The latent space of a Generative Adversarial Network (GAN) has been shown to
encode rich semantics within some subspaces. To identify these subspaces,
researchers typically analyze the statistical information from a collection of
synthesized data, and the identified subspaces tend to control image attributes
globally (i.e., manipulating an attribute causes the change of an entire
image). By contrast, this work introduces low-rank subspaces that enable more
precise control of GAN generation. Concretely, given an arbitrary image and a
region of interest (e.g., eyes of face images), we manage to relate the latent
space to the image region with the Jacobian matrix and then use low-rank
factorization to discover steerable latent subspaces. There are three
distinguishable strengths of our approach that can be aptly called LowRankGAN.
First, compared to analytic algorithms in prior work, our low-rank
factorization of Jacobians is able to find the low-dimensional representation
of attribute manifold, making image editing more precise and controllable.
Second, low-rank factorization naturally yields a null space of attributes such
that moving the latent code within it only affects the outer region of
interest. Therefore, local image editing can be simply achieved by projecting
an attribute vector into the null space without relying on a spatial mask as
existing methods do. Third, our method can robustly work with a local region
from one image for analysis yet well generalize to other images, making it much
easy to use in practice. Extensive experiments on state-of-the-art GAN models
(including StyleGAN2 and BigGAN) trained on various datasets demonstrate the
effectiveness of our LowRankGAN.

使用低秩矩阵分解的方法可以更好地控制 GAN 生成的图像特征，同时局部的图像编辑控制可以通过该方法实现。

GAN 中的低秩子空间

Low-Rank Subspaces in GANs

This paper considers the problem of completing a matrix with many missing
entries under the assumption that the columns of the matrix belong to a union
of multiple low-rank subspaces. This generalizes the standard low-rank matrix
completion problem to situations in which the matrix rank can be quite high or
even full rank. Since the columns belong to a union of subspaces, this problem
may also be viewed as a missing-data version of the subspace clustering
problem. Let X be an n x N matrix whose (complete) columns lie in a union of at
most k subspaces, each of rank <= r < n, and assume N >> kn. The main result of
the paper shows that under mild assumptions each column of X can be perfectly
recovered with high probability from an incomplete version so long as at least
CrNlog^2(n) entries of X are observed uniformly at random, with C>1 a constant
depending on the usual incoherence conditions, the geometrical arrangement of
subspaces, and the distribution of columns over the subspaces. The result is
illustrated with numerical experiments and an application to Internet distance
matrix completion and topology identification.

该文章研究了一个矩阵完成问题，该问题假设矩阵的列属于多个低秩子空间的并集，这将标准的低秩矩阵完成问题推广到了矩阵秩可以相当高或甚至是完全秩的情况。文章的主要结果表明，每一列可以从不完整的版本中完美地恢复出来，只要在均匀随机地观察至少 CrNlog^2 (n) 个条目的情况下，其中 C>1 是依赖于非相干条件、子空间的几何排列和列在子空间中的分布的常数，结果通过数字实验和应用于互联网距离矩阵完成和拓扑识别来加以说明。