  Manifold learning (ML) aims to seek low-dimensional embedding from
high-dimensional data. The problem is challenging on real-world datasets,
especially with under-sampling data, and we find that previous methods perform
poorly in this case. Generally, ML methods first transform input data into a
low-dimensional embedding space to maintain the data's geometric structure and
subsequently perform downstream tasks therein. The poor local connectivity of
under-sampling data in the former step and inappropriate optimization
objectives in the latter step leads to two problems: structural distortion and
underconstrained embedding. This paper proposes a novel ML framework named Deep
Local-flatness Manifold Embedding (DLME) to solve these problems. The proposed
DLME constructs semantic manifolds by data augmentation and overcomes the
structural distortion problem using a smoothness constrained based on a local
flatness assumption about the manifold. To overcome the underconstrained
embedding problem, we design a loss and theoretically demonstrate that it leads
to a more suitable embedding based on the local flatness. Experiments on three
types of datasets (toy, biological, and image) for various downstream tasks
(classification, clustering, and visualization) show that our proposed DLME
outperforms state-of-the-art ML and contrastive learning methods.
