Median Matrix Completion: from Embarrassment to Optimality

• URL: http://arxiv.org/abs/2006.10400v1
• Date: Thu, 18 Jun 2020 10:01:22 GMT
• Title: Median Matrix Completion: from Embarrassment to Optimality
• Authors: Weidong Liu, Xiaojun Mao, Raymond K. W. Wong
• Abstract summary: We consider matrix completion with absolute deviation loss and obtain an estimator of the median matrix. Despite several appealing properties of median, the non-smooth absolute deviation loss leads to computational challenge. We propose a novel refinement step, which turns such inefficient estimators into a rate (near-optimal) matrix completion procedure.
• Score: 16.667260586938234
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we consider matrix completion with absolute deviation loss and obtain an estimator of the median matrix. Despite several appealing properties of median, the non-smooth absolute deviation loss leads to computational challenge for large-scale data sets which are increasingly common among matrix completion problems. A simple solution to large-scale problems is parallel computing. However, embarrassingly parallel fashion often leads to inefficient estimators. Based on the idea of pseudo data, we propose a novel refinement step, which turns such inefficient estimators into a rate (near-)optimal matrix completion procedure. The refined estimator is an approximation of a regularized least median estimator, and therefore not an ordinary regularized empirical risk estimator. This leads to a non-standard analysis of asymptotic behaviors. Empirical results are also provided to confirm the effectiveness of the proposed method.

Related papers

• Distributed Semi-Supervised Sparse Statistical Inference [6.685997976921953]
  A debiased estimator is a crucial tool in statistical inference for high-dimensional model parameters. Traditional methods require computing a debiased estimator on every machine. An efficient multi-round distributed debiased estimator, which integrates both labeled and unlabelled data, is developed.
  arXiv  Detail & Related papers  (2023-06-17T17:30:43Z)
• A Generalized Latent Factor Model Approach to Mixed-data Matrix Completion with Entrywise Consistency [3.299672391663527]
  Matrix completion is a class of machine learning methods that concerns the prediction of missing entries in a partially observed matrix. We formulate it as a low-rank matrix estimation problem under a general family of non-linear factor models. We propose entrywise consistent estimators for estimating the low-rank matrix.
  arXiv  Detail & Related papers  (2022-11-17T00:24:47Z)
• Matrix Completion via Non-Convex Relaxation and Adaptive Correlation Learning [90.8576971748142]
  We develop a novel surrogate that can be optimized by closed-form solutions. We exploit upperwise correlation for completion, and thus an adaptive correlation learning model.
  arXiv  Detail & Related papers  (2022-03-04T08:50:50Z)
• Faster Algorithms and Constant Lower Bounds for the Worst-Case Expected Error [0.3997680012976965]
  The goal is to design estimators that minimize the worst-case expected error. Chen, Valiant and Valiant show that, when data values are $ell_infty$-normalized, there is a time algorithm to compute an estimator for the mean. In this paper we design provably efficient algorithms for approximating the optimal semilinear estimator based on online convex optimization.
  arXiv  Detail & Related papers  (2021-12-27T18:47:25Z)
• Learning to Estimate Without Bias [57.82628598276623]
  Gauss theorem states that the weighted least squares estimator is a linear minimum variance unbiased estimation (MVUE) in linear models. In this paper, we take a first step towards extending this result to non linear settings via deep learning with bias constraints. A second motivation to BCE is in applications where multiple estimates of the same unknown are averaged for improved performance.
  arXiv  Detail & Related papers  (2021-10-24T10:23:51Z)
• Near-optimal inference in adaptive linear regression [60.08422051718195]
  Even simple methods like least squares can exhibit non-normal behavior when data is collected in an adaptive manner. We propose a family of online debiasing estimators to correct these distributional anomalies in at least squares estimation. We demonstrate the usefulness of our theory via applications to multi-armed bandit, autoregressive time series estimation, and active learning with exploration.
  arXiv  Detail & Related papers  (2021-07-05T21:05:11Z)
• Sparse PCA via $l_{2,p}$-Norm Regularization for Unsupervised Feature Selection [138.97647716793333]
  We propose a simple and efficient unsupervised feature selection method, by combining reconstruction error with $l_2,p$-norm regularization. We present an efficient optimization algorithm to solve the proposed unsupervised model, and analyse the convergence and computational complexity of the algorithm theoretically.
  arXiv  Detail & Related papers  (2020-12-29T04:08:38Z)
• A Scalable, Adaptive and Sound Nonconvex Regularizer for Low-rank Matrix Completion [60.52730146391456]
  We propose a new non scalable low-rank regularizer called "nuclear Frobenius norm" regularizer, which is adaptive and sound. It bypasses the computation of singular values and allows fast optimization by algorithms. It obtains state-of-the-art recovery performance while being the fastest in existing matrix learning methods.
  arXiv  Detail & Related papers  (2020-08-14T18:47:58Z)
• Matrix Completion with Quantified Uncertainty through Low Rank Gaussian Copula [30.84155327760468]
  This paper proposes a framework for missing value imputation with quantified uncertainty. The time required to fit the model scales linearly with the number of rows and the number of columns in the dataset. Empirical results show the method yields state-of-the-art imputation accuracy across a wide range of data types.
  arXiv  Detail & Related papers  (2020-06-18T19:51:42Z)
• Robust Matrix Completion with Mixed Data Types [0.0]
  We consider the problem of recovering a structured low rank matrix with partially observed entries with mixed data types. Most approaches assume that there is only one underlying distribution and the low rank constraint is regularized by the matrix Schatten Norm. We propose a computationally feasible statistical approach with strong recovery guarantees along with an algorithmic framework suited for parallelization to recover a low rank matrix with partially observed entries for mixed data types in one step.
  arXiv  Detail & Related papers  (2020-05-25T21:35:10Z)
• Relative Error Bound Analysis for Nuclear Norm Regularized Matrix Completion [101.83262280224729]
  We develop a relative error bound for nuclear norm regularized matrix completion. We derive a relative upper bound for recovering the best low-rank approximation of the unknown matrix.
  arXiv  Detail & Related papers  (2015-04-26T13:12:16Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.