Preface
Our world is full of data, and these data often appear in high-dimensional structures,
with each dimension describing a unique attribute. Examples include data in social
sciences, medicines, pharmacology, and environmental monitoring, just to name a
few. To make sense of the multi-dimensional data, advanced computational tools,
which directly work with tensor rather than first converting a tensor to a matrix, are
needed to unveil the hidden patterns of the data. This is where tensor decomposi-
tion models come into play. Due to the remarkable representation capability, tensor
decomposition models have led to state-of-the-art performances in many domains,
including social network mining, image processing, array signal processing, and
wireless communications.
Previous research on tensor decompositions mainly approached from an optimiza-
tion perspective, which unfortunately does not come with the capability of tensor
rank learning and requires heavy hyper-parameter tuning. While these two tasks are
important in complexity control and avoiding overfitting, they are often overlooked
or downplayed in current research, and assumed can be achieved by trivial opera-
tions, or somehow can be obtained from other methods. In reality, estimating the
tensor rank and a proper set of hyper-parameters usually involve exhaustive search.
This requires running the same algorithm many times, effectively increasing the
computational complexity in actual model deployment.
Another path for model learning is Bayesian methods. They provide a natural
recipe for the integration of tensor rank learning, automatic hyper-parameter deter-
mination, and tensor decomposition. Due to this unique capability, Bayesian models
and inference trigger a recent interest in tensor decompositions for signal processing
and machine learning. From these recent works, Bayesian models show comparable
or even better performance than optimization-based counterparts.
However, Bayesian methods are very different from optimization methods, with
the former learning distributions of the unknown parameters, and the latter learning
a point estimate. The process of building the models and inference algorithm deriva-
tions are fundamentally different as well. This leads to a barrier between the two
groups of researchers working on similar problems but starting from different
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