In 3D scene understanding, deep learning models rely on large models and extensive training to capture basic geometric structures that are present in the 3D data. However, existing methods lack explicit mechanisms to incorporate geometric information, such as learnable primitive shapes, often necessitating large models and more training data which in turn increases cost and can limit generalization. We introduce GIBLy, a lightweight geometric inductive bias layer that integrates learnable geometric priors into 3D segmentation pipelines. GIBLy enhances existing architectures -- whether MLP-based, convolution-based, or transformer-based -- by providing features aligned with simple geometric shapes (and thus human-interpretable) that improve segmentation performance with minimal computational overhead. We validate our approach across multiple 3D semantic segmentation benchmarks, demonstrating consistent performance gains, including up to +11.5% mIoU on TS40K with PTV3, while adding only 58K extra parameters. Our results highlight the benefit of explicitly encoding geometric structure to support accurate and efficient 3D scene understanding, with a lightweight add-on layer

Low-dimensional embeddings are widely used as visual summaries of high-dimensional data and to enable downstream scientific discoveries. Yet, popular nonlinear dimension reduction methods, such as t-SNE and UMAP, are often selected based on visual appeal alone and without rigorous quantitative validation. A major reason is that manifold embeddings typically do not provide an out-of-sample map nor an inverse back to the original feature space; this makes held-out validation, the gold standard in supervised learning, all but impossible. To address these challenges, we develop a novel framework, MEDAL (Manifold Embedding Distillation via Autoencoder Learning), which distills a fitted manifold embedding into a reusable encoder--decoder model. MEDAL trains a constrained autoencoder whose bottleneck exactly matches any teacher embedding while the decoder reconstructs the original input; this yields an explicit map for new samples, an approximate inverse, and a pointwise reconstruction-based measure of distortion in the manifold space. This converts static manifold embeddings into models that can be evaluated on held-out data, enabling quantitative validation including comparing different dimension reduction methods as well as hyperparameter tuning. Across multiple benchmark and scientific case studies, we show that MEDAL enables held-out validation to determine optimal manifold embeddings and hyperparameters, reveals biologically coherent regions that are difficult to preserve in two dimensional embeddings, and detects distribution shift when new samples are mapped into a fixed reference manifold. MEDAL provides a general validation wrapper to any existing dimension reduction technique that will improve the rigor and

Context or prompt-level reweighting has emerged as a central algorithmic lever in Reinforcement Learning with Verified Rewards (RLVR) for improving the reasoning capability of large language models, yet the principle determining what constitutes an optimal weighting remains poorly understood. We address this gap by formulating prompt reweighting as a functional derivative of a utility functional defined in the pass-rate function space, yielding a unified optimality framework that accommodates existing schemes, including REINFORCE and GRPO. Building on this optimality framework, we propose a distribution-aware prompt reweighting approach, called CurveRL, based on a quantile coordinate transform, in which the weight assigned to each prompt depends not on the absolute value of pass rates but on its rank and density to reflect the distributional structure of the pass rates in the learning dynamics. Extensive experiments across multiple benchmarks demonstrate that our proposed CurveRL consistently outperforms GRPO and other RLVR baselines. Our study identifies context-distribution control as a principled axis for analyzing and designing prompt-reweighted RLVR algorithms. The code is released in https://github.com/zhyzmath/CurveRL.

Time series forecasting drives operational decisions in areas like finance, transportation, and energy. While supervised learning approaches achieve strong performance, they require domain-specific training, feature engineering, and ongoing maintenance. Large-scale foundation models have recently emerged as a zero-shot alternative, avoiding task-specific training much like LLMs. In this work, we evaluate foundation models against standard supervised approaches. Rather than focusing solely on aggregate accuracy, we analyze performance across four operational regimes: periodic human-centric systems, physically constrained processes, stochastic financial markets, and heterogeneous demand forecasting. Our results characterize optimal deployment areas. Foundation models perform well in domains with transferable periodic structures and are efficient for cold-start or long-tail scenarios. Conversely, supervised specialists maintain higher precision in systems governed by strict physical constraints. In financial domains, newer foundation models are rapidly closing the performance gap with supervised specialists. We further quantify trade-offs in inference latency, data drift adaptability, and deployment constraints. Finally, we propose a Complexity Router that assigns each series to the optimal model class using empirical features. We demonstrate that this selective routing achieves higher accuracy and significantly lower inference costs compared to deploying a universal foundation model, providing a practical framework for balancing generalization and efficiency.

We audit alignment-induced shifts in residual-stream activations of three open-weight instruction-tuned LLMs (Llama-3.1-8B-Instruct, Gemma-2-9B-it, Qwen-2.5-7B-Instruct) using the effective rank of the alignment modification matrix on safety-relevant inputs, rho_eps := rank_eps(M_Ds)/d, which formalizes the single-refusal-direction observation of Arditi et al. (2024) as a continuous quantity. The paper has three contributions. (1) Confound-controlled measurement: a four-variant decomposition (M_naive, M_template, M_aligned, M_DiD) separates chat-template formatting, alignment-stage shift, and the refusal-mediating direction, and recovers the Arditi refusal direction on M_DiD at |cos| in {0.77, 0.86, 0.50} (Llama/Gemma/Qwen); chat-template-controlled rho_eps is {0.0029, 0.0048, 0.0044}, and the centered SVD residual is 4-7x larger. (2) Constructive calibration on a 3-layer MLP across rho_eps in {0.008, 0.17, 0.33, 0.40} exhibits a sweet-spot vs. brittle distinction: mild rank-maximization (lambda=5) buys ablation robustness, while strong regularization at the same nominal rho_eps (lambda=50) does not. rho_eps is a diagnostic for fragility, not a target whose mechanical inflation buys robustness. (3) Limits of rank-based diagnostics: (a) not safety-specific (LRH baseline is 2-3x the safety value); (b) SVD principal ordering does not match causal ordering (Llama u_2 inert despite ranking second; cumulative ablation non-monotone at k=5); (c) the spectral-gap hypothesis required to upgrade the O(rho_eps * d) achievability bound to a matching Mirsky-route lower bound fails empirically (1/90 Llama layer-reference pairs, 0/36 MLP combinations) and structurally (kappa_lb <= 2/(eps * r)). The matching lower bound remains an open problem.

Low-light, long-exposure defocus deblurring remains a challenging problem due to the simultaneous presence of severe blur and complex biased noise. Existing methods typically rely on simplified noise assumptions, which limits their effectiveness under realistic imaging conditions. In this work, we propose Physen-Noise2Noise, a self-supervised deblurring framework guided by the physical model of defocus imaging, which leverages noisy multi-frame observations without requiring clean reference images. Unlike conventional Noise2Noise-based approaches that assume zero-mean noise, we derive a frequency-domain constraint inherent to the defocus imaging process and incorporate it into the learning framework via a learnable noise bias parameter. In addition, a multi-frame noisy initialization strategy is introduced to suppress complex biased noise prior to deblurring, providing a more stable starting point for reconstruction. This formulation explicitly models biased noise and enables joint bias correction and high-frequency detail recovery during training. Furthermore, we develop a pretrain-finetune variant to enhance robustness and generalization under challenging noise conditions. Extensive experiments on both simulation and real-world datasets demonstrate that the proposed method consistently outperforms state-of-the-art self-supervised approaches for defocus deblurring in the presence of complex biased noise.

We study inference-time alignment for diffusion-based generative models, aiming to steer a base model toward high-reward outputs without updating its weights. Recent Sequential Monte Carlo (SMC)-based steering methods approximate reward-tilted target distributions in a principled way, but their proposals remain largely tied to the base sampler. Since reward information is mainly used after propagation through particle reweighting and resampling, these methods can require large particle budgets and suffer from weight degeneracy and high-variance estimates. One way to reduce variance and improve particle efficiency is to iteratively learn twisting functions that provide look-ahead guidance, as in twisted SMC. However, existing learnable twisting methods are developed mainly for classical sequential inference and can be unstable when applied to diffusion-based alignment with high-dimensional state spaces and terminal, noisy, or black-box rewards. We propose Trust-Region Iterative Twisted Sequential Monte Carlo (TRI-TSMC), a trust-region framework for learning twisting functions in SMC-based inference-time alignment. Each iteration computes an exact KL-constrained update in path space, which admits a closed-form solution by tempered importance reweighting, and projects this target back to the parameterized twisted family by weighted maximum likelihood. Theoretically, we formalize the value-function interpretation of the optimal twisting function and show that it yields a zero-variance sampler. We prove that the trust-region update follows an escort path toward the target distribution, that the weighted maximum-likelihood update is a forward-KL projection, and that the path reduces residual importance-weight variance. Empirically, TRI-TSMC improves primary alignment objectives on discrete diffusion text generation and text-to-image generation under matched inference-time budgets.

World models for partially observed environments must imagine multiple compatible hidden futures and steer between them under counterfactual actions. Joint Embedding Predictive Architectures (JEPAs) do this in latent space, but a vector-valued latent has no internal structure for carrying the belief over hidden continuations through blind rollout. We introduce the Unitary World Model JEPA (UWM-JEPA), a JEPA world model with a density-matrix latent on a joint system-environment space and a learned unitary predictor. The construction preserves the joint-state spectrum exactly during rollout, so the predictor itself cannot dissipate the represented uncertainty. On a hidden-velocity indicator task requiring five-step forward simulation under a given action sequence with the target observation masked, UWM-JEPA reaches 0.77 accuracy and degrades monotonically as actions are perturbed; a parameter-matched LSTM-JEPA trained under the same counterfactual-target objective and action head collapses to majority-class accuracy (0.53) under every action condition. Under blind rollout, UWM-JEPA loses fewer than ten points of probe R^2 at short horizons while vector-latent baselines lose forty-one and sixty-eight; both nevertheless tie on a held-out context probe, locating the separation in the predictor rather than the encoder. Action sensitivity itself requires training against counterfactual rather than teacher-forced targets, a finding that applies beyond the unitary parameterisation. For JEPA world models to imagine under partial observability, latent geometry and predictor dynamics matter, not frozen context-encoding capacity alone.

Reconstructing PDE solutions from sparse observations is a core challenge in scientific computing. We present FM4PDE, a flow-matching generative framework that learns the joint distribution of PDE coefficients (or initial states) and solutions (or final states), enabling both forward simulation and inverse recovery with limited paired data. At inference, sampling is guided by a composite loss that enforces agreement with sparse measurements and reduces the PDE residual; we support deterministic, stochastic, and hybrid samplers. We provide error guarantees for these guided procedures. For the deterministic optimizer, a coercivity condition ensures trajectory boundedness and a phase-wise contraction yields logarithmic complexity in the target accuracy. For the stochastic sampler, we introduce adaptive guidance and assume dissipativity of the velocity field to obtain uniform moment bounds independent of the noise-floor parameter. This leads to polynomial-time error bounds, and a matching lower bound shows constant guidance induces an unavoidable positive bias, motivating adaptivity. A hybrid deterministic-stochastic analysis is also provided. Experiments on static and time-dependent benchmark PDEs demonstrate competitive accuracy and faster inference than diffusion-based generative models.

The ability of deep neural networks to learn hierarchical features is widely regarded as a key mechanism underlying their success in high-dimensional learning. Existing theory partially supports this view by establishing approximation rates based on parameter counts and sample complexity guarantees for compositional models without incurring the curse of dimensionality (CoD). To study overparameterized regimes, where the number of parameters exceeds the sample size, we develop a framework that measures complexity via the parameter norm. Within this approach, we establish approximation rates and excess risk bounds for learning sparse compositional functions whose compositional structure is represented by directed acyclic graphs (DAGs), using Frobenius norm-constrained deep neural networks. Our results have broad applicability since every function that is efficiently Turing computable admits sparse compositional representations. In particular, we cover a range of representative models, including multi-index models, binary tree structures, and general compositional architectures. The rates we derive show that deep networks can exploit the compositional structure of the target functions, effectively avoiding the CoD through hierarchical representations.