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Tensor networks (TNs) have been gaining interest as multiway data analysis tools owing to their ability to tackle the curse of dimensionality and to represent tensors as smaller-scale interconnections of their intrinsic features. However,…
Originating in quantum physics, tensor networks (TNs) have been widely adopted as exponential machines and parameter decomposers for recognition tasks. Typical TN models, such as Matrix Product States (MPS), have not yet achieved successful…
The era of exascale computing opens new venues for innovations and discoveries in many scientific, engineering, and commercial fields. However, with the exaflops also come the extra-large high-dimensional data generated by high-performance…
Tensor train (TT) decomposition is a powerful representation for high-order tensors, which has been successfully applied to various machine learning tasks in recent years. However, since the tensor product is not commutative, permutation of…
Multipartite quantum scenarios are a significant and challenging resource in quantum information science. Tensors provide a powerful framework for representing multipartite quantum systems. In this work, we introduce the role of…
Tensor networks provide compact and scalable representations of high-dimensional data, enabling efficient computation in fields such as quantum physics, numerical partial differential equations (PDEs), and machine learning. This paper…
In this paper, we introduce a type of tensor neural network. For the first time, we propose its numerical integration scheme and prove the computational complexity to be the polynomial scale of the dimension. Based on the tensor product…
Tensor networks, a class of variational quantum many-body wave functions have attracted considerable research interest across many disciplines, including classical machine learning. Recently, Aizpurua et al. demonstrated explainable anomaly…
Many real-world datasets are represented as tensors, i.e., multi-dimensional arrays of numerical values. Storing them without compression often requires substantial space, which grows exponentially with the order. While many tensor…
Tensor Networks are non-trivial representations of high-dimensional tensors, originally designed to describe quantum many-body systems. We show that Tensor Networks are ideal vehicles to connect quantum mechanical concepts to machine…
Tensor network methods provide a scalable solution to represent high-dimensional data. However, their efficacy is often limited by static, expert-defined structures that fail to adapt to evolving data correlations. We address this…
Categorical quantum mechanics (CQM) and the theory of quantum groups rely heavily on the use of structures that have both an algebraic and co-algebraic component, making them well-suited for manipulation using diagrammatic techniques.…
Producing large complex simulation datasets can often be a time and resource consuming task. Especially when these experiments are very expensive, it is becoming more reasonable to generate synthetic data for downstream tasks. Recently,…
We solve high-dimensional steady-state Fokker-Planck equations on the whole space by applying tensor neural networks. The tensor networks are a linear combination of tensor products of one-dimensional feedforward networks or a linear…
Classically simulating quantum circuits is crucial when developing or testing quantum algorithms. Due to the underlying exponential complexity, efficient data structures are key for performing such simulations. To this end, tensor networks…
This survey provides an overview of common applications, both implicit and explicit, of "tensors" and "tensor products" in the fields of data science and statistics. One goal is to reconcile seemingly distinct usages of the term "tensor" in…
Tensor networks provide a powerful framework for compressing multi-dimensional data. The optimal tensor network structure for a given data tensor depends on both data characteristics and specific optimality criteria, making tensor network…
Since its introduction by Gauss, Matrix Algebra has facilitated understanding of scientific problems, hiding distracting details and finding more elegant and efficient ways of computational solving. Today's largest problems, which often…
Tensor networks are efficient representations of high-dimensional tensors which have been very successful for physics and mathematics applications. We demonstrate how algorithms for optimizing such networks can be adapted to supervised…
Deep neural networks (DNNs) have enabled impressive breakthroughs in various artificial intelligence (AI) applications recently due to its capability of learning high-level features from big data. However, the current demand of DNNs for…