Thomas Lukasiewicz

Simon Frieder, Luca Pinchetti, Alexis Chevalier et al. · cambridge

We investigate the mathematical capabilities of two iterations of ChatGPT (released 9-January-2023 and 30-January-2023) and of GPT-4 by testing them on publicly available datasets, as well as hand-crafted ones, using a novel methodology. In contrast to formal mathematics, where large databases of formal proofs are available (e.g., the Lean Mathematical Library), current datasets of natural-language mathematics, used to benchmark language models, either cover only elementary mathematics or are very small. We address this by publicly releasing two new datasets: GHOSTS and miniGHOSTS. These are the first natural-language datasets curated by working researchers in mathematics that (1) aim to cover graduate-level mathematics, (2) provide a holistic overview of the mathematical capabilities of language models, and (3) distinguish multiple dimensions of mathematical reasoning. These datasets also test whether ChatGPT and GPT-4 can be helpful assistants to professional mathematicians by emulating use cases that arise in the daily professional activities of mathematicians. We benchmark the models on a range of fine-grained performance metrics. For advanced mathematics, this is the most detailed evaluation effort to date. We find that ChatGPT can be used most successfully as a mathematical assistant for querying facts, acting as a mathematical search engine and knowledge base interface. GPT-4 can additionally be used for undergraduate-level mathematics but fails on graduate-level difficulty. Contrary to many positive reports in the media about GPT-4 and ChatGPT's exam-solving abilities (a potential case of selection bias), their overall mathematical performance is well below the level of a graduate student. Hence, if your goal is to use ChatGPT to pass a graduate-level math exam, you would be better off copying from your average peer!

Katherine M. Collins, Albert Q. Jiang, Simon Frieder et al. · cambridge

There is much excitement about the opportunity to harness the power of large language models (LLMs) when building problem-solving assistants. However, the standard methodology of evaluating LLMs relies on static pairs of inputs and outputs, and is insufficient for making an informed decision about which LLMs and under which assistive settings can they be sensibly used. Static assessment fails to account for the essential interactive element in LLM deployment, and therefore limits how we understand language model capabilities. We introduce CheckMate, an adaptable prototype platform for humans to interact with and evaluate LLMs. We conduct a study with CheckMate to evaluate three language models (InstructGPT, ChatGPT, and GPT-4) as assistants in proving undergraduate-level mathematics, with a mixed cohort of participants from undergraduate students to professors of mathematics. We release the resulting interaction and rating dataset, MathConverse. By analysing MathConverse, we derive a taxonomy of human behaviours and uncover that despite a generally positive correlation, there are notable instances of divergence between correctness and perceived helpfulness in LLM generations, amongst other findings. Further, we garner a more granular understanding of GPT-4 mathematical problem-solving through a series of case studies, contributed by expert mathematicians. We conclude with actionable takeaways for ML practitioners and mathematicians: models that communicate uncertainty respond well to user corrections, and are more interpretable and concise may constitute better assistants. Interactive evaluation is a promising way to navigate the capability of these models; humans should be aware of language models' algebraic fallibility and discern where they are appropriate to use.

Eleonora Giunchiglia, Mihaela Catalina Stoian, Thomas Lukasiewicz · oxford

In recent years, there has been an increasing interest in exploiting logically specified background knowledge in order to obtain neural models (i) with a better performance, (ii) able to learn from less data, and/or (iii) guaranteed to be compliant with the background knowledge itself, e.g., for safety-critical applications. In this survey, we retrace such works and categorize them based on (i) the logical language that they use to express the background knowledge and (ii) the goals that they achieve.

Zehua Cheng, Wei Dai, Jiahao Sun et al.

Autoregressive chain-of-thought (CoT) reasoning in large language models (LLMs) is fundamentally forward-directed: each step conditions only on prior tokens. This unidirectional inductive bias renders even capable models susceptible to error snowballing, wherein a single logical or arithmetic mistake in an early step irreversibly corrupts the entire reasoning chain. We introduce Teleological Reasoning Infilling (\TRI{}), a training framework that endows decoder-only transformers with a native \emph{goal-conditioned bridging} capability. The key insight is to reframe erroneous reasoning segments as fill-in-the-middle (FIM) tasks: given a verified prefix premise $P$, a verified downstream milestone $S$, and the original query $Q$, the model must synthesise the logical bridge $M$ that connects $P$ to $S$ rigorously and completely. To achieve this with standard causal architectures, we introduce a Prefix-Suffix-Middle (PSM) sequence rearrangement with three non-overlapping sentinel tokens, enabling $M$ to attend to both $P$ and $S$ without any structural modification to the self-attention mechanism. Training proceeds in two stages: (i) Supervised Fine-Tuning (SFT) on symbolically verified $(P, S, M)$ triples extracted from formal mathematics corpora, and (ii) Direct Preference Optimisation (DPO) with a deterministic symbolic verifier (Lean 4 / Python) as the sole reward oracle, eliminating LLM-judge sycophancy. At inference, TRI operates as a surgical repair module within a dual-system loop: a causal draft model generates an initial trace, the verifier pinpoints failures, and TRI infills only the damaged segment, leaving verified sections intact. Comprehensive experiments on three benchmarks demonstrate that TRI achieves state-of-the-art performance across all tasks, while reducing per-problem token expenditure by 31.2%.

Louis Mahon, Thomas Lukasiewicz

Online deep clustering refers to the joint use of a feature extraction network and a clustering model to assign cluster labels to each new data point or batch as it is processed. While faster and more versatile than offline methods, online clustering can easily reach the collapsed solution where the encoder maps all inputs to the same point and all are put into a single cluster. Successful existing models have employed various techniques to avoid this problem, most of which require data augmentation or which aim to make the average soft assignment across the dataset the same for each cluster. We propose a method that does not require data augmentation, and that, differently from existing methods, regularizes the hard assignments. Using a Bayesian framework, we derive an intuitive optimization objective that can be straightforwardly included in the training of the encoder network. Tested on four image datasets and one human-activity recognition dataset, it consistently avoids collapse more robustly than other methods and leads to more accurate clustering. We also conduct further experiments and analyses justifying our choice to regularize the hard cluster assignments. Code is available at https://github.com/Lou1sM/online_hard_clustering.

Eleonora Giunchiglia, Mihaela Cătălina Stoian, Salman Khan et al. · oxford

Neural networks have proven to be very powerful at computer vision tasks. However, they often exhibit unexpected behaviours, violating known requirements expressing background knowledge. This calls for models (i) able to learn from the requirements, and (ii) guaranteed to be compliant with the requirements themselves. Unfortunately, the development of such models is hampered by the lack of datasets equipped with formally specified requirements. In this paper, we introduce the ROad event Awareness Dataset with logical Requirements (ROAD-R), the first publicly available dataset for autonomous driving with requirements expressed as logical constraints. Given ROAD-R, we show that current state-of-the-art models often violate its logical constraints, and that it is possible to exploit them to create models that (i) have a better performance, and (ii) are guaranteed to be compliant with the requirements themselves.

Tommaso Salvatori, Ankur Mali, Christopher L. Buckley et al. · uw

Artificial intelligence (AI) is rapidly becoming one of the key technologies of this century. The majority of results in AI thus far have been achieved using deep neural networks trained with a learning algorithm called error backpropagation, always considered biologically implausible. To this end, recent works have studied learning algorithms for deep neural networks inspired by the neurosciences. One such theory, called predictive coding (PC), has shown promising properties that make it potentially valuable for the machine learning community: it can model information processing in different areas of the brain, can be used in control and robotics, has a solid mathematical foundation in variational inference, and performs its computations asynchronously. Inspired by such properties, works that propose novel PC-like algorithms are starting to be present in multiple sub-fields of machine learning and AI at large. Here, we survey such efforts by first providing a broad overview of the history of PC to provide common ground for the understanding of the recent developments, then by describing current efforts and results, and concluding with a large discussion of possible implications and ways forward.

Jianfeng Wang, Thomas Lukasiewicz, Daniela Massiceti et al. · oxford

Semi-supervised learning (SSL) has been widely explored in recent years, and it is an effective way of leveraging unlabeled data to reduce the reliance on labeled data. In this work, we adjust neural processes (NPs) to the semi-supervised image classification task, resulting in a new method named NP-Match. NP-Match is suited to this task for two reasons. Firstly, NP-Match implicitly compares data points when making predictions, and as a result, the prediction of each unlabeled data point is affected by the labeled data points that are similar to it, which improves the quality of pseudo-labels. Secondly, NP-Match is able to estimate uncertainty that can be used as a tool for selecting unlabeled samples with reliable pseudo-labels. Compared with uncertainty-based SSL methods implemented with Monte Carlo (MC) dropout, NP-Match estimates uncertainty with much less computational overhead, which can save time at both the training and the testing phases. We conducted extensive experiments on four public datasets, and NP-Match outperforms state-of-the-art (SOTA) results or achieves competitive results on them, which shows the effectiveness of NP-Match and its potential for SSL.

Eleonora Giunchiglia, Fergus Imrie, Mihaela van der Schaar et al. · oxford

In the recent years, machine learning has made great advancements that have been at the root of many breakthroughs in different application domains. However, it is still an open issue how make them applicable to high-stakes or safety-critical application domains, as they can often be brittle and unreliable. In this paper, we argue that requirements definition and satisfaction can go a long way to make machine learning models even more fitting to the real world, especially in critical domains. To this end, we present two problems in which (i) requirements arise naturally, (ii) machine learning models are or can be fruitfully deployed, and (iii) neglecting the requirements can have dramatic consequences. We show how the requirements specification can be fruitfully integrated into the standard machine learning development pipeline, proposing a novel pyramid development process in which requirements definition may impact all the subsequent phases in the pipeline, and viceversa.

Tommaso Salvatori, Yuhang Song, Yordan Yordanov et al. · oxford

Predictive coding networks are neuroscience-inspired models with roots in both Bayesian statistics and neuroscience. Training such models, however, is quite inefficient and unstable. In this work, we show how by simply changing the temporal scheduling of the update rule for the synaptic weights leads to an algorithm that is much more efficient and stable than the original one, and has theoretical guarantees in terms of convergence. The proposed algorithm, that we call incremental predictive coding (iPC) is also more biologically plausible than the original one, as it it fully automatic. In an extensive set of experiments, we show that iPC constantly performs better than the original formulation on a large number of benchmarks for image classification, as well as for the training of both conditional and masked language models, in terms of test accuracy, efficiency, and convergence with respect to a large set of hyperparameters.

Luca Pinchetti, Tommaso Salvatori, Yordan Yordanov et al. · oxford

A large amount of recent research has the far-reaching goal of finding training methods for deep neural networks that can serve as alternatives to backpropagation (BP). A prominent example is predictive coding (PC), which is a neuroscience-inspired method that performs inference on hierarchical Gaussian generative models. These methods, however, fail to keep up with modern neural networks, as they are unable to replicate the dynamics of complex layers and activation functions. In this work, we solve this problem by generalizing PC to arbitrary probability distributions, enabling the training of architectures, such as transformers, that are hard to approximate with only Gaussian assumptions. We perform three experimental analyses. First, we study the gap between our method and the standard formulation of PC on multiple toy examples. Second, we test the reconstruction quality on variational autoencoders, where our method reaches the same reconstruction quality as BP. Third, we show that our method allows us to train transformer networks and achieve a performance comparable with BP on conditional language models. More broadly, this method allows neuroscience-inspired learning to be applied to multiple domains, since the internal distributions can be flexibly adapted to the data, tasks, and architectures used.

Jianfeng Wang, Daniela Massiceti, Xiaolin Hu et al. · oxford

Semi-supervised semantic segmentation involves assigning pixel-wise labels to unlabeled images at training time. This is useful in a wide range of real-world applications where collecting pixel-wise labels is not feasible in time or cost. Current approaches to semi-supervised semantic segmentation work by predicting pseudo-labels for each pixel from a class-wise probability distribution output by a model. If the predicted probability distribution is incorrect, however, this leads to poor segmentation results, which can have knock-on consequences in safety critical systems, like medical images or self-driving cars. It is, therefore, important to understand what a model does not know, which is mainly achieved by uncertainty quantification. Recently, neural processes (NPs) have been explored in semi-supervised image classification, and they have been a computationally efficient and effective method for uncertainty quantification. In this work, we move one step forward by adapting NPs to semi-supervised semantic segmentation, resulting in a new model called NP-SemiSeg. We experimentally evaluated NP-SemiSeg on the public benchmarks PASCAL VOC 2012 and Cityscapes, with different training settings, and the results verify its effectiveness.

Luca Pinchetti, Chang Qi, Oleh Lokshyn et al.

In this work, we tackle the problems of efficiency and scalability for predictive coding networks (PCNs) in machine learning. To do so, we propose a library, called PCX, that focuses on performance and simplicity, and use it to implement a large set of standard benchmarks for the community to use for their experiments. As most works in the field propose their own tasks and architectures, do not compare one against each other, and focus on small-scale tasks, a simple and fast open-source library and a comprehensive set of benchmarks would address all these concerns. Then, we perform extensive tests on such benchmarks using both existing algorithms for PCNs, as well as adaptations of other methods popular in the bio-plausible deep learning community. All this has allowed us to (i) test architectures much larger than commonly used in the literature, on more complex datasets; (ii)~reach new state-of-the-art results in all of the tasks and datasets provided; (iii)~clearly highlight what the current limitations of PCNs are, allowing us to state important future research directions. With the hope of galvanizing community efforts towards one of the main open problems in the field, scalability, we release code, tests, and benchmarks. Link to the library: https://github.com/liukidar/pcx

Jianfeng Wang, Xiaolin Hu, Thomas Lukasiewicz

Semi-supervised learning (SSL) has been widely explored in recent years, and it is an effective way of leveraging unlabeled data to reduce the reliance on labeled data. In this work, we adjust neural processes (NPs) to the semi-supervised image classification task, resulting in a new method named NP-Match. NP-Match is suited to this task for two reasons. Firstly, NP-Match implicitly compares data points when making predictions, and as a result, the prediction of each unlabeled data point is affected by the labeled data points that are similar to it, which improves the quality of pseudo-labels. Secondly, NP-Match is able to estimate uncertainty that can be used as a tool for selecting unlabeled samples with reliable pseudo-labels. Compared with uncertainty-based SSL methods implemented with Monte-Carlo (MC) dropout, NP-Match estimates uncertainty with much less computational overhead, which can save time at both the training and the testing phases. We conducted extensive experiments on five public datasets under three semi-supervised image classification settings, namely, the standard semi-supervised image classification, the imbalanced semi-supervised image classification, and the multi-label semi-supervised image classification, and NP-Match outperforms state-of-the-art (SOTA) approaches or achieves competitive results on them, which shows the effectiveness of NP-Match and its potential for SSL. The codes are at https://github.com/Jianf-Wang/NP-Match

Zhongbin Xie, Vid Kocijan, Thomas Lukasiewicz et al. · oxford

Bias-measuring datasets play a critical role in detecting biased behavior of language models and in evaluating progress of bias mitigation methods. In this work, we focus on evaluating gender bias through coreference resolution, where previous datasets are either hand-crafted or fail to reliably measure an explicitly defined bias. To overcome these shortcomings, we propose a novel method to collect diverse, natural, and minimally distant text pairs via counterfactual generation, and construct Counter-GAP, an annotated dataset consisting of 4008 instances grouped into 1002 quadruples. We further identify a bias cancellation problem in previous group-level metrics on Counter-GAP, and propose to use the difference between inconsistency across genders and within genders to measure bias at a quadruple level. Our results show that four pre-trained language models are significantly more inconsistent across different gender groups than within each group, and that a name-based counterfactual data augmentation method is more effective to mitigate such bias than an anonymization-based method.

Myeongjun Erik Jang, Thomas Lukasiewicz

ChatGPT has gained a huge popularity since its introduction. Its positive aspects have been reported through many media platforms, and some analyses even showed that ChatGPT achieved a decent grade in professional exams, adding extra support to the claim that AI can now assist and even replace humans in industrial fields. Others, however, doubt its reliability and trustworthiness. This paper investigates the trustworthiness of ChatGPT and GPT-4 regarding logically consistent behaviour, focusing specifically on semantic consistency and the properties of negation, symmetric, and transitive consistency. Our findings suggest that while both models appear to show an enhanced language understanding and reasoning ability, they still frequently fall short of generating logically consistent predictions. We also ascertain via experiments that prompt designing, few-shot learning and employing larger large language models (LLMs) are unlikely to be the ultimate solution to resolve the inconsistency issue of LLMs.

Bowen Li, Philip H. S. Torr, Thomas Lukasiewicz

We introduce a memory-driven semi-parametric approach to text-to-image generation, which is based on both parametric and non-parametric techniques. The non-parametric component is a memory bank of image features constructed from a training set of images. The parametric component is a generative adversarial network. Given a new text description at inference time, the memory bank is used to selectively retrieve image features that are provided as basic information of target images, which enables the generator to produce realistic synthetic results. We also incorporate the content information into the discriminator, together with semantic features, allowing the discriminator to make a more reliable prediction. Experimental results demonstrate that the proposed memory-driven semi-parametric approach produces more realistic images than purely parametric approaches, in terms of both visual fidelity and text-image semantic consistency.

Jianfeng Wang, Thomas Lukasiewicz

Recently, several Bayesian deep learning methods have been proposed for semi-supervised medical image segmentation. Although they have achieved promising results on medical benchmarks, some problems are still existing. Firstly, their overall architectures belong to the discriminative models, and hence, in the early stage of training, they only use labeled data for training, which might make them overfit to the labeled data. Secondly, in fact, they are only partially based on Bayesian deep learning, as their overall architectures are not designed under the Bayesian framework. However, unifying the overall architecture under the Bayesian perspective can make the architecture have a rigorous theoretical basis, so that each part of the architecture can have a clear probabilistic interpretation. Therefore, to solve the problems, we propose a new generative Bayesian deep learning (GBDL) architecture. GBDL belongs to the generative models, whose target is to estimate the joint distribution of input medical volumes and their corresponding labels. Estimating the joint distribution implicitly involves the distribution of data, so both labeled and unlabeled data can be utilized in the early stage of training, which alleviates the potential overfitting problem. Besides, GBDL is completely designed under the Bayesian framework, and thus we give its full Bayesian formulation, which lays a theoretical probabilistic foundation for our architecture. Extensive experiments show that our GBDL outperforms previous state-of-the-art methods in terms of four commonly used evaluation indicators on three public medical datasets.

Maxime Kayser, Cornelius Emde, Oana-Maria Camburu et al.

Most deep learning algorithms lack explanations for their predictions, which limits their deployment in clinical practice. Approaches to improve explainability, especially in medical imaging, have often been shown to convey limited information, be overly reassuring, or lack robustness. In this work, we introduce the task of generating natural language explanations (NLEs) to justify predictions made on medical images. NLEs are human-friendly and comprehensive, and enable the training of intrinsically explainable models. To this goal, we introduce MIMIC-NLE, the first, large-scale, medical imaging dataset with NLEs. It contains over 38,000 NLEs, which explain the presence of various thoracic pathologies and chest X-ray findings. We propose a general approach to solve the task and evaluate several architectures on this dataset, including via clinician assessment.

Beren Millidge, Yuhang Song, Tommaso Salvatori et al.

How the brain performs credit assignment is a fundamental unsolved problem in neuroscience. Many `biologically plausible' algorithms have been proposed, which compute gradients that approximate those computed by backpropagation (BP), and which operate in ways that more closely satisfy the constraints imposed by neural circuitry. Many such algorithms utilize the framework of energy-based models (EBMs), in which all free variables in the model are optimized to minimize a global energy function. However, in the literature, these algorithms exist in isolation and no unified theory exists linking them together. Here, we provide a comprehensive theory of the conditions under which EBMs can approximate BP, which lets us unify many of the BP approximation results in the literature (namely, predictive coding, equilibrium propagation, and contrastive Hebbian learning) and demonstrate that their approximation to BP arises from a simple and general mathematical property of EBMs at free-phase equilibrium. This property can then be exploited in different ways with different energy functions, and these specific choices yield a family of BP-approximating algorithms, which both includes the known results in the literature and can be used to derive new ones.

Bowen Li, Thomas Lukasiewicz

Story visualization aims to generate a sequence of images to narrate each sentence in a multi-sentence story with a global consistency across dynamic scenes and characters. Current works still struggle with output images' quality and consistency, and rely on additional semantic information or auxiliary captioning networks. To address these challenges, we first introduce a new sentence representation, which incorporates word information from all story sentences to mitigate the inconsistency problem. Then, we propose a new discriminator with fusion features and further extend the spatial attention to improve image quality and story consistency. Extensive experiments on different datasets and human evaluation demonstrate the superior performance of our approach, compared to state-of-the-art methods, neither using segmentation masks nor auxiliary captioning networks.

Beren Millidge, Yuhang Song, Tommaso Salvatori et al.

Predictive coding (PC) is an influential theory in computational neuroscience, which argues that the cortex forms unsupervised world models by implementing a hierarchical process of prediction error minimization. PC networks (PCNs) are trained in two phases. First, neural activities are updated to optimize the network's response to external stimuli. Second, synaptic weights are updated to consolidate this change in activity -- an algorithm called \emph{prospective configuration}. While previous work has shown how in various limits, PCNs can be found to approximate backpropagation (BP), recent work has demonstrated that PCNs operating in this standard regime, which does not approximate BP, nevertheless obtain competitive training and generalization performance to BP-trained networks while outperforming them on tasks such as online, few-shot, and continual learning, where brains are known to excel. Despite this promising empirical performance, little is understood theoretically about the properties and dynamics of PCNs in this regime. In this paper, we provide a comprehensive theoretical analysis of the properties of PCNs trained with prospective configuration. We first derive analytical results concerning the inference equilibrium for PCNs and a previously unknown close connection relationship to target propagation (TP). Secondly, we provide a theoretical analysis of learning in PCNs as a variant of generalized expectation-maximization and use that to prove the convergence of PCNs to critical points of the BP loss function, thus showing that deep PCNs can, in theory, achieve the same generalization performance as BP, while maintaining their unique advantages.

Zhongbin Xie, Thomas Lukasiewicz

The increasingly large size of modern pretrained language models not only makes them inherit more human-like biases from the training corpora, but also makes it computationally expensive to mitigate such biases. In this paper, we investigate recent parameter-efficient methods in combination with counterfactual data augmentation (CDA) for bias mitigation. We conduct extensive experiments with prefix tuning, prompt tuning, and adapter tuning on different language models and bias types to evaluate their debiasing performance and abilities to preserve the internal knowledge of a pre-trained model. We find that the parameter-efficient methods (i) are effective in mitigating gender bias, where adapter tuning is consistently the most effective one and prompt tuning is more suitable for GPT-2 than BERT, (ii) are less effective when it comes to racial and religious bias, which may be attributed to the limitations of CDA, and (iii) can perform similarly to or sometimes better than full fine-tuning with improved time and memory efficiency, as well as maintain the internal knowledge in BERT and GPT-2, evaluated via fact retrieval and downstream fine-tuning.

Wenting Xu, Zhenghua Xu, Junyang Chen et al.

To reduce doctors' workload, deep-learning-based automatic medical report generation has recently attracted more and more research efforts, where deep convolutional neural networks (CNNs) are employed to encode the input images, and recurrent neural networks (RNNs) are used to decode the visual features into medical reports automatically. However, these state-of-the-art methods mainly suffer from three shortcomings: (i) incomprehensive optimization, (ii) low-order and unidimensional attention mechanisms, and (iii) repeated generation. In this article, we propose a hybrid reinforced medical report generation method with m-linear attention and repetition penalty mechanism (HReMRG-MR) to overcome these problems. Specifically, a hybrid reward with different weights is employed to remedy the limitations of single-metric-based rewards. We also propose a search algorithm with linear complexity to approximate the best weight combination. Furthermore, we use m-linear attention modules to explore high-order feature interactions and to achieve multi-modal reasoning, while a repetition penalty applies penalties to repeated terms during the model's training process. Extensive experimental studies on two public datasets show that HReMRG-MR greatly outperforms the state-of-the-art baselines in terms of all metrics. We also conducted a series of ablation experiments to prove the effectiveness of all our proposed components. We also performed a reward search toy experiment to give evidence that our proposed search approach can significantly reduce the search time while approximating the best performance.

Lei Sha, Yuhang Song, Yordan Yordanov et al. · oxford

Transformers have become an indispensable module for text generation models since their great success in machine translation. Previous works attribute the~success of transformers to the query-key-value dot-product attention, which provides a robust inductive bias by the fully connected token graphs. However, we found that self-attention has a severe limitation. When predicting the (i+1)-th token, self-attention only takes the i-th token as an information collector, and it tends to give a high attention weight to those tokens similar to itself. Therefore, most of the historical information that occurred before the i-th token is not taken into consideration. Based on this observation, in this paper, we propose a new architecture, called bird-eye transformer(BET), which goes one step further to improve the performance of transformers by reweighting self-attention to encourage it to focus more on important historical information. We have conducted experiments on multiple text generation tasks, including machine translation (2 datasets) and language models (3 datasets). These experimental~results show that our proposed model achieves a better performance than the baseline transformer architectures on~all~datasets. The code is released at: \url{https://sites.google.com/view/bet-transformer/home}.

Louis Mahon, Thomas Lukasiewicz

Existing image complexity metrics cannot distinguish meaningful content from noise. This means that white noise images, which contain no meaningful information, are judged as highly complex. We present a new image complexity metric through hierarchical clustering of patches. We use the minimum description length principle to determine the number of clusters and designate certain points as outliers and, hence, correctly assign white noise a low score. The presented method has similarities to theoretical ideas for measuring meaningful complexity. We conduct experiments on seven different sets of images, which show that our method assigns the most accurate scores to all images considered. Additionally, comparing the different levels of the hierarchy of clusters can reveal how complexity manifests at different scales, from local detail to global structure. We then present ablation studies showing the contribution of the components of our method, and that it continues to assign reasonable scores when the inputs are modified in certain ways, including the addition of Gaussian noise and the lowering of the resolution.

Myeongjun Jang, Frank Mtumbuka, Thomas Lukasiewicz

The logical negation property (LNP), which implies generating different predictions for semantically opposite inputs, is an important property that a trustworthy language model must satisfy. However, much recent evidence shows that large-size pre-trained language models (PLMs) do not satisfy this property. In this paper, we perform experiments using probing tasks to assess PLM's LNP understanding. Unlike previous studies that only examined negation expressions, we expand the boundary of the investigation to lexical semantics. Through experiments, we observe that PLMs violate the LNP frequently. To alleviate the issue, we propose a novel intermediate training task, names meaning-matching, designed to directly learn a meaning-text correspondence, instead of relying on the distributional hypothesis. Through multiple experiments, we find that the task enables PLMs to learn lexical semantic information. Also, through fine-tuning experiments on 7 GLUE tasks, we confirm that it is a safe intermediate task that guarantees a similar or better performance of downstream tasks. Finally, we observe that our proposed approach outperforms our previous counterparts despite its time and resource efficiency.

Myeongjun Jang, Bodhisattwa Prasad Majumder, Julian McAuley et al.

While recent works have been considerably improving the quality of the natural language explanations (NLEs) generated by a model to justify its predictions, there is very limited research in detecting and alleviating inconsistencies among generated NLEs. In this work, we leverage external knowledge bases to significantly improve on an existing adversarial attack for detecting inconsistent NLEs. We apply our attack to high-performing NLE models and show that models with higher NLE quality do not necessarily generate fewer inconsistencies. Moreover, we propose an off-the-shelf mitigation method to alleviate inconsistencies by grounding the model into external background knowledge. Our method decreases the inconsistencies of previous high-performing NLE models as detected by our attack.

Lei Sha, Oana-Maria Camburu, Thomas Lukasiewicz

Explaining the predictions of AI models is paramount in safety-critical applications, such as in legal or medical domains. One form of explanation for a prediction is an extractive rationale, i.e., a subset of features of an instance that lead the model to give its prediction on that instance. For example, the subphrase ``he stole the mobile phone'' can be an extractive rationale for the prediction of ``Theft''. Previous works on generating extractive rationales usually employ a two-phase model: a selector that selects the most important features (i.e., the rationale) followed by a predictor that makes the prediction based exclusively on the selected features. One disadvantage of these works is that the main signal for learning to select features comes from the comparison of the answers given by the predictor to the ground-truth answers. In this work, we propose to squeeze more information from the predictor via an information calibration method. More precisely, we train two models jointly: one is a typical neural model that solves the task at hand in an accurate but black-box manner, and the other is a selector-predictor model that additionally produces a rationale for its prediction. The first model is used as a guide for the second model. We use an adversarial technique to calibrate the information extracted by the two models such that the difference between them is an indicator of the missed or over-selected features. In addition, for natural language tasks, we propose a language-model-based regularizer to encourage the extraction of fluent rationales. Experimental results on a sentiment analysis task, a hate speech recognition task as well as on three tasks from the legal domain show the effectiveness of our approach to rationale extraction.

Ruizhi Wang, Xiangtao Wang, Zhenghua Xu et al.

In clinical scenarios, multiple medical images with different views are usually generated at the same time, and they have high semantic consistency. However, the existing medical report generation methods cannot exploit the rich multi-view mutual information of medical images. Therefore, in this work, we propose the first multi-view medical report generation model, called MvCo-DoT. Specifically, MvCo-DoT first propose a multi-view contrastive learning (MvCo) strategy to help the deep reinforcement learning based model utilize the consistency of multi-view inputs for better model learning. Then, to close the performance gaps of using multi-view and single-view inputs, a domain transfer network is further proposed to ensure MvCo-DoT achieve almost the same performance as multi-view inputs using only single-view inputs.Extensive experiments on the IU X-Ray public dataset show that MvCo-DoT outperforms the SOTA medical report generation baselines in all metrics.

Billy Byiringiro, Tommaso Salvatori, Thomas Lukasiewicz

Predictive coding is a message-passing framework initially developed to model information processing in the brain, and now also topic of research in machine learning due to some interesting properties. One of such properties is the natural ability of generative models to learn robust representations thanks to their peculiar credit assignment rule, that allows neural activities to converge to a solution before updating the synaptic weights. Graph neural networks are also message-passing models, which have recently shown outstanding results in diverse types of tasks in machine learning, providing interdisciplinary state-of-the-art performance on structured data. However, they are vulnerable to imperceptible adversarial attacks, and unfit for out-of-distribution generalization. In this work, we address this by building models that have the same structure of popular graph neural network architectures, but rely on the message-passing rule of predictive coding. Through an extensive set of experiments, we show that the proposed models are (i) comparable to standard ones in terms of performance in both inductive and transductive tasks, (ii) better calibrated, and (iii) robust against multiple kinds of adversarial attacks.

Tommaso Salvatori, Luca Pinchetti, Amine M'Charrak et al.

Recently, there has been extensive research on the capabilities of biologically plausible algorithms. In this work, we show how one of such algorithms, called predictive coding, is able to perform causal inference tasks. First, we show how a simple change in the inference process of predictive coding enables to compute interventions without the need to mutilate or redefine a causal graph. Then, we explore applications in cases where the graph is unknown, and has to be inferred from observational data. Empirically, we show how such findings can be used to improve the performance of predictive coding in image classification tasks, and conclude that such models are able to perform simple end-to-end causal inference tasks.

Hexiang Zhang, Zhenghua Xu, Dan Yao et al.

Analysis of X-ray images is one of the main tools to diagnose breast cancer. The ability to quickly and accurately detect the location of masses from the huge amount of image data is the key to reducing the morbidity and mortality of breast cancer. Currently, the main factor limiting the accuracy of breast mass detection is the unequal focus on the mass boxes, leading the network to focus too much on larger masses at the expense of smaller ones. In the paper, we propose the multi-head feature pyramid module (MHFPN) to solve the problem of unbalanced focus of target boxes during feature map fusion and design a multi-head breast mass detection network (MBMDnet). Experimental studies show that, comparing to the SOTA detection baselines, our method improves by 6.58% (in AP@50) and 5.4% (in TPR@50) on the commonly used INbreast dataset, while about 6-8% improvements (in AP@20) are also observed on the public MIAS and BCS-DBT datasets.

Louis Mahon, Thomas Lukasiewicz

There has been much recent research on human activity re\-cog\-ni\-tion (HAR), due to the proliferation of wearable sensors in watches and phones, and the advances of deep learning methods, which avoid the need to manually extract features from raw sensor signals. A significant disadvantage of deep learning applied to HAR is the need for manually labelled training data, which is especially difficult to obtain for HAR datasets. Progress is starting to be made in the unsupervised setting, in the form of deep HAR clustering models, which can assign labels to data without having been given any labels to train on, but there are problems with evaluating deep HAR clustering models, which makes assessing the field and devising new methods difficult. In this paper, we highlight several distinct problems with how deep HAR clustering models are evaluated, describing these problems in detail and conducting careful experiments to explicate the effect that they can have on results. We then discuss solutions to these problems, and suggest standard evaluation settings for future deep HAR clustering models. Additionally, we present a new deep clustering model for HAR. When tested under our proposed settings, our model performs better than (or on par with) existing models, while also being more efficient and better able to scale to more complex datasets by avoiding the need for an autoencoder.

Bowen Li, Thomas Lukasiewicz

Story visualization aims to generate a sequence of images to narrate each sentence in a multi-sentence story, where the images should be realistic and keep global consistency across dynamic scenes and characters. Current works face the problem of semantic misalignment because of their fixed architecture and diversity of input modalities. To address this problem, we explore the semantic alignment between text and image representations by learning to match their semantic levels in the GAN-based generative model. More specifically, we introduce dynamic interactions according to learning to dynamically explore various semantic depths and fuse the different-modal information at a matched semantic level, which thus relieves the text-image semantic misalignment problem. Extensive experiments on different datasets demonstrate the improvements of our approach, neither using segmentation masks nor auxiliary captioning networks, on image quality and story consistency, compared with state-of-the-art methods.

Bowen Li, Thomas Lukasiewicz

In this paper, we introduce novel lightweight generative adversarial networks, which can effectively capture long-range dependencies in the image generation process, and produce high-quality results with a much simpler architecture. To achieve this, we first introduce a long-range module, allowing the network to dynamically adjust the number of focused sampling pixels and to also augment sampling locations. Thus, it can break the limitation of the fixed geometric structure of the convolution operator, and capture long-range dependencies in both spatial and channel-wise directions. Also, the proposed long-range module can highlight negative relations between pixels, working as a regularization to stabilize training. Furthermore, we propose a new generation strategy through which we introduce metadata into the image generation process to provide basic information about target images, which can stabilize and speed up the training process. Our novel long-range module only introduces few additional parameters and is easily inserted into existing models to capture long-range dependencies. Extensive experiments demonstrate the competitive performance of our method with a lightweight architecture.

Zihang Xu, Zhenghua Xu, Shuo Zhang et al.

Deep learning based semi-supervised learning (SSL) methods have achieved strong performance in medical image segmentation, which can alleviate doctors' expensive annotation by utilizing a large amount of unlabeled data. Unlike most existing semi-supervised learning methods, adversarial training based methods distinguish samples from different sources by learning the data distribution of the segmentation map, leading the segmenter to generate more accurate predictions. We argue that the current performance restrictions for such approaches are the problems of feature extraction and learning preference. In this paper, we propose a new semi-supervised adversarial method called Patch Confidence Adversarial Training (PCA) for medical image segmentation. Rather than single scalar classification results or pixel-level confidence maps, our proposed discriminator creates patch confidence maps and classifies them at the scale of the patches. The prediction of unlabeled data learns the pixel structure and context information in each patch to get enough gradient feedback, which aids the discriminator in convergent to an optimal state and improves semi-supervised segmentation performance. Furthermore, at the discriminator's input, we supplement semantic information constraints on images, making it simpler for unlabeled data to fit the expected data distribution. Extensive experiments on the Automated Cardiac Diagnosis Challenge (ACDC) 2017 dataset and the Brain Tumor Segmentation (BraTS) 2019 challenge dataset show that our method outperforms the state-of-the-art semi-supervised methods, which demonstrates its effectiveness for medical image segmentation.

Yikuan Li, Mohammad Mamouei, Shishir Rao et al.

Most machine learning (ML) models are developed for prediction only; offering no option for causal interpretation of their predictions or parameters/properties. This can hamper the health systems' ability to employ ML models in clinical decision-making processes, where the need and desire for predicting outcomes under hypothetical investigations (i.e., counterfactual reasoning/explanation) is high. In this research, we introduce a new representation learning framework (i.e., partial concept bottleneck), which considers the provision of counterfactual explanations as an embedded property of the risk model. Despite architectural changes necessary for jointly optimising for prediction accuracy and counterfactual reasoning, the accuracy of our approach is comparable to prediction-only models. Our results suggest that our proposed framework has the potential to help researchers and clinicians improve personalised care (e.g., by investigating the hypothetical differential effects of interventions)

Louis Mahon, Lei Shah, Thomas Lukasiewicz

Recent years have seen growing interest in learning disentangled representations, in which distinct features, such as size or shape, are represented by distinct neurons. Quantifying the extent to which a given representation is disentangled is not straightforward; multiple metrics have been proposed. In this paper, we identify two failings of existing metrics, which mean they can assign a high score to a model which is still entangled, and we propose two new metrics, which redress these problems. We then consider the task of compositional generalization. Unlike prior works, we treat this as a classification problem, which allows us to use it to measure the disentanglement ability of the encoder, without depending on the decoder. We show that performance on this task is (a) generally quite poor, (b) correlated with most disentanglement metrics, and (c) most strongly correlated with our newly proposed metrics.

Xiangtao Wang, Ruizhi Wang, Biao Tian et al.

Existing self-supervised learning methods based on contrastive learning and masked image modeling have demonstrated impressive performances. However, current masked image modeling methods are mainly utilized in natural images, and their applications in medical images are relatively lacking. Besides, their fixed high masking strategy limits the upper bound of conditional mutual information, and the gradient noise is considerable, making less the learned representation information. Motivated by these limitations, in this paper, we propose masked patches selection and adaptive masking strategy based self-supervised medical image segmentation method, named MPS-AMS. We leverage the masked patches selection strategy to choose masked patches with lesions to obtain more lesion representation information, and the adaptive masking strategy is utilized to help learn more mutual information and improve performance further. Extensive experiments on three public medical image segmentation datasets (BUSI, Hecktor, and Brats2018) show that our proposed method greatly outperforms the state-of-the-art self-supervised baselines.

Ruizhi Wang, Xiangtao Wang, Jie Zhou et al.

In clinical scenarios, multiple medical images with different views are usually generated simultaneously, and these images have high semantic consistency. However, most existing medical report generation methods only consider single-view data. The rich multi-view mutual information of medical images can help generate more accurate reports, however, the dependence of multi-view models on multi-view data in the inference stage severely limits their application in clinical practice. In addition, word-level optimization based on numbers ignores the semantics of reports and medical images, and the generated reports often cannot achieve good performance. Therefore, we propose a cross-modal consistent multi-view medical report generation with a domain transfer network (C^2M-DoT). Specifically, (i) a semantic-based multi-view contrastive learning medical report generation framework is adopted to utilize cross-view information to learn the semantic representation of lesions; (ii) a domain transfer network is further proposed to ensure that the multi-view report generation model can still achieve good inference performance under single-view input; (iii) meanwhile, optimization using a cross-modal consistency loss facilitates the generation of textual reports that are semantically consistent with medical images. Extensive experimental studies on two public benchmark datasets demonstrate that C^2M-DoT substantially outperforms state-of-the-art baselines in all metrics. Ablation studies also confirmed the validity and necessity of each component in C^2M-DoT.

Xiangtao Wang, Ruizhi Wang, Thomas Lukasiewicz et al.

Self-supervised masked image modeling (MIM) methods have shown promising performances on analyzing natural images. However, directly applying such methods to medical image segmentation tasks still cannot achieve satisfactory results. The challenges arise from the facts that (i) medical images are inherently more complex compared to natural images, and the subjects in medical images often exhibit more distinct contour features; (ii) moreover, the conventional high and fixed masking ratio in MIM is likely to mask the background, limiting the scope of learnable information. To address these problems, we propose a new self-supervised medical image segmentation framework, called Adjustable Masking Lesion Patches (AMLP), which employs Masked Patch Selection~(MPS) strategy to identify patches with high probabilities of containing lesions to help model achieve precise lesion reconstruction. To improve the categorization of patches in MPS, we further introduce Relative Reconstruction Loss (RRL) to better learn hard-to-reconstruct lesion patches. Then, Category Consistency Loss (CCL) is proposed to refine patch categorization based on reconstruction difficulty, enhancing difference between lesions and backgrounds. Moreover, an Adjustable Masking Ratio (AMR) strategy is proposed to gradually increase the masking ratio over training to expand~the scope of learnable mutual information. Extensive~experiments on two medical segmentation datasets demonstrate the superior performances of the proposed AMLP w.r.t. the SOTA self-supervised methods; the results prove that AMLP effectively addresses the challenges of applying masked modeling to medical images and capturing accurate lesion details that are crucial for segmentation tasks.

Myeongjun Erik Jang, Thomas Lukasiewicz

The non-humanlike behaviour of contemporary pre-trained language models (PLMs) is a leading cause undermining their trustworthiness. A striking phenomenon of such faulty behaviours is the generation of inconsistent predictions, which produces logically contradictory results, such as generating different predictions for texts delivering the same meaning or violating logical properties. Previous studies exploited data augmentation or implemented specialised loss functions to alleviate the issue. However, their usage is limited, because they consume expensive training resources for large-sized PLMs and can only handle a certain consistency type. To this end, we propose a practical approach that alleviates the inconsistent behaviour issue by fundamentally improving PLMs' meaning awareness. Based on the conceptual role theory, our method allows PLMs to capture accurate meaning by learning precise interrelationships between concepts from word-definition pairs in a dictionary. Next, we propose an efficient parameter integration technique that updates only a few additional parameters to combine the learned interrelationship with PLMs' pre-trained knowledge. Our experimental results reveal that the approach can concurrently improve multiple types of consistency, enables efficient knowledge integration, and easily applies to other languages.

Yordan Yordanov, Matteo Forasassi, Bayar Menzat et al.

While state-of-the-art language models (LMs) surpass the vast majority of humans in certain domains, their reasoning remains largely opaque, undermining trust in their output. Furthermore, while autoregressive LMs can output explicit reasoning, their true reasoning process is opaque, which introduces risks like deception and hallucination. In this work, we introduce the Prototype Transformer (ProtoT) -- an autoregressive LM architecture based on prototypes (parameter vectors), posed as an alternative to the standard self-attention-based transformers. ProtoT works by means of two-way communication between the input sequence and the prototypes, and we show that this leads to the prototypes automatically capturing nameable concepts (e.g. "woman") during training. They provide the potential to interpret the model's reasoning and allow for targeted edits of its behavior. Furthermore, by design, the prototypes create communication channels that aggregate contextual information at different time scales, aiding interpretability. In terms of computation scalability, ProtoT scales linearly with sequence length vs the quadratic scalability of SOTA self-attention transformers. Compared to baselines, ProtoT scales well with model and data size, and performs well on text generation and downstream tasks (GLUE). ProtoT exhibits robustness to input perturbations on par or better than some baselines, but differs from them by providing interpretable pathways showing how robustness and sensitivity arises. Reaching close to the performance of state-of-the-art architectures, ProtoT paves the way to creating well-performing autoregressive LMs interpretable by design.

Zehua Cheng, Wei Dai, Jiahao Sun et al.

Large Language Models (LLMs) generate realistic synthetic data but offer no guarantee that their outputs respect the causal mechanisms governing the target domain. We introduce CausalSynth, a framework that decouples causal structure generation from semantic realization, yielding synthetic data that is both causally valid and linguistically rich. The framework operates in three phases. First, a Structural Causal Model (SCM) - a tuple of structural equations defined over a directed acyclic graph (DAG) generates causal skeletons, i.e., variable assignments that satisfy the Global Markov Property of the governing DAG, via ancestral sampling. Second, an LLM acts as a constrained \emph{realizer}, a conditional translator that maps each skeleton to a high-dimensional observation such as a clinical note or a transaction log. Third, an Iterative Consistency Verification module detects structural violations through deterministic extraction and feeds targeted corrections back to the LLM, forming a closed-loop refinement process. We identify the Semantic Backdoor problem the systematic tendency of LLMs to override imposed causal facts with pre-training priors -- and prove that our iterative mechanism reduces the resulting selection bias relative to standard rejection sampling. On three causal benchmarks (ASIA, ALARM, and MIMIC-Struct), CausalSynth preserved conditional independencies with false-positive rates near the nominal $α=0.05$ level and achieved realizability rates above 96% with 70B-parameter LLM backbones. The framework additionally supports principled interventional and counterfactual generation through noise retention and graph mutilation.

Andres Felipe Lerma-Pineda, Philipp Petersen, Simon Frieder et al.

We study the problem of approximating and estimating classification functions that have their decision boundary in the $RBV^2$ space. Functions of $RBV^2$ type arise naturally as solutions of regularized neural network learning problems and neural networks can approximate these functions without the curse of dimensionality. We modify existing results to show that every $RBV^2$ function can be approximated by a neural network with bounded weights. Thereafter, we prove the existence of a neural network with bounded weights approximating a classification function. And we leverage these bounds to quantify the estimation rates. Finally, we present a numerical study that analyzes the effect of different regularity conditions on the decision boundaries.

Magnus Sesodia, Alina Petrova, John Armour et al.

Legal systems worldwide continue to struggle with overwhelming caseloads, limited judicial resources, and growing complexities in legal proceedings. Artificial intelligence (AI) offers a promising solution, with Legal Judgment Prediction (LJP) -- the practice of predicting a court's decision from the case facts -- emerging as a key research area. However, existing datasets often formulate the task of LJP unrealistically, not reflecting its true difficulty. They also lack high-quality annotation essential for legal reasoning and explainability. To address these shortcomings, we introduce AnnoCaseLaw, a first-of-its-kind dataset of 471 meticulously annotated U.S. Appeals Court negligence cases. Each case is enriched with comprehensive, expert-labeled annotations that highlight key components of judicial decision making, along with relevant legal concepts. Our dataset lays the groundwork for more human-aligned, explainable LJP models. We define three legally relevant tasks: (1) judgment prediction; (2) concept identification; and (3) automated case annotation, and establish a performance baseline using industry-leading large language models (LLMs). Our results demonstrate that LJP remains a formidable task, with application of legal precedent proving particularly difficult. Code and data are available at https://github.com/anonymouspolar1/annocaselaw.

Zehua Cheng, Wei Dai, Wenhu Zhang et al.

A user pointing their phone at a supermarket shelf and asking "Which soda has the least sugar?" poses a difficult challenge for current visual Al assistants. Such queries require not only object recognition, but explicit set-based reasoning such as filtering, comparison, and aggregation. Standard endto-end MLLMs often fail at these tasks because they lack an explicit mechanism for compositional logic. We propose treating visual reasoning as Visual Program Synthesis, where the model first generates a symbolic program that is executed by a separate engine grounded in visual scenes. We also introduce Set-VQA, a new benchmark designed specifically for evaluating set-based visual reasoning. Experiments show that our approach significantly outperforms state-of-the-art baselines on complex reasoning tasks, producing more systematic and transparent behavior while substantially improving answer accuracy. These results demonstrate that program-driven reasoning provides a principled alternative to black-box visual-language inference.

Jianfeng Wang, Amine M'Charrak, Luk Koska et al.

Recent vision and multimodal foundation backbones, such as Transformer families and state-space models like Mamba, have achieved remarkable progress, enabling unified modeling across images, text, and beyond. Despite their empirical success, these architectures remain far from the computational principles of the human brain, often demanding enormous amounts of training data while offering limited interpretability. In this work, we propose the Vision Hopfield Memory Network (V-HMN), a brain-inspired foundation backbone that integrates hierarchical memory mechanisms with iterative refinement updates. Specifically, V-HMN incorporates local Hopfield modules that provide associative memory dynamics at the image patch level, global Hopfield modules that function as episodic memory for contextual modulation, and a predictive-coding-inspired refinement rule for iterative error correction. By organizing these memory-based modules hierarchically, V-HMN captures both local and global dynamics in a unified framework. Memory retrieval exposes the relationship between inputs and stored patterns, making decisions more interpretable, while the reuse of stored patterns improves data efficiency. This brain-inspired design therefore enhances interpretability and data efficiency beyond existing self-attention- or state-space-based approaches. We conducted extensive experiments on public computer vision benchmarks, and V-HMN achieved competitive results against widely adopted backbone architectures, while offering better interpretability, higher data efficiency, and stronger biological plausibility. These findings highlight the potential of V-HMN to serve as a next-generation vision foundation model, while also providing a generalizable blueprint for multimodal backbones in domains such as text and audio, thereby bridging brain-inspired computation with large-scale machine learning.

Louis Mahon, Thomas Lukasiewicz

Deep neural networks (DNNs) offer a means of addressing the challenging task of clustering high-dimensional data. DNNs can extract useful features, and so produce a lower dimensional representation, which is more amenable to clustering techniques. As clustering is typically performed in a purely unsupervised setting, where no training labels are available, the question then arises as to how the DNN feature extractor can be trained. The most accurate existing approaches combine the training of the DNN with the clustering objective, so that information from the clustering process can be used to update the DNN to produce better features for clustering. One problem with this approach is that these ``pseudo-labels'' produced by the clustering algorithm are noisy, and any errors that they contain will hurt the training of the DNN. In this paper, we propose selective pseudo-label clustering, which uses only the most confident pseudo-labels for training the~DNN. We formally prove the performance gains under certain conditions. Applied to the task of image clustering, the new approach achieves a state-of-the-art performance on three popular image datasets. Code is available at https://github.com/Lou1sM/clustering.