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Block Tensors and their Role in Multidimensional Data Analysis

Hendrik Kosmidis^{* *}Correspondence: Hendrik Kosmidis, Department of Applied Mathematics, University of Nottingham, Nottingham, United Kingdom, Email:

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Description

Block tensors are mathematical constructs that extend the concept of block matrices into higher dimensions. They are pivotal in various disciplines, including applied mathematics, physics and computer science due to their ability to represent and process multi-dimensional data efficiently. By leveraging the inherent structure of block tensors computational methods can be significantly optimized enabling faster and more accurate solutions to complex problems.

A block tensor is a higher-order tensor partitioned into smaller sub-tensors or "blocks." This structure introduces an additional layer of organization allowing the tensor to capture complex relationships within data. For example, a third-order tensor can be divided into blocks along each of its three dimensions each block containing a smaller subset of the data.

Advantages of using block tensors

Below are some key advantages:

Data organization: Block tensors facilitate a hierarchical representation of data making it easier to manage large datasets.

Computational efficiency: Algorithms can exploit the block structure to reduce complexity particularly for sparse or structured data.

Modularity: Operations can often be performed block-by-block simplifying implementation.

Scalability: Block tensors are well-suited for parallel computation as independent blocks can be processed simultaneously.

Applications of block tensors

Block tensors are widely used in various scientific and engineering fields due to their ability to represent structured and multidimensional data efficiently.

Quantum computing and physics: Block tensors are extensively used in quantum mechanics especially in tensor network representations such as Matrix Product States (MPS) and Tensor Train (TT) decompositions. These representations help simulate quantum systems efficiently handling the exponential growth of the state space.

Machine learning: In deep learning block tensors find applications in tensor decomposition techniques which reduce the dimensionality of weights and inputs in neural networks. This can significantly decrease the memory and computational overhead enabling real-time applications on resource-constrained devices.

Computational biology: Block tensors are employed in bioinformatics to analyse high-dimensional datasets such as those arising in genomics and proteomics. They allow for efficient storage and computation facilitating insights into biological processes.

Signal processing: Multidimensional signal processing often uses block tensors to model data such as videos, hyperspectral images and communication signals. The block structure aids in denoising compression and feature extraction.

Computational techniques

The efficient handling of block tensors requires specialized algorithms and tools. Key computational techniques include:

Tensor decompositions: Tensor decomposition methods such as Canonical Polyadic (CP) and Tucker decompositions can be adapted to block tensors. These decompositions approximate the tensor using a smaller set of factors preserving the block structure.

Block tensor operations: Operations like addition, multiplication and contraction are fundamental to block tensor computations. These operations are optimized by exploiting the block structure reducing the overall computational cost.

Parallel and distributed computing: Block tensors are naturally well-suited for parallel and distributed computing. Frameworks such as MPI and OpenMP facilitate the allocation of individual blocks across multiple processors making it possible to efficiently manage and process large-scale tensors.

Software libraries: Several software libraries support block tensor computations including Tensor flow, PyTorch and specialized tensor computation frameworks like Tensely and Scikit-Tensor. These libraries provide pre-implemented routines for tensor decomposition, manipulation and visualization.

Block tensors represent a potential mathematical framework for managing and processing multidimensional data in a structured and efficient manner. By partitioning data into smaller manageable sub-tensors block tensors provide a hierarchical structure that improve data organization computational efficiency and scalability. Their modular nature makes them particularly advantageous for tasks requiring parallel computation as independent blocks can be processed simultaneously significantly reducing runtime and enabling the handling of large-scale datasets.

The versatility of block tensors extends across numerous applications from quantum mechanics to machine learning, computational biology and signal processing. In quantum physics, block tensors facilitate the efficient simulation of complex systems. In machine learning they reduce computational overhead enabling real-time deployment of models. Similarly, in computational biology and signal processing block tensors optimize the storage and analysis of high-dimensional data leading to critical insights and improved data modelling.