scholarly journals Quantum Divide-and-Conquer Anchoring for Separable Non-negative Matrix Factorization

2018 ◽  
Author(s):  
Yuxuan Du ◽  
Tongliang Liu ◽  
Yinan Li ◽  
Runyao Duan ◽  
Dacheng Tao
Keyword(s):  
Matrix Factorization ◽  
Computational Cost ◽  
Selection Procedure ◽  
Quantum Algorithm ◽  
Divide And Conquer ◽  
Plausible Assumption ◽  
Separability Assumption ◽  
Data Points ◽  
Fixed Rank ◽  
Non Negative Matrix Factorization

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-WH^Τ ||^2. 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.

scholarly journals A Quantum-inspired Classical Algorithm for Separable Non-negative Matrix Factorization

2019 ◽  
Author(s):  
Zhihuai Chen ◽  
Yinan Li ◽  
Xiaoming Sun ◽  
Pei Yuan ◽  
Jialin Zhang
Keyword(s):  
Image Processing ◽  
Polynomial Time ◽  
Matrix Factorization ◽  
Text Analysis ◽  
Low Rank ◽  
Nonnegative Matrices ◽  
Classical Algorithm ◽  
Data Points ◽  
Conical Hull ◽  
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 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.

scholarly journals Fast SVM Trained by Divide-and-Conquer Anchors

2017 ◽  
Author(s):  
Meng Liu ◽  
Chang Xu ◽  
Chao Xu ◽  
Dacheng Tao
Keyword(s):  
Matrix Factorization ◽  
State Of The Art ◽  
Training Data ◽  
Divide And Conquer ◽  
Training Time ◽  
Learning Tasks ◽  
Multiple Datasets ◽  
Original Dataset ◽  
Representative Points ◽  
Non Negative Matrix Factorization

Supporting vector machine (SVM) is the most frequently used classifier for machine learning tasks. However, its training time could become cumbersome when the size of training data is very large. Thus, many kinds of representative subsets are chosen from the original dataset to reduce the training complexity. In this paper, we propose to choose the representative points which are noted as anchors obtained from non-negative matrix factorization (NMF) in a divide-and-conquer framework, and then use the anchors to train an approximate SVM. Our theoretical analysis shows that the solving the DCA-SVM can yield an approximate solution close to the primal SVM. Experimental results on multiple datasets demonstrate that our DCA-SVM is faster than the state-of-the-art algorithms without notably decreasing the accuracy of classification results.

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