用于带约束秩一投影的Grassmann核近似的随机特征
Random features for Grassmannian kernel approximation with bounded rank-one projections
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中文总结 AI 辅助
该研究提出带约束秩一投影的随机特征，用于近似Grassmann核，可降低计算内存开销，在合成数据和ETH-80分类任务上表现良好，为经典Grassmann核提供可扩展替代方案。
中文摘要 AI 辅助
我们提出了一族随机特征映射，用于在低维子空间（即Grassmann流形）上构建可扩展的核机器。当数据类别或簇能被少数样本的张成空间很好地描述时，这类表示非常有用。经典的Grassmann核，包括投影核和Binet-Cauchy核，需要完整的Gram矩阵，这会给大型高维子空间数据集带来难以承受的计算和内存开销。我们通过基于子空间投影矩阵的秩一投影，再结合周期或二值的有界非线性变换来控制所得分布，解决了这一局限。我们证明，随机特征空间中的内积可近似定义明确的、仅依赖子空间之间主角度的旋转不变Grassmann核。当特征数量相对于固有子空间维度足够大时，该近似以高概率在所有固定维子空间上一致成立。对于周期变换，近似核具有闭式表达式，可在逆Binet-Cauchy与高斯型模式之间调节；二值变换则产生紧凑的1位子空间特征，尽管目前尚无已知的闭式核。基于随机快速傅里叶变换的结构化秩一投影进一步降低了计算量，且不牺牲实际精度。在合成数据和ETH-80分类任务上的实验表明，这些特征能准确保留Grassmann几何结构，同时减少计算、内存和存储开销。因此，秩一嵌入为经典Grassmann核提供了一种实用且可扩展的替代方案。
英文摘要
We propose a family of random feature maps for scalable kernel machines on low-dimensional subspaces, ie on the Grassmannian manifold. Such representations are useful when data classes or clusters are well described by the span of a few samples. Classical Grassmannian kernels, including the projection and Binet-Cauchy kernels, require full Gram matrices, which leads to prohibitive computational and memory costs for large high-dimensional subspace datasets. We address this limitation using random features based on rank-one projections of subspace projection matrices followed by bounded non-linear transforms, either periodic or binary, to control the resulting distributions. We show that inner products in the random feature space approximate well-defined rotation-invariant Grassmannian kernels that depend only on the principal angles between subspaces. When the number of features is sufficiently large relative to the intrinsic subspace dimension, the approximation holds uniformly over all fixed-dimensional subspaces with high probability. For periodic transforms, the approximated kernel has a closed-form expression with tunable behaviour between inverse Binet-Cauchy and Gaussian-type regimes. Binary transforms yield compact one-bit subspace features, although no closed-form kernel is known. Structured rank-one projections based on randomised fast Fourier transforms further reduce computation without sacrificing practical accuracy. Experiments on synthetic data and ETH-80 classification tasks show that these features accurately preserve Grassmannian geometry while reducing computation, memory, and storage. Rank-one embeddings therefore provide a practical and scalable alternative to classical Grassmannian kernels.
发表机构
- UCLouvain（天主教鲁汶大学）
- INMA
- ICTEAM
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