cs.LG · 2026-07-08 · No. 47

Machine Learning, 2026-07-08.

40 new papers in cs.LG. Titles, authors, abstracts. Links to arXiv. Want this in your inbox every morning? Subscribe →

01 — The papers

40 entries
  1. 01

    Graph Convolutional Attention: A Spectral Perspective on Graph Denoising and Diffusion

    Shervin Khalafi, Igor Krawczuk, Sergio Rozada, Charilaos Kanatsoulis, Antonio G Marques, Alejandro Ribeiro

    cs.LG · cs.AI

    Denoising graphs is a fundamental problem in graph learning and the core operation of graph diffusion models. Attention-based architectures like graph transformers have recently shown promise in denoising graphs. However, our principled understanding of attention-based graph denoising remains limited, making it unclear whether standard attention is the right mechanism for this task. Here we show that, under a denoising objective, linear...

    arxiv.org/abs/2607.06546 · PDF

  2. 02

    GraphBU: MILP Instance Generation with Graph-Native Block Units

    Xiaolei Guo, Chenyu Zhou, Jianghao Lin, Dongdong Ge

    cs.LG · math.OC

    Mixed-integer linear programming (MILP) instances used for solver development are hard to obtain when models come from private or application-specific pipelines. A generator must keep the structure that solvers and learned policies rely on. Existing general generators usually choose their generation unit from a formulation template, summary statistics, local graph edits, or blocks found after recombination. These units do not explicitly...

    arxiv.org/abs/2607.06532 · PDF

  3. 03

    EntroPath: Maximum Entropy Path Ensemble Embedding for Manifold Learning

    Przemysław Rola

    cs.LG · q-bio.QM · stat.ML

    We introduce EntroPath, a manifold learning method that recovers geodesic geometry from data graphs through ensembles of diffusion paths. Many existing graph-based embeddings rely either on locally normalised random walks or on shortest-path distances. The former can concentrate diffusion in densely sampled regions, while the latter are sensitive to spurious shortcut edges in the graph. EntroPath instead builds its dissimilarities from the...

    arxiv.org/abs/2607.06497 · PDF

  4. 04

    TILDE: TILt-based Distributional Erasure for Concept Unlearning

    Naveen George, Naoki Murata, Yuhta Takida, Konda Reddy Mopuri, Yuki Mitsufuji

    cs.LG · cs.AI · cs.CV

    Concept unlearning in text-to-image diffusion models is critical for safe and practical deployment: with rising privacy concerns, copyright disputes, trademark constraints, and safety regulations, deployed systems must be able to suppress unwanted concepts after training. Existing methods often remove the target concept effectively, but practical unlearning also requires an equally fundamental property: the unlearned model should retain...

    arxiv.org/abs/2607.06432 · PDF

  5. 05

    Physics-Informed Neural Embeddings of PDE Solution Families

    Raul Jimenez, Svitlana Mayboroda, Pavlos Protopapas, Leonid Sarieddine, David N. Spergel, Pedro Tarancón-Álvarez

    cs.LG · math.NA · physics.comp-ph

    We introduce a physics-informed framework for learning finite-dimensional embeddings of solution families of partial differential equations. The method uses a multihead Physics-Informed Neural Network in which a shared body learns a latent manifold representing the solution space, while linear heads reconstruct individual solutions associated with different initial conditions. A head-orthogonalization penalty removes degeneracies in the...

    arxiv.org/abs/2607.06348 · PDF

  6. 06

    Quantitative Gaussian-Process limits of Tensor Programs

    Andrea Agazzi, Eloy Mosig García, Dario Trevisan

    cs.LG · math.PR · stat.ML

    We study the infinite-width Gaussian-process limit of random neural networks through the lens of tensor programs, and we provide a quantitative convergence theory in Wasserstein distance. Our main result gives explicit finite-width error bounds, of order inverse square-root of the widths between finite-network executions and their Gaussian-process limits. The framework is architecture-agnostic and covers feed-forward models together with...

    arxiv.org/abs/2607.06290 · PDF

  7. 07

    Canopy: A Heterograph Foundation Model for Metabolic Engineering

    Jake Bowden, Laurence Legon, Satnam Surae

    cs.LG

    Designing microbial strains that produce high-value chemicals at commercially viable titers remains a central challenge in metabolic engineering. Existing computational approaches either rely on stoichiometric constraint-based models that cannot learn from experimental data, or apply tabular machine learning to hand-crafted features that discard the relational structure of biological knowledge. We present Canopy, a heterogeneous graph...

    arxiv.org/abs/2607.06224 · PDF

  8. 08

    X-FEMR: A Token-level Explainable Approach for Electronic Health Records Foundation Models using Transformer-based Models

    Jie Huang, Pengfei Yin, Zihan Xu, Daniel Capurro, Mike Conway, Ting Dang

    cs.LG · cs.AI

    Foundation Models for Electronic Health Records (FEMRs) are pretrained on large-scale structured patient data, enabling them to convert longitudinal patient trajectories into generalizable representations for diverse clinical prediction tasks. Despite their effectiveness, FEMRs remain black-box models, raising concerns about bias, interpretability, and clinical trust. To address this, we propose the first token-level explainability approach...

    arxiv.org/abs/2607.06163 · PDF

  9. 09

    Leveraging Extragradient for Effective Sharpness-Aware Minimization in Deep Learning

    Yao Fu, Chunxia Zhang, Junmin Liu, Yihang Jin, Haishan Ye, Yuanao Yang

    cs.LG · math.PR

    Generalization remains a pivotal challenge in deep learning, where traditional optimizers like Stochastic Gradient Descent (SGD) often converge to sharp minima, leading to overfitting and reduced performance on unseen data. Building on Sharpness-Aware Minimization (SAM), for seeking flat minima associated with improved generalization, we propose the Extragradient-Inspired Sharpness-Aware Minimization (EISAM), a novel optimizer that enhances...

    arxiv.org/abs/2607.06151 · PDF

  10. 10

    Self-Supervised Implicit CEST Reconstruction via Physics-Informed Lorentz Encoding

    Dexuan Li, Yupeng Wu, Chenglong Wang, Hanlin Liu, Hui Zhen, Jianqi Li, Guang Yang

    cs.LG · cs.AI · physics.med-ph

    Multi-Pool Chemical Exchange Saturation Transfer (CEST) MRI provides valuable metabolic information but is clinically limited by long acquisition times. Although sparse sampling reduces scanning time, reconstructing high-resolution Z-spectra from limited data remains an ill-posed inverse problem. Conventional interpolation and generic Implicit Neural Rep-resentations (INRs) often lack physical constraints, leading to spectral artifacts and...

    arxiv.org/abs/2607.06132 · PDF

  11. 11

    x-Prediction Is All You Need:Training-Free Accelerated Generation via Endpoint Decodability

    Xin Peng, Ang Gao

    cs.LG · cs.AI

    Diffusion and flow matching models generate high-quality samples, but their ODE samplers often need tens to hundreds of neural function evaluations (NFEs). This remains a practical challenge for released checkpoints, since many accelerators require additional design choices and training cost through retraining, distillation, or trajectory redesign. We investigate a different route based on $x$-prediction. During sampling, standard affine...

    arxiv.org/abs/2607.06114 · PDF

  12. 12

    Modeling Normal Is All You Need: Joint Latent Clustering for Anomaly Detection in Multimodal Cyber-Physical Systems

    Alexander Apartsin, Yehudit Aperstein

    cs.LG

    Faults on a cyber-physical system (CPS) are too rare and unrepresentative to characterise, or even to select a model on, so detection must instead model normal behaviour; the standard point-adjusted evaluation, however, rewards detectors that never do. CPS normal behaviour is the union of many imbalanced, curved, thin-fringed operating regimes rather than a single blob; we state this structure as ten assumptions (A1-A10), abbreviated Massive,...

    arxiv.org/abs/2607.06094 · PDF

  13. 13

    Scalable Perturbation Learning for Online Self-Supervised Echo State Networks

    Taiki Yamada, Kantaro Fujiwara

    cs.LG · cs.NE

    Intelligent systems should not only solve tasks but also adapt under real-world constraints. Autonomous adaptation via self-supervised learning, sequential adaptation via online learning, and memory-efficient implementation via perturbation-based learning are important requirements for such systems. However, these requirements are generally in tension for high-dimensional systems, because perturbation-based learning suffers from variance that...

    arxiv.org/abs/2607.06079 · PDF

  14. 14

    SplineNet: An Isogeometric Deep Learning Method for Complex Shells

    Shizhou Luo, Xiaodong Wei

    cs.LG · math.NA

    We present a novel isogeometric deep learning method, termed SplineNet, for the seamless design and analysis of shell structures with complex geometries. The proposed approach is built upon watertight spline representations, e.g., analysis-suitable unstructured T-splines, and features exact geometric descriptions of Computer-Aided Design (CAD) models in neural networks. Bézier extraction is used to build the network architecture, where...

    arxiv.org/abs/2607.06026 · PDF

  15. 15

    Learning When to Automate: Queue Control in Human-AI Service Systems

    Giovanni Montanari, Marco Scarsini, Vianney Perchet

    cs.LG · math.OC

    We study a human-AI service system in which tasks arrive sequentially and are processed through a two-stage architecture: an automated chatbot followed, when necessary, by a human agent. We consider $T$ sequentially arriving tasks, each belonging to one of $K$ heterogeneous types. For each task the decision maker chooses how many resources to allocate to the chatbot, whose type-dependent success probabilities are initially unknown. Tasks not...

    arxiv.org/abs/2607.06017 · PDF

  16. 16

    Stability Annealing Selects the Implicit Bias of Smoothed Sign Descent: A Rate-Indexed Barrier Path on Separable Data

    Xiangwu Wang, Chengwei Cao, Yicheng Song, Ran Bi, Peilin Yu

    cs.LG · math.OC

    Adaptive gradient methods can favor max-margin separators that differ from gradient descent, yet a fixed positive numerical stability constant eventually changes the update geometry again. This paper studies the rate-controlled middle case for full-batch linear classification on separable data. For memoryless stability-annealed smoothed-sign descent with weighted exponential loss, we prove that the normalized iterates converge to the...

    arxiv.org/abs/2607.06013 · PDF

  17. 17

    Learning Sparsest Linear Causal DAGs with Latent Confounders via Higher-Order Cumulants

    Ming Cai, Hisayuki Hara

    cs.LG

    Recovering the exact directed acyclic graph (DAG) in linear non-Gaussian acyclic models with latent confounders (LvLiNGAM) remains a challenging problem. Although LvLiNGAM is identifiable only up to an observational equivalence class, each equivalence class is characterized by a unique sparsest DAG. Recovering the sparsest DAG from finite samples, however, remains difficult. Although existing methods are asymptotically consistent, they do not...

    arxiv.org/abs/2607.05984 · PDF

  18. 18

    Drift Happens: An Empirical Study of Neural Architecture Robustness to Temporal Distribution Shift

    Robin Holzinger, Riccardo Colletti

    cs.LG

    Real-world data distributions evolve over time, inducing temporal distribution shift that can substantially degrade the reliability of deployed machine learning systems. However, the extent to which architectural choices and their associated inductive biases affect temporal robustness remains insufficiently understood. We present a systematic empirical comparison of temporal robustness across three heterogeneous, time-indexed domains...

    arxiv.org/abs/2607.05908 · PDF

  19. 19

    More Convincing, Not More Correct: Self-Play Reward Hacking of Reference-Free LLM Judges

    Chenyu Zhou

    cs.LG

    Training a language model against its own reference-free judgments (the premise of self-rewarding, self-play, and LLM-as-a-judge pipelines) assumes a model's verdict on a shown answer tracks correctness. We show it fails structurally: conditioned on a candidate, a judge scores plausibility, not correctness, leaving false-positive basins a policy learns to exploit. We measure this with a hidden-anchor audit: a held-out, cross-source...

    arxiv.org/abs/2607.05904 · PDF

  20. 20

    K-ABENA: K-Adaptive Backpropagation with Error-based N-exclusion Algorithm : (Compensated Loss-Based Sample Exclusion with Unbiased Gradient Estimation)

    Jean-Francois Bonbhel

    cs.LG · cs.AI · cs.CL

    We present K-ABENA (K-Adaptive Backpropagation with Error-based N-exclusion Algorithm), a selective gradient computation framework that reduces per-iteration training cost by excluding a fraction of low-loss ("minor") observations from the backward pass. Its canonical form (v3) combines a defensive-mixture sampling design over the minor set with Horvitz-Thompson inverse-probability reweighting, yielding a design-unbiased Horvitz-Thompson...

    arxiv.org/abs/2607.05903 · PDF

  21. 21

    Auditing of Unlearning Algorithms

    Sahasrajit Sarmasarkar, Anastasia Koloskova, Sanmi Koyejo

    cs.LG · cs.CR

    Evaluating whether unlearning algorithms truly remove training data influence remains an open challenge. We propose a practical auditor that computes data-dependent lower bounds on the unlearning parameter $\varepsilon$ using membership inference attacks. Evaluating multiple unlearning algorithms, we find a sharp separation: algorithms with rigorous guarantees, such as model clipping and rewind-to-delete, achieve very small $\varepsilon$...

    arxiv.org/abs/2607.05898 · PDF

  22. 22

    No Subspace to Track: Non-Identifiability and Optimizer State in Low-Rank Training

    Noel Thomas

    cs.LG · math.OC · stat.ML

    Memory-efficient optimizers such as GaLore train large language models by projecting gradients onto a rank-r subspace recomputed every T steps, assuming this subspace is a slowly drifting object that can be tracked. We show that beyond a small reproducible core, there is no such object. Two estimates of the top-r subspace computed at the same step from disjoint minibatches disagree as much as estimates computed T steps apart (0.73 vs 0.74 of...

    arxiv.org/abs/2607.05872 · PDF

  23. 23

    Differentially Private Natural Gradient Descent

    Pan Li, Kai Chen, Shuai Chang, Shengzhi Zhang, Peizhuo Lv, Jinwen He

    cs.LG · cs.AI

    Under a fixed privacy budget, the utility of differentially private (DP) training is ultimately determined by its optimization efficiency. Standard first-order DP optimizers such as DP-SGD rely solely on local gradients and ignore the underlying loss curvature. This geometric blindness causes severe zigzagging in ill-conditioned landscapes, squandering precious privacy budgets on inefficient iterations. Practitioners are thus trapped in a...

    arxiv.org/abs/2607.05866 · PDF

  24. 24

    Strategic Bargaining in Multi-Buyer Markets: Reinforcement Learning from Verifiable Rewards for LLM Negotiations

    Shuze Daniel Liu, Claire Chen, Jiabao Sean Xiao, Xin Chen, David Simchi-Levi

    cs.LG · cs.GT

    Negotiation is a fundamental strategic interaction in management science, characterized by agents attempting to reach agreements while protecting private information, such as reservation costs and hidden valuations. A prevalent yet complex scenario involves a single seller negotiating concurrently with multiple buyers, each possessing heterogeneous, private budgets. In such settings, constrained by a limited number of communication turns, the...

    arxiv.org/abs/2607.05863 · PDF

  25. 25

    Unsupervised Anomaly Detection of Information Operations Users via Behavioral and Language Patterns

    Sishun Liu, Sajal Halder, Ke Deng, Yan Wang, Xiuzhen Zhang

    cs.LG · cs.AI

    Information Operations on social media networks have been identified as a significant threat to democracy and modern society, but they are challenging and expensive to detect by humans. Existing supervised IO detection methods fail to capture the dynamic nature of evolving IO user behavior, while existing unsupervised approaches rely on oversimplified assumptions of coordination among IO users that may not exist in practice. To overcome the...

    arxiv.org/abs/2607.05855 · PDF

  26. 26

    AbICL: In-Context Learning for Antigen-Specific Antibody Affinity Ranking

    Zhiyuan Chen, Jing Hu, Junzhe Wang, Yueyang Huang, Xinyi Yang, Zhaoyang Wang, Feng Zhu

    cs.LG · cs.AI · cs.CE · q-bio.QM

    Accurate ranking of antibody candidates according to their binding affinity is essential for therapeutic antibody discovery. However, existing methods treat affinity comparisons independently and ignore the contextual information encoded in other labeled comparisons, limiting their ability to capture antigen-specific binding landscapes. For many target antigens, a small number of experimentally characterized affinity comparisons are often...

    arxiv.org/abs/2607.05846 · PDF

  27. 27

    Decision-Focused Scenario Generation and Selection for Efficient and Robust Grid Dispatch

    Yangze Zhou, Yihong Zhou, Thomas Morstyn, Yi Wang

    cs.LG · cs.AI

    The increasing uncertainty from flexible demand and renewable generation has made distributionally robust optimization (DRO) an important tool for robust power system dispatch. DRO relies on forecast scenarios to construct ambiguity sets, but conventional scenario generation pipelines are often trained in an accuracy-oriented manner and may neglect spatial correlations among uncertainties. This mismatch can produce ambiguity sets that are...

    arxiv.org/abs/2607.05830 · PDF

  28. 28

    Level-Crossing Density as a Mesh-Free High-Frequency Auxiliary Loss for Implicit Neural Representations

    Gunner Levi Howe

    cs.LG

    The Minkowski functionals of a field's excursion sets -- area, boundary measure, and Euler characteristic -- describe its level-set morphology; the Euler characteristic is the cheapest handle on topology. We derive smooth Monte-Carlo estimators for all three of a continuous neural field, evaluated at scattered points via the co-area formula and Gauss-Bonnet, using only autodiff: no grid, no complex, no persistence. The estimator is accurate...

    arxiv.org/abs/2607.05815 · PDF

  29. 29

    Heckman-Corrected Epistemic Uncertainty: Selection on Unobservables Defeats Importance Weighting

    Gunner Levi Howe

    cs.LG

    Training data for machine learning is routinely collected by a selection process the model never sees: loans are observed only when granted, outcomes only when a test was ordered. The standard fixes -- importance weighting, covariate-shift correction, MAR imputation -- assume selection is ignorable given observables. Econometrics solved the harder case in 1979: Heckman's two-equation model jointly fits a probit selection equation and an...

    arxiv.org/abs/2607.05806 · PDF

  30. 30

    Two Sides of the Same Coin: Learning the Backdoor to Remove the Backdoor

    Qi Zhao, Christian Wressnegger

    cs.LG

    The community has recently developed various training-time defenses to counter neural backdoors introduced through data poisoning. In light of the observation that a model learns poisonous samples responsible for the backdoor easier than benign samples, these approaches either use a fixed threshold of the training loss for splitting or iteratively learn a reference model as an oracle for identifying benign samples. In particular, the latter...

    arxiv.org/abs/2607.05748 · PDF

  31. 31

    Multimodal Molecular Representation Learning with Graph Neural Networks, Deep & Cross Networks, and SMILES Embeddings

    Qiwei Han, Chi Zhou, Ruobing Wang, Zheng Ma

    cs.LG

    Molecular property prediction often relies on isolated data modalities, where continuous 3D graph neural networks (GNNs) struggle to efficiently capture long-range topological dependencies and exact macroscopic heuristics. In this work, we introduce a parameter-efficient Tri-Branch Modular Fusion Neural Network that synthesizes three orthogonal modalities: 3D spatial geometry (SchNet), discrete topological grammar (SMILES via ChemBERTa), and...

    arxiv.org/abs/2607.05736 · PDF

  32. 32

    Low-Overhead Error-Corrected QCNNs Using Bivariate Bicycle Codes

    Alejandro Rosales, Animesh Yadav

    cs.LG · quant-ph

    Quantum convolutional neural networks (QCNNs) combine the power of quantum computing and classical CNN for computational speedup in classification tasks. However, noise levels on state-of-the-art quantum devices remain too high for practical QCNN execution. In addition, despite the reliable surface code providing a method for error rates below a threshold value, they have a prohibitively large qubit cost. Recently introduced bivariate bicycle...

    arxiv.org/abs/2607.05724 · PDF

  33. 33

    FourTune: Towards Fully 4-Bit Efficient Post-Training for Diffusion Models

    Bowen Xue, Zihan Min, Xingyang Li, Zhekai Zhang, Haocheng Xi, Lvmin Zhang, Maneesh Agrawala, Jun-Yan Zhu, Song Han,...

    cs.LG · cs.CV

    Diffusion models have become a dominant paradigm for high-quality generative modeling, while post-training is essential for adapting them to diverse downstream applications. However, post-training of large diffusion models is still challenging due to the prohibitive memory footprints and slow training speed, which existing parameter-efficient fine-tuning methods only partially address. To overcome these limitations, we propose FourTune, an...

    arxiv.org/abs/2607.05711 · PDF

  34. 34

    LLM-Driven Neural Network Generation with Same-Family Architecture Guidance: Disentangling Transfer and Adaptation

    Kabir Dev Paul Baghel, Radu Timofte, Dmitry Ignatov

    cs.LG · cs.CV

    Large language models (LLMs) can generate neural-network modifications, but unrestricted generation is often invalid or harmful. This paper studies a narrower setting: improving a weak target model using a stronger same-family source model from a neural-network database. We propose a source-guided candidate-generation protocol with non-source controls, source-conditioned candidates, and a no-LLM hp_copy ablation under equal evaluation...

    arxiv.org/abs/2607.05704 · PDF

  35. 35

    Deep Reinforcement Learning for Dynamic Battery Management of Autonomous Order Pickers

    Taniya Shaji, Abhay Sobhanan, Christof Defryn

    cs.LG · cs.MA · math.OC

    Battery charging of Autonomous Mobile Robots (AMRs) in warehouses is a critical operational challenge that heavily impacts both order processing times and throughput. In this study, we address the dynamic AMR charging problem under stochastic order arrivals, where robots must learn optimal charging decisions. Traditional fixed-rule heuristics often prove suboptimal in dynamic environments and fail to account for multi-AMR coordination,...

    arxiv.org/abs/2607.05683 · PDF

  36. 36

    Orthogonal Dendritic Intrinsic Networks: An Architecture for Significance-Ordered, Orthogonal Latent Spaces

    Jeanie Schreiber, Tyrus Berry, Zeeshan Ahmed

    cs.LG · math.OC

    Principal Component Analysis or PCA-like properties (orthogonality, variance ranking) are seldom realized in deep autoencoder architectures. In this work, we present ODIN (Orthogonal Dendritic Intrinsic Network), a novel autoencoder architecture that recovers PCA-like latent structure in a fully non-linear regime. By incorporating a set of geometric constraints directly into the training objective, ODIN encourages latent dimensions to be...

    arxiv.org/abs/2607.05653 · PDF

  37. 37

    Domain-Adaptive Climate Downscaling Under Temporal Distribution Shift

    Shuochen Wang, Nishant Yadav, Auroop R. Ganguly

    cs.LG · physics.ao-ph

    Deep-learning-based climate downscaling aims to learn relationships from historical low-resolution (LR) and high-resolution (HR) climate data to generate HR climate projections. However, this setting faces a temporal out-of-distribution (OOD) challenge: models trained on historical data are commonly applied to future projections whose distributions may differ substantially from the training period. This study investigates temporal OOD shift...

    arxiv.org/abs/2607.05645 · PDF

  38. 38

    Intuitionistic Fuzzy Graph Embedded Random Vector Functional Link with Multiview Learning

    Vrushank Ahire, Yogesh Kumar, M. A. Ganaie

    cs.LG

    Random Vector Functional Link (RVFL) networks are popular due to their fast training and universal approximation capabilities. However, RVFL models face challenges in preserving geometric relationships and utilizing multiple feature views effectively. To address these limitations we propose the Intuitionistic Fuzzy Graph Embedded Random Vector Functional Link with Multiview Learning (IFGRVFL-MV) model. The proposed approach comprises three...

    arxiv.org/abs/2607.05635 · PDF

  39. 39

    Safe Bayesian Optimization with Counterfactual Policies

    Katherine Avery, Bruno Castro da Silva, David Jensen

    cs.LG · cs.AI

    In many decision-making settings, new interventions are acceptable only if they do not reduce outcomes below some established threshold. For example, in clinical medicine, new treatments are often acceptable only if they do not worsen outcomes relative to an established standard of care. Safe Bayesian optimization maximizes an objective subject to safety constraints. In the setting that we consider here, safety is defined relative to a known...

    arxiv.org/abs/2607.05620 · PDF

  40. 40

    A Coin Flip Per Token: Bernoulli Sparse Steering of Large Language Models

    Nima Eshraghi, Lovedeep Gondara, Yuqing Huang, Sagarika Suresh, Leizer Teran, Jithin Pradeep, Xiaotong Xu, Fanny Chevalier

    cs.LG

    Activation steering via sparse autoencoders (SAEs) enables behavioral control of large language models without task-specific fine-tuning, but standard methods apply the steering signal at every generated token, incurring constant per-token perturbation that risks degrading fluency. We ask: is dense intervention necessary? We introduce Stochastic Token Steering (STS), which gates each token independently with probability $p$, and Stochastic...

    arxiv.org/abs/2607.05615 · PDF

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