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A gradient descent algorithm on the grassman manifold for matrix completion
, 2009
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Sparse Bayesian methods for lowrank matrix estimation. arXiv:1102.5288v1 [stat.ML
, 2011
"... Abstract—Recovery of lowrank matrices has recently seen significant ..."
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Abstract—Recovery of lowrank matrices has recently seen significant
Matrix estimation by universal singular value thresholding
, 2012
"... Abstract. Consider the problem of estimating the entries of a large matrix, when the observed entries are noisy versions of a small random fraction of the original entries. This problem has received widespread attention in recent times, especially after the pioneering works of Emmanuel Candès and ..."
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Abstract. Consider the problem of estimating the entries of a large matrix, when the observed entries are noisy versions of a small random fraction of the original entries. This problem has received widespread attention in recent times, especially after the pioneering works of Emmanuel Candès and collaborators. This paper introduces a simple estimation procedure, called Universal Singular Value Thresholding (USVT), that works for any matrix that has ‘a little bit of structure’. Surprisingly, this simple estimator achieves the minimax error rate up to a constant factor. The method is applied to solve problems related to low rank matrix estimation, blockmodels, distance matrix completion, latent space models, positive definite matrix completion, graphon estimation, and generalized Bradley–Terry models for pairwise comparison. 1.
Localization from Incomplete Noisy Distance Measurements
"... Abstract—We consider the problem of positioning a cloud of points in the Euclidean space R d, from noisy measurements of a subset of pairwise distances. This task has applications in various areas, such as sensor network localizations, NMR spectroscopy of proteins, and molecular conformation. Also, ..."
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Abstract—We consider the problem of positioning a cloud of points in the Euclidean space R d, from noisy measurements of a subset of pairwise distances. This task has applications in various areas, such as sensor network localizations, NMR spectroscopy of proteins, and molecular conformation. Also, it is closely related to dimensionality reduction problems and manifold learning, where the goal is to learn the underlying global geometry of a data set using measured local (or partial) metric information. Here we propose a reconstruction algorithm based on a semidefinite programming approach. For a random geometric graph model and uniformly bounded noise, we provide a precise characterization of the algorithm’s performance: In the noiseless case, we find a radius r0 beyond which the algorithm reconstructs the exact positions (up to rigid transformations). In the presence of noise, we obtain upper and lower bounds on the reconstruction error that match up to a factor that depends only on the dimension d, and the average degree of the nodes in the graph. I.
Lowrank matrix completion with noisy observations: A quantitative comparison
 in The 47th Annual Allerton Conference on Communication, Control, and Computing
, 2009
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Refining hopcount for localisation in wireless sensor networks
 International Journal of Sensor Networks
"... Abstract: Distance estimation is a crucial component in localisation for wireless sensor networks. Among the estimation methods, hopcount is widely used in situations where only connectivity information is available. However, hopcount is integervalued, implying crude distance estimation. In this ..."
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Abstract: Distance estimation is a crucial component in localisation for wireless sensor networks. Among the estimation methods, hopcount is widely used in situations where only connectivity information is available. However, hopcount is integervalued, implying crude distance estimation. In this paper, we refine hopcount to achieve better distance estimation. This is done by estimating neighbour distance and then approximating nonneighbour distance by the length of the shortest path. To estimate neighbour distance, we propose three estimators and show that they have negligible bias. We also show that the variance of the estimators is related to node density. The final refined hopcounts are further studied by simulations. Results verify the improvement on distance estimation and show that existing localisation methods can benefit from the improvement in various scenarios.
EVLoc: Integrating electronic and visual signals for accurate localization
 In Proc. of ACM MobiHoc
, 2012
"... Nowadays, an increasing number of objects can be represented by their wireless electronic identifiers. For example, people can be recognized by their phone numbers or their phones ’ WiFi MAC addresses and products can be identified by their RFID numbers. Localizing objects with electronic identifier ..."
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Nowadays, an increasing number of objects can be represented by their wireless electronic identifiers. For example, people can be recognized by their phone numbers or their phones ’ WiFi MAC addresses and products can be identified by their RFID numbers. Localizing objects with electronic identifiers is increasingly important as our lives become increasingly “digitalized”. However, traditional wireless localization techniques cannot meet the fast growing needs of accurate and cost efficient localization. Some of these techniques require expensive hardware to achieve high accuracy, which is impractical for massive deployment. Others, such as WiFi RSSI based localization, are inaccurate and not robust to environmental noise. In this paper, we propose a new localization technique called EVLoc. In EVLoc, we use visual signals to help improve the accuracy of wireless localization. Our technique fully leverages visual signals’ high accuracy and electronic signals ’ pervasiveness. To effectively couple these two signals, we design an EV match engine to find the correspondence between an object’s electronic identifier and its visual appearance. We implement our technique on mobile devices and evaluate it in realworld scenarios. The localization error is less than 1 m. We also evaluate our approach using large scale simulations. The results show that our approach is accurate and robust.
Gossip PCA
"... Eigenvectors of data matrices play an important role in many computational problems, ranging from signal processing to machine learning and control. For instance, algorithms that compute positions of the nodes of a wireless network on the basis of pairwise distance measurements require a few leading ..."
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Eigenvectors of data matrices play an important role in many computational problems, ranging from signal processing to machine learning and control. For instance, algorithms that compute positions of the nodes of a wireless network on the basis of pairwise distance measurements require a few leading eigenvectors of the distances matrix. While eigenvector calculation is a standard topic in numerical linear algebra, it becomes challenging under severe communication or computation constraints, or in absence of central scheduling. In this paper we investigate the possibility of computing the leading eigenvectors of a large data matrix through gossip algorithms. The proposed algorithm amounts to iteratively multiplying a vector by independent random sparsification of the original matrix and averaging the resulting normalized vectors. This can be viewed as a generalization of gossip algorithms for consensus, but the resulting dynamics is significantly more intricate. Our analysis is based on controlling the convergence to stationarity of the associated KestenFurstenberg Markov chain.
Calibration Using Matrix Completion with Application to Ultrasound Tomography
, 2011
"... We study the calibration process in circular ultrasound tomography devices where the sensor positions deviate from the circumference of a perfect circle. This problem arises in a variety of applications in signal processing ranging from breast imaging to sensor network localization. We introduce a n ..."
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We study the calibration process in circular ultrasound tomography devices where the sensor positions deviate from the circumference of a perfect circle. This problem arises in a variety of applications in signal processing ranging from breast imaging to sensor network localization. We introduce a novel method of calibration/localization based on the timeofflight (ToF) measurements between sensors when the enclosed medium is homogeneous. In the presence of all the pairwise ToFs, one can easily estimate the sensor positions using multidimensional scaling (MDS) method. In practice however, due to the transitional behaviour of the sensors and the beam form of the transducers, the ToF measurements for closeby sensors are unavailable. Further, random malfunctioning of the sensors leads to random missing ToF measurements. On top of the missing entries, in practice an unknown time delay is also added to the measurements. In this work, we incorporate the fact that a matrix defined from all the ToF measurements is of rank at most four. In order to estimate the missing ToFs, we apply a stateoftheart lowrank matrix completion algorithm, OPTSPACE. To find the correct positions of the sensors (our ultimate goal) we then apply MDS. We show analytic bounds on the overall error of the whole process in the presence of noise and hence deduce its robustness. Finally, we confirm the functionality of our method in practice by simulations mimicking the measurements of a circular ultrasound tomography device.
Ultrasound Tomography Calibration Using Structured Matrix Completion
"... Calibration of ultrasound tomography devices is a challenging problem and of highly practical interest in medical and seismic imaging. This work addresses the position calibration problem in circular apertures where sensors are arranged on a circular ring and act both as transmitters and receivers. ..."
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Calibration of ultrasound tomography devices is a challenging problem and of highly practical interest in medical and seismic imaging. This work addresses the position calibration problem in circular apertures where sensors are arranged on a circular ring and act both as transmitters and receivers. We introduce a new method of calibration based on the timeofflight (ToF) measurements between sensors when the enclosed medium is homogeneous. Knowing all the pairwise ToFs, one can find the positions of the sensors using multidimensional scaling (MDS) method. In practice, however, we are facing two major sources of loss. One is due to the transitional behaviour of the sensors, which makes the ToF measurements for closeby sensors unavailable. The other is due to the random malfunctioning of the sensors, that leads to random missing ToF measurements. On top of the missing entries, since in practice the impulse response of the piezoelectric and the time origin in the measurement procedure are not present, a time mismatch is also added to the measurements. In this work, we first show that a matrix defined from all the ToF measurements is of rank at most four. In order to estimate the structured and random missing entries, utilizing the fact that the matrix in question is shown to be lowrank, we apply a stateoftheart lowrank matrix completion algorithm. Then we use MDS in order to find the correct positions of the sensors. To confirm the functionality of our method in practice, simulations mimicking the measurements of an ultrasound tomography device are performed.