Deep networks are successfully used as classification models yielding
state-of-the-art results when trained on a large number of labeled samples.
These models, however, are usually much less suited for semi-supervised
problems because of their tendency to overfit easily when trained on small
amounts of data. In this work we will explore a new training objective that is
targeting a semi-supervised regime with only a small subset of labeled data.
This criterion is based on a deep metric embedding over distance relations
within the set of labeled samples, together with constraints over the
embeddings of the unlabeled set. The final learned representations are
discriminative in euclidean space, and hence can be used with subsequent
nearest-neighbor classification using the labeled samples.

本文提出了一种基于深度度量嵌入的半监督学习算法，利用少量标注数据和无标注数据的相对距离关系约束，得到在欧氏空间内区分力强的分类器，并可用于基于最近邻分类的应用。

度量嵌入的半监督深度学习

Semi-supervised deep learning by metric embedding

For $p\in (1,\infty)$ let $\mathscr{P}_p(\mathbb{R}^3)$ denote the metric
space of all $p$-integrable Borel probability measures on $\mathbb{R}^3$,
equipped with the Wasserstein $p$ metric $\mathsf{W}_p$. We prove that for
every $\varepsilon>0$, every $\theta\in (0,1/p]$ and every finite metric space
$(X,d_X)$, the metric space $(X,d_{X}^{\theta})$ embeds into
$\mathscr{P}_p(\mathbb{R}^3)$ with distortion at most $1+\varepsilon$. We show
that this is sharp when $p\in (1,2]$ in the sense that the exponent $1/p$
cannot be replaced by any larger number. In fact, for arbitrarily large $n\in
\mathbb{N}$ there exists an $n$-point metric space $(X_n,d_n)$ such that for
every $\alpha\in (1/p,1]$ any embedding of the metric space $(X_n,d_n^\alpha)$
into $\mathscr{P}_p(\mathbb{R}^3)$ incurs distortion that is at least a
constant multiple of $(\log n)^{\alpha-1/p}$. These statements establish that
there exists an Alexandrov space of nonnegative curvature, namely
$\mathscr{P}_{\! 2}(\mathbb{R}^3)$, with respect to which there does not exist
a sequence of bounded degree expander graphs. It also follows that
$\mathscr{P}_{\! 2}(\mathbb{R}^3)$ does not admit a uniform, coarse, or
quasisymmetric embedding into any Banach space of nontrivial type. Links to
several longstanding open questions in metric geometry are discussed, including
the characterization of subsets of Alexandrov spaces, existence of expanders,
the universality problem for $\mathscr{P}_{\! 2}(\mathbb{R}^k)$, and the metric
cotype dichotomy problem.

证明了有关 Wasserstein 度量下的度量嵌入问题及度量几何、扩张问题和非负曲率的问题。