We provide several new depth-based separation results for feed-forward neural
networks, proving that various types of simple and natural functions can be
better approximated using deeper networks than shallower ones, even if the
shallower networks are much larger. This includes indicators of balls and
ellipses; non-linear functions which are radial with respect to the $L_1$ norm;
and smooth non-linear functions. We also show that these gaps can be observed
experimentally: Increasing the depth indeed allows better learning than
increasing width, when training neural networks to learn an indicator of a unit
ball.

本文提供了一些新的基于深度的前馈神经网络分离结果，证明了各种类型的简单自然函数可以更好地用深层网络逼近比更浅的更大的网络，这包括指示球和椭圆体的指示器，$L_1$ 范数下径向非线性函数，以及平滑的非线性函数。我们还展示了这些差距的实验观察结果：当训练神经网络学习一个单位球的指示器时，增加深度比增加宽度更容易收敛学习。

神经网络中逼近自然函数的深度 - 宽度权衡

Depth-Width Tradeoffs in Approximating Natural Functions with Neural  Networks

In this paper, we consider the problem of finding dense intrinsic
correspondence between manifolds using the recently introduced functional
framework. We pose the functional correspondence problem as matrix completion
with manifold geometric structure and inducing functional localization with the
$L_1$ norm. We discuss efficient numerical procedures for the solution of our
problem. Our method compares favorably to the accuracy of state-of-the-art
correspondence algorithms on non-rigid shape matching benchmarks, and is
especially advantageous in settings when only scarce data is available.

使用函数框架，在矩阵完成和使用流形几何结构进行函数定位的过程中，研究了在流形之间寻找密集本征对应的问题，并探讨了其解决方案的有效数值方法。该方法在非刚性形状匹配基准测试上的准确性优于最先进的对应算法，并在仅有少量数据的情况下具有特殊优势。