Method for spatial domain identification in spatial transcriptomics based on hyperbolic graph neural networks
By constructing a spatial graph based on a hyperbolic graph neural network and introducing a multi-manifold encoder, using attention mechanisms and Pareto optimization, combined with a Gaussian mixture model clustering algorithm, the problem of insufficient hierarchical relationship resolution in spatial transcriptome data analysis in existing technologies is solved, achieving higher recognition accuracy and real-time performance.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- JILIN UNIVERSITY
- Filing Date
- 2024-04-15
- Publication Date
- 2026-07-07
AI Technical Summary
Existing spatial transcriptome data analysis methods cannot fully resolve the internal structural characteristics of high-dimensional data, especially hierarchical relationships, leading to data distortion. Furthermore, their reliance on prior human knowledge results in poor performance when the data distribution does not meet certain criteria.
We employ a hyperbolic graph neural network approach, which involves constructing a spatial graph, introducing multi-manifold encoders in Euclidean and hyperbolic spaces, fusing features using an attention mechanism, optimizing the balance loss function through Pareto, and combining a Gaussian mixture model clustering algorithm to identify the spatial domain.
It improves the accuracy and real-time performance of spatial domain recognition, reduces reliance on expert knowledge, shortens recognition time, and has higher accuracy than existing models.
Smart Images
Figure CN118280442B_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of data processing technology, and particularly relates to a spatial domain identification method in spatial transcriptomics based on hyperbolic graph neural networks. Background Technology
[0002] Spatial transcriptomics technology captures gene expression information while preserving the spatial location information of each site. This characteristic gives it a significant advantage in tasks such as describing cellular spatial patterns and identifying information exchange between cells. However, gene expression information is highly dimensional; in reality, the features of each site can reach tens of thousands of dimensions, making the analysis of the underlying data patterns a challenge.
[0003] Existing methods can be broadly categorized into three types: statistical / probabilistic model-based methods, deep learning-based methods, and graph neural network-based methods. Statistical / probabilistic model-based methods, such as using Hidden Markov Random Fields (HMRFs), apply prior knowledge that "first-order neighbors should be more similar" to smooth site features. Deep learning-based methods use tissue image slices and spatial location information to smooth features. A major limitation of the first two types is that they directly determine the influence weights between different sites based on artificially proposed prior knowledge, which cannot be adjusted subsequently. When the data distribution does not satisfy the prior knowledge, the effectiveness is significantly reduced. Graph neural network-based methods, by constructing a neighbor node relationship graph, use graph neural networks, especially graph convolutional networks, to adaptively learn the influence weights between neighbor nodes during training. Compared to the first two types, they are more flexible and perform better and more stably on different datasets. Although graph structure modeling of spatial transcriptome data can well represent its spatial patterns, existing research only focuses on how to embed high-dimensional data into low-dimensional Euclidean space, failing to fully analyze the internal structural characteristics of the data, especially hierarchical relationships, resulting in data distortion. Hyperbolic space, as a manifold space with constant negative curvature, better preserves complex structural relationships due to its non-uniform metric tensor. Furthermore, multi-view learning can fuse information from different sources, resulting in fused feature vectors containing more information and thus achieving superior and robust performance. Summary of the Invention
[0004] The purpose of this invention is to provide a spatial domain identification method in spatial transcriptomics based on hyperbolic graph neural networks, aiming to solve the problems mentioned in the background art.
[0005] To achieve the above objectives, the present invention provides the following technical solution:
[0006] Spatial domain identification methods in spatial transcriptomics based on hyperbolic graph neural networks include the following steps:
[0007] Step S1: Construct a spatial map;
[0008] Step S2: Introduce operations of multiple manifold encoders in Euclidean and hyperbolic spaces for spatial transcriptome datasets;
[0009] Step S3: Use an attention mechanism to fuse features from multiple manifolds;
[0010] Step S4: Multiple loss functions constructed in step S3 are optimized using Pareto to balance the differences between the original features and the decoded reconstructed features.
[0011] Step S5: Identify the spatial domain using a clustering algorithm based on Gaussian mixture model.
[0012] Furthermore, in step S2, the manifold encoder in Euclidean space is an Euclidean encoder, and the manifold encoder in hyperbolic space includes a Poincare encoder and a Lorentz encoder.
[0013] Furthermore, the specific process of step S2 is as follows:
[0014] Step S21, Hyperbolic Initialization Layer: The exponential function exp is used to project the node features in Euclidean space onto a specific hyperbolic space model, including the Poincaré ball model and the Lorentz model. The logarithmic function log is used to perform the opposite operation.
[0015] The definitions of exponential and logarithmic functions under the Poincaré ball model are as follows:
[0016]
[0017]
[0018] Where c represents curvature. Let denote Möbius, where λ is a coefficient, v is a vector in the original space, y is a vector in hyperbolic space, and x represents the projection position.
[0019] The definitions of exponential and logarithmic functions under the Lorentz model are as follows:
[0020]
[0021]
[0022] in,<x,y> Represents the dot product of vectors. Represents operations in the Lorentz space;
[0023] The feature vectors projected onto the Poincaré ball model and the Lorentz model are calculated using the following formula:
[0024]
[0025]
[0026] in, Let represent the feature vectors in Euclidean space, Poincaré space, and Lorentz space, respectively. This indicates that the projection starts from the origin of the coordinate system.
[0027] Step S22, Hyperbolic Feature Transformation: The features in hyperbolic space are projected onto the corresponding tangent space using a logarithmic function. Matrix-vector multiplication in Euclidean space is then performed on the tangent space. Finally, the feature vectors are projected onto the corresponding hyperbolic space using an exponential function, completing the matrix-vector multiplication operation. The process of matrix-vector multiplication in Poincaré and Lorentz spaces is shown in the following equations:
[0028]
[0029]
[0030] Where M represents a matrix and n represents the feature dimension of the original Euclidean space;
[0031] Implement matrix-vector addition using parallel transmission to unify the representation of eigenvectors under the Poincaré ball model and the Lorentz model. The specific process is as follows:
[0032]
[0033] in, The bias term is represented; the parallel transmission definitions under the Poincaré ball model and the Lorentz model are as follows:
[0034]
[0035]
[0036] Where gyr represents the gyroscope transformation operation;
[0037] Step S23, Hyperbolic Neighbor Aggregation: In a graph neural network, the adjacency matrix is calculated as follows:
[0038]
[0039] Among them, A ijd represents the weight of the edge between node i and node j. i d represents the degree of node i. j Indicates the degree of node j;
[0040] Message aggregation is achieved using an addition strategy:
[0041]
[0042] in, Indicates that node i is in space The eigenvectors below, Indicate the neighboring nodes of node i;
[0043] Step S24, Nonlinear Activation: First, project the hyperbolic feature vector onto the corresponding tangent space, perform nonlinear activation in the tangent space, and then project it back. The process is as follows:
[0044]
[0045]
[0046] in, This indicates that when the input curvature is c l-1 The output curvature is c l Activation function;
[0047] The overall process of obtaining the hyperbolic graph neural network layers is represented as follows:
[0048]
[0049]
[0050]
[0051] in, express The eigenvector W of node i in the l-th layer of space after eigentransformation l This represents the weight coefficient of the l-th layer. Let b represent the feature vector of node i in layer (l-1), where l represents the layer number of the network, and b represents the feature vector of node i in layer (l-1). l This represents the bias weight coefficient of the l-th layer. This represents the inactive feature vector of node i in the l-th layer.
[0052] Furthermore, the specific operation of step S3 is as follows:
[0053] An attention coefficient vector v is multiplied by the concatenated vector of vectors from each manifold to obtain an attention score matrix B. This score matrix is then normalized. The calculation process for the score matrix is as follows:
[0054]
[0055]
[0056] Where T represents the matrix transpose operation, Let represent the eigenvectors of node i in Euclidean space, Poincaré space, and Lorentz space, respectively.
[0057] Update the feature using a damping coefficient λ:
[0058]
[0059] Where A represents the adjacency matrix. Represents a matrix vector in Poincaré space.
[0060] Furthermore, the specific operation of step S4 is as follows:
[0061] For a scenario involving two vectors, assuming the sum of their weights is 1, the optimization objective is:
[0062]
[0063] Where, θ g θ represents the global weight parameter. 1 This represents the local weight parameters applicable to the first task. This indicates the first loss function pair The resulting gradient vector, The second loss function represents the pair The resulting gradient vector, θ 2 This represents the local weight parameters applicable to the second task;
[0064] The weight coefficient γ of the first loss function in the above formula is calculated as follows:
[0065]
[0066] Where T represents the matrix transpose operation;
[0067] For scenarios with multiple vectors, the Frank-Wolfe algorithm is used to calculate the weight coefficients of each loss function:
[0068] α:=(1-η)·α+η·e t
[0069]
[0070]
[0071] in, This represents graph-structured data, where η represents the increment of the weight coefficient, α represents the cumulative weight coefficient, and e t Let K represent a one-hot vector, where 1 is at position t and 0 is at all other positions. K represents the number of tasks. This represents the k-th task. Let θ represent the loss function for the t-th task. t Let T represent the local weight parameters applicable to the t-th task, and T denote the matrix transpose operation.
[0072] Furthermore, in step S5, the spatial domain is identified using the mclust algorithm, specifically as follows:
[0073] The mclust algorithm assumes that the data distribution is composed of multiple Gaussian distributions. Each Gaussian distribution is considered as a cluster. By feeding the fused low-dimensional features into the mclust algorithm, the number of spatial domains is specified by the actual number of true labels, thus obtaining the actual clustering result, i.e., the spatial domain.
[0074] Compared with the prior art, the beneficial effects of the present invention are:
[0075] This invention is of great significance in computer-aided spatial domain recognition. It not only replaces the time-consuming manual segmentation and annotation of spatial domains by biomedical scientists, shortening the recognition time, but also does not rely on the knowledge level of experts and can control errors to a certain extent, ensuring both real-time performance and accuracy. Furthermore, the accuracy of this invention is superior to existing spatial domain recognition models. Attached Figure Description
[0076] Figure 1 This is a flowchart of the present invention.
[0077] Figure 2 This is a schematic diagram of the spatial diagram for the present invention.
[0078] Figure 3 This is a schematic diagram of the multi-manifold encoder of the present invention.
[0079] Figure 4 This is a schematic diagram illustrating the use of an attention mechanism for multi-manifold fusion in this invention.
[0080] Figure 5 This is a schematic diagram of the Pareto optimization of the two vectors in this invention.
[0081] Figure 6 This is a schematic diagram illustrating the Pareto optimization of multiple vectors in this invention.
[0082] Figure 7 This is a schematic diagram of the spatial domain identification of the present invention.
[0083] Figure 8This is a comparison diagram of the present invention with other SOTA spatial domain identification methods in Example 1.
[0084] Figure 9 This is a comparison chart of the predicted partitioning results and the actual partitioning results in Example 1. Detailed Implementation
[0085] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.
[0086] The specific implementation of the present invention will be described in detail below with reference to specific embodiments.
[0087] like Figure 1 The illustration shows a spatial domain identification method in spatial transcriptomics based on hyperbolic graph neural networks, provided by an embodiment of the present invention, comprising the following steps:
[0088] Step S1: Construct a spatial map;
[0089] In step S1, the spatial graph consists of edges and nodes. The feature vector of each node is the input gene expression information, and the edges are constructed from positional information. In the spatial graph, edges only exist between adjacent nodes. For a given node, its k nearest neighbors are considered, and this distance is defined as the Euclidean distance of the spatial location. The specific construction process is as follows... Figure 2 As shown.
[0090] Step S2: Introduce operations of multiple manifold encoders in Euclidean and hyperbolic spaces for spatial transcriptome datasets;
[0091] In step S2, the manifold encoder in Euclidean space is a manifold encoder, and the manifold encoder in hyperbolic space includes the Poincare encoder and the Lorentz encoder.
[0092] Because the metric tensor in hyperbolic geometry is mutable, matrix-vector addition, matrix-vector multiplication, or nonlinear activation operations cannot be directly performed in hyperbolic space. Therefore, we introduce operations on hyperbolic graph neural networks for use on spatial transcriptome datasets. The overall process is as follows: Figure 3 As shown. Specifically includes:
[0093] Step S21: Hyperbolic initialization layer;
[0094] To input node features from Euclidean space into a hyperbolic graph network, we first need to project them onto a specific hyperbolic space model. Commonly used hyperbolic space models include the Poincaré ball model and the Lorentz model. We use the exponential function `exp` to project the features from Euclidean space onto hyperbolic space, and the logarithmic function `log` to perform the opposite operation. Specifically, the definitions of the exponential and logarithmic functions in the Poincaré ball model are as follows:
[0095]
[0096]
[0097] Where c represents curvature. Let denote Möbius, where λ is a coefficient, v is a vector in the original space, y is a vector in hyperbolic space, and x represents the projection position.
[0098] The definitions of exponential and logarithmic functions under the Lorentz model are as follows:
[0099]
[0100]
[0101] in,<x,y> Represents the dot product of vectors. This represents operations in the Lorentz space.
[0102] The feature vectors projected onto the Poincaré ball model and the Lorentz model are calculated using the following formula:
[0103]
[0104]
[0105] in, Let represent the feature vectors in Euclidean space, Poincaré space, and Lorentz space, respectively. This indicates that the projection starts from the origin of the coordinate system; in order to adapt to the hierarchical characteristics of different types of data, the curvature c is set as a trainable parameter.
[0106] Step S22: Hyperbolic feature conversion;
[0107] To achieve matrix-vector multiplication, this invention employs a solution from hyperbolic graph neural networks. First, a logarithmic function is used to project the features in hyperbolic space onto its tangent space, which is a vector space homeomorphic to Euclidean space. Therefore, matrix-vector multiplication in Euclidean space can be performed on this tangent space. Then, an exponential function is used to project this feature vector back onto the corresponding hyperbolic space, thus completing the matrix-vector multiplication operation. The process of matrix-vector multiplication in Poincaré and Lorentz spaces is shown in the following equations:
[0108]
[0109]
[0110] Where M represents a matrix, and n represents the feature dimension in the original Euclidean space;
[0111] The same approach can be used for matrix-vector addition. This invention uses parallel transmission to unify the representation of the eigenvectors from the two hyperbolic models into a single representation. The specific process is as follows:
[0112]
[0113] in, The bias term is represented; the parallel transmission definitions under the Poincaré ball model and the Lorentz model are as follows:
[0114]
[0115]
[0116] Where gyr represents the gyroscope transformation operation;
[0117] Step S23: Hyperbolic neighbor aggregation;
[0118] In graph neural networks, each node is considered equally important. To mitigate the impact of node degree, the weights are normalized using the node's degree. Furthermore, to avoid oversmoothing, a high-weight self-loop is added to each node. The adjacency matrix is calculated as follows:
[0119]
[0120] Among them, A ij d represents the weight of the edge between node i and node j. i d represents the degree of node i. j Indicates the degree of node j;
[0121] Message aggregation is achieved using an addition strategy:
[0122]
[0123] in, Indicates that node i is in space The eigenvectors below, Indicate the neighboring nodes of node i;
[0124] Step S24: Nonlinear activation;
[0125] First, the hyperbolic feature vector is projected onto the corresponding tangent space, then nonlinear activation is performed in the tangent space, and finally projected back. The process is as follows:
[0126]
[0127]
[0128] in, This indicates that when the input curvature is c l-1 The output curvature is c l Activation function;
[0129] Based on the above operations, the overall process of the hyperbolic graph neural network layer is represented as follows:
[0130]
[0131]
[0132]
[0133] in, express The eigenvector W of node i in the l-th layer of space after eigentransformation l This represents the weight coefficient of the l-th layer. Let b represent the feature vector of node i in layer (l-1), where l represents the layer number of the network, and b represents the feature vector of node i in layer (l-1). l This represents the bias weight coefficient of the l-th layer. This represents the inactive feature vector of node i in the l-th layer.
[0134] Step S3: Use an attention mechanism to fuse features from multiple manifolds;
[0135] To acquire information from different manifolds, this invention uses an attention mechanism to fuse features from multiple manifolds. The entire process is as follows: Figure 4 As shown.
[0136] First, an attention coefficient vector v is multiplied by the concatenated vector of vectors from each manifold to obtain an attention score matrix B. Then, the score matrix is normalized. The calculation process for the score matrix is as follows:
[0137]
[0138]
[0139] Where T represents the matrix transpose operation, Let represent the eigenvectors of node i in Euclidean space, Poincaré space, and Lorentz space, respectively.
[0140] To reduce the risk of overfitting, a damping coefficient λ is used to update the features:
[0141]
[0142] Where A represents the adjacency matrix. Represents a matrix vector in Poincaré space.
[0143] Step S4: Multiple loss functions constructed in step S3 are optimized using Pareto to balance the differences between the original features and the decoded reconstructed features.
[0144] from Figure 4 As can be seen, the differences between the original features and the decoded reconstructed features construct four reconstruction loss functions, and optimizing these losses constitutes a multi-objective optimization problem. Balancing these loss functions has always been a challenge. In this invention, we employ a multi-gradient descent algorithm to determine the weight of each loss function, ultimately achieving a Pareto optimum. Specifically, we design a shared-weight graph convolutional network encoder, which plays two roles: during the push-forward process, this encoder reduces the input features to a lower dimension and further aggregates neighbor information; during the back-out process, the gradients generated by each loss function are recorded. These gradients are used to calculate the weight coefficients of each loss function.
[0145] We treat the gradient generated by each loss function as a vector in the parameter space, and these four vectors form a convex hull. Reaching the Pareto optimum is equivalent to continuously moving in the direction that minimizes the norm of the sum vector. Now consider a simple scenario: there are only two vectors, assuming their weights sum to 1, then the optimization objective is:
[0146]
[0147] Where, θ g θ represents the global weight parameter. 1 This represents the local weight parameters applicable to the first task. This indicates the first loss function pair The resulting gradient vector, The second loss function represents the pair The resulting gradient vector, θ 2 This represents the local weight parameters applicable to the second task;
[0148] To minimize the norm of the sum vector, we considered the following: Figure 5 Based on the three conditions shown, the weight coefficient γ of the first loss function is calculated as follows:
[0149]
[0150] Where T represents the matrix transpose operation;
[0151] For scenarios with multiple vectors, such as Figure 6 As shown, the weight coefficients of each loss function are calculated using the Frank-Wolfe algorithm:
[0152] α:=(1-η)·α+η·e t #(twenty four)
[0153]
[0154]
[0155] in, This represents graph-structured data, where η represents the increment of the weight coefficient, α represents the cumulative weight coefficient, and e t Let K represent a one-hot vector, where 1 is at position t and 0 is at all other positions. K represents the number of tasks. This represents the k-th task. Let θ represent the loss function for the t-th task. t Let T represent the local weight parameters applicable to the t-th task, and T denote the matrix transpose operation.
[0156] Step S5: Identify the spatial domain using a clustering algorithm based on Gaussian mixture model;
[0157] like Figure 7 As shown, this invention uses the mclust algorithm, a clustering algorithm based on Gaussian mixture models, to identify spatial domains. The mclust algorithm assumes that the data distribution is composed of multiple Gaussian distributions, and each of the decomposed Gaussian distributions is considered a cluster. By feeding the fused low-dimensional features into the mclust algorithm, the number of spatial domains is specified by the actual number of true labels, thus obtaining the actual clustering result, i.e., the spatial domain.
[0158] In addition, to achieve greater spatial continuity, this invention also designs an operation to enhance the clustering effect: making each node belong to the class with the largest number of nodes under a specified radius.
[0159] Example 1: Spatial Domain Recognition of DLPFC Dataset;
[0160] The DLPFC dataset is a publicly available dataset that records gene expression and spatial location information from 12 slices of the prefrontal cortex of the brain. Each dataset contains 3460 to 4789 loci and 10725 to 36601 genes. Experts divided each slice into seven layers—Layer 1, Layer 2, Layer 3, Layer 4, Layer 5, Layer 6, and WM—as ground truth labels. Training was stopped after 2000 epochs, and ARI and NMI were used as evaluation metrics. The final model's prediction metrics are as follows: Figure 8 As shown in the figure. The comparison between the predicted partitioning results and the actual partitioning is shown in the figure. Figure 9 As shown.
[0161] As can be seen, in the publicly available 10x Visium technology test on the prefrontal cortex, i.e. the DLPFC dataset, the accuracy of the present invention is superior to existing spatial domain recognition models.
[0162] The working principle of this invention is:
[0163] This spatial domain identification method in spatial transcriptomics based on hyperbolic graph neural networks employs a hyperbolic graph convolutional neural network. It projects features originally in Euclidean space onto hyperbolic space, then performs information transfer and aggregation within hyperbolic space to obtain feature vectors with hyperbolic manifold properties. Simultaneously, a standard graph convolutional neural network embeds the original features into a low-dimensional Euclidean space, and an attention mechanism fuses feature vectors from different manifolds. The fused vector is fed into a decoder structure composed of a multilayer perceptron, resulting in a reconstructed vector with the same shape as the original input. The reconstruction loss is calculated by determining the difference between the original and reconstructed vectors, and this loss is used to train the model until a fit is achieved. Pareto optimization is used to balance multiple loss functions. The low-dimensional features obtained after model fitting are then fed into a model-based clustering algorithm for clustering; each cluster is referred to as a spatial domain.
[0164] The above are merely preferred embodiments of the present invention. It should be noted that those skilled in the art can make several modifications and improvements without departing from the concept of the present invention, and these should also be considered within the scope of protection of the present invention. These modifications and improvements will not affect the effectiveness of the implementation of the present invention or the practicality of the patent.
Claims
1. A spatial domain identification method in spatial transcriptomics based on hyperbolic graph neural networks, characterized in that, Includes the following steps: Step S1: Construct a spatial map; In step S1, the spatial graph consists of edges and nodes; wherein, the feature vector of a node is the input gene expression information, and the edges are constructed from position information; Step S2: Introduce operations of multiple manifold encoders in Euclidean and hyperbolic spaces for spatial transcriptome datasets; In step S2, the manifold encoder in Euclidean space is a manifold encoder, and the manifold encoder in hyperbolic space includes a Poincare encoder and a Lorentz encoder. The specific process of step S2 is as follows: Step S21, Hyperbolic Initialization Layer: The exponential function exp is used to project the node features in Euclidean space onto a specific hyperbolic space model, including the Poincaré ball model and the Lorentz model. The logarithmic function log is used to perform the opposite operation. Step S22, Hyperbolic Feature Transformation: Use a logarithmic function to project the features in hyperbolic space onto the corresponding tangent space, perform matrix-vector multiplication in Euclidean space on the tangent space, and use an exponential function to project the feature vectors onto the corresponding hyperbolic space to complete the matrix-vector multiplication operation. Step S23, Hyperbolic Neighbor Aggregation: In a graph neural network, the adjacency matrix is calculated as follows: in, Represents a node With nodes The weight of the edges between them. Represents a node The degree, Represents a node The degree; Message aggregation is achieved using an addition strategy: in, Indicates that node i is in space The eigenvectors below, Indicate the neighboring nodes of node i; Step S24, Nonlinear Activation: First, project the hyperbolic feature vector onto the corresponding tangent space, perform nonlinear activation in the tangent space, and then project it back. The process is as follows: in, Indicates that when the input curvature is The output curvature is Activation function; The overall process of obtaining the hyperbolic graph neural network layers is represented as follows: in, express Space under the first The feature vector after feature transformation of layer node i Indicates the first Layer weight coefficients, Indicates the first The feature vector of layer node i l Indicates the number of layers in the network. Indicates the first Layer bias weight coefficient, Indicates the first The inactive feature vector of layer node i; Step S3: Use an attention mechanism to fuse features from multiple manifolds; Step S4: Multiple loss functions constructed in step S3 are optimized using Pareto to balance the differences between the original features and the decoded reconstructed features. Step S5: Identify the spatial domain using a clustering algorithm based on Gaussian mixture model.
2. The spatial domain identification method in spatial transcriptomics based on hyperbolic graph neural networks according to claim 1, characterized in that, In step S21, the definitions of the exponential and logarithmic functions under the Poincaré ball model are as follows: in, Indicates curvature. It represents Möbiuska. It is a coefficient. It is a vector in the original space. It is a vector in hyperbolic space. Indicates the projection position; The definitions of exponential and logarithmic functions under the Lorentz model are as follows: in, Represents the dot product of vectors. This represents operations in the Lorentz space; The feature vectors projected onto the Poincaré ball model and the Lorentz model are calculated using the following formula: in, , , Let represent the feature vectors in Euclidean space, Poincaré space, and Lorentz space, respectively. This indicates that the projection starts from the origin of the coordinate system. In step S22, the matrix-vector multiplication operation in Poincaré space and Lorentz space is as follows: in, Represents a matrix. n Represents the characteristic dimensions of primitive Euclidean space; Implement matrix-vector addition using parallel transmission to unify the representation of eigenvectors under the Poincaré ball model and the Lorentz model. The specific process is as follows: in, The bias term is represented; the parallel transmission definitions under the Poincaré ball model and the Lorentz model are as follows: in, This indicates a gyroscope transformation operation.
3. The spatial domain identification method in spatial transcriptomics based on hyperbolic graph neural networks according to claim 1, characterized in that, The specific operation of step S3 is as follows: An attention coefficient vector Multiplying this by the concatenated vector from each manifold yields an attention score matrix. The fraction matrix is then normalized, and the calculation process for the fraction matrix is as follows: in, This represents the matrix transpose operation. Representing nodes respectively Feature vectors in Euclidean space, Poincaré space, and Lorentz space; Update the feature using a damping coefficient λ: in, Represents the adjacency matrix. Let represent the characteristic matrix in the Poincaré space.
4. The spatial domain identification method in spatial transcriptomics based on hyperbolic graph neural networks according to claim 1, characterized in that, The specific operation of step S4 is as follows: For a scenario involving two vectors, assuming the sum of their weights is 1, the optimization objective is: in, Represents the global weight parameters. This represents the local weight parameters applicable to the first task. This indicates the first loss function pair The resulting gradient vector, The second loss function represents the pair The resulting gradient vector, This represents the local weight parameters applicable to the second task; The weight coefficient γ of the first loss function in the above formula is calculated as follows: in, This represents the matrix transpose operation; For scenarios with multiple vectors, the Frank-Wolfe algorithm is used to calculate the weight coefficients of each loss function: in, Represents graph structure data, This represents the increment of the weighting coefficient. This represents the cumulative weighting coefficient. Let K represent a one-hot vector, where 1 is at position t and 0 is at all other positions. K represents the number of tasks. This represents the k-th task. Let represent the loss function for the t-th task. This represents the local weight parameters applicable to the t-th task. This represents the matrix transpose operation.
5. The spatial domain identification method in spatial transcriptomics based on hyperbolic graph neural networks according to claim 1, characterized in that, In step S5, the spatial domain is identified using the mclust algorithm, specifically as follows: The mclust algorithm assumes that the data distribution is composed of multiple Gaussian distributions. Each Gaussian distribution is considered as a cluster. By feeding the fused low-dimensional features into the mclust algorithm, the number of spatial domains is specified by the actual number of true labels, thus obtaining the actual clustering result, i.e., the spatial domain.