We study the estimation of a planted signal hidden in a recently introduced
nested matrix-tensor model, which is an extension of the classical spiked
rank-one tensor model, motivated by multi-view clustering. Prior work has
theoretically examined the performance of a tensor-based approach, which relies
on finding a best rank-one approximation, a problem known to be computationally
hard. A tractable alternative approach consists in computing instead the best
rank-one (matrix) approximation of an unfolding of the observed tensor data,
but its performance was hitherto unknown. We quantify here the performance gap
between these two approaches, in particular by deriving the precise algorithmic
threshold of the unfolding approach and demonstrating that it exhibits a
BBP-type transition behavior. This work is therefore in line with recent
contributions which deepen our understanding of why tensor-based methods
surpass matrix-based methods in handling structured tensor data.

在本研究中，我们研究了嵌套矩阵 - 张量模型中隐藏的嵌入信号的估计，该模型是经过拓展的经典脉冲秩一张量模型，其动机来源于多视图聚类。我们量化了在通过计算观测张量数据的展开的最佳秩一（矩阵）近似而不是寻找最佳秩一近似的情况下这两种方法之间的性能差距，特别是通过推导出展开方法的精确算法门限并展示其呈现 BBP 类型的转换行为。因此，这项工作与最近的研究成果相一致，深化了我们对为什么张量方法优于矩阵方法处理结构化张量数据的理解。

多视角聚类中嵌套矩阵 - 张量模型的性能差距

Performance Gaps in Multi-view Clustering under the Nested Matrix-Tensor  Model

Machine learning (ML) and tensor-based methods have been of significant
interest for the scientific community for the last few decades. In a previous
work we presented a novel tensor-based system identification framework to ease
the computational burden of tensor-only architectures while still being able to
achieve exceptionally good performance. However, the derived approach only
allows to process real-valued problems and is therefore not directly applicable
on a wide range of signal processing and communications problems, which often
deal with complex-valued systems. In this work we therefore derive two new
architectures to allow the processing of complex-valued signals, and show that
these extensions are able to surpass the trivial, complex-valued extension of
the original architecture in terms of performance, while only requiring a
slight overhead in computational resources to allow for complex-valued
operations.

本文介绍两种新架构来处理复数信号，以提高系统识别性能，同时只需略微增加计算资源投入。

低秩张量分解下的复数自适应系统辨识

Complex-valued Adaptive System Identification via Low-Rank Tensor  Decomposition

Graphs emerge in almost every real-world application domain, ranging from
online social networks all the way to health data and movie viewership
patterns. Typically, such real-world graphs are big and dynamic, in the sense
that they evolve over time. Furthermore, graphs usually contain multi-aspect
information i.e. in a social network, we can have the "means of communication"
between nodes, such as who messages whom, who calls whom, and who comments on
whose timeline and so on.
How can we model and mine useful patterns, such as communities of nodes in
that graph, from such multi-aspect graphs? How can we identify dynamic patterns
in those graphs, and how can we deal with streaming data, when the volume of
data to be processed is very large? In order to answer those questions, in this
thesis, we propose novel tensor-based methods for mining static and dynamic
multi-aspect graphs. In general, a tensor is a higher-order generalization of a
matrix that can represent high-dimensional multi-aspect data such as
time-evolving networks, collaboration networks, and spatio-temporal data like
Electroencephalography (EEG) brain measurements.
The thesis is organized in two synergistic thrusts: First, we focus on static
multi-aspect graphs, where the goal is to identify coherent communities and
patterns between nodes by leveraging the tensor structure in the data. Second,
as our graphs evolve dynamically, we focus on handling such streaming updates
in the data without having to re-compute the decomposition, but incrementally
update the existing results.

基于张量的方法可以对从社交网络到健康数据和电影观看模式等各种领域中涌现出的大规模、多层面、动态的图数据进行挖掘，包括发现静态和动态图数据中的凝聚社区和模式。

使用可扩展的流式张量分解对多方面图进行建模和挖掘

Modeling and Mining Multi-Aspect Graphs With Scalable Streaming Tensor Decomposition

The data in many disciplines such as social networks, web analysis, etc. is
link-based, and the link structure can be exploited for many different data
mining tasks. In this paper, we consider the problem of temporal link
prediction: Given link data for times 1 through T, can we predict the links at
time T+1? If our data has underlying periodic structure, can we predict out
even further in time, i.e., links at time T+2, T+3, etc.? In this paper, we
consider bipartite graphs that evolve over time and consider matrix- and
tensor-based methods for predicting future links. We present a weight-based
method for collapsing multi-year data into a single matrix. We show how the
well-known Katz method for link prediction can be extended to bipartite graphs
and, moreover, approximated in a scalable way using a truncated singular value
decomposition. Using a CANDECOMP/PARAFAC tensor decomposition of the data, we
illustrate the usefulness of exploiting the natural three-dimensional structure
of temporal link data. Through several numerical experiments, we demonstrate
that both matrix- and tensor-based techniques are effective for temporal link
prediction despite the inherent difficulty of the problem. Additionally, we
show that tensor-based techniques are particularly effective for temporal data
with varying periodic patterns.

本文考虑了基于时间的链接预测问题，研究了基于矩阵和张量的方法，并探讨了在具有不同周期结构的数据中利用张量分解的优点。通过多个实验，证明了这两种方法对于时间链接预测具有很好的效果。