Research on Matrix Completion part2(Machine Learning core)
- Energy-modified Leverage Sampling for Radio Map Construction via Matrix Completion(arXiv)
Authors: Hao Sun, Junting Chen
Abstract: This paper explores an energy-modified leverage sampling strategy for matrix completion in radio map construction. The main goal is to address potential identifiability issues in matrix completion with sparse observations by using a probabilistic sampling approach. Although conventional leverage sampling is commonly employed for designing sampling patterns, it often assigns high sampling probability to locations with low received signal strength (RSS) values, leading to a low sampling efficiency. Theoretical analysis demonstrates that the leverage score produces pseudo images of sources, and in the regions around the source locations, the leverage probability is asymptotically consistent with the RSS. Based on this finding, an energy-modified leverage probability-based sampling strategy is investigated for efficient sampling. Numerical demonstrations indicate that the proposed sampling strategy can decrease the normalized mean squared error (NMSE) of radio map construction by more than 10% for both matrix completion and interpolation-assisted matrix completion schemes, compared to conventional methods.
2. Matrix Domination: Convergence of a Genetic Algorithm Metaheuristic with the Wisdom of Crowds to Solve the NP-Complete Problem(arXiv)
Authors: Shane Storm Strachan
Abstract: This research explores the application of a genetic algorithm metaheuristic enriched by the wisdom of crowds in order to address the NP-Complete matrix domination problem (henceforth: TMDP) which is itself a constraint on related problems applied in graphs. Matrix domination involves accurately placing a subset of cells, referred to as dominators, within a matrix with the goal of their dominating the remainder of the cells. This research integrates the exploratory nature of a genetic algorithm with the wisdom of crowds to find more optimal solutions with user-defined parameters to work within computational complexity considerations and gauge performance mainly with a fitness evaluation function and a constraining function to combat the stochastic nature of genetic algorithms. With this, I propose a novel approach to MDP with a genetic algorithm that incorporates the wisdom of crowds, emphasizing collective decision-making in the selection process, and by exploring concepts of matrix permutations and their relevance in finding optimal solutions. Results demonstrate the potential of this convergence to generate efficient solutions, optimizing the trade-off between the number of dominators and their strategic placements within the matrices while efficiently ensuring consistent and complete matrix domination. △ Less
3. Statistical Inference For Noisy Matrix Completion Incorporating Auxiliary Information(arXiv)
Authors: Shujie Ma, Po-Yao Niu, Yichong Zhang, Yinchu Zhu
Abstract: This paper investigates statistical inference for noisy matrix completion in a semi-supervised model when auxiliary covariates are available. The model consists of two parts. One part is a low-rank matrix induced by unobserved latent factors; the other part models the effects of the observed covariates through a coefficient matrix which is composed of high-dimensional column vectors. We model the observational pattern of the responses through a logistic regression of the covariates, and allow its probability to go to zero as the sample size increases. We apply an iterative least squares (LS) estimation approach in our considered context. The iterative LS methods in general enjoy a low computational cost, but deriving the statistical properties of the resulting estimators is a challenging task. We show that our method only needs a few iterations, and the resulting entry-wise estimators of the low-rank matrix and the coefficient matrix are guaranteed to have asymptotic normal distributions. As a result, individual inference can be conducted for each entry of the unknown matrices. We also propose a simultaneous testing procedure with multiplier bootstrap for the high-dimensional coefficient matrix. This simultaneous inferential tool can help us further investigate the effects of covariates for the prediction of missing entries
