Simplicial map neural networks (SMNNs) are topology-based neural networks
with interesting properties such as universal approximation capability and
robustness to adversarial examples under appropriate conditions. However, SMNNs
present some bottlenecks for their possible application in high dimensions.
First, no SMNN training process has been defined so far. Second, SMNNs require
the construction of a convex polytope surrounding the input dataset. In this
paper, we propose a SMNN training procedure based on a support subset of the
given dataset and a method based on projection to a hypersphere as a
replacement for the convex polytope construction. In addition, the
explainability capacity of SMNNs is also introduced for the first time in this
paper.

本文首次提出了基于支持子集以及基于超球的投影法代替凸多面体构建的 Simplicial map neural networks（SMNNs）的训练过程，同时也首次引入 SMNNs 的可解释性能力。

简单图神经网络中的可解释性

Explainability in Simplicial Map Neural Networks

We give a deterministic nearly-linear time algorithm for approximating any
point inside a convex polytope with a sparse convex combination of the
polytope's vertices. Our result provides a constructive proof for the
Approximate Carath\'{e}odory Problem, which states that any point inside a
polytope contained in the $\ell_p$ ball of radius $D$ can be approximated to
within $\epsilon$ in $\ell_p$ norm by a convex combination of only $O\left(D^2
p/\epsilon^2\right)$ vertices of the polytope for $p \geq 2$. We also show that
this bound is tight, using an argument based on anti-concentration for the
binomial distribution.
Along the way of establishing the upper bound, we develop a technique for
minimizing norms over convex sets with complicated geometry; this is achieved
by running Mirror Descent on a dual convex function obtained via Sion's
Theorem.
As simple extensions of our method, we then provide new algorithms for
submodular function minimization and SVM training. For submodular function
minimization we obtain a simplification and (provable) speed-up over Wolfe's
algorithm, the method commonly found to be the fastest in practice. For SVM
training, we obtain $O(1/\epsilon^2)$ convergence for arbitrary kernels; each
iteration only requires matrix-vector operations involving the kernel matrix,
so we overcome the obstacle of having to explicitly store the kernel or compute
its Cholesky factorization.

本文提出了一种快速算法来处理凸多面体中点的问题，并在此基础上对子模函数最小化和 SVM 训练提供了新算法。