Sparse sampling and reconstruction for freeform surface based on low-rank matrix completion

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.

A Riemannian rank-adaptive method for low-rank matrix completion

The low-rank matrix completion problem can be solved by Riemannian optimization on a fixed-rank manifold. However, a drawback of the known approaches is that the rank parameter has to be fixed a priori. In this paper, we consider the optimization problem on the set of bounded-rank matrices. We propose a Riemannian rank-adaptive method, which consists of fixed-rank optimization, rank increase step and rank reduction step. We explore its performance applied to the low-rank matrix completion problem. Numerical experiments on synthetic and real-world datasets illustrate that the proposed rank-adaptive method compares favorably with state-of-the-art algorithms. In addition, it shows that one can incorporate each aspect of this rank-adaptive framework separately into existing algorithms for the purpose of improving performance.

Fast Construction of the Radio Map Based on the Improved Low-Rank Matrix Completion and Recovery Method for an Indoor Positioning System

With the development of information technology, indoor positioning technology has been rapidly evolving. Due to the advantages of high positioning accuracy, low cost, and wide coverage simultaneously, received signal strength- (RSS-) based WLAN indoor positioning technology has become one of the mainstream technologies. A radio map is the basis for the realization of the WLAN positioning system. However, by reasons of the huge workload of RSS collection, the instability of wireless signal strength, and the disappearance of signals caused by the occlusion of people and objects, the construction of a radio map is time-consuming and inefficient. In order to rapidly deploy the WLAN indoor positioning system, an improved low-rank matrix completion method is proposed to construct the radio map. Firstly, we evenly arrange a small number of reference points (RP) in the positioning area and collect RSS data on the RP to construct the radio map. Then, the low-rank matrix completion method is used to fill a small amount of data in the radio map into a complete database. The Frobenius parameter (F-parameter) is introduced into the traditional low-rank matrix completion model to control the instability of the model solution when filling the data. To solve the noise problem caused by environment and equipment, a low-rank matrix recovery algorithm is used to eliminate noise. The experimental results show that the improved algorithm achieves the expected goal.

Predicting the Effects of Drug Combinations Using Probabilistic Matrix Factorization

Drug development is costly and time-consuming, and developing novel practical strategies for creating more effective treatments is imperative. One possible solution is to prescribe drugs in combination. Synergistic drug combinations could allow lower doses of each constituent drug, reducing adverse reactions and drug resistance. However, it is not feasible to sufficiently test every combination of drugs for a given illness to determine promising synergistic combinations. Since there is a finite amount of time and resources available for finding synergistic combinations, a model that can identify synergistic combinations from a limited subset of all available combinations could accelerate development of therapeutics. By applying recommender algorithms, such as the low-rank matrix completion algorithm Probabilistic Matrix Factorization (PMF), it may be possible to identify synergistic combinations from partial information of the drug interactions. Here, we use PMF to predict the efficacy of two-drug combinations using the NCI ALMANAC, a robust collection of pairwise drug combinations of 104 FDA-approved anticancer drugs against 60 common cancer cell lines. We find that PMF is able predict drug combination efficacy with high accuracy from a limited set of combinations and is robust to changes in the individual training data. Moreover, we propose a new PMF-guided experimental design to detect all synergistic combinations without testing every combination.

Simultaneous completion and spatiotemporal grouping of corrupted motion tracks

Given an unordered list of 2D or 3D point trajectories corrupted by noise and partial observations, in this paper we introduce a framework to simultaneously recover the incomplete motion tracks and group the points into spatially and temporally coherent clusters. This advances existing work, which only addresses partial problems and without considering a unified and unsupervised solution. We cast this problem as a matrix completion one, in which point tracks are arranged into a matrix with the missing entries set as zeros. In order to perform the double clustering, the measurement matrix is assumed to be drawn from a dual union of spatiotemporal subspaces. The bases and the dimensionality for these subspaces, the affinity matrices used to encode the temporal and spatial clusters to which each point belongs, and the non-visible tracks, are then jointly estimated via augmented Lagrange multipliers in polynomial time. A thorough evaluation on incomplete motion tracks for multiple-object typologies shows that the accuracy of the matrix we recover compares favorably to that obtained with existing low-rank matrix completion methods, specially under noisy measurements. In addition, besides recovering the incomplete tracks, the point trajectories are directly grouped into different object instances, and a number of semantically meaningful temporal primitive actions are automatically discovered.

Simulation comparisons between Bayesian and de-biased estimators in low-rank matrix completion

In this paper we perform numerous numerical studies for the problem of low-rank matrix completion. We compare the Bayesian approaches and a recently introduced de-biased estimator which provides a useful way to build confidence intervals of interest. From a theoretical viewpoint, the de-biased estimator comes with a sharp minimax-optimal rate of estimation error whereas the Bayesian approach reaches this rate with an additional logarithmic factor. Our simulation studies show originally interesting results that the de-biased estimator is just as strong as the Bayesian estimators. Moreover, Bayesian approaches are much more stable and can outperform the de-biased estimator in the case of small samples. In additions, we also find that the coverage rate of the confidence intervals revealed by the de-biased estimator for an entry is absolutely lower than of the considered credible interval. These suggest further theoretical studies on the estimation error and the concentration for Bayesian methods as they are being quite limited up to present.

Predicting the effects of drug combinations using probabilistic matrix factorization

Drug development is costly and time-consuming, and developing novel practical strategies for creating more effective treatments is imperative. One possible solution is to prescribe drugs in combination. Synergistic drug combinations could allow lower doses of each constituent drug, reducing adverse reactions and drug resistance. However, it is not feasible to sufficiently test every combination of drugs for a given illness to determine promising synergistic combinations. Since there is a finite amount of time and resources available for finding synergistic combinations, a model that can identify synergistic combinations from a limited subset of all available combinations could accelerate development of therapeutics. By applying recommender algorithms, such as the low-rank matrix completion algorithm Probabilistic Matrix Factorization (PMF), it may be possible to identify synergistic combinations from partial information of the drug interactions. Here, we use PMF to predict the efficacy of two-drug combinations using the NCI ALMANAC, a robust collection of pairwise drug combinations of 104 FDA-approved anticancer drugs against 60 common cancer cell lines. We find that PMF is able predict drug combination efficacy with high accuracy from a limited set of combinations and is robust to changes in the individual training data. Moreover, we propose a new PMF-guided experimental design to detect all synergistic combinations without testing every combination.