge-scale datasets. In this paper, inspired by recent development on dequantizing techniques, we propose a new classical algorithm for separable NMF problem. Our new algorithm runs in polynomial time in the rank and logarithmic in the size of input matrices, which achieves an exponential speedup in the low-rank setting

Quantum differentially private sparse regression learning

Differentially private (DP) learning, which aims to accurately extract patterns from the given dataset without exposing individual information, is an important subfield in machine learning and has been extensively explored. However, quantum algorithms that could preserve privacy, while outperform their classical counterparts, are still lacking. The difficulty arises from the distinct priorities in DP and quantum machine learning, i.e., the former concerns a low utility bound while the latter pursues a low runtime cost. These varied goals request that the proposed quantum DP algorithm should achieve the runtime speedup over the best known classical results while preserving the optimal utility bound. The Lasso estimator is broadly employed to tackle the high dimensional sparse linear regression tasks. The main contribution of this paper is devising a quantum DP Lasso estimator to earn the runtime speedup with the privacy preservation, i.e., the runtime complexity is $\tilde{O}(N^{3/2}\sqrt{d})$ with a nearly optimal utility bound $\tilde{O}(1/N^{2/3})$, where $N$ is the sample size and $d$ is the data dimension with $N\ll d$. Since the optimal classical (private) Lasso takes $\Omega(N+d)$ runtime, our proposal achieves quantum speedups when $N<O(d^{1/3})$. There are two key components in our algorithm. First, we extend the Frank-Wolfe algorithm from the classical Lasso to the quantum scenario, {where the proposed quantum non-private Lasso achieves a quadratic runtime speedup over the optimal classical Lasso.} Second, we develop an adaptive privacy mechanism to ensure the privacy guarantee of the non-private Lasso. Our proposal opens an avenue to design various learning tasks with both the proven runtime speedups and the privacy preservation