传感器网络定位、欧氏距离矩阵完备化及图形实现

张量完备化的一个新的凸松弛

A New Convex Relaxation for Tensor Completion

我们研究了从一组线性测量中学习张量的问题。这个问题的一个突出的方法是基于迹范数正则化的推广，它已被广泛用于学习低秩矩阵，张量设置。在本文中，我们强调了这种方法的一些局限性，并提出了另一种凸松弛的欧几里德球。然后，我们描述了一种技术来解决相关的正则化问题，这是建立在交替方向的方法的乘数。在一个合成数据集和两个真实数据集上的实验表明，该方法在估计误差方面明显优于张量迹范数正则化，同时保持计算易处理。

与有界秩正半定矩阵补有关的一个新的图参数

A new graph parameter related to bounded rank positive semidefinite matrix completions

图$G$的Gram维数$\gd(G)$是最小整数$k\ge 1$，使得任何部分实对称矩阵，其元素被指定在$G$的对角线上和对应于$G$的边的非对角线位置处，可以被完成为秩至多为$k$的正半定矩阵（假设存在半正定完成）。对于任何固定的$k$，满足$\gd(G) \le k$的图类是子闭的，因此它可以用有限的禁止子表来刻画.我们发现，唯一的最小禁止未成年人是$K_{k+1}$的$k\le 3$和有两个最小的禁止未成年人：$K_5$和$K_{2,2,2}$的$k=4$。我们还显示了一些密切的联系，以欧几里德实现的图形和图形参数$ν^=(G)$的\cite{H03}。特别地，我们对具有$\gd(G)\le 4$的图的刻画暗示了Belk和Connelly \cite{Belk,BC}的3-可实现图和van der Holst \cite{H03}的具有$ν^=(G) \le 4$的图的禁止子刻画.

Exact Linear Convergence Rate Analysis for Low-Rank Symmetric Matrix Completion via Gradient Descent

Factorization-based gradient descent is a scalable and efficient algorithm for solving low-rank matrix completion. Recent progress in structured non-convex optimization has offered global convergence guarantees for gradient descent under certain statistical assumptions on the low-rank matrix and the sampling set. However, while the theory suggests gradient descent enjoys fast linear convergence to a global solution of the problem, the universal nature of the bounding technique prevents it from obtaining an accurate estimate of the rate of convergence. In this paper, we perform a local analysis…

Quartic First-Order Methods for Low-Rank Minimization

We study a generalized nonconvex Burer-Monteiro formulation for low-rank minimization problems. We use recent results on non-Euclidean first order methods to provide efficient and scalable algorithms. Our approach uses geometries induced by quartic kernels on matrix spaces; for unconstrained cases we introduce a novel family of Gram kernels that considerably improves numerical performances. Numerical experiments for Euclidean distance matrix completion and symmetric nonnegative matrix factorization show that our algorithms scale well and reach state of the art performance when compared to spec…

Grassmannian 内的欧几里得距离优化

Euclidean Distance Optimization Within the Grassmannian

本文研究 Grassmannian 子簇上的欧几里得距离优化问题，定义并计算其 ED 度与 GD 度，并给出若干几何子簇的公式。

Robust Matrix Completion with Heavy-tailed Noise

This paper studies low-rank matrix completion in the presence of heavy-tailed and possibly asymmetric noise, where we aim to estimate an underlying low-rank matrix given a set of highly incomplete noisy entries. Though the matrix completion problem has attracted much attention in the past decade, there is still lack of theoretical understanding when the observations are contaminated by heavy-tailed noises. Prior theory falls short of explaining the empirical results and is unable to capture the optimal dependence of the estimation error on the noise level. In this paper, we adopt an adaptive H…

Noisy Matrix Completion: Understanding Statistical Guarantees for Convex Relaxation via Nonconvex Optimization

This paper studies noisy low-rank matrix completion: given partial and noisy entries of a large low-rank matrix, the goal is to estimate the underlying matrix faithfully and efficiently. Arguably one of the most popular paradigms to tackle this problem is convex relaxation, which achieves remarkable efficacy in practice. However, the theoretical support of this approach is still far from optimal in the noisy setting, falling short of explaining its empirical success. We make progress towards demystifying the practical efficacy of convex relaxation vis-à-vis random noise. When the rank and the …