Harmonized System (HS) code classification is a hierarchically structured and regulationconstrained task, often complicated by short and noisy product descriptions.Misclassification can lead to tariff misapplication, regulatory violations, or delayed customs clearance; predictions therefore need to be both semantically appropriate and hierarchically valid.While large language models (LLMs) show strong semantic understanding, their unconstrained generation is poorly aligned with these requirements, often producing non-existent or hierarchically inconsistent codes.We propose HSGraphAgent, a knowledge-graph-guided LLM framework that formulates HS classification as a stepwise, regulation-aware reasoning process over an explicit HS knowledge graph.By encoding hierarchical containment relations and regulatory exclusion rules, and enforcing them through a Select-Redirect mechanism, HSGraphAgent constrains inference to legally valid paths while producing explicit and traceable reasoning trajectories.Experiments on taxonomy-wide 4-digit and fine-grained 6-digit HS benchmarks demonstrate consistent improvements over direct generation and retrievalaugmented baselines, with particularly strong gains in fine-grained and regulation-sensitive classification settings.

Tubular structure segmentation (TSS) is important for various applications, such as hemodynamic analysis and route navigation. Despite significant progress in TSS, domain shifts remain a major challenge, leading to performance degradation in unseen target domains. Unlike other segmentation tasks, TSS is more sensitive to domain shifts, as changes in topological structures can compromise segmentation integrity, and variations in local features distinguishing foreground from background (e.g., texture and contrast) may further disrupt topological continuity. To address these challenges, we propose Topology-enhanced Test-Time Adaptation (TopoTTA), the first test-time adaptation framework designed specifically for TSS. TopoTTA consists of two stages: Stage 1 adapts models to cross-domain topological discrepancies using the proposed Topological Meta Difference Convolutions (TopoMDCs), which enhance topological representation without altering pre-trained parameters; Stage 2 improves topological continuity by a novel Topology Hard sample Generation (TopoHG) strategy and prediction alignment on hard samples with pseudo-labels in the generated pseudo-break regions. Extensive experiments across four scenarios and ten datasets demonstrate TopoTTA's effectiveness in handling topological distribution shifts, achieving an average improvement of 31.81% in clDice. TopoTTA also serves as a plug-and-play TTA solution for CNN-based TSS models.

Despite recent advances in stereo matching, the extension to intricate underwater settings remains unexplored, primarily owing to: 1) the reduced visibility, low contrast, and other adverse effects of underwater images; 2) the difficulty in obtaining ground truth data for training deep learning models, i.e. simultaneously capturing an image and estimating its corresponding pixel-wise depth information in underwater environments. To enable further advance in underwater stereo matching, we introduce a large synthetic dataset called UWStereo. Our dataset includes 29,568 synthetic stereo image pairs with dense and accurate disparity annotations for left view. We design four distinct underwater scenes filled with diverse objects such as corals, ships and robots. We also induce additional variations in camera model, lighting, and environmental effects. In comparison with existing underwater datasets, UWStereo is superior in terms of scale, variation, annotation, and photo-realistic image quality. To substantiate the efficacy of the UWStereo dataset, we undertake a comprehensive evaluation compared with eleven state-of-the-art algorithms as benchmarks. The results indicate that current models still struggle to generalize to new domains. Hence, we design a new strategy that learns to reconstruct cross domain masked images before stereo matching training and integrate a cross view attention enhancement module that aggregates long-range content information to enhance the generalization ability.

Many neuropsychiatric disorders are considered to be associated with abnormalities in the functional connectivity networks of the brain. The research on the classification of functional connectivity can therefore provide new perspectives for understanding the pathology of disorders and contribute to early diagnosis and treatment. Functional connectivity exhibits a nature of dynamically changing over time, however, the majority of existing methods are unable to collectively reveal the spatial topology and time-varying characteristics. Furthermore, despite the efforts of limited spatial-temporal studies to capture rich information across different spatial scales, they have not delved into the temporal characteristics among different scales. To address above issues, we propose a novel Multi-Scale Spatial-Temporal Attention Networks (MSSTAN) to exploit the multi-scale spatial-temporal information provided by functional connectome for classification. To fully extract spatial features of brain regions, we propose a Topology Enhanced Graph Transformer module to guide the attention calculations in the learning of spatial features by incorporating topology priors. A Multi-Scale Pooling Strategy is introduced to obtain representations of brain connectome at various scales. Considering the temporal dynamic characteristics between dynamic functional connectome, we employ Locality Sensitive Hashing attention to further capture long-term dependencies in time dynamics across multiple scales and reduce the computational complexity of the original attention mechanism. Experiments on three brain fMRI datasets of MDD and ASD demonstrate the superiority of our proposed approach. In addition, benefiting from the attention mechanism in Transformer, our results are interpretable, which can contribute to the discovery of biomarkers. The code is available at https://github.com/LIST-KONG/MSSTAN.

In the existing spectral GNNs, polynomial-based methods occupy the mainstream in designing a filter through the Laplacian matrix. However, polynomial combinations factored by the Laplacian matrix naturally have limitations in message passing (e.g., over-smoothing). Furthermore, most existing spectral GNNs are based on polynomial bases, which struggle to capture the high-frequency parts of the graph spectral signal. Additionally, we also find that even increasing the polynomial order does not change this situation, which means polynomial-based models have a natural deficiency when facing high-frequency signals. To tackle these problems, we propose WaveNet, which aims to effectively capture the high-frequency part of the graph spectral signal from the perspective of wavelet bases through reconstructing the message propagation matrix. We utilize Multi-Resolution Analysis (MRA) to model this question, and our proposed method can reconstruct arbitrary filters theoretically. We also conduct node classification experiments on real-world graph benchmarks and achieve superior performance on most datasets. Our code is available at https://github.com/Bufordyang/WaveNet

Mainstream backdoor attack methods typically demand substantial tuning data for poisoning, limiting their practicality and potentially degrading the overall performance when applied to Large Language Models (LLMs). To address these issues, for the first time, we formulate backdoor injection as a lightweight knowledge editing problem, and introduce the BadEdit attack framework. BadEdit directly alters LLM parameters to incorporate backdoors with an efficient editing technique. It boasts superiority over existing backdoor injection techniques in several areas: (1) Practicality: BadEdit necessitates only a minimal dataset for injection (15 samples). (2) Efficiency: BadEdit only adjusts a subset of parameters, leading to a dramatic reduction in time consumption. (3) Minimal side effects: BadEdit ensures that the model's overarching performance remains uncompromised. (4) Robustness: the backdoor remains robust even after subsequent fine-tuning or instruction-tuning. Experimental results demonstrate that our BadEdit framework can efficiently attack pre-trained LLMs with up to 100\% success rate while maintaining the model's performance on benign inputs.

Recently, deep learning models have been widely applied in program understanding tasks, and these models achieve state-of-the-art results on many benchmark datasets. A major challenge of deep learning for program understanding is that the effectiveness of these approaches depends on the quality of their datasets, and these datasets often contain noisy data samples. A typical kind of noise in program understanding datasets is label noise, which means that the target outputs for some inputs are incorrect.

Software vulnerabilities pose serious threats to the security of modern software systems. Deep Learning-based Automated Vulnerability Repair (AVR) has gained attention as a potential solution to accelerate the remediation of vulnerabilities. However, recent studies indicate that existing AVR approaches often only generate patches, which may not align with developers' current repair practices or expectations. In this paper, we introduce VulAdvisor, an automated approach that generates natural language suggestions to guide developers or AVR tools in repairing vulnerabilities. VulAdvisor comprises two main components: oracle extraction and suggestion learning. To address the challenge of limited historical data, we propose an oracle extraction method facilitating ChatGPT to construct a comprehensive and high-quality dataset. For suggestion learning, we take the supervised fine-tuning CodeT5 model as the basis, integrating local context into Multi-Head Attention and introducing a repair action loss, to improve the relevance and meaningfulness of the generated suggestions. Extensive experiments on a large-scale dataset from real-world C/C++ projects demonstrate the effectiveness of VulAdvisor, surpassing several alternatives in terms of both lexical and semantic metrics. Moreover, we show that the generated suggestions enhance the patch generation capabilities of existing AVR tools. Human evaluations further validate the quality and utility of VulAdvisor's suggestions, confirming their potential to improve software vulnerability repair practices.

Code models have made significant advancements in code intelligence by encoding knowledge about programming languages. While previous studies have explored the capabilities of these models in learning code syntax, there has been limited investigation on their ability to understand code semantics. Additionally, existing analyses assume that the number of edges between nodes at the abstract syntax tree (AST) is related to syntax distance, and also often require transforming the high-dimensional space of deep learning models to a low-dimensional one, which may introduce inaccuracies. To study how code models represent code syntax and semantics, we conduct a comprehensive analysis of seven code models, including four representative code pre-trained models (CodeBERT, GraphCodeBERT, CodeT5, and UnixCoder) and three large language models (LLMs) (StarCoder, CodeLlama and CodeT5+). We design four probing tasks to assess the models’ capacities in learning both code syntax and semantics. These probing tasks reconstruct code syntax and semantics structures (AST, control dependence graph (CDG), data dependence graph (DDG), and control flow graph (CFG)) in the representation space. These structures are core concepts for code understanding. We also investigate the syntax token role in each token representation and the long dependency between the code tokens. Additionally, we analyze the distribution of attention weights related to code semantic structures. Through extensive analysis, our findings highlight the strengths and limitations of different code models in learning code syntax and semantics. The results demonstrate that these models excel in learning code syntax, successfully capturing the syntax relationships between tokens and the syntax roles of individual tokens. However, their performance in encoding code semantics varies. CodeT5 and CodeBERT demonstrate proficiency in capturing control and data dependencies, whereas UnixCoder shows weaker performance in this aspect. We do not observe LLMs generally performing much better than pre-trained models. The shallow layers of LLMs perform better than their deep layers. The investigation of attention weights reveals that different attention heads play distinct roles in encoding code semantics. Our research findings emphasize the need for further enhancements in code models to better learn code semantics. This study contributes to the understanding of code models’ abilities in syntax and semantics analysis. Our findings provide guidance for future improvements in code models, facilitating their effective application in various code-related tasks.

Learning about the cognitive state of the brain has always been a popular topic. Based on the fact that fluctuations of brain signals and functional connectome (FC) relate to specific human behaviors, deep learning based methods have shown promising results on the prediction of such behaviors by analyzing biological signals. Existing methods either model from static perspectives or apply spatial-temporal graph convolution to extract dynamic properties. However, the static information and dynamic information can reflect global brain activities and local brain activities respectively. Thus, we propose BrainNetFormer to incorporate both static and dynamic properties for human behavior prediction. To be specific, a spatial cross attention module and a temporal cross attention module are introduced for information fusion. In addition, since a specific behavior of subjects can be decomposed into a series of subtasks, we introduce a sub-task regularization loss to assist in training and empower the model to recognize subtasks at each moment. Experiments on the HCP-Task dataset demonstrate the superior performance of the proposed model.

Contextual batched bandit (CBB) is a setting where a batch of rewards is observed from the environment at the end of each episode, but the rewards of the non-executed actions are unobserved, resulting in partial-information feedback. Existing approaches for CBB often ignore the rewards of the non-executed actions, leading to underutilization of feedback information. In this paper, we propose an efficient approach called Sketched Policy Updating with Imputed Rewards (SPUIR) that completes the unobserved rewards using sketching, which approximates the full-information feedbacks. We formulate reward imputation as an imputation regularized ridge regression problem that captures the feedback mechanisms of both executed and non-executed actions. To reduce time complexity, we solve the regression problem using randomized sketching. We prove that our approach achieves an instantaneous regret with controllable bias and smaller variance than approaches without reward imputation. Furthermore, our approach enjoys a sublinear regret bound against the optimal policy. We also present two extensions, a rate-scheduled version and a version for nonlinear rewards, making our approach more practical. Experimental results show that SPUIR outperforms state-of-the-art baselines on synthetic, public benchmark, and real-world datasets.

Learning vector representations for programs is a critical step in applying deep learning techniques for program understanding tasks. Various neural network models are proposed to learn from tree-structured program representations, e.g., abstract syntax tree (AST) and concrete syntax tree (CST). However, most neural architectures either fail to capture long-range dependencies which are ubiquitous in programs, or cannot learn effective representations for syntax tree nodes, making them incapable of performing the node-level prediction tasks, e.g., bug localization. In this paper, we propose Tree-Transformer, a novel recursive tree-structured neural network to learn the vector representations for source codes. We propose a multi-head attention mechanism to model the dependency between siblings and parent-children node pairs. Moreover, we propose a bi-directional propagation strategy to allow node information passing in two directions, bottom-up and top-down along trees. In this way, Tree-Transformer can learn the information of the node features as well as the global contextual information. The extensive experimental results show that our Tree-Transformer significantly outperforms the existing tree-based and graph-based program representation learning approaches in both the tree-level and node-level prediction tasks.

The pathophysiology of major depressive disorder (MDD) has been demonstrated to be highly associated with the dysfunctional integration of brain activity. Existing studies only fuse multi-connectivity information in a one-shot approach and ignore the temporal property of functional connectivity. A desired model should utilize the rich information in multiple connectivities to help improve the performance. In this study, we develop a multi-connectivity representation learning framework to integrate multi-connectivity topological representation from structural connectivity, functional connectivity and dynamic functional connectivities for automatic diagnosis of MDD. Briefly, structural graph, static functional graph and dynamic functional graphs are first computed from the diffusion magnetic resonance imaging (dMRI) and resting state functional magnetic resonance imaging (rsfMRI). Secondly, a novel Multi-Connectivity Representation Learning Network (MCRLN) approach is developed to integrate the multiple graphs with modules of structural-functional fusion and static-dynamic fusion. We innovatively design a Structural-Functional Fusion (SFF) module, which decouples graph convolution to capture modality-specific features and modality-shared features separately for an accurate brain region representation. To further integrate the static graphs and dynamic functional graphs, a novel Static-Dynamic Fusion (SDF) module is developed to pass the important connections from static graphs to dynamic graphs via attention values. Finally, the performance of the proposed approach is comprehensively examined with large cohorts of clinical data, which demonstrates its effectiveness in classifying MDD patients. The sound performance suggests the potential of the MCRLN approach for the clinical use in diagnosis. The code is available at https://github.com/LIST-KONG/MultiConnectivity-master.

Numerous evidence has demonstrated the pathophysiology of a number of mental disorders is intimately associated with abnormal changes of dysfunctional integration of brain network. Functional connectome (FC) exhibits a strong discriminative power for mental disorder identification. However, existing methods are insufficient for modeling both spatial correlation and temporal dynamics of FC. In this study, we propose a novel Spatio-Temporal Attention Graph Convolution Network (STAGCN) for FC classification. In spatial domain, we develop attention enhanced graph convolutional network to take advantage of brain regions’ topological features. Moreover, a novel multi-head self-attention approach is proposed to capture the temporal relationships among different dynamic FC. Extensive experiments on two tasks of mental disorder diagnosis demonstrate the superior performance of the proposed STAGCN.

Prediction of brain functional activity is of great significance for neuroscience research. The brain functional activities at different regions are highly related, and their relationships can be captured with functional connectivity and structural connectivity. The existing works are challenging to integrate two connectivity information for functional activity prediction. In this paper, we propose a Temporal Cross-Graph Network (TCGN) for predicting brain functional activity, which can comprehensively exploit multi-modal spatial dependence and temporal patterns. In particular, a novel cross-graph convolution module is developed to capture the spatial features of brain structural and functional connectivity. A temporal fusion module is designed to learn the pattern of dynamic functional connectivity to guide the prediction. Specially, a multi-task loss function is proposed to incorporate functional activity and dynamic functional connectivity. Extensive experiments on the Human Connectome Project dataset demonstrate the effectiveness of the proposed framework.

Code representation, which transforms programs into vectors with semantics, is essential for source code processing. We have witnessed the effectiveness of incorporating structural information (i.e., graph) into code representations in recent years. Specifically, the abstract syntax tree (AST) and the AST-augmented graph of the program contain much structural and semantic information, and most existing studies apply them for code representation. The graph adopted by existing approaches is homogeneous, i.e., it discards the type information of the edges and the nodes lying within AST. That may cause plausible obstruction to the representation model. In this paper, we propose to leverage the type information in the graph for code representation. To be specific, we propose the heterogeneous program graph (HPG), which provides the types of the nodes and the edges explicitly. Furthermore, we employ the heterogeneous graph transformer (HGT) architecture to generate representations based on HPG, considering the type of information during processing. With the additional types in HPG, our approach can capture complex structural information, produce accurate and delicate representations, and finally perform well on certain tasks. Our in-depth evaluations upon four classic datasets for two typical tasks (i.e., method name prediction and code classification) demonstrate that the heterogeneous types in HPG benefit the representation models. Our proposed HPG+HGT also outperforms the SOTA baselines on the subject tasks and datasets.

Given a stream of money transactions between accounts in a bank, how can we accurately detect money laundering agent accounts and suspected behaviors in real-time? Money laundering agents try to hide the origin of illegally obtained money by dispersive multiple small transactions and evade detection by smart strategies. Therefore, it is challenging to accurately catch such fraudsters in an unsupervised manner. Existing approaches do not consider the characteristics of those agent accounts and are not suitable to the streaming settings. Therefore, we propose MonLAD and MonLAD-W to detect money laundering agent accounts in a transaction stream by keeping track of their residuals and other features; we devise AnoScore algorithm to find anomalies based on the robust measure of statistical deviation. Experimental results show that MonLAD outperforms the state-of-the-art baselines on real-world data and finds various suspicious behavior patterns of money laundering. Additionally, several detected suspected accounts have been manually-verified as agents in real money laundering scenario.

Nearest Neighbor Search (NNS) has recently drawn a rapid growth of interest because of its core role in high-dimensional vector data management in data science and AI applications. The interest is fueled by the success of neural embedding, where deep learning models transform unstructured data into semantically correlated feature vectors for data analysis, e.g., recommending popular items. Among several categories of methods for fast NNS, graph-based approximate nearest neighbor search algorithms have led to the best-in-class search performance on a wide range of real-world datasets. While prior works improve graph-based NNS search efficiency mainly through exploiting the structure of the graph with sophisticated heuristic rules, in this work, we show that the frequency distributions of edge visits for graph-based NNS can be highly skewed. This finding leads to the study of pruning unnecessary edges to avoid redundant computation during graph traversal by utilizing the query distribution, an important yet under-explored aspect of graph-based NNS. In particular, we formulate graph pruning as a discrete optimization problem, and introduce a graph optimization algorithm GraSP that improves the search efficiency of similarity graphs by learning to prune redundant edges. GraSP enhances an existing similarity graph with a probabilistic model. It then performs a novel subgraph sampling and iterative refinement optimization to explicitly maximize search efficiency when removing a subset of edges in expectation over a graph for a large set of training queries. The evaluation shows that GraSP consistently improves the search efficiency on real-world datasets, providing up to 2.24X faster search speed than state-of-the-art methods without losing accuracy.

Coded distributed computing (CDC) is an efficient method to reduce the communication load in general distributed computing frameworks such as MapReduce. In these systems, one usually needs to split the data set into disjoint files and design several output functions to complete a computational task. Li et al. provided some CDC schemes achieving optimal communication load. However, as the number of computing nodes increases, the numbers of input files and output functions of such schemes grow too fast to be applied in practice. In this paper, several infinite families of cascaded CDC schemes, where each output function is computed multiple times, are constructed by taking advantage of symmetric designs. Most importantly, the numbers of input files and output functions of each new scheme are linear with the number of computing nodes and the communication load of each new scheme approximates to that of the CDC scheme derived by Li et al. when the number of nodes becomes large.

Learning distributed representation of source code requires modelling its syntax and semantics. Recent state-of-the-art models leverage highly structured source code representations, such as the syntax trees and paths therein. In this paper, we investigate two representative path encoding methods shown in previous research work and integrate them into the attention module of Transformer. We draw inspiration from the ideas of positional encoding and modify them to incorporate these path encoding. Specifically, we encode both the pairwise path between tokens of source code and the path from the leaf node to the tree root for each token in the syntax tree. We explore the interaction between these two kinds of paths by integrating them into the unified Transformer framework. The detailed empirical study for path encoding methods also leads to our novel state-of-the-art representation model TPTrans, which finally outperforms strong baselines. Extensive experiments and ablation studies on code summarization across four different languages demonstrate the effectiveness of our approaches. We release our code at \url{https://github.com/AwdHanPeng/TPTrans}.

The user feedbacks could be delayed in many streaming recommendation scenarios. As an example, the user feedbacks to a recommended coupon consist of the immediate feedback on the click event and the delayed feedback on the resultant conversion. The delayed feedbacks pose a challenge of training recommendation models using instances with incomplete labels. When being applied to real products, the challenge becomes more severe as the streaming recommendation models need to be retrained very frequently and the training instances need to be collected over very short time scales. Existing approaches either simply ignore the unobserved feedbacks or heuristically adjust the feedbacks on a static instance set, resulting in biases in the training data and hurting the accuracy of the learned recommenders. In this paper, we propose a novel and theoretic sound counterfactual approach to adjusting the user feedbacks and learning the recommendation models, called CBDF (Counterfactual Bandit with Delayed Feedback). CBDF formulates the streaming recommendation with delayed feedback as a problem of sequential decision making and models it with a batched bandit. To deal with the issue of delayed feedback, at each iteration (episode), a counterfactual importance sampling model is employed to re-weight the original feedbacks and generate the modified rewards. Based on the modified rewards, a batched bandit is learned for conducting online recommendation at the next iteration. Theoretical analysis showed that the modified rewards are statistically unbiased, and the learned bandit policy enjoys a sub-linear regret bound. Experimental results demonstrated that CBDF can outperform the state-of-the-art baselines on a synthetic dataset, the Criteo dataset, and a dataset from Tencent's WeChat app.

Code clones are semantically similar code fragments pairs that are syntactically similar or different. Detection of code clones can help to reduce the cost of software maintenance and prevent bugs. Numerous approaches of detecting code clones have been proposed previously, but most of them focus on detecting syntactic clones and do not work well on semantic clones with different syntactic features. To detect semantic clones, researchers have tried to adopt deep learning for code clone detection to automatically learn latent semantic features from data. Especially, to leverage grammar information, several approaches used abstract syntax trees (AST) as input and achieved significant progress on code clone benchmarks in various programming languages. However, these AST-based approaches still can not fully leverage the structural information of code fragments, especially semantic information such as control flow and data flow. To leverage control and data flow information, in this paper, we build a graph representation of programs called flow-augmented abstract syntax tree (FA-AST). We construct FA-AST by augmenting original ASTs with explicit control and data flow edges. Then we apply two different types of graph neural networks (GNN) on FA-AST to measure the similarity of code pairs. As far as we have concerned, we are the first to apply graph neural networks on the domain of code clone detection. We apply our FA-AST and graph neural networks on two Java datasets: Google Code Jam and BigCloneBench. Our approach outperforms the state-of-the-art approaches on both Google Code Jam and BigCloneBench tasks.

With the rapid development of wireless communication technology, various indoor location‐based services (ILBSs) have gradually penetrated into daily life. Although many other methods have been proposed to be applied to ILBS in the past decade, WiFi‐based positioning techniques with a wide range of infrastructure have attracted attention in the field of wireless transmission. In this survey, the authors divide WiFi‐based indoor positioning techniques into the active positioning technique and the passive positioning technique based on whether the target carries certain devices. After reviewing a large number of excellent papers in the related field, the authors make a detailed summary of these two types of positioning techniques. In addition, they also analyse the challenges and future development trends in the current technological environment.

Learning representation for source code is a foundation of many program analysis tasks. In recent years, neural networks have already shown success in this area, but most existing models did not make full use of the unique structural information of programs. Although abstract syntax tree (AST)-based neural models can handle the tree structure in the source code, they cannot capture the richness of different types of substructure in programs. In this article, we propose a modular tree network that dynamically composes different neural network units into tree structures based on the input AST. Different from previous tree-structural neural network models, a modular tree network can capture the semantic differences between types of AST substructures. We evaluate our model on two tasks: program classification and code clone detection. Our model achieves the best performance compared with state-of-the-art approaches in both tasks, showing the advantage of leveraging more elaborate structure information of the source code.

Neural language models (NLMs) have recently gained a renewed interest by achieving state-of-the-art performance across many natural language processing (NLP) tasks. However, NLMs are very computationally demanding largely due to the computational cost of the decoding process, which consists of a softmax layer over a large vocabulary.We observe that in the decoding of many NLP tasks, only the probabilities of the top-K hypotheses need to be calculated preciously and K is often much smaller than the vocabulary size. This paper proposes a novel softmax layer approximation algorithm, called Fast Graph Decoder (FGD), which quickly identifies, for a given context, a set of K words that are most likely to occur according to a NLM. We demonstrate that FGD reduces the decoding time by an order of magnitude while attaining close to the full softmax baseline accuracy on neural machine translation and language modeling tasks. We also prove the theoretical guarantee on the softmax approximation quality.