Motivated by the problem of learning a linear regression model whose
parameter is a large fixed-rank non-symmetric matrix, we consider the
optimization of a smooth cost function defined on the set of fixed-rank
matrices. We adopt the geometric framework of optimization on Riemannian
quotient manifolds. We study the underlying geometries of several well-known
fixed-rank matrix factorizations and then exploit the Riemannian quotient
geometry of the search space in the design of a class of gradient descent and
trust-region algorithms. The proposed algorithms generalize our previous
results on fixed-rank symmetric positive semidefinite matrices, apply to a
broad range of applications, scale to high-dimensional problems and confer a
geometric basis to recent contributions on the learning of fixed-rank
non-symmetric matrices. We make connections with existing algorithms in the
context of low-rank matrix completion and discuss relative usefulness of the
proposed framework. Numerical experiments suggest that the proposed algorithms
compete with the state-of-the-art and that manifold optimization offers an
effective and versatile framework for the design of machine learning algorithms
that learn a fixed-rank matrix.

采用 Riemannian 余维度流形上的优化几何方法及其梯度下降和信任区域算法，对学习大型固定秩非对称矩阵的线性回归模型进行了研究，推广了固定秩对称正定矩阵的一般结果，可用于机器学习算法的设计，数值实验表明，与现有算法竞争并提供了一种有效且灵活的算法，用于学习固定秩矩阵。

固定秩矩阵分解与黎曼低秩优化

Fixed-rank matrix factorizations and Riemannian low-rank optimization

This paper introduces a new metric and mean on the set of positive
semidefinite matrices of fixed-rank. The proposed metric is derived from a
well-chosen Riemannian quotient geometry that generalizes the reductive
geometry of the positive cone and the associated natural metric. The resulting
Riemannian space has strong geometrical properties: it is geodesically
complete, and the metric is invariant with respect to all transformations that
preserve angles (orthogonal transformations, scalings, and pseudoinversion). A
meaningful approximation of the associated Riemannian distance is proposed,
that can be efficiently numerically computed via a simple algorithm based on
SVD. The induced mean preserves the rank, possesses the most desirable
characteristics of a geometric mean, and is easy to compute.

本文介绍了一个派生于黎曼商几何的指标和均值，用于一组固定秩的半正定矩阵。从积极锥的缩紧几何和相关流畅自然度量，提出了所提出的指标。所得到的黎曼空间具有强的几何特性：它是测地线完成的，并且度量在保持角度的所有变换（正交变换、比例变换和伪逆变换）下是不变的。提出了与所关联的黎曼距离的有意义的近似，可以通过基于 SVD 的简单算法有效地数值计算。其中的几何平均保留着秩，具有最理想的几何平均特性并易于计算。