We consider the problem of learning low-rank tensors from partial
observations with structural constraints, and propose a novel factorization of
such tensors, which leads to a simpler optimization problem. The resulting
problem is an optimization problem on manifolds. We develop first-order and
second-order Riemannian optimization algorithms to solve it. The duality gap
for the resulting problem is derived, and we experimentally verify the
correctness of the proposed algorithm. We demonstrate the algorithm on
nonnegative constraints and Hankel constraints.

提出了一种基于流形的张量低秩分解方法，利用 Riemannian 优化算法解决了存在结构约束和部分观测时的优化问题，并在非负约束和 Hankel 约束下得到了验证。

结构化低秩张量学习

Structured Low-Rank Tensor Learning

We establish theoretical recovery guarantees of a family of Riemannian
optimization algorithms for low rank matrix recovery, which is about recovering
an $m\times n$ rank $r$ matrix from $p < mn$ number of linear measurements. The
algorithms are first interpreted as iterative hard thresholding algorithms with
subspace projections. Based on this connection, we show that provided the
restricted isometry constant $R_{3r}$ of the sensing operator is less than
$C_\kappa /\sqrt{r}$, the Riemannian gradient descent algorithm and a restarted
variant of the Riemannian conjugate gradient algorithm are guaranteed to
converge linearly to the underlying rank $r$ matrix if they are initialized by
one step hard thresholding. Empirical evaluation shows that the algorithms are
able to recover a low rank matrix from nearly the minimum number of
measurements necessary.

本文研究了低秩矩阵恢复的一类黎曼优化算法，证明了当感知算子的受限等距常数小于 Cκ/√r 时，算法能够从几乎最小的测量中恢复低秩矩阵。

低秩矩阵恢复的黎曼优化保证

Guarantees of Riemannian Optimization for Low Rank Matrix Recovery

Optimization on manifolds is a rapidly developing branch of nonlinear
optimization. Its focus is on problems where the smooth geometry of the search
space can be leveraged to design efficient numerical algorithms. In particular,
optimization on manifolds is well-suited to deal with rank and orthogonality
constraints. Such structured constraints appear pervasively in machine learning
applications, including low-rank matrix completion, sensor network
localization, camera network registration, independent component analysis,
metric learning, dimensionality reduction and so on. The Manopt toolbox,
available at www.manopt.org, is a user-friendly, documented piece of software
dedicated to simplify experimenting with state of the art Riemannian
optimization algorithms. We aim particularly at reaching practitioners outside
our field.

论文主要介绍了流形上的优化方法在处理机器学习应用中出现的结构性约束时的优越表现，介绍了基于 Riemannian 优化算法的 Manopt 工具箱，并旨在让优化领域之外的从业者更容易地尝试最新的算法。