A Counterexample to the Tang Zhang Schatten Norm Conjecture and Sharp Positive Results

For $m\geq 2$, let $c_p(m)$ be the all-dimensional best constant in $$ \left\|\sum_{k=1}^m A_k\right\|_p \leq c_p(m)\left\|\sum_{k=1}^m |A_k|\right\|_p. $$ Tang and Zhang conjectured an explicit formula for every finite $p>1$. We disprove the conjecture with two explicit real $2\times 2$ rank-one matrices at $p=3/2$. The comparison is certified by seven strict rational inequalities and, in particular, places the attained ratio above $207/200$, while the conjectured constant lies below $207/200$. On the positive side, we prove the conjectured sharp bound for every family of rank-at-most-one summands when $2\leq p<\infty$, and classify all equality cases. We also prove the corresponding endpoint statement for $p=\infty$. Finally, for arbitrary complex matrices, we establish the conjectured sharp constant in the case $m=2$, $p=4$.

On the Impossibility of Unbiased and Length-Invariant Policy Optimization with Outcome Rewards

Group Relative Policy Optimization (GRPO) is the dominant reinforcement learning algorithm for training reasoning capabilities in large language models, notably adopted by DeepSeek-R1. The recent improvement Dr. GRPO (COLM 2025) identifies the response-level length bias caused by per-trajectory length normalization in GRPO and proposes removing this normalization, claiming the resulting optimizer is "unbiased." We show that this claim is incomplete. Specifically, we establish an impossibility theorem: under the standard outcome reward + GRPO setting, no length-based weighting scheme can simultaneously achieve the following two properties. (P1) Gradient unbiasedness: the gradient estimator is an unbiased estimate of the true policy gradient. (P2) Length invariance: each trajectory's effective contribution to the gradient is independent of its token length. GRPO approximately satisfies P2 but violates P1; Dr. GRPO satisfies P1 but violates P2. We characterize the complete tradeoff spectrum via the parametric family f_alpha(L) = L^{alpha - 1}, where alpha = 0 recovers GRPO, alpha = 1 recovers Dr. GRPO, and provide quantitative analysis showing that Dr. GRPO's length bias can cause longer trajectories to dominate gradient updates by a factor proportional to the length ratio. Our results reveal that neither algorithm is universally "done right"; they occupy opposite ends of a fundamental and unavoidable tradeoff.

HELM: Harness-Enhanced Long-horizon Memory for Vision-Language-Action Manipulation

Vision-Language-Action (VLA) models fail systematically on long-horizon manipulation tasks despite strong short-horizon performance. We show that this failure is not resolved by extending context length alone in the current reactive execution setting; instead, it stems from three recurring execution-loop deficiencies: the memory gap, the verification gap, and the recovery gap. We present HELM, a model-agnostic framework that addresses these deficiencies with three components: an Episodic Memory Module (EMM) that retrieves key task history via CLIP-indexed keyframes, a learned State Verifier (SV) that predicts action failure before execution from observation, action, subgoal, and memory-conditioned context, and a Harness Controller (HC) that performs rollback and replanning. The SV is the core learning contribution: it consistently outperforms rule-based feasibility checks and ensemble uncertainty baselines, and its effectiveness depends critically on access to episodic memory. On LIBERO-LONG, HELM improves task success rate by 23.1 percentage points over OpenVLA (58.4% to 81.5%), while extending the context window to H=32 yields only a 5.4-point gain and same-budget LoRA adaptation remains 12.2 points below HELM. HELM also improves long-horizon performance on CALVIN and substantially boosts recovery success under controlled perturbations. Ablations and mechanism analyses isolate the contribution of each component, and we release LIBERO-Recovery as a perturbation-injection protocol for evaluating failure recovery in long-horizon manipulation.

Difference Feedback: Generating Multimodal Process-Level Supervision for VLM Reinforcement Learning

Vision--language models (VLMs) are increasingly aligned via Group Relative Policy Optimization (GRPO)-style training. However, relying solely on terminal outcome rewards yields sparse credit assignment in multi-step reasoning, weakening the linkage between visual evidence and intermediate steps and often causing unstable optimization and visual hallucinations. We propose Differential Feedback, which automatically constructs token/step-level supervision masks by repairing erroneous reasoning trajectories, explicitly marking the key positions that require correction. Without costly large-scale step-by-step human annotations, our method enables process-level visual alignment and can be seamlessly integrated into existing GRPO-like frameworks. Experiments on multimodal reasoning benchmarks including MMMStar and MathVista show an average 3% improvement under matched compute budgets. Our approach offers an effective, low-cost solution for accurate vision--reasoning process alignment.

PoseStreamer: A Multi-modal Framework for 3D Tracking of Unseen Moving Objects

Six degree of freedom (6DoF) pose estimation for novel objects is a critical task in computer vision, yet it faces significant challenges in high-speed and low-light scenarios where standard RGB cameras suffer from motion blur. While event cameras offer a promising solution due to their high temporal resolution, current 6DoF pose estimation methods typically yield suboptimal performance in high-speed object moving scenarios. To address this gap, we propose PoseStreamer, a robust multi-modal 6DoF pose estimation framework designed specifically on high-speed moving scenarios. Our approach integrates three core components: an Adaptive Pose Memory Queue that utilizes historical orientation cues for temporal consistency, an Object-centric 2D Tracker that provides strong 2D priors to boost 3D center recall, and a Ray Pose Filter for geometric refinement along camera rays. Furthermore, we introduce MoCapCube6D, a novel multi-modal dataset constructed to benchmark performance under rapid motion. Extensive experiments demonstrate that PoseStreamer not only achieves superior accuracy in high-speed moving scenarios, but also exhibits strong generalizability as a template-free framework for unseen moving objects.

PoseStreamer: A Multi-modal Framework for 6DoF Pose Estimation of Unseen Moving Objects

Six degree of freedom (6DoF) pose estimation for novel objects is a critical task in computer vision, yet it faces significant challenges in high-speed and low-light scenarios where standard RGB cameras suffer from motion blur. While event cameras offer a promising solution due to their high temporal resolution, current 6DoF pose estimation methods typically yield suboptimal performance in high-speed object moving scenarios. To address this gap, we propose PoseStreamer, a robust multi-modal 6DoF pose estimation framework designed specifically on high-speed moving scenarios. Our approach integrates three core components: an Adaptive Pose Memory Queue that utilizes historical orientation cues for temporal consistency, an Object-centric 2D Tracker that provides strong 2D priors to boost 3D center recall, and a Ray Pose Filter for geometric refinement along camera rays. Furthermore, we introduce MoCapCube6D, a novel multi-modal dataset constructed to benchmark performance under rapid motion. Extensive experiments demonstrate that PoseStreamer not only achieves superior accuracy in high-speed moving scenarios, but also exhibits strong generalizability as a template-free framework for unseen moving objects.

Multi-Layer GRPO: Enhancing Reasoning and Self-Correction in Large Language Models

The Group Relative Policy Optimization (GRPO) algorithm has demonstrated considerable success in enhancing the reasoning capabilities of large language models (LLMs), as evidenced by DeepSeek-R1. However, the absence of intermediate supervision in GRPO frequently leads to inefficient exploration dynamics. A single error in a complex reasoning chain can invalidate the entire solution, resulting in abrupt reward vanishing and compromising training stability.To address these challenges, we propose MGRPO (Multi-layer GRPO). MGRPO operates in two layers: the first layer employs standard GRPO to generate an initial response. This response, along with the original query, is then fed into a second-layer GRPO process. This second layer is specifically trained to identify and correct errors in the initial response, effectively creating a self-correction loop. This mechanism provides implicit process-level supervision by rewarding successful error correction, without requiring an explicit, densely-annotated reward model. Experimental results on several mathematical reasoning benchmarks demonstrate that MGRPO significantly outperforms standard GRPO, achieving superior performance by fostering both reasoning and self-correction abilities.