164 research outputs found
Low-Rank Matrices on Graphs: Generalized Recovery & Applications
Many real world datasets subsume a linear or non-linear low-rank structure in
a very low-dimensional space. Unfortunately, one often has very little or no
information about the geometry of the space, resulting in a highly
under-determined recovery problem. Under certain circumstances,
state-of-the-art algorithms provide an exact recovery for linear low-rank
structures but at the expense of highly inscalable algorithms which use nuclear
norm. However, the case of non-linear structures remains unresolved. We revisit
the problem of low-rank recovery from a totally different perspective,
involving graphs which encode pairwise similarity between the data samples and
features. Surprisingly, our analysis confirms that it is possible to recover
many approximate linear and non-linear low-rank structures with recovery
guarantees with a set of highly scalable and efficient algorithms. We call such
data matrices as \textit{Low-Rank matrices on graphs} and show that many real
world datasets satisfy this assumption approximately due to underlying
stationarity. Our detailed theoretical and experimental analysis unveils the
power of the simple, yet very novel recovery framework \textit{Fast Robust PCA
on Graphs
Fast Robust PCA on Graphs
Mining useful clusters from high dimensional data has received significant
attention of the computer vision and pattern recognition community in the
recent years. Linear and non-linear dimensionality reduction has played an
important role to overcome the curse of dimensionality. However, often such
methods are accompanied with three different problems: high computational
complexity (usually associated with the nuclear norm minimization),
non-convexity (for matrix factorization methods) and susceptibility to gross
corruptions in the data. In this paper we propose a principal component
analysis (PCA) based solution that overcomes these three issues and
approximates a low-rank recovery method for high dimensional datasets. We
target the low-rank recovery by enforcing two types of graph smoothness
assumptions, one on the data samples and the other on the features by designing
a convex optimization problem. The resulting algorithm is fast, efficient and
scalable for huge datasets with O(nlog(n)) computational complexity in the
number of data samples. It is also robust to gross corruptions in the dataset
as well as to the model parameters. Clustering experiments on 7 benchmark
datasets with different types of corruptions and background separation
experiments on 3 video datasets show that our proposed model outperforms 10
state-of-the-art dimensionality reduction models. Our theoretical analysis
proves that the proposed model is able to recover approximate low-rank
representations with a bounded error for clusterable data
Graph Learning and Its Applications: A Holistic Survey
Graph learning is a prevalent domain that endeavors to learn the intricate
relationships among nodes and the topological structure of graphs. These
relationships endow graphs with uniqueness compared to conventional tabular
data, as nodes rely on non-Euclidean space and encompass rich information to
exploit. Over the years, graph learning has transcended from graph theory to
graph data mining. With the advent of representation learning, it has attained
remarkable performance in diverse scenarios, including text, image, chemistry,
and biology. Owing to its extensive application prospects, graph learning
attracts copious attention from the academic community. Despite numerous works
proposed to tackle different problems in graph learning, there is a demand to
survey previous valuable works. While some researchers have perceived this
phenomenon and accomplished impressive surveys on graph learning, they failed
to connect related objectives, methods, and applications in a more coherent
way. As a result, they did not encompass current ample scenarios and
challenging problems due to the rapid expansion of graph learning. Different
from previous surveys on graph learning, we provide a holistic review that
analyzes current works from the perspective of graph structure, and discusses
the latest applications, trends, and challenges in graph learning.
Specifically, we commence by proposing a taxonomy from the perspective of the
composition of graph data and then summarize the methods employed in graph
learning. We then provide a detailed elucidation of mainstream applications.
Finally, based on the current trend of techniques, we propose future
directions.Comment: 20 pages, 7 figures, 3 table
Recommending on graphs: a comprehensive review from a data perspective
Recent advances in graph-based learning approaches have demonstrated their
effectiveness in modelling users' preferences and items' characteristics for
Recommender Systems (RSS). Most of the data in RSS can be organized into graphs
where various objects (e.g., users, items, and attributes) are explicitly or
implicitly connected and influence each other via various relations. Such a
graph-based organization brings benefits to exploiting potential properties in
graph learning (e.g., random walk and network embedding) techniques to enrich
the representations of the user and item nodes, which is an essential factor
for successful recommendations. In this paper, we provide a comprehensive
survey of Graph Learning-based Recommender Systems (GLRSs). Specifically, we
start from a data-driven perspective to systematically categorize various
graphs in GLRSs and analyze their characteristics. Then, we discuss the
state-of-the-art frameworks with a focus on the graph learning module and how
they address practical recommendation challenges such as scalability, fairness,
diversity, explainability and so on. Finally, we share some potential research
directions in this rapidly growing area.Comment: Accepted by UMUA
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