近似消失理想的条件梯度
Conditional Gradients for the Approximate Vanishing Ideal
AI总结：
提出PCGAVI算法，通过成对条件梯度求解凸优化问题构建稀疏生成元，捕捉数据多项式结构并产生鲁棒特征映射，且提供了理论学习保证。
摘要：
点集 $X\subseteq \mathbb{R}^n$ 的消失理想是在所有点 $\mathbf{x} \in X$ 上求值为 $0$ 的多项式集合，它可以通过一组称为生成元的有限多项式集合进行高效表示。为了适应数据集中的噪声，我们引入了成对条件梯度近似消失理想算法（PCGAVI），用于构建近似消失理想的生成元集合。所构建的生成元能够捕捉数据中的多项式结构，并由此产生一个特征映射，例如可将其与线性分类器结合用于监督学习。在 PCGAVI 中，我们通过使用成对条件梯度算法求解受约束的凸优化问题来构建生成元集合。因此，PCGAVI 不仅构建的生成元数量少，而且具有稀疏性，使得相应的特征变换具有鲁棒性且紧凑。此外，我们为 PCGAVI 推导出了多项学习保证，使得该算法在理论上比相关的生成元构建方法更有依据。
英文摘要：
The vanishing ideal of a set of points $X\subseteq \mathbb{R}^n$ is the set of polynomials that evaluate to $0$ over all points $\mathbf{x} \in X$ and admits an efficient representation by a finite set of polynomials called generators. To accommodate the noise in the data set, we introduce the pairwise conditional gradients approximate vanishing ideal algorithm (PCGAVI) that constructs a set of generators of the approximate vanishing ideal. The constructed generators capture polynomial structures in data and give rise to a feature map that can, for example, be used in combination with a linear classifier for supervised learning. In PCGAVI, we construct the set of generators by solving constrained convex optimization problems with the pairwise conditional gradients algorithm. Thus, PCGAVI not only constructs few but also sparse generators, making the corresponding feature transformation robust and compact. Furthermore, we derive several learning guarantees for PCGAVI that make the algorithm theoretically better motivated than related generator-constructing methods.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
