论文检索

Applications · Robotics

Jianzhe Gao, Rui Liu, Yuxuan Xu, Tongtong Cao, Yingxue Zhang, Zhanguang Zhang, Sida Peng, Yi Yang, Wenguan Wang

Vision-Language Navigation (VLN) requires an agent to navigate 3D environments following natural language instructions. During navigation, existing agents commonly encounter perceptual uncertainty, such as insufficient evidence for reliable grounding or ambiguity in interpreting spatial cues, yet they typically ignore such information when predicting actions. In this work, we explicitly model three forms of perceptual uncertainty (i.e., geometric, semantic, and appearance uncertainty) and integrate them into the agent’s observation space to enable informed decision-making. Concretely, our agent first constructs a Semantic Gaussian Map (SGM), composed of differentiable 3D Gaussian primitives initialized from panoramic observations, that encodes both the geometric structure and semantic content of the environment. On top of SGM, geometric uncertainty is estimated through variational perturbations of Gaussian position and scale to assess structural reliability; semantic uncertainty is captured by perturbing Gaussian semantic attributes to reveal ambiguous interpretations; and appearance uncertainty is characterized by Fisher Information, which measures the sensitivity of rendered observations to Gaussian-level variations. These uncertainties are incorporated into SGM, extending it into a unified 3D Value Map, which grounds them as affordances and constraints that support reliable navigation. Comprehensive evaluations across multiple VLN benchmarks show the effectiveness of our agent.

Optimization · Large Scale, Parallel and Distributed

Alexander Tyurin, Danil Sivtsov

We propose a new unifying framework, Birch SGD, for analyzing and designing distributed SGD methods. The central idea is to represent each method as a weighted directed tree, referred to as a computation tree. Leveraging this representation, we introduce a general theoretical result that reduces convergence analysis to studying the geometry of these trees. This perspective yields a purely graph-based interpretation of optimization dynamics, offering a new and intuitive foundation for method development. Using Birch SGD, we design eight new methods and analyze them alongside previously known ones, with at least six of the new methods shown to have optimal computational time complexity. Our research leads to two key insights: (i) all methods share the same iteration rate of $\mathcal{O}\left(\frac{(R + 1) L \Delta}{\varepsilon} + \frac{\sigma^2 L \Delta}{\varepsilon^2}\right)$, where $R$ the maximum ``tree distance'' along the main branch of a tree; and (ii) different methods exhibit different trade-offs---for example, some update iterates more frequently, improving practical performance, while others are more communication-efficient or focus on other aspects. Birch SGD serves as a unifying framework for navigating these trade-offs. We believe these results provide a unified foundation for understanding, analyzing, and designing efficient asynchronous and parallel optimization methods.

Computer Vision · Image and Video Generation

Chutian Meng, Fan Ma, Chi Zhang, Jiaxu Miao, Yi Yang, Yueting Zhuang

Generating coherent and communicative visual sequences, such as image sequences and videos, remains a significant challenge for current multimodal systems. Despite advances in visual quality and the integration of world knowledge, existing models still struggle to maintain logical flow, often resulting in disjointed actions, fragmented narratives, and unclear storylines. We attribute these issues to the lack of attention to visual logic, a critical yet underexplored dimension of visual sequence generation that we define as the perceptual and causal coherence among characters, actions, and scenes over time. To bridge this gap, we propose a logic-aware multi-image story visualization framework, LogiStory. The framework is built around the central innovation of explicitly modeling visual logic in story visualization. To realize this idea, we design a multi-agent system that grounds roles, extracts causal chains, and verifies story-level consistency, transforming narrative coherence from an implicit byproduct of image generation into an explicit modeling objective. This design effectively bridges structured story planning with visual generation, enhancing both narrative clarity and visual quality in story visualization. Furthermore, to evaluate the generation capacity, we construct LogicTale, a benchmark comprising richly annotated stories, emphasizing causal reasoning, and visual logic interpretability. We establish comprehensive automatic and human evaluation protocols designed to measure both visual logic and perceptual quality. Experiments demonstrate that our approach significantly improves the narrative logic of generated visual stories. This work provides a foundational step towards modeling and enforcing visual logic in general image sequence and video generation tasks.

General Machine Learning · Everything Else

Satchel Grant, Simon Jerome Han, Alexa Tartaglini, Christopher Potts

A common approach to mechanistic interpretability is to causally manipulate model representations via targeted interventions in order to understand what those representations encode. Here we ask whether such interventions create out-of-distribution (divergent) representations, and whether this raises concerns about how faithful their resulting explanations are to the target model in its natural state. First, we demonstrate theoretically and empirically that common causal intervention techniques often do shift internal representations away from the natural distribution of the target model. Then, we provide a theoretical analysis of two cases of such divergences: "harmless" divergences that occur in the behavioral null-space of the layer(s) of interest, and "pernicious" divergences that activate hidden network pathways and cause dormant behavioral changes. Finally, in an effort to mitigate the pernicious cases, we apply and modify the Counterfactual Latent (CL) loss from Grant (2025) allowing representations from causal interventions to remain closer to the natural distribution, reducing the likelihood of harmful divergences while preserving the interpretive power of the interventions. Together, these results highlight a path towards more reliable interpretability methods.

Deep Learning · Theory

Ouns El Harzli, Bernardo Grau, Artur Garcez, Ian Horrocks, Tarek R. Besold

Fibring of modal logics is a well-established formalism for combining countable families of modal logics into a single fibred language with common semantics, characterized by fibred models. Inspired by this formalism, fibring of neural networks was introduced as a neurosymbolic framework for combining learning and reasoning in neural networks. Fibring of neural networks uses the (pre-)activations of a trained network to evaluate a fibring function computing the weights of another network whose outputs are injected back into the original network. However, the exact correspondence between fibring of neural networks and fibring of modal logics was never formally established. In this paper, we close this gap by formalizing the idea of fibred models compatible with fibred neural networks. Using this correspondence, we then derive non-uniform logical expressiveness results for Graph Neural Networks (GNNs), Graph Attention Networks (GATs) and Transformer encoders. Longer-term, the goal of this paper is to open the way for the use of fibring as a formalism for interpreting the logical theories learnt by neural networks with the tools of computational logic.

Theory · Interpretability and Visualization

Razvan Marinescu, Victoria-Elisabeth Gruber, Diego Fajardo Vargas

We present a systematic study of medical-domain interpretability in Large Language Models (LLMs). We study how the LLMs both represent and process medical knowledge through four different interpretability techniques: (1) UMAP projections of intermediate activations, (2) gradient-based saliency with respect to the model weights, (3) layer lesioning/removal and (4) activation patching. We present knowledge maps of five LLMs which show, at a coarse-resolution, where knowledge about patient's ages, medical symptoms, diseases and drugs is stored in the models. In particular for Llama3.3-70B, we find that most medical knowledge is processed in the first half of the model's layers. In addition, we find several interesting phenomena: (i) age is often encoded in a non-linear and sometimes discontinuous manner at intermediate layers in the models, (ii) the disease progression representation is non-monotonic and circular at certain layers of the model, (iii) in Llama, drugs cluster better by medical specialty rather than mechanism of action, especially for Llama and (iv) Gemma-27B and MedGemma-27B have activations that collapse at intermediate layers but recover by the final layers. These results can guide future research on fine-tuning, un-learning or de-biasing LLMs for medical tasks by suggesting at which layers in the model these techniques should be applied. We attached our source code to the paper for reproducibility.

Applications · Physics

Jin Lee, Ziming Liu, Xinling Yu, Yixuan Wang, Haewon Jeong, Murphy Niu, Zheng Zhang

We introduce Kolmogorov–Arnold Neural Operator (KANO), a dual‑domain neural operator jointly parameterized by both spectral and spatial bases with intrinsic symbolic interpretability. We theoretically demonstrate that KANO overcomes the pure-spectral bottleneck of Fourier Neural Operator (FNO): KANO remains expressive over a generic position-dependent dynamics for any physical input, whereas FNO stays practical only to spectrally sparse operators and strictly imposes fast-decaying input Fourier tail. We verify our claims empirically on position-dependent differential operators, for which KANO robustly generalizes but FNO fails to. In the quantum Hamiltonian learning benchmark, KANO reconstructs ground‑truth Hamiltonians in closed-form symbolic representations accurate to the fourth decimal place in coefficients and attains $\approx6\times10^{-6}$ state infidelity from projective measurement data, substantially outperforming that of the FNO trained with ideal full wave function data, $\approx1.5\times10^{-2}$, by orders of magnitude.

Theory · Interpretability and Visualization

Jiayu Yang, Yuxuan Fan, Songning Lai, Shengen Wu, Jiaqi Tang, Chun Kang, Zhijiang Guo, Yutao Yue

LLMs require efficient knowledge editing (KE) to update factual information, yet existing methods exhibit significant performance decay in multi-hop factual recall. This failure is particularly acute when edits involve intermediate implicit subjects within reasoning chains. Through causal analysis, we reveal that this limitation stems from an oversight of how chained knowledge is dynamically represented and utilized at the neuron level. We discover that during multi-hop reasoning, implicit subjects function as query neurons, which sequentially activate corresponding value neurons across transformer layers to accumulate information toward the final answer—a dynamic prior KE work has overlooked. Guided by this insight, we propose ACE (Attribution-Controlled Knowledge Editing), a framework that leverages neuron-level attribution to identify and edit these critical query-value (Q-V) pathways. Ace provides a mechanistically grounded solution for multi-hop KE, empirically outperforming state-of-the-art methods by 9.44% on GPT-J and 37.46% on Qwen3-8B. Our analysis further reveals more fine-grained activation patterns in Qwen3 and demonstrates that the semantic interpretability of value neurons is orchestrated by query-driven accumulation. These findings establish a new pathway for advancing KE capabilities based on the principled understanding of internal reasoning mechanisms.

Theory · Interpretability and Visualization

Lucas Prieto, Edward Stevinson, Melih Barsbey, Tolga Birdal, Pedro Mediano

A central idea in mechanistic interpretability is that neural networks represent more features than they have dimensions, arranging them in superposition to form an over-complete basis. This framing has been influential, motivating dictionary learning approaches such as sparse autoencoders. However, superposition has mostly been studied in idealized settings where features are sparse and uncorrelated. In these settings, superposition is typically understood as introducing interference that must be minimized geometrically and filtered out by non-linearities such as ReLUs, yielding local structures like regular polytopes. We show that this account is incomplete for realistic data by introducing Bag-of-Words Superposition (BOWS), a controlled setting to encode binary bag-of-words representations of internet text in superposition. Using BOWS, we find that when features are correlated, interference can be constructive rather than just noise to be filtered out. This is achieved by arranging features according to their co-activation patterns, making interference between active features constructive, while still using ReLUs to avoid false positives. We show that this kind of arrangement is more prevalent in models trained with weight decay and naturally gives rise to semantic clusters and cyclical structures which have been observed in real language models yet were not explained by the standard picture of superposition. Code for this paper can be found at: https://github.com/LucasPrietoAl/correlations-feature-geometry.

Computer Vision · Vision Models & Multimodal

Vardaan Pahuja, Samuel Stevens, Alyson East, Sydne Record, Yu Su

Morphological traits are physical characteristics of biological organisms that provide vital clues on how organisms interact with their environment. Yet extracting these traits remains a slow, expert-driven process, limiting their use in large-scale ecological studies. A major bottleneck is the absence of high-quality datasets linking biological images to trait-level annotations. In this work, we demonstrate that sparse autoencoders trained on foundation-model features yield monosemantic, spatially grounded neurons that consistently activate on meaningful morphological parts. Leveraging this property, we introduce a trait annotation pipeline that localizes salient regions and uses vision-language prompting to generate interpretable trait descriptions. Using this approach, we construct Bioscan-Traits, a dataset of 80K trait annotations spanning 19K insect images from BIOSCAN-5M. Human evaluation confirms the biological plausibility of the generated morphological descriptions. We assess design sensitivity through a comprehensive ablation study, systematically varying key design choices and measuring their impact on the quality of the resulting trait descriptions. By annotating traits with a modular pipeline rather than prohibitively expensive manual efforts, we offer a scalable way to inject biologically meaningful supervision into foundation models, enable large-scale morphological analyses, and bridge the gap between ecological relevance and machine-learning practicality.

Applications · Neuroscience, Cognitive Science

Yule Wang, Joseph Yu, Chengrui Li, Weihan Li, Anqi Wu

Understanding how neural populations in higher visual areas encode object-centered visual information remains a key challenge in computational neuroscience. Prior work has examined representational alignment between artificial neural networks and the visual cortex, but these findings are indirect and provide limited insight into the structure of neural population coding. Decoding-based methods can recover semantic features from neural activity, yet they do not reveal how these features are organized. This leaves an open question: how feature-specific visual information is distributed across neural populations, and whether it forms structured, semantically meaningful subspaces. To address this, we introduce MIG-Vis, a method that leverages diffusion models to visualize and validate visual-semantic attributes encoded in neural latent subspaces. MIG-Vis first learns a group-wise disentangled neural latent representation using a variational autoencoder, and then applies mutual information (MI)-guided diffusion synthesis to visualize the visual-semantic features encoded in each latent group. We validate MIG-Vis on multi-session neural spiking datasets from the inferior temporal (IT) cortex of two macaques. The synthesized results show that MIG-Vis identifies neural latent groups with clear semantic selectivity to diverse visual features, including object pose, inter-category variation, and intra-category content. These findings provide direct and interpretable evidence of structured semantic representation in higher visual cortex.

Applications · Language, Speech and Dialog

Thibault Formal, Maxime Louis, Hervé Déjean, Stéphane Clinchant

Sparse autoencoders (SAEs) provide a powerful mechanism for decomposing the dense representations produced by Large Language Models (LLMs) into interpretable latent features. We posit that SAEs constitute a natural foundation for Learned Sparse Retrieval (LSR), whose objective is to encode queries and documents into high-dimensional sparse representations optimized for efficient retrieval. In contrast to existing LSR approaches that project input sequences into the vocabulary space, SAE-based representations offer the potential to produce more semantically structured, expressive, and language-agnostic features. By leveraging recently released open-source SAEs, we show that their latent features can serve as effective indexing units for representing documents and queries for sparse retrieval. Our experiments demonstrate that SAE-based LSR models consistently outperform their vocabulary-based counterparts in multilingual and out-of-domain settings. Finally, we introduce SPLARE, a 7B-parameter multilingual retrieval model capable of producing generalizable sparse latent embeddings for a wide range of languages and domains, achieving top results on MMTEB’s multilingual and English retrieval tasks. We also release a more efficient 2B-parameter variant, offering strong performance with a significantly lighter footprint.

Theory · Interpretability and Visualization

J. Ben Tamo, Daniel Carlander-Reuterfelt, Jonathan Rubin, Oleg Poliannikov, Dezhi Hong, Mingxian Wang

Despite multilingual pretraining, large language models often struggle with non-English tasks, particularly in language control--the ability to respond in the intended language. We identify and characterize two key failure modes: the *multilingual transfer bottleneck* (correct language, incorrect task response) and the *language consistency bottleneck* (correct task response, wrong language). To systematically surface these issues, we design a four-scenario evaluation protocol spanning MMLU, MGSM, and XQuAD benchmarks. To probe these issues with interpretability, we extend logit lens analysis to track language probabilities layer by layer and compute cross-lingual semantic similarity of hidden states. The results reveal a three-phase internal structure: early layers align inputs into shared semantic space, middle layers perform task reasoning, and late layers drive language-specific generation. Guided by these insights, we introduce *selective fine-tuning* of only the final layers responsible for language control. On Qwen-3-32B and Bloom-7.1B, this method achieves over 98% language consistency across six languages while fine-tuning only 3–5% of parameters, without sacrificing task accuracy. Importantly, this result is nearly identical to that of full-scope fine-tuning (e.g., $>98\%$ language consistency for both methods across all prompt scenarios) but uses a fraction of the computational resources. To the best of our knowledge, this is the first approach to leverage *layer-localization of language control* for efficient multilingual adaptation.

Social Aspects · Accountability, Transparency and Interpretability

Maty Bohacek, Thomas Fel, Maneesh Agrawala, Ekdeep Singh Lubana

Despite their impressive performance, generative image models trained on large-scale datasets frequently fail to produce images with seemingly simple concepts -- e.g., human hands or objects appearing in groups of four -- that are reasonably expected to appear in the training data. These failure modes have largely been documented anecdotally, leaving open the question of whether they reflect idiosyncratic anomalies or more structural limitations of these models. To address this, we introduce a systematic approach for identifying and characterizing "conceptual blindspots" -- concepts present in the training data but absent or misrepresented in a model's generations. Our method leverages sparse autoencoders (SAEs) to extract interpretable concept embeddings, enabling a quantitative comparison of concept prevalence between real and generated images. We train an archetypal SAE (RA-SAE) on DINOv2 features with 32,000 concepts -- the largest such SAE to date -- enabling fine-grained analysis of conceptual disparities. Applied to four popular generative models (Stable Diffusion 1.5/2.1, PixArt, and Kandinsky), our approach reveals specific suppressed blindspots (e.g., bird feeders, DVD discs, and whitespaces on documents) and exaggerated blindspots (e.g., wood background texture and palm trees). At the individual datapoint level, we further isolate memorization artifacts -- instances where models reproduce highly specific visual templates seen during training. Overall, we propose a theoretically grounded framework for systematically identifying conceptual blindspots in generative models by assessing their conceptual fidelity with respect to the underlying data-generating process.

Social Aspects · Accountability, Transparency and Interpretability

Zhaoyi Li, Jiatong Li, Gangwei Jiang, Linqi Song, Defu Lian, Ying Wei

Chain-of-thought (CoT) reasoning has become the standard paradigm for enabling Large Language Models (LLMs) to solve complex problems. However, recent studies reveal a sharp performance drop in *reasoning hop generalization* scenarios, where the required number of reasoning steps exceeds training distributions while the underlying algorithm remains unchanged. The internal mechanisms driving this failure remain poorly understood. In this work, we conduct a systematic study on tasks from multiple domains, and find that errors concentrate at token positions of a few critical error types, rather than being uniformly distributed. Closer inspection reveals that these token-level erroneous predictions stem from internal *competition mechanisms*: certain attention heads, termed *erroneous processing heads* (ep heads), tip the balance by amplifying incorrect reasoning trajectories while suppressing correct ones. Notably, removing individual ep heads during inference can often restore the correct predictions. Motivated by these insights, we propose test-time correction of reasoning, a lightweight intervention method that dynamically identifies and deactivates ep heads in the reasoning process. Extensive experiments across different tasks and LLMs show that it consistently improves reasoning hop generalization, highlighting both its effectiveness and potential.

Zhenlong Yuan, Chengxuan Qian, Jing Tang, Rui Chen, Zijian Song, Lei Sun, Xiangxiang Chu, Yujun Cai, Dapeng Zhang, Shuo Li

Vision–Language–Action (VLA) models in autonomous driving systems have recently demonstrated transformative potential by integrating multimodal perception with decision-making capabilities. However, the interpretability and coherence of the decision process and the plausibility of action sequences remain largely underexplored. To address these issues, we propose AutoDrive-R², a novel VLA framework that enhances both reasoning and self-reflection capabilities of autonomous driving systems through chain-of-thought (CoT) processing and reinforcement learning (RL). Specifically, we first propose an innovative CoT dataset named nuScenesR²-6K for supervised fine-tuning, which effectively builds cognitive bridges between input information and output trajectories through a four-step logical chain with self-reflection for validation. Moreover, to maximize both reasoning and self-reflection during the RL stage, we further employ the Group Relative Policy Optimization (GRPO) algorithm within a physics-grounded reward framework that incorporates spatial alignment, vehicle dynamic, and temporal smoothness criteria to ensure reliable and realistic trajectory planning. Extensive evaluation results across both nuScenes and Waymo datasets demonstrates the state-of-the-art performance and robust generalization capacity of our proposed method.

Social Aspects · Accountability, Transparency and Interpretability

Ekdeep Singh Lubana, Can Rager, Sai Sumedh R. Hindupur, Valérie Costa, Oam Patel, Sonia Murthy, Thomas Fel, Greta Tuckute, Daniel Wurgaft, Eric Bigelow 等

A central aim of interpretability tools applied to language models is to recover meaningful concepts from model activations. Existing feature extraction methods focus on single activations regardless of the context, implicitly assuming independence (and therefore stationarity). This leaves open whether they can capture the rich temporal and context-sensitive structure in the activations of language models (LMs). Adopting a Bayesian view, we demonstrate that standard Sparse Autoencoders (SAEs) impose priors that assume independence of concepts across time. We then show that LM representations exhibit rich temporal dynamics, including systematic growth in conceptual dimensionality, context-dependent correlations, and pronounced non-stationarity, in direct conflict with the priors of SAEs. This mismatch casts doubt on existing SAEs' ability to reflect temporal structures of interest in the data. We introduce a novel SAE architecture---Temporal SAE---with a temporal inductive bias that decomposes representations at a given time into two parts: a predictable component, which can be inferred from the context, and a residual component, which captures novel information that cannot be captured by the context. Experiments on LLM activations with Temporal SAE demonstrate its ability to correctly parse garden path sentences, identify event boundaries, and more broadly delineate abstract, slow-moving information from novel, fast-moving information, while existing SAEs show significant pitfalls in all the above tasks. Our results underscore the need for inductive biases that match the data in designing robust interpretability tools.

Social Aspects · Accountability, Transparency and Interpretability

Yupu Gu, Rongzhe Wei, Andy Zhu, Pan Li

Knowledge editing (KE) is crucial for making precise modifications to factual knowledge within large language models (LLMs). Existing KE methods, however, are primarily designed for dense architectures, limiting their applicability to the increasingly popular sparse Mixture-of-Experts (MoE) models that power modern scalable LLMs. While MoEs offer remarkable efficiency and capacity scaling, their unique structure introduces new challenges for KE. Naively adapting dense-model editors to MoEs is not only computationally expensive but also induces routing distribution shifts that degrade model stability and consistency. To address these challenges, we introduce MoEEdit, the first systematic framework for routing-stable knowledge editing in MoE LLMs. Our approach reparameterizes expert updates through per-expert null-space projections, ensuring router inputs remain invariant to suppress these shifts, and solves the resulting block-structured optimization with an efficient block coordinate descent (BCD) solver. Experiments demonstrate that MoEEdit achieves state-of-the-art efficacy and generalization, while maintaining high specificity, routing stability, and superior computational and memory efficiency. Our work establishes a robust foundation for scalable and precise knowledge editing in modern sparse LLMs by highlighting the necessity of routing-stable interventions.

Applications · Time Series

Antanas Žilinskas, Robert Shorten, Jakub Marecek

Forecasting rare events in multivariate time-series data is a central challenge in machine learning, complicated by severe class imbalance, long-range dependencies, and distributional uncertainty. We introduce EVEREST, a transformer-based architecture for probabilistic rare-event forecasting that delivers calibrated predictions and tail-aware risk estimation, with auxiliary interpretability through attention-based signal attribution. EVEREST integrates four key components: (i) a learnable attention bottleneck for soft aggregation of temporal dynamics; (ii) an evidential head for estimating aleatoric and epistemic uncertainty via a Normal–Inverse–Gamma distribution; (iii) an extreme-value head that models tail risk using a Generalized Pareto Distribution; and (iv) a lightweight precursor head for early-event detection. These modules are jointly optimised with a composite loss combining focal loss, evidential negative log-likelihood, and a tail-sensitive EVT penalty, and act only at training time; deployment uses a single classification head with no inference overhead. We evaluate EVEREST on a real-world benchmark spanning a decade of space-weather data and demonstrate state-of-the-art performance, including True Skill Statistic (TSS) scores of 0.973, 0.970, and 0.966 at 24, 48, and 72-hour horizons for C-class flares. The model is compact (≈0.81M parameters), efficient to train on commodity hardware, and applicable to other high-stakes domains such as industrial monitoring, weather, and satellite diagnostics. Limitations include reliance on fixed-length inputs and exclusion of image-based modalities, motivating future extensions to streaming and multimodal forecasting.

Theory · Interpretability and Visualization

Rim Assouel, Declan Campbell, Yoshua Bengio, Taylor Webb

To accurately process a visual scene, observers must bind features together to represent individual objects. This capacity is necessary, for instance, to distinguish an image containing a red square and a blue circle from an image containing a blue square and a red circle. Recent work has found that language models solve this ‘binding problem’ via a set of symbol-like, content-independent indices, but it is unclear whether similar mechanisms are employed by Vision Language Models (VLM). This question is especially relevant, given the persistent failures of VLMs on tasks that require binding. Here, we identify a previously unknown set of emergent symbolic mechanisms that support binding specifically in VLMs, via a content-independent, spatial indexing scheme. Moreover, we find that binding errors, when they occur, can be traced directly to failures in these mechanisms. Taken together, these results shed light on the mechanisms that support symbol-like processing in VLMs, and suggest possible avenues for reducing the number of binding failures exhibited by these models.