Ran Li

Weize Chen, Ziming You, Ran Li et al.

The rapid advancement of large language models (LLMs) has paved the way for the development of highly capable autonomous agents. However, existing multi-agent frameworks often struggle with integrating diverse capable third-party agents due to reliance on agents defined within their own ecosystems. They also face challenges in simulating distributed environments, as most frameworks are limited to single-device setups. Furthermore, these frameworks often rely on hard-coded communication pipelines, limiting their adaptability to dynamic task requirements. Inspired by the concept of the Internet, we propose the Internet of Agents (IoA), a novel framework that addresses these limitations by providing a flexible and scalable platform for LLM-based multi-agent collaboration. IoA introduces an agent integration protocol, an instant-messaging-like architecture design, and dynamic mechanisms for agent teaming and conversation flow control. Through extensive experiments on general assistant tasks, embodied AI tasks, and retrieval-augmented generation benchmarks, we demonstrate that IoA consistently outperforms state-of-the-art baselines, showcasing its ability to facilitate effective collaboration among heterogeneous agents. IoA represents a step towards linking diverse agents in an Internet-like environment, where agents can seamlessly collaborate to achieve greater intelligence and capabilities. Our codebase has been released at \url{https://github.com/OpenBMB/IoA}.

Xin Li, Weize Chen, Qizhi Chu et al.

The need to analyze graphs is ubiquitous across various fields, from social networks to biological research and recommendation systems. Therefore, enabling the ability of large language models (LLMs) to process graphs is an important step toward more advanced general intelligence. However, current LLM benchmarks on graph analysis require models to directly reason over the prompts describing graph topology, and are thus limited to small graphs with only a few dozens of nodes. In contrast, human experts typically write programs based on popular libraries for task solving, and can thus handle graphs with different scales. To this end, a question naturally arises: can LLMs analyze graphs like professionals? In this paper, we introduce ProGraph, a manually crafted benchmark containing 3 categories of graph tasks. The benchmark expects solutions based on programming instead of directly reasoning over raw inputs. Our findings reveal that the performance of current LLMs is unsatisfactory, with the best model achieving only 36% accuracy. To bridge this gap, we propose LLM4Graph datasets, which include crawled documents and auto-generated codes based on 6 widely used graph libraries. By augmenting closed-source LLMs with document retrieval and fine-tuning open-source ones on the codes, we show 11-32% absolute improvements in their accuracies. Our results underscore that the capabilities of LLMs in handling structured data are still under-explored, and show the effectiveness of LLM4Graph in enhancing LLMs' proficiency of graph analysis. The benchmark, datasets and enhanced open-source models are available at https://github.com/BUPT-GAMMA/ProGraph.

Shuwei Shao, Zhongcai Pei, Weihai Chen et al.

This work aims to estimate a high-quality depth map from a single RGB image. Due to the lack of depth clues, making full use of the long-range correlation and the local information is critical for accurate depth estimation. Towards this end, we introduce an uncertainty rectified cross-distillation between Transformer and convolutional neural network (CNN) to learn a unified depth estimator. Specifically, we use the depth estimates from the Transformer branch and the CNN branch as pseudo labels to teach each other. Meanwhile, we model the pixel-wise depth uncertainty to rectify the loss weights of noisy pseudo labels. To avoid the large capacity gap induced by the strong Transformer branch deteriorating the cross-distillation, we transfer the feature maps from Transformer to CNN and design coupling units to assist the weak CNN branch to leverage the transferred features. Furthermore, we propose a surprisingly simple yet highly effective data augmentation technique CutFlip, which enforces the model to exploit more valuable clues apart from the vertical image position for depth inference. Extensive experiments demonstrate that our model, termed~\textbf{URCDC-Depth}, exceeds previous state-of-the-art methods on the KITTI, NYU-Depth-v2 and SUN RGB-D datasets, even with no additional computational burden at inference time. The source code is publicly available at \url{https://github.com/ShuweiShao/URCDC-Depth}.

Andrey Ignatov, Radu Timofte, Shuai Liu et al.

The role of mobile cameras increased dramatically over the past few years, leading to more and more research in automatic image quality enhancement and RAW photo processing. In this Mobile AI challenge, the target was to develop an efficient end-to-end AI-based image signal processing (ISP) pipeline replacing the standard mobile ISPs that can run on modern smartphone GPUs using TensorFlow Lite. The participants were provided with a large-scale Fujifilm UltraISP dataset consisting of thousands of paired photos captured with a normal mobile camera sensor and a professional 102MP medium-format FujiFilm GFX100 camera. The runtime of the resulting models was evaluated on the Snapdragon's 8 Gen 1 GPU that provides excellent acceleration results for the majority of common deep learning ops. The proposed solutions are compatible with all recent mobile GPUs, being able to process Full HD photos in less than 20-50 milliseconds while achieving high fidelity results. A detailed description of all models developed in this challenge is provided in this paper.

Xiancheng Wang, Zhibo Zhang, Ran Li et al.

This paper proposes a two-stage pseudo anomaly-guided anomaly detection method (\textbf{T}wo-stage \textbf{P}seudo \textbf{A}nomaly-guided \textbf{A}nomaly \textbf{D}etection, \textbf{TPA-AD}) for axle-box bearing time-series anomaly detection (time series anomaly detection, TSAD) under the setting where only normal samples are available for training. The method first generates pseudo-anomalous windows near the normal boundary using a reconstruction model and per-feature target-error control. It then learns anomaly-sensitive representations through contrastive learning between normal and pseudo-anomalous windows, and finally produces window-level and point-level anomaly scores using k-nearest neighbors (KNN). Compared with existing methods that rely on known fault categories, real anomaly priors, or random anomaly injection, TPA-AD improves the separability of the normal boundary by constructing pseudo-anomalies in boundary neighborhoods and can jointly handle continuous and discrete features in mixed-variable scenarios. The main experiments are conducted on bearing fault detection datasets and degradation-process datasets, with an additional exploratory extension on $13$ public TSAD datasets. The results show that the proposed method yields relatively stable anomaly responses, is sensitive to degradation evolution, and demonstrates a certain degree of broader applicability on public TSAD benchmarks and real high-speed-train-related bearing data.

Zhong Liu, Ran Li, Shuwei Shao et al.

Monocular depth estimation plays a fundamental role in computer vision. Due to the costly acquisition of depth ground truth, self-supervised methods that leverage adjacent frames to establish a supervisory signal have emerged as the most promising paradigms. In this work, we propose two novel ideas to improve self-supervised monocular depth estimation: 1) self-reference distillation and 2) disparity offset refinement. Specifically, we use a parameter-optimized model as the teacher updated as the training epochs to provide additional supervision during the training process. The teacher model has the same structure as the student model, with weights inherited from the historical student model. In addition, a multiview check is introduced to filter out the outliers produced by the teacher model. Furthermore, we leverage the contextual consistency between high-scale and low-scale features to obtain multiscale disparity offsets, which are used to refine the disparity output incrementally by aligning disparity information at different scales. The experimental results on the KITTI and Make3D datasets show that our method outperforms previous state-of-the-art competitors.

Peiye Zhuang, Bliss Chapman, Ran Li et al.

In the experimental sciences, statistical power analyses are often used before data collection to determine the required sample size. However, traditional power analyses can be costly when data are difficult or expensive to collect. We propose synthetic power analyses; a framework for estimating statistical power at various sample sizes, and empirically explore the performance of synthetic power analysis for sample size selection in cognitive neuroscience experiments. To this end, brain imaging data is synthesized using an implicit generative model conditioned on observed cognitive processes. Further, we propose a simple procedure to modify the statistical tests which result in conservative statistics. Our empirical results suggest that synthetic power analysis could be a low-cost alternative to pilot data collection when the proposed experiments share cognitive processes with previously conducted experiments.

Ran Li, Wei Hao, Weiliang Zhao et al.

Large language models (LLMs) present significant risks when used to generate non-factual content and spread disinformation at scale. Detecting such LLM-generated content is crucial, yet current detectors often struggle to generalize in open-world contexts. We introduce Learning2Rewrite, a novel framework for detecting AI-generated text with exceptional generalization to unseen domains. Our method leverages the insight that LLMs inherently modify AI-generated content less than human-written text when tasked with rewriting. By training LLMs to minimize alterations on AI-generated inputs, we amplify this disparity, yielding a more distinguishable and generalizable edit distance across diverse text distributions. Extensive experiments on data from 21 independent domains and four major LLMs (GPT-3.5, GPT-4, Gemini, and Llama-3) demonstrate that our detector outperforms state-of-the-art detection methods by up to 23.04% in AUROC for in-distribution tests, 37.26% for out-of-distribution tests, and 48.66% under adversarial attacks. Our unique training objective ensures better generalizability compared to directly training for classification, when leveraging the same amount of parameters. Our findings suggest that reinforcing LLMs' inherent rewriting tendencies offers a robust and scalable solution for detecting AI-generated text.

Ran Li, Zeyuan Liu, Yinghao chen et al.

Large Language Models (LLMs) have demonstrated strong potential in complex reasoning, yet their progress remains fundamentally constrained by reliance on massive high-quality human-curated tasks and labels, either through supervised fine-tuning (SFT) or reinforcement learning (RL) on reasoning-specific data. This dependence renders supervision-heavy training paradigms increasingly unsustainable, with signs of diminishing scalability already evident in practice. To overcome this limitation, we introduce CPMöbius (CPMobius), a collaborative Coach-Player paradigm for data-free reinforcement learning of reasoning models. Unlike traditional adversarial self-play, CPMöbius, inspired by real world human sports collaboration and multi-agent collaboration, treats the Coach and Player as independent but cooperative roles. The Coach proposes instructions targeted at the Player's capability and receives rewards based on changes in the Player's performance, while the Player is rewarded for solving the increasingly instructive tasks generated by the Coach. This cooperative optimization loop is designed to directly enhance the Player's mathematical reasoning ability. Remarkably, CPMöbius achieves substantial improvement without relying on any external training data, outperforming existing unsupervised approaches. For example, on Qwen2.5-Math-7B-Instruct, our method improves accuracy by an overall average of +4.9 and an out-of-distribution average of +5.4, exceeding RENT by +1.5 on overall accuracy and R-zero by +4.2 on OOD accuracy.

Zhili Cheng, Yuge Tu, Ran Li et al. · pku, tsinghua

Multimodal Large Language Models (MLLMs) have shown significant advancements, providing a promising future for embodied agents. Existing benchmarks for evaluating MLLMs primarily utilize static images or videos, limiting assessments to non-interactive scenarios. Meanwhile, existing embodied AI benchmarks are task-specific and not diverse enough, which do not adequately evaluate the embodied capabilities of MLLMs. To address this, we propose EmbodiedEval, a comprehensive and interactive evaluation benchmark for MLLMs with embodied tasks. EmbodiedEval features 328 distinct tasks within 125 varied 3D scenes, each of which is rigorously selected and annotated. It covers a broad spectrum of existing embodied AI tasks with significantly enhanced diversity, all within a unified simulation and evaluation framework tailored for MLLMs. The tasks are organized into five categories: navigation, object interaction, social interaction, attribute question answering, and spatial question answering to assess different capabilities of the agents. We evaluated the state-of-the-art MLLMs on EmbodiedEval and found that they have a significant shortfall compared to human level on embodied tasks. Our analysis demonstrates the limitations of existing MLLMs in embodied capabilities, providing insights for their future development. We open-source all evaluation data and simulation framework at https://github.com/thunlp/EmbodiedEval.

Huanwen Liang, Jingxian Xu, Yuanji Zhang et al.

Fetal abdominal malformations are serious congenital anomalies that require accurate diagnosis to guide pregnancy management and reduce mortality. Although AI has demonstrated significant potential in medical diagnosis, its application to prenatal abdominal anomalies remains limited. Most existing studies focus on image-level classification and rely on standard plane localization, placing less emphasis on case-level diagnosis. In this paper, we develop a case-level multiple instance learning (MIL)-based method, free of standard plane localization, for classifying fetal abdominal anomalies in prenatal ultrasound. Our contribution is three-fold. First, we adopt a mixture-of-attention-experts module (MoAE) to weight different attention heads for various planes. Secondly, we propose a medical-knowledge-driven feature selection module (MFS) to align image features with medical knowledge, performing self-supervised image token selection at the case-level. Finally, we propose a prompt-based prototype learning (PPL) to enhance the MFS. Extensively validated on a large prenatal abdominal ultrasound dataset containing 2,419 cases, with a total of 24,748 images and 6 categories, our proposed method outperforms the state-of-the-art competitors. Codes are available at:https://github.com/LL-AC/AAcls.

Ran Li, Shimin Di, Yuchen Liu et al.

Previous study suggest that powerful Large Language Models (LLMs) trained with Reinforcement Learning with Verifiable Rewards (RLVR) only refines reasoning path without improving the reasoning capacity in math tasks while supervised-finetuning(SFT) with distillation can. We study this from the view of Scientific information extraction (SciIE) where LLMs and reasoning LLMs underperforms small Bert-based models. SciIE require both the reasoning and memorization. We argue that both SFT and RLVR can refine the reasoning path and improve reasoning capacity in a simple way based on SciIE. We propose two-stage training with 1. MimicSFT, using structured reasoning templates without needing high-quality chain-of-thought data, 2. R$^2$GRPO with relevance and rule-induced rewards. Experiments on scientific IE benchmarks show that both methods can improve the reasoning capacity. R$^2$GRPO with mimicSFT surpasses baseline LLMs and specialized supervised models in relation extraction. Our code is available at https://github.com/ranlislz/R2GRPO.

Anjiang Wei, Jiannan Cao, Ran Li et al. · stanford

As large language models (LLMs) become integral to code-related tasks, a central question emerges: Do LLMs truly understand program semantics? We introduce EquiBench, a new benchmark for evaluating LLMs through equivalence checking, i.e., determining whether two programs produce identical outputs for all possible inputs. Unlike prior code generation benchmarks, this task directly tests a model's ability to reason about program semantics. EquiBench consists of 2400 program pairs across four languages and six categories. These pairs are generated through program analysis, compiler scheduling, and superoptimization, ensuring high-confidence labels, nontrivial difficulty, and full automation. We evaluate 19 state-of-the-art LLMs and find that in the most challenging categories, the best accuracies are 63.8% and 76.2%, only modestly above the 50% random baseline. Further analysis reveals that models often rely on syntactic similarity rather than exhibiting robust reasoning about program semantics, highlighting current limitations. Our code and dataset are publicly available at https://github.com/Anjiang-Wei/equibench

Ran Li, Pan Xiao, Kaushik Dutta et al.

Fluorescence Microcopy Calcium Imaging is a fundamental tool to in-vivo record and analyze large scale neuronal activities simultaneously at a single cell resolution. Automatic and precise detection of behaviorally relevant neuron activity from the recordings is critical to study the mapping of brain activity in organisms. However a perpetual bottleneck to this problem is the manual segmentation which is time and labor intensive and lacks generalizability. To this end, we present a Bayesian Deep Learning Framework to detect neuronal activities in 4D spatio-temporal data obtained by light sheet microscopy. Our approach accounts for the use of temporal information by calculating pixel wise correlation maps and combines it with spatial information given by the mean summary image. The Bayesian framework not only produces probability segmentation maps but also models the uncertainty pertaining to active neuron detection. To evaluate the accuracy of our framework we implemented the test of reproducibility to assert the generalization of the network to detect neuron activity. The network achieved a mean Dice Score of 0.81 relative to the synthetic Ground Truth obtained by Otsu's method and a mean Dice Score of 0.79 between the first and second run for test of reproducibility. Our method successfully deployed can be used for rapid detection of active neuronal activities for behavioural studies.

Xu Li, Ruiqi Sun, Jiameng Lv et al.

Strong gravitational lensing is a powerful tool for investigating dark matter and dark energy properties. With the advent of large-scale sky surveys, we can discover strong lensing systems on an unprecedented scale, which requires efficient tools to extract them from billions of astronomical objects. The existing mainstream lens-finding tools are based on machine learning algorithms and applied to cut-out-centered galaxies. However, according to the design and survey strategy of optical surveys by CSST, preparing cutouts with multiple bands requires considerable efforts. To overcome these challenges, we have developed a framework based on a hierarchical visual Transformer with a sliding window technique to search for strong lensing systems within entire images. Moreover, given that multi-color images of strong lensing systems can provide insights into their physical characteristics, our framework is specifically crafted to identify strong lensing systems in images with any number of channels. As evaluated using CSST mock data based on an Semi-Analytic Model named CosmoDC2, our framework achieves precision and recall rates of 0.98 and 0.90, respectively. To evaluate the effectiveness of our method in real observations, we have applied it to a subset of images from the DESI Legacy Imaging Surveys and media images from Euclid Early Release Observations. 61 new strong lensing system candidates are discovered by our method. However, we also identified false positives arising primarily from the simplified galaxy morphology assumptions within the simulation. This underscores the practical limitations of our approach while simultaneously highlighting potential avenues for future improvements.

Qintao Du, Ran Li, Weiyi Lv et al.

Conservation Voltage Reduction (CVR) relies on the effective coordination of slow-acting devices, such as OLTCs and CBs, and fast-acting devices, such as SVGs and PV inverters, typically implemented through a hierarchical multi-stage Volt-Var Control (VVC) spanning day-ahead scheduling, intra-day dispatch, and real-time control. However, existing sequential methods fail to account for the cas-cading impact of forecast errors on multi-stage decision-making. This oversight results in suboptimal day-ahead schedules for OLTCs and CBs that hinder the ef-fective coordination with fast-acting SVGs and inverters, inevitably driving a trade-off between real-time voltage security and CVR efficiency. To improve the Pareto front of this trade-off, this paper proposes a novel bi-level multi-timescale forecasting (Bi-MTF) framework for multi-stage VVC optimization. By integrating the downstream multi-stage VVC optimization into the upstream forecasting mod-els training, the decision-focused forecasting models are able to learn the trade-offs across temporal horizons. To solve the computationally challenging bi-level for-mulation, a modified sensitivity-driven integer L-shaped method is developed. It utilizes a hybrid gradient feedback mechanism that integrates numerical sensitivity analysis for discrete variables with analytical dual information for continuous fore-cast parameters to ensure tractability. Numerical results on a modified IEEE 33-bus system demonstrate that the proposed approach yields superior energy savings and operational safety compared to conventional MSE-based sequential paradigms. Specifically, as the capacity of fast-acting devices increases, the energy savings of the proposed method rise from 2.74% to 3.41%, which is far superior to the 1.50% to 1.76% achieved by conventional MSE-based sequential paradigms.

Rennai Qiu, Chen Qian, Ran Li et al.

Recent advancements in Large Language Models (LLMs) and autonomous agents have demonstrated remarkable capabilities across various domains. However, standalone agents frequently encounter limitations when handling complex tasks that demand extensive interactions and substantial computational resources. Although Multi-Agent Systems (MAS) alleviate some of these limitations through collaborative mechanisms like task decomposition, iterative communication, and role specialization, they typically remain resource-unaware, incurring significant inefficiencies due to high token consumption and excessive execution time. To address these limitations, we propose a resource-aware multi-agent system -- Co-Saving (meaning that multiple agents collaboratively engage in resource-saving activities), which leverages experiential knowledge to enhance operational efficiency and solution quality. Our key innovation is the introduction of "shortcuts" -- instructional transitions learned from historically successful trajectories -- which allows to bypass redundant reasoning agents and expedite the collective problem-solving process. Experiments for software development tasks demonstrate significant advantages over existing methods. Specifically, compared to the state-of-the-art MAS ChatDev, our method achieves an average reduction of 50.85% in token usage, and improves the overall code quality by 10.06%.

Ran Li, Lingshu Zeng

Pseudo-random number generators (PRNGs) are high-nonlinear processes, and they are key blocks in optimization of Large language models. Transformers excel at processing complex nonlinear relationships. Thus it is reasonable to generate high-quality pseudo-random numbers based on transformers. In this paper, we explore this question from both theoretical and practical perspectives, highlighting the potential benefits and implications of Transformer in PRNGs. We theoretically demonstrate that decoder-only Transformer models with Chain-of-Thought can simulate both the Linear Congruential Generator (LCG) and Mersenne Twister (MT) PRNGs. Based on this, we conclude that the log-precision decoder-only Transformer can represent non-uniform $\text{AC}^0$. Our simulative theoretical findings are validated through experiments. The random numbers generated by Transformer-based PRNGs successfully pass the majority of NIST tests, whose heat maps exhibit clear statistical randomness. Finally, we assess their capability in prediction attacks.

Jiawen Weng, Zeke Xia, Ran Li et al.

Due to the advantages of privacy-preserving, Federated Learning (FL) is widely used in distributed machine learning systems. However, existing FL methods suffer from low-inference performance caused by data heterogeneity. Specifically, due to heterogeneous data, the optimization directions of different local models vary greatly, making it difficult for the traditional FL method to get a generalized global model that performs well on all clients. As one of the state-of-the-art FL methods, the mutation-based FL method attempts to adopt a stochastic mutation strategy to guide the model training towards a well-generalized area (i.e., flat area in the loss landscape). Specifically, mutation allows the model to shift within the solution space, providing an opportunity to escape areas with poor generalization (i.e., sharp area). However, the stochastic mutation strategy easily results in diverse optimal directions of mutated models, which limits the performance of the existing mutation-based FL method. To achieve higher performance, this paper proposes a novel mutation-based FL approach named FedQP, utilizing a quadratic programming strategy to regulate the mutation directions wisely. By biasing the model mutation towards the direction of gradient update rather than traditional random mutation, FedQP can effectively guide the model to optimize towards a well-generalized area (i.e., flat area). Experiments on multiple well-known datasets show that our quadratic programming-guided mutation strategy effectively improves the inference accuracy of the global model in various heterogeneous data scenarios.

Shuwei Shao, Ran Li, Zhongcai Pei et al.

Depth estimation attracts widespread attention in the computer vision community. However, it is still quite difficult to recover an accurate depth map using only one RGB image. We observe a phenomenon that existing methods tend to fail in different cases, caused by differences in network architecture, loss function and so on. In this work, we investigate into the phenomenon and propose to integrate the strengths of multiple weak depth predictor to build a comprehensive and accurate depth predictor, which is critical for many real-world applications, e.g., 3D reconstruction. Specifically, we construct multiple base (weak) depth predictors by utilizing different Transformer-based and convolutional neural network (CNN)-based architectures. Transformer establishes long-range correlation while CNN preserves local information ignored by Transformer due to the spatial inductive bias. Therefore, the coupling of Transformer and CNN contributes to the generation of complementary depth estimates, which are essential to achieve a comprehensive depth predictor. Then, we design mixers to learn from multiple weak predictions and adaptively fuse them into a strong depth estimate. The resultant model, which we refer to as Transformer-assisted depth ensembles (TEDepth). On the standard NYU-Depth-v2 and KITTI datasets, we thoroughly explore how the neural ensembles affect the depth estimation and demonstrate that our TEDepth achieves better results than previous state-of-the-art approaches. To validate the generalizability across cameras, we directly apply the models trained on NYU-Depth-v2 to the SUN RGB-D dataset without any fine-tuning, and the superior results emphasize its strong generalizability.

Liangyong Yu, Ran Li, Xiangrui Zeng et al.

Motivation: Cryo-Electron Tomography (cryo-ET) visualizes structure and spatial organization of macromolecules and their interactions with other subcellular components inside single cells in the close-to-native state at sub-molecular resolution. Such information is critical for the accurate understanding of cellular processes. However, subtomogram classification remains one of the major challenges for the systematic recognition and recovery of the macromolecule structures in cryo-ET because of imaging limits and data quantity. Recently, deep learning has significantly improved the throughput and accuracy of large-scale subtomogram classification. However often it is difficult to get enough high-quality annotated subtomogram data for supervised training due to the enormous expense of labeling. To tackle this problem, it is beneficial to utilize another already annotated dataset to assist the training process. However, due to the discrepancy of image intensity distribution between source domain and target domain, the model trained on subtomograms in source domainmay perform poorly in predicting subtomogram classes in the target domain. Results: In this paper, we adapt a few shot domain adaptation method for deep learning based cross-domain subtomogram classification. The essential idea of our method consists of two parts: 1) take full advantage of the distribution of plentiful unlabeled target domain data, and 2) exploit the correlation between the whole source domain dataset and few labeled target domain data. Experiments conducted on simulated and real datasets show that our method achieves significant improvement on cross domain subtomogram classification compared with baseline methods.

Xiaoyan Wang, Ran Li, Bowei Yan et al.

We propose a framework for constructing and analyzing multiclass and multioutput classification metrics, i.e., involving multiple, possibly correlated multiclass labels. Our analysis reveals novel insights on the geometry of feasible confusion tensors -- including necessary and sufficient conditions for the equivalence between optimizing an arbitrary non-decomposable metric and learning a weighted classifier. Further, we analyze averaging methodologies commonly used to compute multioutput metrics and characterize the corresponding Bayes optimal classifiers. We show that the plug-in estimator based on this characterization is consistent and is easily implemented as a post-processing rule. Empirical results on synthetic and benchmark datasets support the theoretical findings.