Many machine learning applications are naturally formulated as optimization
problems on Riemannian manifolds. The main idea behind Riemannian optimization
is to maintain the feasibility of the variables while moving along a descent
direction on the manifold. This results in updating all the variables at every
iteration. In this work, we provide a general framework for developing
computationally efficient coordinate descent (CD) algorithms on matrix
manifolds that allows updating only a few variables at every iteration while
adhering to the manifold constraint. In particular, we propose CD algorithms
for various manifolds such as Stiefel, Grassmann, (generalized) hyperbolic,
symplectic, and symmetric positive (semi)definite. While the cost per iteration
of the proposed CD algorithms is low, we further develop a more efficient
variant via a first-order approximation of the objective function. We analyze
their convergence and complexity, and empirically illustrate their efficacy in
several applications.

提出了在矩阵流形上开发计算效率高的坐标下降（CD）算法的一般框架，从而允许在每次迭代中仅更新少数变量，并符合流形约束。通过一阶目标函数的近似实现了更高效的变体，分析了它们的收敛性和复杂性，并在多个应用中验证了它们的有效性。

矩阵流形上的 Riemannian 坐标下降算法

Riemannian coordinate descent algorithms on matrix manifolds

Cognitive Diagnosis (CD) algorithms receive growing research interest in
intelligent education. Typically, these CD algorithms assist students by
inferring their abilities (i.e., their proficiency levels on various knowledge
concepts). The proficiency levels can enable further targeted skill training
and personalized exercise recommendations, thereby promoting students' learning
efficiency in online education. Recently, researchers have found that building
and incorporating a student-exercise bipartite graph is beneficial for
enhancing diagnostic performance. However, there are still limitations in their
studies. On one hand, researchers overlook the heterogeneity within edges,
where there can be both correct and incorrect answers. On the other hand, they
disregard the uncertainty within edges, e.g., a correct answer can indicate
true mastery or fortunate guessing. To address the limitations, we propose
Adaptive Semantic-aware Graph-based Cognitive Diagnosis model (ASG-CD), which
introduces a novel and effective way to leverage bipartite graph information in
CD. Specifically, we first map students, exercises, and knowledge concepts into
a latent representation space and combine these latent representations to
obtain student abilities and exercise difficulties. After that, we propose a
Semantic-aware Graph Neural Network Layer to address edge heterogeneity. This
layer splits the original bipartite graph into two subgraphs according to edge
semantics, and aggregates information based on these two subgraphs separately.
To mitigate the impact of edge uncertainties, we propose an Adaptive Edge
Differentiation Layer that dynamically differentiates edges, followed by
keeping reliable edges and filtering out uncertain edges. Extensive experiments
on three real-world datasets have demonstrated the effectiveness of ASG-CD.

通过引入学生 - 习题二分图信息，我们提出了一种自适应语义感知图神经网络层并应用于认知诊断模型 (ASG-CD)，以解决其研究中的限制，并在三个真实数据集上进行了广泛实验证明了 ASG-CD 的有效性。