rank matrix
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2022 ◽  
Vol 13 (1) ◽  
Author(s):  
George C. Linderman ◽  
Jun Zhao ◽  
Manolis Roulis ◽  
Piotr Bielecki ◽  
Richard A. Flavell ◽  
...  

AbstractA key challenge in analyzing single cell RNA-sequencing data is the large number of false zeros, where genes actually expressed in a given cell are incorrectly measured as unexpressed. We present a method based on low-rank matrix approximation which imputes these values while preserving biologically non-expressed genes (true biological zeros) at zero expression levels. We provide theoretical justification for this denoising approach and demonstrate its advantages relative to other methods on simulated and biological datasets.


2022 ◽  
Vol 2146 (1) ◽  
pp. 012038
Author(s):  
Pengyi Tian ◽  
Dinggen Xu ◽  
Xiuyuan Zhang

Abstract Most of the current image fusion algorithms directly process the original image, neglect the analysis of the main components of the image, and have a great influence on the effect of image fusion. In this paper, the main component analysis method is used to decompose the image, divided into low rank matrix and sparse matrix, introduced compression perception technology and NSST transformation algorithm to process the two types of matrix, according to the corresponding fusion rules to achieve image fusion, through experimental results: this algorithm has greater mutual information compared with traditional algorithms, structural information similarity and average gradient.


Author(s):  
Kang Gu ◽  
Sheng Chen ◽  
Xiaoyu You ◽  
Yifei Li ◽  
Jianwei Cui ◽  
...  

Abstract The coordinate measuring machine (CMM) becomes an extensive and effective method for high precision inspection of free-form surfaces due to its ability to measure complex and irregular surfaces. Sampling strategy and surface restoration method have an important influence on the efficiency and precision of CMM. In this paper, a sparse sampling strategy and surface reconstruction method for free-form surfaces based on low-rank matrix completion (LRMC) is proposed. In this method, the free-form surface is sampled randomly with uniform distribution in the cartesian coordinate system to obtain sparse sampling points, and then optimizes the scanning path to obtain the shortest path through all measurement points, and finally, the LRMC algorithm based on alternating root mean square prop was used to reconstruct the surface with high precision. The simulation and experimental results show that under the premise of ensuring accuracy, the number of sampling points is greatly reduced and the measurement efficiency is greatly improved.


2021 ◽  
Author(s):  
Dimitris Bertsimas ◽  
Ryan Cory-Wright ◽  
Jean Pauphilet

Many central problems throughout optimization, machine learning, and statistics are equivalent to optimizing a low-rank matrix over a convex set. However, although rank constraints offer unparalleled modeling flexibility, no generic code currently solves these problems to certifiable optimality at even moderate sizes. Instead, low-rank optimization problems are solved via convex relaxations or heuristics that do not enjoy optimality guarantees. In “Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints,” Bertsimas, Cory-Wright, and Pauphilet propose a new approach for modeling and optimizing over rank constraints. They generalize mixed-integer optimization by replacing binary variables z that satisfy z2 =z with orthogonal projection matrices Y that satisfy Y2 = Y. This approach offers the following contributions: First, it supplies certificates of (near) optimality for low-rank problems. Second, it demonstrates that some of the best ideas in mixed-integer optimization, such as decomposition methods, cutting planes, relaxations, and random rounding schemes, admit straightforward extensions to mixed-projection optimization.


2021 ◽  
Author(s):  
Hang Xu ◽  
Song Li ◽  
Junhong Lin

Abstract Many problems in data science can be treated as recovering a low-rank matrix from a small number of random linear measurements, possibly corrupted with adversarial noise and dense noise. Recently, a bunch of theories on variants of models have been developed for different noises, but with fewer theories on the adversarial noise. In this paper, we study low-rank matrix recovery problem from linear measurements perturbed by $\ell_1$-bounded noise and sparse noise that can arbitrarily change an adversarially chosen $\omega$-fraction of the measurement vector. For Gaussian measurements with nearly optimal number of measurements, we show that the nuclear-norm constrained least absolute deviation (LAD) can successfully estimate the ground-truth matrix for any $\omega<0.239$. Similar robust recovery results are also established for an iterative hard thresholding algorithm applied to the rank-constrained LAD considering geometrically decaying step-sizes, and the unconstrained LAD based on matrix factorization as well as its subgradient descent solver.


2021 ◽  
Author(s):  
Yangyang Ge ◽  
Zhimin Wang ◽  
Wen Zheng ◽  
Yu Zhang ◽  
Xiangmin Yu ◽  
...  

Abstract Quantum singular value thresholding (QSVT) algorithm, as a core module of many mathematical models, seeks the singular values of a sparse and low rank matrix exceeding a threshold and their associated singular vectors. The existing all-qubit QSVT algorithm demands lots of ancillary qubits, remaining a huge challenge for realization on near-term intermediate-scale quantum computers. In this paper, we propose a hybrid QSVT (HQSVT) algorithm utilizing both discrete variables (DVs) and continuous variables (CVs). In our algorithm, raw data vectors are encoded into a qubit system and the following data processing is fulfilled by hybrid quantum operations. Our algorithm requires O[log(MN)] qubits with O(1) qumodes and totally performs O(1) operations, which significantly reduces the space and runtime consumption.


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