Topological data analysis (TDA) is a rapidly evolving field in applied mathematics and data science that leverages tools from topology to uncover robust, shape-driven, and explainable insights in complex datasets. The main workhorse is persistent homology, a technique rooted in algebraic topology. Paired with topological deep learning (TDL) or topological machine learning, persistent homology has achieved tremendous success in a wide variety of applications in science, engineering, medicine, and industry. However, persistent homology has many limitations due to its high-level abstraction, insensitivity to non-topological changes, and restriction to point cloud data. This paper presents a comprehensive review of TDA and TDL beyond persistent homology. It analyzes how persistent topological Laplacians and Dirac operators provide spectral representations to capture both topological invariants and homotopic evolution. Other formulations are presented in terms of sheaf theory, Mayer topology, and interaction topology. For data on differentiable manifolds, techniques rooted in differential topology, such as persistent de Rham cohomology, persistent Hodge Laplacian, and Hodge decomposition, are reviewed. For one-dimensional (1D) curves embedded in 3-space, approaches from geometric topology are discussed, including multiscale Gauss-link integrals, persistent Jones polynomials, and persistent Khovanov homology. This paper further discusses the appropriate selection of topological tools for different input data, such as point clouds, sequential data, data on manifolds, curves embedded in 3-space, and data with additional non-geometric information. A review is also given of various topological representations, software packages, and machine learning vectorizations. Finally, this review ends with concluding remarks.

As a generalization of graphs and simplicial complexes, hypergraphs have various applications in chemistry, biology, computer science and data science. Recently, the first well-defined hypergraph-based embedded homology and persistent (embedded) homology models have been proposed and achieved great success in drug design. However, the further applications of these models have been significantly hindered by the lack of efficient computational algorithms. In this paper, we propose an algorithm for embedded homology and persistent (embedded) homology of hypergraphs over field coefficient. One of the key issues for hypergraph-based homology models is the absence of proper boundary operators as in traditional simplicial complex-based models. A supremum/infimum chain complex has been proposed in the embedded homology model with well-defined boundary operators. Here we give detailed algorithms for transforming a filtered hypergraph to its filtered infimum chain complex and supremum chain complex. In this way, the general algorithm for chain complex can be used directly on the supremum/infimum chain complexes. Our algorithm has been validated on hypergraph models of protein structures from the protein data bank. It has been found that our algorithm is very robust and efficient.

Aiming at the problem that the high classification feature dimensionality of the back propagation neural network (BPNN) leads to slow convergence speed and the initial weight and threshold sensitivity of the BPNN lead to the problem of easy convergence to the local optimum. A novel BPNN optimized by rough set and particle swarm algorithm (RS-PSO-BPNN) for remanufacturing service provider classification and selection is proposed. First, the attribute reduction method of rough set theory is used to preprocess the classification features of remanufacturing service providers, redundant attributes are deleted from the decision table, and the input feature dimension is reduced; then the PSO algorithm is used to optimize the network Initial weight and threshold. Finally, the proposed method is used for the selection and optimization of remanufacturing service providers. The results show that the proposed RS-PSO-BPNN has higher classification accuracy and efficiency for the problem, which provides scientific decision supports for remanufacturing service provider selection.