Principal components analysis (PCA) is a well-known technique for
approximating a tabular data set by a low rank matrix. Here, we extend the idea
of PCA to handle arbitrary data sets consisting of numerical, Boolean,
categorical, ordinal, and other data types. This framework encompasses many
well known techniques in data analysis, such as nonnegative matrix
factorization, matrix completion, sparse and robust PCA, $k$-means, $k$-SVD,
and maximum margin matrix factorization. The method handles heterogeneous data
sets, and leads to coherent schemes for compressing, denoising, and imputing
missing entries across all data types simultaneously. It also admits a number
of interesting interpretations of the low rank factors, which allow clustering
of examples or of features. We propose several parallel algorithms for fitting
generalized low rank models, and describe implementations and numerical
results.

本文将 PCA 技术扩展到处理包含数字、布尔、分类、有序等多种数据类型的任意数据集，提出了一种处理异构数据集的通用低秩模型，并为其提供了一些基于并行算法的实现。

广义低秩模型

Generalized Low Rank Models

Scientific practice typically involves repeatedly studying a system, each
time trying to unravel a different perspective. In each study, the scientist
may take measurements under different experimental conditions (interventions,
manipulations, perturbations) and measure different sets of quantities
(variables). The result is a collection of heterogeneous data sets coming from
different data distributions. In this work, we present algorithm COmbINE, which
accepts a collection of data sets over overlapping variable sets under
different experimental conditions; COmbINE then outputs a summary of all causal
models indicating the invariant and variant structural characteristics of all
models that simultaneously fit all of the input data sets. COmbINE converts
estimated dependencies and independencies in the data into path constraints on
the data-generating causal model and encodes them as a SAT instance. The
algorithm is sound and complete in the sample limit. To account for conflicting
constraints arising from statistical errors, we introduce a general method for
sorting constraints in order of confidence, computed as a function of their
corresponding p-values. In our empirical evaluation, COmbINE outperforms in
terms of efficiency the only pre-existing similar algorithm; the latter
additionally admits feedback cycles, but does not admit conflicting constraints
which hinders the applicability on real data. As a proof-of-concept, COmbINE is
employed to co-analyze 4 real, mass-cytometry data sets measuring
phosphorylated protein concentrations of overlapping protein sets under 3
different interventions.

该论文提出了一种算法，称为 COmbINE，可以处理来自不同数据分布下的重叠变量集合的数据集合，将数据的依赖关系和独立性转化为路径约束，并将它们编码为一个 SAT 实例，并将所有模型的不变和变异结构特征作为输出，方便地进行分析研究。

基于约束的多重干预交叠变量集合因果发现

Constraint-based Causal Discovery from Multiple Interventions over  Overlapping Variable Sets

Joint analysis of data from multiple sources has the potential to improve our
understanding of the underlying structures in complex data sets. For instance,
in restaurant recommendation systems, recommendations can be based on rating
histories of customers. In addition to rating histories, customers' social
networks (e.g., Facebook friendships) and restaurant categories information
(e.g., Thai or Italian) can also be used to make better recommendations. The
task of fusing data, however, is challenging since data sets can be incomplete
and heterogeneous, i.e., data consist of both matrices, e.g., the person by
person social network matrix or the restaurant by category matrix, and
higher-order tensors, e.g., the "ratings" tensor of the form restaurant by meal
by person.
In this paper, we are particularly interested in fusing data sets with the
goal of capturing their underlying latent structures. We formulate this problem
as a coupled matrix and tensor factorization (CMTF) problem where heterogeneous
data sets are modeled by fitting outer-product models to higher-order tensors
and matrices in a coupled manner. Unlike traditional approaches solving this
problem using alternating algorithms, we propose an all-at-once optimization
approach called CMTF-OPT (CMTF-OPTimization), which is a gradient-based
optimization approach for joint analysis of matrices and higher-order tensors.
We also extend the algorithm to handle coupled incomplete data sets. Using
numerical experiments, we demonstrate that the proposed all-at-once approach is
more accurate than the alternating least squares approach.

本文提出了一种耦合矩阵和张量分解 (CMTF) 方法，通过 all-at-once 优化的方法来融合包含不同矩阵和高阶张量的异构数据集，并可以处理不完整数据集，实现更精准的数据分析。