#01Jul 21, 2026
math.NA
1-Lipschitz Neural Networks on Hadamard Manifolds
Davide Murari, Marta Ghirardelli, Ben Adcock and 3 more
Controlling the Lipschitz constant of a neural network is a standard way to promote robustness and stability. Most existing constraining strategies are designed for Euclidean spaces. In this work, we construct and analyze a class of 1-Lipschitz neural networks on Hadamard manifolds. Our layers are of gradient-descent type, $1$-Lipschitz, and quasi-$α$-firmly nonexpansive. The core building blocks of the proposed architecture are Busemann functions, and we exploit the properties of Busemann gradient flows to design $1$-Lipschitz geometry-preserving layers. We provide explicit constructions and examples for hyperbolic manifolds and the manifold of symmetric positive definite (SPD) matrices. We test the proposed architecture in two numerical experiments: robust classification on the Poincaré disk and masked-Wishart covariance reconstruction. On the Poincaré disk, the proposed networks yield robust classifiers under hyperbolic perturbations. On the SPD manifold, we train SPD-valued denoisers and adopt them as a Plug-and-Play prior for a masked-Wishart covariance reconstruction problem. We show improved results from the nonexpansive denoiser over static, data-only, and Log-Euclidean denoising baselines, and empirically test its convergence properties.
#02Jul 21, 2026
cs.AI
Associative Emotional Learning in Convolutional Neural Networks
Seowung Leem, Andreas Keil, Mingzhou Ding and 1 more
Associative emotional learning enables organisms to adaptively link pleasant or unpleasant outcomes to the presence of predictive stimuli. Whereas computational models such as the Rescorla-Wagner model have shed light on this important function, the limitations of these models are also known, especially when they are applied to neural data. The advent of deep neural networks has opened another avenue for modeling associative emotional learning. In this work we proposed a deep neural network model of visual valence processing, consisting of a visual module that encodes complex natural scenes and a module that recognizes their emotional significance in terms of valence, a key dimension of emotion, and tested a novel Pavlovian learning paradigm on the model. The results showed that with learning, the model reproduced several observations from human associative learning studies, including association formation and generalization, and that the neural representations of the conditioned and the unconditioned stimuli became increasingly aligned both at the single unit and at the neural population level. Comparison between the model and human experimental data provided further validation of our approach. This study thus suggests that deep neural network models, when combined with appropriate learning algorithms, can be used to model behavioral and neural signatures of associative emotion/valence learning.
#03Jul 21, 2026
cs.CV
ExpertVerse: A General-Purpose Benchmark for Expert-Level Reasoning in Knowledge-Intensive Visual Synthesis
Yuan Wang, Yongchao Du, Mengting Chen and 3 more
Recent advances in multimodal generative models have enabled instruction-based image generation to move beyond semantic manipulation to knowledge-driven visual reasoning. However, these methods focus on explicit commonsense reasoning, shallow causal understanding, and direct knowledge recall, failing at knowledge-intensive generation. We develop \textbf{ExpertVerse}, a capability-centric benchmark to evaluate generative models via knowledge-intensive lens. ExpertVerse stratifies reasoning generation across an orthogonal taxonomy of \textit{9 cognitive capabilities} and \textit{8 expert disciplines}, yielding \textit{58 sub-disciplines}. We curate 1,611 expert-annotated instances covering single-image editing, multi-image composition, and text-to-image generation. We further develop an automated workflow to produce \textbf{ExpertVerse-100K}, a large-scale dataset with reasoning traces and knowledge-anchored rationale annotations. Based on this, we train \textbf{KnowThinker} with RL fine-tuning, a VLM reasoning engine with world knowledge that jointly generates thinking processes and refined instructions. Towards the cross-modal credit misalignment and multi-objective gradient conflicts in multi-reward optimization, we propose a tailored Bootstrapped Pareto Policy Optimization (BPPO), which synergizes Bootstrapping Reward Rectification (BRR) and Conflict-Aware Pareto Advantage Fusion (CPAF). Extensive results of both open-source and proprietary models exposes critical reasoning deficits, highlighting imperative for knowledge-intensive benchmarks towards next-generation visual generation.
#04Jul 21, 2026
cs.LG
Off-Context GRPO: Learning to Reason on Hard Problems using Privileged Information
Priyank Agrawal, Ankur Samanta, Shervin Ghasemlou and 4 more
Reinforcement learning with verifiable rewards (RLVR) improves reasoning in large language models. Yet, typical RLVR approaches fail on difficult problems: when a model cannot generate any correct solutions, it receives \textit{zero} learning signal. Providing privileged guidance during training, such as solution prefixes, can help overcome this learning cliff by steering the model towards {correct solutions with non-zero reward}. {We call these rollouts \textit{off-context}: they are generated from a training prompt that contains privileged guidance, while the target objective is defined by the original prompt without that guidance.} {We introduce} Off-Context GRPO (OC-GRPO), a minimally modified variant of GRPO that uses guided rollouts but applies an importance-corrected objective to steer the update back toward the original unguided objective, avoiding the mismatch that destabilizes uncorrected guided training. Empirically, our algorithm achieves a 3.9\% absolute improvement (13.8\% relative gain) over vanilla GRPO on average across standard mathematical reasoning benchmarks with negligible additional cost.
#05Jul 21, 2026
cs.LG
ISO: An RLVR-Native Optimization Stack
Hanqing Zhu, Wenyan Cong, Zhizhou Sha and 8 more
Reinforcement learning with verifiable rewards (RLVR) is rapidly advancing the reasoning capabilities of language models, yet the optimization layer that converts reward feedback into weight-space updates remains poorly understood. Building on our prior analysis (Zhu et al., 2025), we study this missing layer through the singular structure of model weights and identify spectral inheritance: RLVR can reuse the base model's weight spectra while acquiring new behavior through changes in the associated input and output singular frames. We operationalize spectral inheritance as Isospectral Optimization (ISO), an RLVR-native, fixed-spectrum optimization framework with complementary offline and online instantiations. Offline, ISO-Merger combines the frame changes of shared-base specialists into a single fixed-spectrum model, requiring no post-merge data, rollouts, gradient updates, or on-policy distillation (OPD). It recovers complementary specialist capabilities and achieves the strongest aggregate performance among the compared data-free merging methods. Online, ISO-Optimizer applies a chosen base optimizer, including AdamW and Muon, to the frame variables while keeping the base spectra fixed. Across reasoning and coding tasks ranging from 1.5B to 8B parameters, ISO-Optimizer improves accuracy in the reported runs and reaches matched scores with substantially fewer training steps. On Qwen3-8B-Base, AdamW reaches an aggregate accuracy of 0.495 after 270 training steps. ISO-AdamW reaches the same accuracy after only 100 training steps and improves further to 0.509 after 210 training steps. Together, ISO offers a concrete answer to RLVR's missing optimization layer: rather than inheriting pre-training optimization wholesale, design post-training around the structure of reward-driven adaptation: inherit the spectrum, optimize the frames.