Unsupervised data representation and visualization using tools from topology
is an active and growing field of Topological Data Analysis (TDA) and data
science. Its most prominent line of work is based on the so-called Mapper
graph, which is a combinatorial graph whose topological structures (connected
components, branches, loops) are in correspondence with those of the data
itself. While highly generic and applicable, its use has been hampered so far
by the manual tuning of its many parameters-among these, a crucial one is the
so-called filter: it is a continuous function whose variations on the data set
are the main ingredient for both building the Mapper representation and
assessing the presence and sizes of its topological structures. However, while
a few parameter tuning methods have already been investigated for the other
Mapper parameters (i.e., resolution, gain, clustering), there is currently no
method for tuning the filter itself. In this work, we build on a recently
proposed optimization framework incorporating topology to provide the first
filter optimization scheme for Mapper graphs. In order to achieve this, we
propose a relaxed and more general version of the Mapper graph, whose
convergence properties are investigated. Finally, we demonstrate the usefulness
of our approach by optimizing Mapper graph representations on several datasets,
and showcasing the superiority of the optimized representation over arbitrary
ones.

使用拓扑学工具进行无监督数据表示和可视化是拓扑数据分析和数据科学领域中一个活跃且快速发展的研究方向。本研究提出了一种优化方案，通过对 Mapper 图的筛选器进行参数调优，构建了更加优化的数据表示，其收敛性质也得到了研究。实验证明，该方法在多个数据集上优于任意其他表示方法。

数据表征的拓扑优化可微化映射

Differentiable Mapper For Topological Optimization Of Data  Representation

Accelerating the inference speed of CNNs is critical to their deployment in
real-world applications. Among all the pruning approaches, those implementing a
sparsity learning framework have shown to be effective as they learn and prune
the models in an end-to-end data-driven manner. However, these works impose the
same sparsity regularization on all filters indiscriminately, which can hardly
result in an optimal structure-sparse network. In this paper, we propose a
Saliency-Adaptive Sparsity Learning (SASL) approach for further optimization. A
novel and effective estimation of each filter, i.e., saliency, is designed,
which is measured from two aspects: the importance for the prediction
performance and the consumed computational resources. During sparsity learning,
the regularization strength is adjusted according to the saliency, so our
optimized format can better preserve the prediction performance while zeroing
out more computation-heavy filters. The calculation for saliency introduces
minimum overhead to the training process, which means our SASL is very
efficient. During the pruning phase, in order to optimize the proposed
data-dependent criterion, a hard sample mining strategy is utilized, which
shows higher effectiveness and efficiency. Extensive experiments demonstrate
the superior performance of our method. Notably, on ILSVRC-2012 dataset, our
approach can reduce 49.7% FLOPs of ResNet-50 with very negligible 0.39% top-1
and 0.05% top-5 accuracy degradation.

本文提出了一种基于 saliency 的 CNN 精度和资源消耗综合考虑的自适应稀疏学习方法（SASL），通过调整正则化强度实现更优化的结构稀疏网络，实验结果表明在 ILSVRC-2012 数据集上，SASL 可以将 ResNet-50 的 FLOP 降低 49.7％，同时只有 0.39％的 top-1 准确度下降和 0.05％的 top-5 准确度下降。