Linear latent variable models such as principal component analysis (PCA),
independent component analysis (ICA), canonical correlation analysis (CCA), and
factor analysis (FA) identify latent directions (or loadings) either ordered or
unordered. The data is then projected onto the latent directions to obtain
their projected representations (or scores). For example, PCA solvers usually
rank the principal directions by explaining the most to least variance, while
ICA solvers usually return independent directions unordered and often with
single sources spread across multiple directions as multiple sub-sources, which
is of severe detriment to their usability and interpretability.
This paper proposes a general framework to enhance latent space
representations for improving the interpretability of linear latent spaces.
Although the concepts in this paper are language agnostic, the framework is
written in Python. This framework automates the clustering and ranking of
latent vectors to enhance the latent information per latent vector, as well as,
the interpretation of latent vectors. Several innovative enhancements are
incorporated including latent ranking (LR), latent scaling (LS), latent
clustering (LC), and latent condensing (LCON).
For a specified linear latent variable model, LR ranks latent directions
according to a specified metric, LS scales latent directions according to a
specified metric, LC automatically clusters latent directions into a specified
number of clusters, while, LCON automatically determines an appropriate number
of clusters into which to condense the latent directions for a given metric.
Additional functionality of the framework includes single-channel and
multi-channel data sources, data preprocessing strategies such as Hankelisation
to seamlessly expand the applicability of linear latent variable models (LLVMs)
to a wider variety of data.
The effectiveness of LR, LS, and LCON are showcased on two crafted
foundational problems with two applied latent variable models, namely, PCA and
ICA.

本文提出了一个通用框架来增强线性潜在空间表示，以提高其可解释性和可用性，并展示了该框架在 PCA 和 ICA 上的效果。

潜空间明晰度与解释性增强（LS-PIE）框架

Latent Space Perspicacity and Interpretation Enhancement (LS-PIE)  Framework

Dimensionality reduction techniques play an essential role in data analytics,
signal processing and machine learning. Dimensionality reduction is usually
performed in a preprocessing stage that is separate from subsequent data
analysis, such as clustering or classification. Finding reduced-dimension
representations that are well-suited for the intended task is more appealing.
This paper proposes a joint factor analysis and latent clustering framework,
which aims at learning cluster-aware low-dimensional representations of matrix
and tensor data. The proposed approach leverages matrix and tensor
factorization models that produce essentially unique latent representations of
the data to unravel latent cluster structure -- which is otherwise obscured
because of the freedom to apply an oblique transformation in latent space. At
the same time, latent cluster structure is used as prior information to enhance
the performance of factorization. Specific contributions include several
custom-built problem formulations, corresponding algorithms, and discussion of
associated convergence properties. Besides extensive simulations, real-world
datasets such as Reuters document data and MNIST image data are also employed
to showcase the effectiveness of the proposed approaches.

本文提出了一种联合因子分析和潜在聚类的框架，旨在学习矩阵和张量数据的聚类感知低维表示。该方法利用矩阵和张量分解模型来揭示潜在聚类结构，并通过潜在聚类结构作为先验信息来提高因子分解的性能。