Large Language Models (LLMs) have enabled embodied agents to perform high-level planning, yet grounding language-based plans into the physical world remains challenging. While feedback-based closed-loop planning is the dominant paradigm, existing methods typically rely on a single scalar feedback signal, which conflates physically infeasible actions that violate hard constraints with strategically sub-optimal but valid choices. We identify this structural limitation as Signal Confounding, which leads to ambiguous feedback and unreliable plan correction. To address this issue, we propose Decoupled Dual-Critic Guidance (DDCG), a framework that explicitly disentangles feedback into two complementary signals: a Feasibility Critic (C_F) that enforces physical and logical executability constraints, and a Quality Critic (C_Q) that evaluates the strategic value of feasible actions. This decoupled guidance enables precise error attribution during planning without requiring parameter updates to the LLM planner. We provide a theoretical interpretation of DDCG as critic-regularized planning with hard feasibility constraints. Experiments on two embodied planning benchmarks, VirtualHome and ScienceWorld, show that DDCG consistently improves plan executability and task success, particularly in long-horizon settings, highlighting the importance of structured feedback for reliable LLM-based planning.

Motivated by the widespread interest in the cross-lingual transfer of NLP models from high resource to low resource languages, research on Cross-lingual word embeddings (CLWEs) has gained much popularity over the years. Among the most successful and attractive CLWE models are the unsupervised CLWE models. These unsupervised CLWE models pose the alignment task as a Wasserstein-Procrustes problem aiming to estimate a permutation matrix and an orthogonal matrix jointly. Most existing unsupervised CLWE models resort to Optimal Transport (OT) based methods to estimate the permutation matrix. However, linear programming algorithms and approximate OT solvers via Sinkhorn for computing the permutation matrix scale cubically and quadratically, respectively, in the input size. This makes it impractical and infeasible to compute OT distances exactly for larger sample size, resulting in a poor approximation quality of the permutation matrix and subsequently a less robust learned transfer function or mapper. This paper proposes an unsupervised projection-based CLWE model called quantized Wasserstein Procrustes (qWP) that jointly estimates a permutation matrix and an orthogonal matrix. qWP relies on a quantization step to estimate the permutation matrix between two probability distributions or measures. This approach substantially improves the approximation quality of empirical OT solvers given fixed computational cost. We demonstrate that qWP achieves state-of-the-art results on the Bilingual lexicon Induction (BLI) task.