Dimension reduction techniques usually lose information in the sense that
reconstructed data are not identical to the original data. However, we argue
that it is possible to have reconstructed data identically distributed as the
original data, irrespective of the retained dimension or the specific mapping.
This can be achieved by learning a distributional model that matches the
conditional distribution of data given its low-dimensional latent variables.
Motivated by this, we propose Distributional Principal Autoencoder (DPA) that
consists of an encoder that maps high-dimensional data to low-dimensional
latent variables and a decoder that maps the latent variables back to the data
space. For reducing the dimension, the DPA encoder aims to minimise the
unexplained variability of the data with an adaptive choice of the latent
dimension. For reconstructing data, the DPA decoder aims to match the
conditional distribution of all data that are mapped to a certain latent value,
thus ensuring that the reconstructed data retains the original data
distribution. Our numerical results on climate data, single-cell data, and
image benchmarks demonstrate the practical feasibility and success of the
approach in reconstructing the original distribution of the data. DPA
embeddings are shown to preserve meaningful structures of data such as the
seasonal cycle for precipitations and cell types for gene expression.

通过学习一个分布模型以匹配数据的条件分布，Dimensional Principal Autoencoder (DPA) 旨在实现重建数据与原始数据完全分布相同，从而实现降维和重建的成功。

分布式主要自编码器

Distributional Principal Autoencoders

Time series classification is of significant importance in monitoring
structural systems. In this work, we investigate the use of supervised machine
learning classification algorithms on simulated data based on a physical system
with two states: Intact and Broken. We provide a comprehensive discussion of
the preprocessing of temporal data, using measures of statistical dispersion
and dimension reduction techniques. We present an intuitive baseline method and
discuss its efficiency. We conclude with a comparison of the various methods
based on different performance metrics, showing the advantage of using machine
learning techniques as a tool in decision making.

通过监测物理系统的仿真数据，我们使用监督机器学习分类算法对时间序列进行分类，并讨论了时间序列数据预处理、统计分散度和降维技术。我们提出了一种直观的基准方法，并讨论了其效率，并通过不同性能指标的比较，展示了使用机器学习技术在决策中的优势。

海底工程异常检测的有监督时间序列分类

Supervised Time Series Classification for Anomaly Detection in Subsea  Engineering

Persistence diagrams are used as signatures of point cloud data assumed to be
sampled from manifolds, and represent their topology in a compact fashion.
Further, two given clouds of points can be compared by directly comparing their
persistence diagrams using the bottleneck distance, d_B. But one potential
drawback of this pipeline is that point clouds sampled from topologically
similar manifolds can have arbitrarily large d_B values when there is a large
degree of scaling between them. This situation is typical in dimension
reduction frameworks that are also aiming to preserve topology.
We define a new scale-invariant distance between persistence diagrams termed
normalized bottleneck distance, d_N, and study its properties. In defining d_N,
we also develop a broader framework called metric decomposition for comparing
finite metric spaces of equal cardinality with a bijection. We utilize metric
decomposition to prove a stability result for d_N by deriving an explicit bound
on the distortion of the associated bijective map. We then study two popular
dimension reduction techniques, Johnson-Lindenstrauss (JL) projections and
metric multidimensional scaling (mMDS), and a third class of general
biLipschitz mappings. We provide new bounds on how well these dimension
reduction techniques preserve homology with respect to d_N. For a JL map f that
transforms input X to f(X), we show that d_N(dgm(X),dgm(f(X)) < e, where dgm(X)
is the Vietoris-Rips persistence diagram of X, and 0 < e < 1 is the tolerance
up to which pairwise distances are preserved by f. For mMDS, we present new
bounds for both d_B and d_N between persistence diagrams of X and its
projection in terms of the eigenvalues of the covariance matrix. And for
k-biLipschitz maps, we show that d_N is bounded by the product of (k^2-1)/k and
the ratio of diameters of X and f(X).

本文提出了一种新的 scale-invariant 度量，即标准化瓶颈距离 d_N，并介绍了称为 metric decomposition 的框架，用于比较具有双射的相等基数的有限度量空间，研究了多种降维技术如何在保持拓扑结构方面表现良好。