We investigate a geometric generalization of trifference, a concept introduced by Elias [1] in the study of zero-error channel capacity. In the discrete setting, a code C⊆{0,1,2}n is trifferent if for any three distinct codewords x,y,z∈C, there exists a coordinate i∈[n] where xi,yi,zi are mutually distinct. Determining the maximum size of such codes remains a central open problem; the classical upper bound |C|≤2⋅(3/2)n from [1], proved via a simple ‘pruning’ argument, has resisted significant improvement.Motivated by the search for new techniques, and in line with the vectorial extensions of other classical combinatorial notions, we introduce the concept of vector trifferent codes. Consider C⊆(S2)n, where the alphabet is the unit sphere S2={v∈R3:‖v‖=1}. We say the code C is vector trifferent if for any three distinct x,y,z∈C, there is an index i where the vectors xi,yi,zi are mutually orthogonal. A direct reduction of the vectorial problem to the discrete setting appears infeasible, making it difficult to replicate Elias's pruning argument. Nevertheless, we develop a new method to establish the following upper bound for the size of a vector trifferent code:|C|≤(2+o(1))⋅(3/2)n. Interestingly, our approach, when modified and adapted back to the discrete setting, yields a polynomial improvement to Elias's bound: |C|≲n−1/4⋅(3/2)n. This improvement arises from a technique that parallels, but is not identical to, the method of [2]. However, it still falls short of the sharper n−2/5 factor obtained there. We also generalize the concept of vector trifferent codes to richer alphabets and prove a vectorial version of the Fredman-Komlós theorem [3] for general k-separating codes.
We propose FlowRL: matching the full reward distribution via flow balancing instead of solely maximizing rewards in large language model (LLM) reinforcement learning (RL). Recent advanced reasoning models adopt reward-maximizing methods (e.g., PPO and GRPO), which tend to over-optimize dominant reward signals while neglecting less frequent but valid reasoning paths, thus reducing diversity. In contrast, we transform scalar rewards into a normalized target distribution using a learnable partition function, and then minimize the reverse KL divergence between the policy and the target distribution. We implement this idea as a flow-balanced optimization method that promotes diverse exploration and generalizable reasoning trajectories. We conduct experiments on both math and code reasoning tasks: FlowRL achieves a significant average improvement of $10.0\%$ over GRPO and $5.1\%$ over PPO on math benchmarks, and performs consistently better on code reasoning tasks. These results highlight reward distribution-matching as a key step toward efficient exploration and diverse reasoning in LLM reinforcement learning.
We study view-invariant imitation learning by explicitly conditioning policies on camera extrinsics. Using Plücker embeddings of per-pixel rays, we show that conditioning on extrinsics significantly improves generalization across viewpoints for standard behavior cloning policies, including ACT, Diffusion Policy, and SmolVLA. To evaluate policy robustness under realistic viewpoint shifts, we introduce six manipulation tasks in robosuite and ManiSkill that pair "fixed" and "randomized" scene variants, decoupling background cues from camera pose. Our analysis reveals that policies without extrinsics often infer camera pose using visual cues from static backgrounds in fixed scenes, a shortcut that collapses when workspace geometry or camera placement shifts. Conditioning on extrinsics restores performance and yields robust RGB-only control without depth. We release the tasks, demonstrations, and code to facilitate reproducibility and future research.
Test-time algorithms that combine the *generative* power of language models with *process verifiers* that assess the quality of partial generations offer a promising lever for eliciting new reasoning capabilities, but the algorithmic design space and computational scaling properties of such approaches are still opaque, and their benefits are far from apparent when one accounts for the cost of learning a high-quality verifier. Our starting point is the observation that seemingly benign errors in a learned verifier can lead to catastrophic failures for standard decoding techniques due to *error amplification* during the course of generation. We then ask: can this be improved with more sophisticated decoding strategies? We introduce a new process-guided test-time sampling algorithm, VGB, which uses theoretically grounded *backtracking* to achieve *provably* better robustness to verifier errors. VGB interprets autoregressive generation as a random walk on a tree of partial completions, with transition probabilities guided by the process verifier and base model; crucially, backtracking occurs probabilistically. This process generalizes the seminal *Sinclair-Jerrum random walk* (Sinclair & Jerrum, 1989) from the literature on approximate counting and sampling in theoretical computer science, and a conceptual contribution of our work is to highlight parallels with this literature. Empirically, we demonstrate on both synthetic and real language modeling tasks that VGB outperforms baselines on a variety of metrics.
Hallucinations remain a persistent challenge for vision-language models (VLMs), which often describe nonexistent objects or fabricate facts. Existing detection methods typically operate after text generation, making intervention both costly and untimely. We investigate whether hallucination risk can instead be predicted before any token is generated by probing a model's internal representations in a single forward pass. Across a diverse set of vision-language tasks and eight modern VLMs, including Llama-3.2-Vision, Gemma-3, Phi-4-VL, and Qwen2.5-VL, we examine three families of internal representations: (i) visual-only features without multimodal fusion, (ii) vision-token representations within the text decoder, and (iii) query-token representations that integrate visual and textual information before generation. Probes trained on these representations achieve strong hallucination-detection performance without decoding, reaching up to 0.93 AUROC on Gemma-3-12B, Phi-4-VL 5.6B, and Molmo 7B. Late query-token states are the most predictive for most models, while visual or mid-layer features dominate in a few architectures (e.g., ~0.79 AUROC for Qwen2.5-VL-7B using visual-only features). These results demonstrate that (1) hallucination risk is detectable pre-generation, (2) the most informative layer and modality vary across architectures, and (3) lightweight probes have the potential to enable early abstention, selective routing, and adaptive decoding to improve both safety and efficiency.
