In artificial-intelligence-aided signal processing, existing deep learning
models often exhibit a black-box structure, and their validity and
comprehensibility remain elusive. The integration of topological methods,
despite its relatively nascent application, serves a dual purpose of making
models more interpretable as well as extracting structural information from
time-dependent data for smarter learning. Here, we provide a transparent and
broadly applicable methodology, TopCap, to capture the most salient topological
features inherent in time series for machine learning. Rooted in
high-dimensional ambient spaces, TopCap is capable of capturing features rarely
detected in datasets with low intrinsic dimensionality. Applying time-delay
embedding and persistent homology, we obtain descriptors which encapsulate
information such as the vibration of a time series, in terms of its variability
of frequency, amplitude, and average line, demonstrated with simulated data.
This information is then vectorised and fed into multiple machine learning
algorithms such as k-nearest neighbours and support vector machine. Notably, in
classifying voiced and voiceless consonants, TopCap achieves an accuracy
exceeding 96% and is geared towards designing topological convolutional layers
for deep learning of speech and audio signals.

利用高维环境空间的 TopCap 方法，通过时间序列的拓扑特征来进行机器学习和信号处理，以提高模型的可解释性和性能，尤其在语音和音频信号的深度学习中效果显著。

拓扑结合机器学习的辅音识别

Topology combined machine learning for consonant recognition

Given a set of $p$ players we consider problems concerning envy-free
allocation of collections of $k$ pieces from a given set of goods or chores. We
show that if $p\le n$ and each player can choose $k$ pieces out of $n$ pieces
of a cake, then there exist a division of the cake and an allocation of the
pieces where at least $\frac{p}{2(k^2-k+1)}$ players get their desired $k$
pieces each. We further show that if $p\le k(n-1)+1$ and each player can choose
$k$ pieces, one from each of $k$ cakes that are divided into $n$ pieces each,
then there exist a division of the cakes and allocation of the pieces where at
least $\frac{p}{2k(k-1)}$ players get their desired $k$ pieces. Finally we
prove that if $p\ge k(n-1)+1$ and each player can choose one shift in each of
$k$ days that are partitioned into $n$ shifts each, then, given that the
salaries of the players are fixed, there exist $n(1+\ln k)$ players covering
all the shifts, and moreover, if $k=2$ then $n$ players suffice. Our proofs
combine topological methods and theorems of F\"uredi, Lov\'asz and Gallai from
hypergraph theory.

本文将枚举的方式应用于一些常见的分配问题，并使用拓扑学方法和 Furedi，Lovasz 和 Gallai 的超图定理进行了证明。

多件公平分割

Fair division with multiple pieces

Extracting useful information from large data sets can be a daunting task.
Topological methods for analyzing data sets provide a powerful technique for
extracting such information. Persistent homology is a sophisticated tool for
identifying such topological features -- connected components, holes, or voids
-- and for determining how such features persist as the data is viewed at
different scales. This paper provides quantum algorithms for calculating Betti
numbers in persistent homology, and for finding eigenvectors and eigenvalues of
the combinatorial Laplacian. The algorithms provide an exponential speedup over
classical algorithms for topological data analysis.

本研究提供了用于持久同调中计算 Betti 数的量子算法，以及用于查找组合拉普拉斯的特征向量和特征值的算法。这种算法比拓扑数据分析的经典算法速度更快。