Recent work has explored methods for learning continuous vector space word
representations reflecting the underlying semantics of words. Simple vector
space arithmetic using cosine distances has been shown to capture certain types
of analogies, such as reasoning about plurals from singulars, past tense from
present tense, etc. In this paper, we introduce a new approach to capture
analogies in continuous word representations, based on modeling not just
individual word vectors, but rather the subspaces spanned by groups of words.
We exploit the property that the set of subspaces in n-dimensional Euclidean
space form a curved manifold space called the Grassmannian, a quotient subgroup
of the Lie group of rotations in n- dimensions. Based on this mathematical
model, we develop a modified cosine distance model based on geodesic kernels
that captures relation-specific distances across word categories. Our
experiments on analogy tasks show that our approach performs significantly
better than the previous approaches for the given task.

本文介绍了一种基于 Grassmannian 的新方法，旨在通过建模单词集合所围成的子空间来捕获连续词向量表示中的类比关系，该方法利用测地核的修改余弦距离模型捕获跨单词类别的关系特定距离，实验结果表明与以前的方法相比，本方法在类比任务上表现显著改善。

利用矩阵流形推理单词嵌入中的语言规律

Reasoning about Linguistic Regularities in Word Embeddings using Matrix  Manifolds

Modeling videos and image-sets as linear subspaces has proven beneficial for
many visual recognition tasks. However, it also incurs challenges arising from
the fact that linear subspaces do not obey Euclidean geometry, but lie on a
special type of Riemannian manifolds known as Grassmannian. To leverage the
techniques developed for Euclidean spaces (e.g, support vector machines) with
subspaces, several recent studies have proposed to embed the Grassmannian into
a Hilbert space by making use of a positive definite kernel. Unfortunately,
only two Grassmannian kernels are known, none of which -as we will show- is
universal, which limits their ability to approximate a target function
arbitrarily well. Here, we introduce several positive definite Grassmannian
kernels, including universal ones, and demonstrate their superiority over
previously-known kernels in various tasks, such as classification, clustering,
sparse coding and hashing.

介绍了几个正定的 Grassmann 核，并证明这些核优于以前已知的核在各种任务中的表现，诸如分类，聚类，稀疏编码和哈希等。

扩展 Grassmann 核家族：一个嵌入视角

Expanding the Family of Grassmannian Kernels: An Embedding Perspective

Many applications in data analysis rely on the decomposition of a data matrix
into a low-rank and a sparse component. Existing methods that tackle this task
use the nuclear norm and L1-cost functions as convex relaxations of the rank
constraint and the sparsity measure, respectively, or employ thresholding
techniques. We propose a method that allows for reconstructing and tracking a
subspace of upper-bounded dimension from incomplete and corrupted observations.
It does not require any a priori information about the number of outliers. The
core of our algorithm is an intrinsic Conjugate Gradient method on the set of
orthogonal projection matrices, the so-called Grassmannian. Non-convex sparsity
measures are used for outlier detection, which leads to improved performance in
terms of robustly recovering and tracking the low-rank matrix. In particular,
our approach can cope with more outliers and with an underlying matrix of
higher rank than other state-of-the-art methods.

提出一种基于 Grassmannian 的内置共轭梯度方法，可以从不完整和受损观测中重构和跟踪上限维度的子空间，用于处理数据分析中的低秩矩阵和稀疏组件，采用非凸稀疏度量进行异常点检测，性能优于其他最先进的方法。