Book Recommendation Method and System Based on Lorentz Diagram Convolutional Network

Abstract

The present invention discloses a book recommendation method and system based on a Lorentz graph convolutional network, belonging to the technical field of intelligent recommendation. The technical problem to be solved by the present invention is how to improve the accuracy of book recommendation. The adopted technical solution is as follows: The method is specifically as follows: Data collection: Obtain the book data interacted by each user; Construct a user-book interaction bipartite graph: Construct the collected Euclidean space data into non-Euclidean space data, that is, graph data. Specifically, construct the interaction data between users and books into a user-book interaction bipartite graph; Divide the data set into a training set and a test set; Construct an LCF model: Initialize the relevant parameters of the Lorentz graph convolutional network model, and input the data of the training set into the Lorentz graph convolutional network model for training until the Lorentz graph convolutional network model converges; Generate prediction results for book recommendation.

Description

Technical Field

[0001] The present invention relates to the technical field of intelligent recommendation, and more specifically, to a book recommendation method and system based on a Lorentz graph convolutional network. Background Art

[0002] In the digital age, the development of information technology has led to the wide application of recommendation systems, which attempt to make personalized recommendations by capturing user behaviors and item features. Collaborative filtering assumes that users with similar preferences prefer similar items, and users who prefer the same item are similar. It is widely used in personalized recommendations, such as book recommendations. Existing research has proposed considering the user-book interaction matrix as a bipartite graph, where nodes represent users or books, and edges represent the interactions between users and books. Then, a graph neural network is applied to it from a graph perspective. In the graph convolution setting, high-order relationships between users and books are extracted through multi-layer neighborhood aggregation to obtain the final representation.

[0003] Although the above graph-based research has achieved excellent performance, most of their work is modeled based on Euclidean space, which may limit the expressive power of the model. First, many complex networks (such as user-item networks) usually exhibit a tree-like hierarchical structure. Second, user preferences and item popularity generally follow a power-law distribution. Taking books as an example, the sales volume of best-selling books may be in the millions, while the sales volume of general books is around a few thousand or less per year. It is worth mentioning that books with average or low sales volume account for the majority. Similarly, user preferences also show such a power-law distribution. Existing research has shown that the Euclidean algorithm ignores the inherent structure of the user-item network, resulting in information loss. Therefore, for user-item data presenting a tree-like hierarchical structure (or power-law distribution), the Euclidean space will have a high distortion. At the same time, the volume of the low-dimensional Euclidean latent space is too small to well accommodate the large number of users and items in the real-world recommendation network. Different from the flat Euclidean space, the hyperbolic space is a curved space with a constant negative curvature. As shown in the appendix Figure 1 it, its volume grows exponentially with the radius, which makes it more suitable than the Euclidean space for modeling user-book networks presenting a hierarchical structure (or power-law distribution). As can be seen from the appendix Figure 1 the Euclidean space is too narrow to well embed the tree-like graph G, while if G is embedded in a hyperbolic space with a larger volume, it will make the nodes easier to separate and distinguish. Therefore, the hyperbolic space is more suitable than the Euclidean space for embedding such a tree-like hierarchical structure.

[0004] Therefore, how to improve the accuracy of book recommendations is a technical problem that needs to be solved urgently at present. Summary of the Invention

[0005] The technical task of the present invention is to provide a book recommendation method and system based on a Lorentz graph convolutional network to solve the problem of how to improve the accuracy of book recommendations.

[0006] The technical task of the present invention is realized in the following way. A book recommendation method based on a Lorentz graph convolutional network is as follows:

[0007] Data collection: Obtain book data interacted by each user;

[0008] Construct a user-book interaction bipartite graph: Construct the collected Euclidean space data into non-Euclidean space data, that is, graph data. Specifically, construct the interaction data between users and books into a user-book interaction bipartite graph;

[0009] Divide the data set into a training set and a test set: Randomly select 80% from the historical interaction data in the data set as the training set for model training; The remaining 20% in the data set is used as the test set to evaluate the generalization ability of the model; Regard each observed user-book interaction data as a positive sample, and pair negative samples for the books not interacted by the user through a negative sampling strategy;

[0010] Construct an LCF (Lorentzian graph convolutional network model for collaborative filtering) model: Initialize the relevant parameters of the Lorentz graph convolutional network model, and input the data of the training set into the Lorentz graph convolutional network model for training until the Lorentz graph convolutional network model converges;

[0011] Generate prediction results and make book recommendations: Jointly utilize multiple embedding vectors learned by multiple graph convolutional layers to capture the embedding influence of different convolutional layers; For the final user and book embedding vectors generated by the Lorentz graph convolutional network model, obtain the prediction score between the user and the book based on the hyperbolic distance, and recommend books of interest to the user according to the prediction score.

[0012] Preferably, the architecture of the LCF model includes,

[0013] A hyperbolic encoding layer for mapping the interaction information between users and books to the hyperbolic space using graph embedding technology based on the constructed user-book bipartite graph;

[0014] A graph convolutional layer for learning the embedding representations of user and book nodes through the Lorentz graph convolutional network;

[0015] A prediction layer for obtaining the representation encoding rich structural information of users and books after L layers of graph convolution, and predicting the preference of users for books based on the hyperbolic distance;

[0016] The hyperbolic loss function layer is used to separate the user-book pairs of positive samples from those of negative samples using a composite loss function, and apply regularizers to user and book nodes to fully utilize the hyperbolic space until the network converges.

[0017] Preferably, the hyperbolic encoding layer is as follows:

[0018] Define the tangent space at point x as a d-dimensional vector space that approximates the hyperbolic manifold at point x The formula is as follows;

[0019]

[0020] where, represents the user-book interaction matrix; d represents the embedding size; β represents the negative reciprocal of the curvature;

[0021] Initialization in Euclidean space is usually based on Gaussian sampling. Apply the hyperbolic Gaussian sampling method to the Lorentz graph convolutional network model (hyperbolic recommendation model); in hyperbolic space, for a given user node u and book node i, use the vector to represent the origin in and use point o as the reference point for performing tangent space operations. First, sample from the multivariate Gaussian distribution on the tangent space of point o to obtain the initial embeddings of user u and book i and and add a "0" element to the first coordinate of the initial embeddings in (same as h i ) constraints; then map the initial embeddings and to the hyperbolic space through the exponential map to obtain the hyperbolic initial embeddings of all user nodes and book nodes. The formula is as follows:

[0022]

[0023]

[0024] where, is the exponential map operation used to map a point from the tangent space to the hyperbolic space; β represents the negative reciprocal of the curvature; k represents the embedding size.

[0025] Preferably, the graph convolutional layer is as follows:

[0026] Perform hyperbolic feature transformation on all nodes; specifically as follows:

[0027] For a given point \(x=(x_0,\ldots,x\) n )\in H\) d,β , transform the values of the last \(n\) coordinates to satisfy the constraints in the equation , ensuring that the transformed node features strictly follow hyperbolic geometry. The formula is as follows:

[0028]

[0029] where \(M:\mathbb{R}\) n \to\mathbb{R}\) m is an \(m\times n\) matrix representing a linear mapping from \(n\) dimensions to \(m\) dimensions; is the logarithmic mapping operation used to map points from the hyperbolic space to the tangent space;

[0030] Use Lorentz matrix-vector multiplication to perform feature transformation on the initial embeddings of users and books to obtain the transformed embedding representation and Specifically:

[0031]

[0032]

[0033] Hyperbolic neighbor aggregation: Use the centroid based on the squared Lorentz distance for neighbor information aggregation; specifically as follows:

[0034] For each node feature (including users and books) Calculate the attention-related score of each neighbor node \(j\) to the central node \(t\); use the self-attention mechanism based on the Lorentz squared distance. The formula is as follows:

[0035]

[0036] where the matrix \(M\) a is a \(d\times d\) matrix used to transform the node features into features with attention;

[0037] To make the attention-related scores comparable and obtain the attention weights, for all neighbors \(N\) of node \(t\) (including itself) t , introduce the softmax function to normalize \(\mu\) tj , and the formula is as follows:

[0038]

[0039] Based on the centroid of each user node embedding and book node embedding perform neighbor information aggregation, that is, minimize the equation as follows:

[0040]

[0041] There is a closed - form solution, so the node embeddings of users and books are updated as:

[0042]

[0043] Lorentz point - wise non - linear activation: For a given point \(x=(x_0,\ldots,x_{ n )\in H d,β we have:

[0044]

[0045] where \(\sigma:\mathbb{R} n \to\mathbb{R} n represents a point - wise non - linear mapping;

[0046] The embeddings of users and books are updated as:

[0047]

[0048]

[0049] More preferably, the prediction layer is specifically as follows:

[0050] Considering the representations generated by each layer, and at the same time using the preferences of users or the attributes of books captured from different perspectives to obtain the prediction score of user \(u\) for target book \(i\), the formula is as follows:

[0051]

[0052] where, and are the user and book representations generated by the \(l\) - th layer respectively; \(L\) is the total number of Lorentz graph convolutional layers passed; Intuitively, in the same embedding space, when the embedding of book \(i\) is closer to that of user \(u\), it means that user \(u\) prefers book \(i\) more than other books.

[0053] More preferably, the hyperbolic loss function layer is specifically as follows:

[0054] Optimization is carried out using the margin ranking loss in the hyperbolic space: For each user \(u\), sample a positive user - book pair \((u,i)\) and a negative user - book pair \((u,j)\), and define the margin ranking loss based on the hyperbolic distance as:

[0055]

[0056] where \(m\) is the margin, a non - negative hyperparameter; Note that the representations obtained from different layers will affect the loss simultaneously, which not only helps to use different semantics captured at different layers, but also reduces the optimization difficulties caused by residual connections;

[0057] Find a root node and align it with the hyperbolic origin. After the root node is aligned, other user nodes and book nodes will naturally be arranged well during the optimization process of the loss function;

[0058] Apply a regularizer to the aligned user and book nodes: To push the embeddings of all user nodes and book nodes in the hyperbolic space away from the origin and make full use of the hyperbolic space, restrict each node in the loss function to have a large norm, that is, apply a regularizer to the user node embeddings and book node embeddings. The formula is as follows:

[0059]

[0060]

[0061] The final loss function is \(L = L g +\alpha L r ;

[0062] where \(\alpha\) is a weight hyperparameter used to balance the values of the two loss functions.

[0063] More preferably, find a root node and align it with the hyperbolic origin. After the root node is aligned, other user nodes and book nodes will naturally be arranged well during the optimization process of the loss function as follows:

[0064] To ensure that the internal structure of the overall nodes is not damaged during the pushing-away process, for each user node embedding and book node embedding Calculate the root node based on the center of the overall node embeddings. The formula is as follows:

[0065]

[0066] where \(N\) is the total number of user and book nodes;

[0067] To always place the root node near the origin, align the root node with the hyperbolic origin, that is, subtract the root node from each node embedding. The formula is as follows:

[0068]

[0069] A book recommendation system based on a Lorentz graph convolutional network, which includes

[0070] A data acquisition module for obtaining book data interacted by each user;

[0071] A bipartite graph construction module for constructing the Euclidean space data collected into non-Euclidean space data, that is, graph data. Specifically, construct the interaction data between users and books into a user-book interaction bipartite graph;

[0072] A dataset partitioning module, which is used to randomly select 80% of the historical interaction data in the dataset as the training set for model training; the remaining 20% in the dataset is used as the test set to evaluate the generalization ability of the model; each observed user-book interaction data is regarded as a positive sample, and negative samples are paired for the books that the user has not interacted with through a negative sampling strategy.

[0073] A model construction module, which is used to construct the architecture diagram of the LCF (Lorentzian graph convolutional network model for collaborative filtering) model, initialize the relevant parameters of the Lorentz graph convolutional network model, and input the data of the training set into the Lorentz graph convolutional network model for training until the Lorentz graph convolutional network model converges.

[0074] A recommendation prediction module, which is used to jointly utilize multiple embedding vectors learned by multiple graph convolutional layers to capture the embedding influence of different convolutional layers; for the final user and book embedding vectors generated by the Lorentz graph convolutional network model, based on the hyperbolic distance, obtain the prediction score between the user and the book, and recommend books of interest to the user according to the prediction score.

[0075] An electronic device, including: a memory and at least one processor;

[0076] Wherein, a computer program is stored on the memory;

[0077] The at least one processor executes the computer program stored in the memory, so that the at least one processor executes the book recommendation method based on the Lorentz graph convolutional network as described above.

[0078] A computer-readable storage medium, in which a computer program is stored, and the computer program can be executed by a processor to implement the book recommendation method based on the Lorentz graph convolutional network as described above.

[0079] The book recommendation method and system based on the Lorentz graph convolutional network of the present invention have the following advantages:

[0080] (1) Traditional recommendation methods usually perform graph convolution of user and item embeddings in the Euclidean space. However, in large-scale recommendation systems, the user-item interaction graph usually presents a tree-like hierarchical structure, while the present invention embeds the user-book interaction graph into the hyperbolic space and uses the geometric characteristics of the hyperbolic space to better improve the recommendation accuracy;

[0081] (2) In traditional recommendation models, the IDs of these users or items are represented as embedding vectors and directly fed into the interaction layer to obtain prediction scores. To achieve better recommendation effects, the present invention obtains interaction information based on the high-order connectivity in the user-book interaction graph to construct the embedding vectors of users and items. Thus, the present invention takes into account the interaction information between users and items and designs a graph convolutional layer, which can ensure that the user and book embeddings do not deviate from the hyperbolic space during the calculation process;

[0082] (3) To make the aggregated node embeddings satisfy mathematical significance and improve the operation efficiency at the same time, the present invention aggregates neighbor information by calculating the centroid based on the squared Lorentz distance;

[0083] (4) Traditional graph convolutional neural networks cannot assign different weights to each neighbor node. For example, when performing convolution, all neighbor nodes are treated equally and cannot assign different weights according to the importance of the nodes. To address this problem, the present invention adopts an attention mechanism, which can assign different weights to different nodes, enabling the model training to rely on paired adjacent nodes rather than a specific graph structure;

[0084] (5) The present invention adopts a composite loss function to separate positive user-item pairs from negative user-item pairs and enable the final embeddings to make full use of the hyperbolic space; in distance-based recommendation models, the margin ranking loss is usually a good choice, which can reduce the distance between the positive item and the user and increase the distance between the negative item and the user at the same time; therefore, the present invention optimizes using the margin ranking loss in the hyperbolic space; at the same time, to make full use of the hyperbolic space, the present invention applies a regularizer to all user and book nodes. BRIEF DESCRIPTION OF THE DRAWINGS

[0085] The present invention will be further described below with reference to the accompanying drawings.

[0086] FIG Figure 1 is a schematic diagram of the hyperbolic space;

[0087] FIG Figure 2 is a user-book interaction bipartite graph;

[0088] FIG Figure 3 is an LCF model architecture diagram. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0089] The book recommendation method and system based on the Lorentz graph convolutional network of the present invention will be described in detail below with reference to the accompanying drawings of the specification and specific embodiments.

[0090] Example 1:

[0091] This embodiment provides a book recommendation method based on a Lorentz graph convolutional network, and the method is as follows:

[0092] S1. Data collection: Obtain the book data interacted by each user; in this embodiment, the dataset uses the Amazon-Book dataset, where the Amazon-Book dataset includes 52,406 user data, 41,264 item data, and 1,861,118 user-item interaction data;

[0093] S2. Construct a user-book interaction bipartite graph: Construct the Euclidean space data collected into non-Euclidean space data, that is, graph data. Specifically, construct the interaction data between users and books into a user-book interaction bipartite graph, as shown in the appendix Figure 2 as follows;

[0094] S3. Divide the dataset into a training set and a test set: Randomly select 80% from the historical interaction data in the dataset as the training set for model training; the remaining 20% in the dataset is used as the test set to evaluate the generalization ability of the model; regard each observed user-book interaction data as a positive sample, and pair negative samples for the books that the user has not interacted with through the negative sampling strategy;

[0095] S4. Construct an LCF (Lorentzian graph convolutional network model for collaborative filtering) model, as shown in the appendix Figure 2 as follows: Initialize the relevant parameters of the Lorentz graph convolutional network model, and input the data of the training set into the Lorentz graph convolutional network model for training until the Lorentz graph convolutional network model converges; based on the user-book bipartite graph constructed in step S2, first use the graph embedding technology to map the interaction information of users and books into the hyperbolic space. Then, learn the embedding representations of user and book nodes through the Lorentz graph convolutional network, that is, first perform hyperbolic feature transformation on all nodes, then use the centroid based on the squared Lorentz distance for neighbor information aggregation, and then perform Lorentz pointwise non-linear activation. After L layers of graph convolution, the obtained representations of users and books encode rich structural information. At the same time, adopt a composite loss function to separate the positive user-book pairs from the negative user-book pairs, and apply regularizers to the user and book nodes to make full use of the hyperbolic space until the network converges.

[0096] S5. Generate prediction results for book recommendations: Jointly utilize multiple embedding vectors learned by multiple graph convolutional layers to capture the embedding impacts of different convolutional layers; for the final user and book embedding vectors generated by the Lorentz graph convolutional network model, obtain the prediction scores between users and books based on the hyperbolic distance, and recommend interesting books to users according to the prediction scores.

[0097] The architecture of the LCF model in this embodiment includes

[0098] A hyperbolic encoding layer for mapping user-book interaction information to the hyperbolic space using graph embedding techniques based on the constructed user-book bipartite graph;

[0099] A graph convolutional layer for learning the embedding representations of user and book nodes through the Lorentz graph convolutional network;

[0100] A prediction layer for obtaining the representation encodings of users and books with rich structural information after L layers of graph convolution and predicting user preferences for books based on the hyperbolic distance;

[0101] A hyperbolic loss function layer for separating positive sample user-book pairs from negative sample user-book pairs using a composite loss function and applying regularizers to user and book nodes to fully utilize the hyperbolic space until the network converges.

[0102] The hyperbolic encoding layer in this embodiment is specifically as follows:

[0103] Define the tangent space at point x as a d-dimensional vector space that approximates the hyperbolic manifold at point x The formula is as follows;

[0104]

[0105] where represents the user-book interaction matrix; d represents the embedding size; β represents the negative reciprocal of the curvature;

[0106] Initialization in Euclidean space is usually based on Gaussian sampling. Apply the hyperbolic Gaussian sampling method to the Lorentz graph convolutional network model (hyperbolic recommendation model); in the hyperbolic space, for a given user node u and book node i, use the vector to represent the origin in and take point o as the reference point for performing tangent space operations. First, sample from the multivariate Gaussian distribution on the tangent space of point o to obtain the initial embeddings of user u and book i and and add a "0" element to the first coordinate of the initial embeddings China (same as h i )'s constraint; then through exponential mapping, the initial embedding and are mapped to the hyperbolic space to obtain the hyperbolic initial embeddings of all user nodes and book nodes. The formula is as follows:

[0107]

[0108]

[0109] where, is the exponential mapping operation, which is used to map a point from the tangent space to the hyperbolic space; β represents the negative reciprocal of the curvature, and k represents the embedding size.

[0110] The graph convolutional layer in this embodiment is specifically as follows:

[0111] Perform hyperbolic feature transformation on all nodes; specifically as follows:

[0112] For a given point x = (x0,..., x n ) ∈ H d,β , transform the values of the last n coordinates to satisfy the constraint in the equation , ensuring that the transformed node features strictly follow hyperbolic geometry. The formula is as follows:

[0113]

[0114] where, M: R n → R m is an m×n matrix, representing a linear mapping from n dimensions to m dimensions; is the logarithmic mapping operation, which is used to map a point from the hyperbolic space to the tangent space;

[0115] Use Lorentz matrix-vector multiplication to perform feature transformation on the initial embeddings of users and books to obtain the transformed embedding representations and Specifically:

[0116]

[0117]

[0118] To extract the high-order interaction information of user-books, hyperbolic message aggregation is usually applied, that is, in each layer, the hidden features of the neighbors of user nodes and book nodes in the previous layer are aggregated. Mean aggregation in Euclidean space is to calculate the weighted average or centroid of the neighborhood features of user and item nodes. The generalization of Euclidean mean aggregation in hyperbolic space is the Frechet mean. However, the Frechet mean is difficult to apply because it has no closed-form solution in hyperbolic space and can only be calculated by the method of gradient descent, which reduces the operation efficiency. In addition, some hyperbolic GCNs usually aggregate neighbor information with the help of the tangent space, which leads to the distortion of the aggregated node features. Therefore, in order to make the aggregated node embeddings satisfy the mathematical meaning and improve the operation efficiency at the same time, this embodiment aggregates neighbor information by calculating the centroid based on the squared Lorentz distance.

[0119] Traditional graph convolutional neural networks cannot assign different weights to each neighbor node. For example, when performing convolution, all neighbor nodes are treated equally and different weights cannot be assigned according to the importance of the nodes. To address this issue, this embodiment adopts an attention mechanism that can assign different weights to different nodes, enabling the model to rely on paired adjacent nodes during training rather than on a specific graph structure.

[0120] Hyperbolic neighbor aggregation: Aggregate neighbor information using the centroid based on the squared Lorentz distance; specifically as follows:

[0121] For each node feature (including users and books) Calculate the attention-related score of each neighbor node j to the central node t; use the self-attention mechanism based on the Lorentz squared distance, and the formula is as follows:

[0122]

[0123] where the matrix M a is a d×d-dimensional matrix used to transform the node features into features with attention;

[0124] To make the attention-related scores easy to compare and obtain the attention weights, for all neighbors N of node t (including itself) t , introduce the softmax function to normalize μ tj , and the formula is as follows:

[0125]

[0126] Based on the centroid of each user node embedding and book node embedding Perform neighbor information aggregation, that is, minimize the following equation:

[0127]

[0128] There is a closed - form solution, so the node embeddings of the user and the book are updated as:

[0129]

[0130] Lorentz point - wise non - linear activation: For a given point \(x=(x_0,\ldots,x\) n )\(\in H\) d,β we have:

[0131]

[0132] where \(\sigma:\mathbb{R}\) n \(\to\mathbb{R}\) n represents a point - wise non - linear mapping;

[0133] The embeddings of the user and the book are updated as:

[0134]

[0135]

[0136] The prediction layer in this embodiment is specifically as follows:

[0137] Considering the representations generated by each layer, and at the same time using the preferences of the user or the attributes of the book captured from different perspectives to obtain the prediction score of user \(u\) for target book \(i\), the formula is as follows:

[0138]

[0139] where, and are the user and book representations generated by the \(l\) - th layer respectively; \(L\) is the total number of Lorentz graph convolutional layers passed; Intuitively, in the same embedding space, when the embedding of book \(i\) is closer to that of user \(u\), it means that user \(u\) prefers book \(i\) more than other books.

[0140] The hyperbolic loss function layer in this embodiment is specifically as follows:

[0141] In the hyperbolic space, it is optimized using the margin ranking loss: For each user \(u\), sample a positive user - book pair \((u,i)\) and a negative user - book pair \((u,j)\), and define the margin ranking loss based on the hyperbolic distance as:

[0142]

[0143] Among them, m is the margin, which is a non - negative hyperparameter; note that the representations obtained from different layers affect the loss simultaneously, which not only helps to use different semantics captured at different layers, but also reduces the optimization difficulties brought by residual connections;

[0144] The capacity of the hyperbolic space grows exponentially with the increase of the radius, so the region far from the origin is much larger than the region close to the origin. In addition, research shows that when the boundary is far from the origin, the closer the embedding is to the boundary, the easier it is to be distinguished. Therefore, to make full use of the hyperbolic space, the overall nodes need to be embedded far from the origin. First, find a root node and align it with the hyperbolic origin. After the root node is aligned, other user nodes and book nodes will be naturally arranged during the optimization process of the loss function; specifically as follows:

[0145] To ensure that the internal structure of the overall nodes is not damaged during the pushing - away process, for each user - node embedding and book - node embedding Calculate the root node based on the center of the overall node embedding, and the formula is as follows:

[0146]

[0147] where N is the total number of user and book nodes;

[0148] To always place the root node close to the origin, align the root node with the hyperbolic origin, that is, subtract the root node from each node embedding, and the formula is as follows:

[0149]

[0150] Then, apply a regularizer to the aligned user and book nodes: To push the embeddings of all user nodes and book nodes in the hyperbolic space away from the origin and make full use of the hyperbolic space, restrict each node in the loss function to have a large norm, that is, apply a regularizer to the user - node embedding and book - node embedding, and the formula is as follows:

[0151]

[0152]

[0153] The final loss function is L = L g +αL r ;

[0154] where α is a weight hyperparameter used to balance the values of the two loss functions.

[0155] This embodiment uses three publicly available datasets, Amazon - Book, Amazon - CD, and Yelp2020, which have been widely used in many works. Table 1 summarizes the statistics of each dataset. There are differences in the number of nodes and sparsity of these three datasets, providing a suitable performance metric for the model. In this embodiment, the historical interactions in each dataset are randomly divided into 80% and 20% for training and testing. In these datasets, this embodiment simulates the implicit feedback setting by applying a threshold ≥4 to convert ratings into binary preferences.

[0156] Table 1 Statistics of Datasets

[0157] Dataset #User #Item #Interactions Density Amazon-CD 22,947 18,395 422,301 0.00100 Amazon-Book 52,406 41,264 1,861,118 0.00086 Yelp2020 71,135 45,063 1,940,014 0.00047

[0158] To effectively evaluate the performance of top - K recommendations and relevance ranking, this embodiment adopts two widely used evaluation metrics: Recall and NDCG. This embodiment regards each observed user - item interaction as a positive instance, and then pairs it with a negative item that the user has not rated before through a negative sampling strategy.

[0159] To fully verify the effectiveness of LCF, this embodiment compares it with the following advanced baseline methods:

[0160] WRMF: WRMF proposes to regard the data as positive and negative examples related to confidence for implicit feedback recommendation.

[0161] VAE - CF: VAE - CF extends variational autoencoders to collaborative filtering for implicit feedback.

[0162] TransCF: TransCF models the strength and heterogeneity of user - item relationships in implicit feedback through the translation mechanism.

[0163] LRML: LRML constructs the latent relationship between users and items by introducing a memory network module to learn the metric distance.

[0164] SML: In addition to user - centered metrics, it symmetrically introduces positive - item - centered metrics to improve recommendation performance.

[0165] NGCF: NGCF is a recommendation framework based on graph neural networks, which explicitly encodes collaborative signals in the form of high - order connections by performing embedding propagation.

[0166] LightGCN: LightGCN simplifies the design of GCN and only includes the most basic neighborhood aggregation part in GCN for collaborative filtering.

[0167] HAVE: HAVE only uses a single hidden layer to solve the standard collaborative filtering problem.

[0168] HGCF: HGCF learns user and item representations in hyperbolic space by aggregating neighborhood information on the tangent space of reference points.

[0169] HRCF: HRCF proposes a geometric-aware hyperbolic regularizer and alleviates the over-smoothing problem caused by hyperbolic aggregation.

[0170] The experimental results are shown in Tables 2 and 3:

[0171] Table 2 Comparison results of Recall on three datasets

[0172]

[0173] Table 3 Comparison results of NDCG on three datasets

[0174]

[0175] The overall experimental results are shown in Tables 2 and 3, where the optimal results are in bold and the sub-optimal results are underlined. Generally speaking, LCF outperforms the baseline methods in terms of Recall@K and NDCG@K metrics on the three datasets, demonstrating the superiority of the method of the present invention. Specifically, it can be observed that the hyperbolic model has achieved a significant performance improvement compared to the Euclidean baseline LightGCN. This is because as the network continues to expand, the power-law distribution of the network becomes more obvious. It can also be noted that for datasets with higher data density, the performance improvement of the hyperbolic model is better. For example, on the Amazon-Book dataset, LCF is 16.1% higher than LightGCN in Recall@10 and 18.7% higher in NDCG@10. This shows that hyperbolic GCNs can be effectively applied in the recommendation setting and are superior to the state-of-the-art Euclidean GCNs models. The experimental results also show that the present invention is superior to other hyperbolic baselines HAVE, HGCF, and HRCF. Although HAVE also learns the representations of users and items in the hyperbolic space, its autoencoder introduces a large number of additional training parameters, which increases the training difficulty and may make it difficult to optimize the model on large datasets. In addition, the autoencoder is optimized using cross-entropy loss, which processes each user-book pair separately and is therefore not conducive to improving objective metrics such as ranking-based NDCG. At the same time, HGCF and HRCF aggregate neighborhood information in the tangent space of the reference point, which may lead to distortion of the aggregated node features. In contrast, the aggregation process of this embodiment guarantees the mathematical meaning of the hyperbolic space and improves the performance of the LCF model.

[0176] Example 2:

[0177] This embodiment provides a book recommendation system based on a Lorentz graph convolutional network, which includes,

[0178] A data acquisition module for obtaining book data interacted by each user;

[0179] A bipartite graph construction module for constructing the Euclidean space data collected into non-Euclidean space data, that is, graph data. Specifically, the interaction data between users and books is constructed into a user-book interaction bipartite graph;

[0180] A dataset division module for randomly selecting 80% of the historical interaction data in the dataset as the training set for model training; the remaining 20% in the dataset is used as the test set to evaluate the generalization ability of the model; each observed user-book interaction data is regarded as a positive sample, and negative samples are paired for the books not interacted by the user through a negative sampling strategy;

[0181] A model construction module for constructing the architecture diagram of the LCF (Lorentzian graph convolutional network model for collaborative filtering), initializing the relevant parameters of the Lorentz graph convolutional network model, and inputting the data of the training set into the Lorentz graph convolutional network model for training until the Lorentz graph convolutional network model converges;

[0182] A recommendation prediction module for jointly utilizing multiple embedding vectors learned by multiple graph convolutional layers to capture the embedding influence of different convolutional layers; for the final user and book embedding vectors generated by the Lorentz graph convolutional network model, obtaining the prediction scores between users and books based on the hyperbolic distance, and recommending interesting books for users according to the prediction scores.

[0183] The specific algorithm of the LCF model is as follows:

[0184] Traditional recommendation methods usually perform graph convolution of user and item embeddings in the Euclidean space. However, in large-scale recommendation systems, the user-item interaction graph usually presents a tree-like hierarchical structure. Therefore, the present invention embeds the user-book interaction graph into the hyperbolic space and utilizes the geometric characteristics of the hyperbolic space to better improve the recommendation accuracy.

[0185] 1. Hyperbolic encoding layer

[0186] First, define the tangent space at point x as a d-dimensional vector space, which approximates the hyperbolic manifold at point x

[0187]

[0188] where is the user-book interaction matrix, d represents the embedding size; β represents the negative reciprocal of the curvature. Initialization in the Euclidean space is usually based on Gaussian sampling. Similarly, apply the hyperbolic Gaussian sampling method to the hyperbolic recommendation model. In the hyperbolic space, for a given user node u and book node i, we use the vector to represent the origin in, and take point o as the reference point for performing tangent space operations. We first sample from the multivariate Gaussian distribution on the tangent space of point o to obtain the initial embeddings and of user u and book i, and add a "0" element to their first coordinates to satisfy the equation in (same as h i ) constraints. Then, map the initial embeddings to the hyperbolic space through the exponential map to obtain the hyperbolic initial embeddings of all user nodes and book nodes:

[0189]

[0190] Among them, is the exponential mapping operation, which is used to map points from the tangent space to the hyperbolic space; β represents the negative reciprocal of the curvature, and k represents the embedding size.

[0191] 2. Graph Convolutional Layer

[0192] In traditional recommendation models, the IDs of these users or items are represented as embedding vectors and directly fed into the interaction layer to obtain prediction scores. However, in the LCF model of this embodiment, in order to achieve better recommendation effects, this embodiment obtains interaction information according to the high-order connectivity in the user-book interaction graph to construct the embedding vectors of users and items. Thus, this embodiment considers the interaction information between users and items and designs a graph convolutional layer, which can ensure that the user and book embeddings do not deviate from the hyperbolic space during the calculation process and consists of the following three steps.

[0193] 1) Hyperbolic Feature Transformation

[0194] Since the arithmetic operations in the Euclidean space cannot be simply transferred to the hyperbolic space for application, and at the same time, the features of the transformed user nodes and book nodes always remain on the hyperbolic manifold. Therefore, this embodiment adopts the Lorentz matrix-vector multiplication. For a given point x=(x0,…,x n )∈H d,β , only the values of the last n coordinates need to be transformed to satisfy the constraints in the equation , ensuring that the transformed node features strictly follow the hyperbolic geometry. The formula is as follows:

[0195]

[0196] where M:R n →R m is an m×n matrix, representing a linear mapping from n dimensions to m dimensions. is the logarithmic mapping operation, which is used to map points from the hyperbolic space to the tangent space. Therefore, the Lorentz matrix-vector multiplication is used to perform feature transformation on the initial embeddings of users and books to obtain the transformed embedding representations and

[0197]

[0198] 2) Hyperbolic Neighbor Aggregation

[0199] To extract the high-order interaction information of user-books, hyperbolic message aggregation is usually applied. That is, in each layer, the hidden features of the neighbors of user nodes and book nodes in the previous layer are aggregated. Mean aggregation in Euclidean space is to calculate the weighted average or centroid of the neighborhood features of user and item nodes. The generalization of Euclidean mean aggregation in hyperbolic space is the Frechet mean. However, the Frechet mean is difficult to apply because it has no closed-form solution in hyperbolic space and can only be calculated by the method of gradient descent, which reduces the operation efficiency. In addition, some hyperbolic GCNs usually rely on the tangent space for neighbor information aggregation, which leads to distortion of the aggregated node features. Therefore, in order to make the aggregated node embeddings satisfy the mathematical meaning and improve the operation efficiency at the same time, in this embodiment, neighbor information aggregation is performed by calculating the centroid based on the squared Lorentz distance.

[0200] Traditional graph convolutional neural networks cannot assign different weights to each neighbor node. For example, when performing convolution, all neighbor nodes are treated equally and different weights cannot be assigned according to the importance of the nodes. To address this problem, this embodiment adopts an attention mechanism that can assign different weights to different nodes, enabling the model to rely on paired adjacent nodes during training rather than on a specific graph structure.

[0201] For each node feature (including users and books) it is necessary to calculate the attention-related scores of each neighbor node j to the central node t. Therefore, this embodiment uses a self-attention mechanism based on the Lorentz squared distance, and the formula is as follows:

[0202]

[0203] where the matrix M a is a d×d-dimensional matrix used to transform the node features into features with attention. To make the attention-related scores easy to compare and obtain the attention weights, for all neighbors N k of node t (including itself), a softmax function is introduced to normalize μ tj , and the formula is as follows:

[0204]

[0205] Subsequently, neighbor information aggregation is performed based on the centroid of each user node embedding and book node embedding, that is, the following equation is minimized:

[0206]

[0207] where represents the squared Lorentz distance, and for points x, y ∈ H d,β, the squared Lorentz distance is defined as: Equation has a closed-form solution, so the node embeddings of users and books can be updated as:

[0208]

[0209] 3) Nonlinear activation

[0210] Nonlinear activation is an important step in neural networks. To ensure that the results still lie in the Lorentz manifold, the Lorentz pointwise nonlinear activation is defined as follows. That is, for a given point x = (x0, …, x n ) ∈ H d,β we have:

[0211]

[0212] where σ: R n → R n represents the pointwise nonlinear mapping. Then the embeddings of users and books are updated as:

[0213]

[0214] 3. Prediction layer

[0215] After L layers of graph convolution, the representations of users and books encode rich structural information. In this embodiment, the hyperbolic distance is used to predict the preference of user u for book i. Additionally, since the embedding expression vectors obtained through different layers have different impacts on user preferences, the generated representations emphasize different semantic and structural relationships. Therefore, this embodiment considers the representations generated by each layer and simultaneously utilizes the preferences of users or the attributes of books captured from different perspectives. The prediction score of user u for target book i is given by the following formula:

[0216]

[0217] where, and are the representations of the user and the book generated at the l-th layer respectively, and L is the total number of Lorentz graph convolution layers passed. Intuitively, in the same embedding space, when the embedding of book i is closer to that of user u, it means that user u prefers book i more than other books.

[0218] 4. Hyperbolic loss function

[0219] This embodiment adopts a composite loss function to separate positive user-item pairs from negative user-item pairs, while enabling the final embeddings to fully utilize the hyperbolic space. In distance-based recommendation models, the margin ranking loss is usually a good choice, which can reduce the distance between positive items and users, while increasing the distance between negative items and users. Therefore, this embodiment uses the margin ranking loss for optimization in the hyperbolic space. Formally, for each user u, a positive user-book pair (u, i) and a negative user-book pair (u, j) are sampled, and the margin ranking loss is defined based on the hyperbolic distance as follows:

[0220]

[0221] where m is the margin, which is a non-negative hyperparameter. Note that the representations obtained from different layers will affect the loss simultaneously, which not only helps to use different semantics captured at different layers, but also reduces the optimization difficulties caused by residual connections.

[0222] The capacity of the hyperbolic space grows exponentially with the increase of the radius, so the region far from the origin is much larger than the region close to the origin. In addition, research shows that when the boundary is far from the origin, the embeddings closer to the boundary are easier to distinguish. Therefore, to fully utilize the hyperbolic space, it is necessary to embed the overall nodes far from the origin. In the present invention, first, a root node is found and aligned with the hyperbolic origin. After the root node is aligned, other user nodes and book nodes will be naturally arranged during the optimization process of the loss function, and then a regularizer is applied to the aligned user and book nodes.

[0223] Specifically, to ensure that the internal structure of the overall nodes is not damaged during the pushing-away process, for each user node embedding and book node embedding First, the root node is calculated based on the center of the overall node embeddings:

[0224]

[0225] where N is the total number of user and book nodes. Then, to always place the root node close to the origin, the root node is aligned with the hyperbolic origin by subtracting the root node from each node embedding:

[0226]

[0227] In addition, to push the embeddings of all user nodes and book nodes in the hyperbolic space away from the origin and fully utilize the hyperbolic space. The present invention restricts each node in the loss function to have a large norm, that is, a regularizer is applied to the user node embeddings and book node embeddings, and the formula is as follows:

[0228]

[0229]

[0230] Therefore, the final loss function is L = L g + αL r , where α is a weight hyperparameter responsible for balancing the values of the two loss functions.

[0231] Example 3:

[0232] The embodiment of the present invention further provides an electronic device, including: a memory and a processor;

[0233] wherein, the memory stores computer-executable instructions;

[0234] The processor executes the computer-executable instructions stored in the memory, so that the processor executes the book recommendation method based on the Lorentz graph convolutional network in any embodiment of the present invention.

[0235] The processor may be a central processing unit (CPU), or may also be other general-purpose processors, digital signal processors (DSPs), application-specific integrated circuits (ASICs), field-programmable gate arrays (FPGAs) or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. The processor may be a microprocessor or the processor may also be any conventional processor, etc.

[0236] The memory can be used to store computer programs and / or modules. The processor realizes various functions of the electronic device by running or executing the computer programs and / or modules stored in the memory, and by calling the data stored in the memory. The memory mainly includes a program storage area and a data storage area. Among them, the program storage area can store an operating system, application programs required for at least one function, etc.; the data storage area can store data created according to the use of the terminal, etc. In addition, the memory may further include high-speed random access memory, and may also include non-volatile memory, such as a hard disk, a memory, a plug-in hard disk, a smart media card (SMC), a secure digital (SD) card, a flash memory card, at least one magnetic disk storage period, a flash memory device, or other volatile solid-state storage devices.

[0237] Example 4:

[0238] An embodiment of the present invention also provides a computer-readable storage medium, in which multiple instructions are stored. The instructions are loaded by a processor to cause the processor to execute the book recommendation method based on the Lorentz graph convolutional network in any embodiment of the present invention. Specifically, a system or device equipped with a storage medium can be provided, on which software program codes for implementing the functions of any one of the above embodiments are stored, and the computer (or CPU or MPU) of the system or device is caused to read and execute the program codes stored in the storage medium.

[0239] In this case, the program code read from the storage medium itself can implement the functions of any one of the above embodiments, so the program code and the storage medium storing the program code constitute a part of the present invention.

[0240] Embodiments of the storage medium for providing program codes include floppy disks, hard disks, magneto-optical disks, optical disks (such as CD-ROM, CD-R, CD-RW, DVD-ROM, DVD-RYM, DVD-RW, DVD+RW), magnetic tapes, non-volatile memory cards, and ROMs. Optionally, the program code can be downloaded from a server computer via a communication network.

[0241] In addition, it should be clear that not only can the functions of any one of the above embodiments be implemented by executing the program code read by the computer, but also by causing an operating system or the like operating on the computer based on the instructions of the program code to complete part or all of the actual operations.

[0242] In addition, it can be understood that the program code read from the storage medium is written into the memory provided in the expansion board inserted into the computer or the memory provided in the expansion unit connected to the computer, and then based on the instructions of the program code, the CPU or the like installed on the expansion board or the expansion unit is caused to execute part and all of the actual operations, so as to implement the functions of any one of the above embodiments.

[0243] Finally, it should be noted that: the above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that: they can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements on some or all of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A book recommendation method based on a Lorentz diagram convolutional network, characterized in that, The method is as follows: Data collection: Obtain the book data of each user interaction; Construct a user-book interaction bipartite graph: Construct the collected Euclidean space data into non-Euclidean space data, that is, graph data. Specifically, construct the interaction data between users and books into a user-book interaction bipartite graph; Divide the dataset into a training set and a test set; Construct an LCF model: Initialize the relevant parameters of the Lorentz graph convolutional network model, and input the data of the training set into the Lorentz graph convolutional network model for training until the Lorentz graph convolutional network model converges; Generate prediction results and perform book recommendations: Jointly utilize the multiple embedding vectors learned by multiple graph convolutional layers to capture the embedding influences of different convolutional layers; For the final user and book embedding vectors generated by the Lorentz graph convolutional network model, obtain the prediction scores between users and books based on the hyperbolic distance, and recommend interesting books to users according to the prediction scores; Among them, the architecture of the LCF model includes A hyperbolic encoding layer, which is used to map the interaction information between users and books to the hyperbolic space using graph embedding technology based on the constructed user-book bipartite graph; A graph convolutional layer, which is used to learn the embedding representations of user and book nodes through the Lorentz graph convolutional network; A prediction layer, which is used to obtain the representation encoding rich structural information of users and books after L layers of graph convolution, and predict the preferences of users for books based on the hyperbolic distance; A hyperbolic loss function layer, which is used to separate the positive sample user-book pairs from the negative sample user-book pairs using a composite loss function, and apply regularizers to user and book nodes to make full use of the hyperbolic space until the network converges; The graph convolutional layer is specifically as follows: Perform hyperbolic feature transformation on all nodes; Specifically as follows: For a given point x = (x0, …, x n ) ∈ H d,β , transform the values of the last n coordinates to satisfy the constraints in the equation , ensuring that the transformed node features strictly follow hyperbolic geometry. The formula is as follows: where, M:R n →R m is an m×n matrix representing a linear mapping from n dimensions to m dimensions; is the logarithmic mapping operation for mapping points from the hyperbolic space to the tangent space; The initial embeddings of the user and the book are subjected to feature transformation using Lorentz matrix-vector multiplication to obtain the transformed embedding representation and Specifically: Hyperbolic neighbor aggregation: Use the centroid based on the squared Lorentz distance for neighbor information aggregation; Specifically as follows: For each node feature Calculate the attention-related score of each neighbor node j to the central node t; use the self-attention mechanism based on the Lorentz squared distance, and the formula is as follows: Among them, the matrix M a is a d×d-dimensional matrix used to transform node features into features with attention; For all neighbors N of node t t , the softmax function is introduced to normalize μ tj as follows: Centroid based on each user node embedding and book node embedding Perform neighbor information aggregation, that is, minimize the following equation: There is a closed-form solution, so the node embeddings of the user and the book are updated to: Lorentz pointwise non - linear activation: For a given point \(x=(x_0,\ldots,x\) n )\in H d,β we have: where, σ:R n →R n represents a pointwise non-linear mapping; The embeddings of users and books are updated as:

2. The book recommendation method based on the Lorentz diagram convolutional network according to claim 1, wherein Dividing the dataset into a training set and a test set is specifically as follows: Randomly select 80% from the historical interaction data in the dataset as the training set for model training; The remaining 20% in the dataset is used as the test set to evaluate the generalization ability of the model; Regard each observed user-book interaction data as a positive sample, and pair negative samples for the books that the user has not interacted with through a negative sampling strategy.

3. The book recommendation method based on the Lorentz diagram convolutional network according to claim 1, wherein The hyperbolic encoding layer is specifically as follows: Define the tangent space at point x as a d-dimensional vector space that approximates the hyperbolic manifold at point x The formula is as follows; Among them, represents the user-book interaction matrix; d represents the embedding size; β represents the negative reciprocal of the curvature; Initialization in Euclidean space is usually based on Gaussian sampling, and the hyperbolic Gaussian sampling method is applied to the Lorentz graph convolutional network model; in hyperbolic space, for a given user node u and book node i, a vector represents the origin in and uses point o as the reference point for performing tangent space operations. First, sample from the multivariate Gaussian distribution on the tangent space of point o to obtain the initial embeddings and of user u and book i, and add a "0" element to the first coordinate of the initial embeddings in (same as h in i ) to satisfy the constraint in and ; then map the initial embeddings and to hyperbolic space through exponential mapping to obtain the hyperbolic initial embeddings of all user nodes and book nodes. The formula is as follows: Among them, is an exponential mapping operation used to map points from the tangent space to the hyperbolic space; β represents the negative reciprocal of the curvature; k represents the embedding size.

4. The book recommendation method based on the Lorentz diagram convolutional network according to claim 1, wherein The prediction layer is specifically as follows: Consider the representations generated by each layer, and at the same time utilize the preferences of users or the attributes of books captured from different perspectives to obtain the prediction score of user u for target book i. The formula is as follows: Among them, and are the user and book representations generated at the l-th layer respectively; L is the total number of Lorentz graph convolution layers passed; Intuitively, in the same embedding space, when the embedding of book i is closer to that of user u, it means that user u prefers book i compared to other books.

5. The book recommendation method based on the Lorentz diagram convolutional network according to claim 1, characterized in that, The hyperbolic loss function layer is specifically as follows: Optimize using the margin ranking loss in the hyperbolic space: For each user u, sample a positive sample user-book pair (u, i) and a negative sample user-book pair (u, j), and define the margin ranking loss based on the hyperbolic distance as: Among them, m is the margin, which is a non-negative hyperparameter; Find a root node and align it with the hyperbolic origin. After the root node is aligned, other user nodes and book nodes will naturally be arranged well along with the optimization process of the loss function; Apply regularizers to the aligned user and book nodes: Limit each node in the loss function to have a large norm, that is, apply regularizers to the user node embeddings and book node embeddings. The formula is as follows: The final loss function is \(L = L\) g +\(\alpha L\) r ;\ where α is a weight hyperparameter used to balance the values of the two loss functions.

6. The book recommendation method based on the Lorentz diagram convolutional network according to claim 5, characterized in that Find a root node and align it with the hyperbolic origin. After the root node is aligned, other user nodes and book nodes will naturally be arranged during the optimization process of the loss function as follows: For each user node embedding and book node embedding The root node is calculated based on the center of the overall node embedding, and the formula is as follows: where N is the total number of user and book nodes; Align the root node with the hyperbolic origin, that is, subtract the root node from each node embedding. The formula is as follows:

7. A book recommendation system based on a Lorentz diagram convolutional network, characterized in that, This system is used to implement the book recommendation method based on the Lorentz graph convolutional network described in any one of claims 1-6; the system includes, a data acquisition module for obtaining book data interacted by each user; a bipartite graph construction module for constructing non-Euclidean space data from the collected Euclidean space data, that is, graph data. Specifically, the interaction data between users and books is constructed into a user-book interaction bipartite graph; a dataset division module for randomly selecting 80% of the historical interaction data in the dataset as the training set for model training; the remaining 20% in the dataset is used as the test set for evaluating the generalization ability of the model; each observed user-book interaction data is regarded as a positive sample, and negative samples are paired for the books not interacted by the user through the negative sampling strategy; a model construction module for constructing an LCF model architecture diagram, initializing the relevant parameters of the Lorentz graph convolutional network model, and inputting the data of the training set into the Lorentz graph convolutional network model for training until the Lorentz graph convolutional network model converges; a recommendation prediction module for jointly using multiple embedding vectors learned by multiple graph convolutional layers to capture the embedding influences of different convolutional layers; for the final user and book embedding vectors generated by the Lorentz graph convolutional network model, obtain the prediction scores between users and books based on the hyperbolic distance, and recommend interesting books for users according to the prediction scores.

8. An electronic device, characterized in that, including: a memory and at least one processor; wherein, a computer program is stored on the memory; the at least one processor executes the computer program stored in the memory, so that the at least one processor executes the book recommendation method based on the Lorentz graph convolutional network described in any one of claims 1 to 6.

9. A computer-readable storage medium, characterized in that, A computer program is stored in the computer-readable storage medium, and the computer program can be executed by a processor to implement the book recommendation method based on the Lorentz graph convolutional network described in any one of claims 1 to 6.

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