Science & Mathematics

We study the gradient flow training dynamics of two-layer multi-layer perceptrons (MLPs) on anisotropic Gaussian data. Even in this simple setting, solving the dynamics in full is notoriously difficult, so we try something different. We propose a "feature ODE" that might be intuitively expected to describe the learning dynamics of a shallow MLP. We find that MLPs trained to learn simple target functions (low-order Hermite polynomials and staircase functions) actually roughly follow the loss trajectories from this feature ODE. This agreement is a surprise because of the size of the simplification and the fact that it does not come from a rigorous chain of approximations. We discuss this surprise and speculate on its reasons and uses.

We study Euclidean embeddings of higher-order line metrics arising from hypergraphs. For a fixed threshold \(s\geq 1\), the \(s\)-line graph \(L_s(H)\) has one vertex for each hyperedge of \(H\), with two hyperedges adjacent when they intersect in at least \(s\) vertices; its shortest-path metric is the native \(s\)-line metric. In the unrestricted setting, every finite graph occurs as the \(s\)-line graph of some hypergraph, so the worst-case Euclidean distortion is \(Θ(\log n)\). We show that \(α\)-acyclicity yields a sharp improvement to \(Θ(\sqrt{\log\log n})\). Although the canonical join tree of an \(α\)-acyclic hypergraph may have arbitrarily large stretch relative to the native metric, we prove that for every \(s\), the \(s\)-line graphs of \(α\)-acyclic hypergraphs are exactly the dually chordal graphs. Using their constant-factor spanning-tree approximations together with classical tree-embedding bounds, we obtain an \(O(\sqrt{\log\log n})\) Euclidean embedding bound. We then show this is asymptotically tight by realizing every finite tree as the \(s\)-line graph of an \(α\)-acyclic hypergraph. Thus, the worst-case distortion is \(Θ(\sqrt{\log\log n})\), including for hypergraphs of rank at most \(3s+1\).

Public leaderboards rank large language models (LLMs) in general, but an engineer has to choose a model for a specific application, architecture and cost. This paper presents a method for application-specific capability profiling of LLMs and its realization as a containerized testbed. The method treats each application as a unit of extension: an Oracle-to-PostgreSQL migration agent, a documentation-grounded chatbot or an enterprise-support automation agent runs in its own container behind a small contract, brings its own dataset, evaluator and owned attributes, and is onboarded together with its dashboards. Measurements accumulate in a versioned, sparse metamodel of every model. The method separates the model under test from judges and agents, freezes and hashes plans, refuses to compare values from different contexts, diagnoses the instrument before the system, and applies a null-agent check and a noise rule whose operating characteristics are quantified by simulation. In the evaluated automation conditions, model choices produced larger differences on selected observable outcomes than framework choices, although the experimental design does not establish a general causal ranking of these factors. The experiments also show that instruction tuning is deployment-specific and that system-reported success can exceed independently observed success: of 216 migration units checked against a real Oracle database, 108 were demonstrably correct. These measurements establish application-specific observations, not a universal model ranking.

Video generation models produce visually compelling motion but systematically fail on fluid phenomena, including smoke, fire, and explosion debris, where outputs lack coherent vortical structure and collapse to bland dynamics that violate the physics of real fluids. Classical solvers achieve high physical fidelity but require detailed 3D scenes and carefully tuned parameters, making them impractical for in-the-wild imagery and video generation. We address this gap by injecting a data-driven fluid prior into a pre-trained video model, trained on a mixture of high-fidelity Houdini simulation data with paired velocity-field ground truth and curated in-the-wild fluid video data for appearance diversity. The core of our approach is a dual stream architecture that predicts per-frame velocity fields alongside RGB, supervised by simulation ground truth. This auxiliary signal regularizes generation toward physically consistent motion and exposes an interpretable physical quantity that we further leverage as a reward for velocity-aware direct preference optimization. Our model produces substantially more realistic fluid dynamics sequences than strong video baselines and optimization-based methods. Project page: https://y-zheng18.github.io/FluidGen/

The rapid expansion and specialization of scientific literature make it increasingly difficult to identify not only what is known, but where new and still supportable knowledge may emerge. We introduce Knowledge Manifold Optimization (KMO), a discovery-oriented extension of the Knowledge Manifold framework that treats scientific discovery as a multiobjective search over a continuous, literature-derived knowledge space. Candidate locations are evaluated using four complementary objectives: persistent knowledge-regime boundary strength, band-pass white-space novelty, semanticgradient magnitude, and evidential support. Rather than collapsing these criteria into a single weighted score, KMO identifies Pareto-optimal Supported Knowledge Gaps, defined as corpus-relative unoccupied regions that remain sufficiently supported by surrounding literature. The method is demonstrated on a 99-paper corpus in aerodynamics and aerodynamic design. On a 21 x 21 grid, exact Pareto analysis produced 19 non-dominated points, of which 16 formed the practical front after excluding points with zero white-space score. These solutions exhibited distinct discovery modes, including methodology bridges, aero-structural topology/FSI integration, functional aerodynamic-device integration, and rear-wake/vehicle flow-control integration. A distance-maximization baseline preferentially selected extrapolative regions with substantially lower support and reliability, demonstrating why emptiness alone is insufficient. Representative Pareto locations were converted into auditable evidence packages and then verbalized by a constrained large language model into candidate research hypotheses. The results show that scientific discovery can be formulated as structured multi-objective search in an explicit knowledge space, while keeping novelty corpus-relative and separating numerical discovery from downstream language generation.

Source labels tell us where a sample came from, but not how that source oriented its coordinates. In a sparse Gaussian multi-source model with source-specific sign gauges, that distinction changes the learning problem. Without calibration, pooling across sources erases the mean signal, and the source-labelled experiment reduces to sparse PCA. With task-independent relative-gauge metadata, synchronization aligns sources and the same observations become a sparse mean problem. We formalize the value of metadata through AlignedMass, which quantifies how much first-order signal a metadata-induced alignment restores. Source labels create only limited AlignedMass, whereas valid calibration can create AlignedMass proportional to the full sample size. This yields a statistical separation in micro-source regimes and, under standard sparse-PCA hardness assumptions, a computational separation as well. Synthetic experiments check the predicted scaling; controlled real-feature and physical sensor studies examine sign cancellation, separately acquired calibration, and reuse across tasks.

Diffusion and flow matching models achieve impressive visual generation quality but remain computationally expensive due to iterative sampling. Caching-based acceleration provides a practical training-free solution by reusing or predicting intermediate features. However, existing methods largely enforce a forward-only information flow: approximations produced for skipped model evaluations are immediately committed to the sampling trajectory and cannot benefit from feedback generated by later full-computation steps. We propose \textbf{PoSTCache} (\emph{Postdictive Smoothing Trajectory Cache}), a training-free and plug-and-play framework that reformulates diffusion caching as revisable trajectory inference. PoSTCache first uses an existing caching strategy to construct forward proposals for skipped steps. At a later full-computation step, the discrepancy between the refreshed prediction and its forward proposal yields a correction signal, which PoSTCache propagates backward to refine and reintegrate the preceding predictions. Viewing full-computation steps as sources of corrective feedback also changes how they should be selected. We therefore introduce the Posterior Variance Reduction (PVR) strategy, which estimates cross-timestep dependencies from offline denoising trajectories and selects anchors according to their expected reduction in surrogate trajectory uncertainty rather than local approximation error alone. Experiments on representative image and video generation models show that PoSTCache consistently improves the quality--efficiency trade-off over representative caching methods.

Robotic simulation, navigation, and inspection need photorealistic and geometrically accurate maps. Radiance fields achieve photorealism, but building them from robot data raises three challenges. A robot's trajectory serves its task, not reconstruction, so images from few viewpoints leave the geometry under-constrained. Residual errors in camera poses and intrinsics blur the map. The map's GPU memory grows with scene size, which limits its level of detail in large scenes. We present \textbf{SpaRV}, a \textbf{Spa}rse \textbf{R}adiance \textbf{V}oxel map rendered by differentiable ray marching, which lets irregular LiDAR scans supervise it alongside images. LiDAR depth and occupancy anchor the under-constrained geometry, and normal and smoothness regularizers keep it smooth between LiDAR points. To correct camera errors, SpaRV alternates map training with photometric optimization of the camera poses and focal lengths. To bound GPU memory in training and rendering, SpaRV holds one submap at a time at full resolution and the rest at coarser octree levels. Experiments show that SpaRV keeps the geometry consistent away from the captured trajectory and sharpens textures by optimizing the camera parameters. On one 24\,GB GPU, SpaRV reconstructs a university campus captured in 63{,}582 images and simulates other cameras and LiDARs. Our code will be released at \url{https://github.com/hku-mars/SpaRV}.

Streaming video world models generate the future chunk by chunk, conditioned on their own past and a stream of actions. Trained on short clips, they drift once a rollout exceeds the training horizon, as prediction errors re-enter the context and compound. A common remedy is to train the model on its own rollouts via distribution matching distillation (DMD). For action-conditioned world models, the teacher becomes a bottleneck: an external video model is agnostic to the actions, whereas the model's own pretrained network drifts in the same manner as the student. We observe that a long rollout need not match a teacher, provided it remains a realistic continuation of its own scene that follows the commanded actions. Realism and scene fidelity can be measured against real-data statistics, which remain valid at any horizon, while action following is preserved by the model's conditional generation loss. We propose Fréchet Forcing, which fine-tunes a world model together with its deployed multi-step sampler under three constraints. Generative anchoring applies the flow-matching loss after a self-generated history ending on real data. Context fidelity, a Fréchet distance (FD) on latent statistics of the free rollout centred on each clip's own ground truth, keeps every rollout faithful to its scene. Reality matching, an FD to real frames in a frozen perceptual space, keeps the generated frames realistic. Because all three targets are derived from short real clips without comparing a rollout against a long recorded future, training requires no teacher, no online critic and no long videos. Fine-tuning MiniWorld-1B, pretrained on 253-frame clips, for at most 36 GPU-hours on 381-frame clips yields rollouts that remain stable beyond 4,000 frames and continue to follow their actions in both robot manipulation and camera-controlled exploration.

This work focuses on reconstructing a continuous implied volatility (IV) surface from a small number of option quotes, a fundamental task for option pricing and risk management. We propose Additive Martingale (AddMart), a few-shot IV surface reconstruction framework that satisfies static no-arbitrage conditions by construction. Rather than predicting IVs directly, AddMart maps sparse quotes to non-negative increment densities of an exponential additive martingale, from which valid risk-neutral marginals and IVs are recovered. By incorporating the no-arbitrage structure directly into its architecture, rather than enforcing it through regularization in the learning objective, AddMart produces IV surfaces that satisfy static no-arbitrage conditions throughout the domain. Extensive experiments on ten real-option datasets demonstrate that AddMart achieves competitive reconstruction accuracy with no material static arbitrage violations, while directly enabling the estimation of VaR, CVaR, and crash probabilities.

Reliable microbial taxonomic inference is challenged by the rapid growth of newly sequenced organisms and incomplete reference coverage. While alignment- and database-driven tools remain essential for reference-backed identification, they are not designed for learned inference over species absent from genome-level training. We introduce HyperTaxa, a taxonomy-aligned representation learning framework for open-set microbial taxonomic inference. HyperTaxa encodes whole genomes as multiscale hyperspectral k-mer transition fields and aligns these representations to a taxonomy-derived semantic space using a two-stage Node2Vec-to-MDS procedure that combines local graph structure with global hierarchical distance preservation. The resulting space supports zero-shot inference over species excluded from genome-level training but represented in the taxonomy, generalized zero-shot inference over seen and unseen species, and cascaded genus-to-species prediction. Across benchmark and large-scale disjoint-species evaluations, HyperTaxa improves taxonomic prediction over pseudo-image and transformer-graph baselines, with the strongest gains at higher taxonomic ranks. Ablations further show that multiscale transition encoding and taxonomy-aware alignment are key contributors to generalization. These results position HyperTaxa as a complementary taxonomy-aware representation and candidate-generation framework for microbial identification under incomplete reference coverage and open-set uncertainty.

Post-Training Quantization (PTQ) enables efficient deployment of Large Language Models (LLMs), yet activation outliers remain a key challenge for low-bit quantization. Existing global transformation methods overlook heterogeneous channel sensitivity to quantization noise, leading to suboptimal accuracy-efficiency trade-offs. To address this, we propose \textbf{Locally Isolated Rotation Quantization (LIRQuant)}, a sensitivity-aware framework that selectively applies transformations. LIRQuant identifies sensitive channels using a noise-weighted sensitivity metric, applying expressive learnable rotations only to these channels and lightweight fixed rotations to the remaining ones. This decoupled design reduces optimization complexity without sacrificing outlier smoothing capability. Extensive experiments across the LLaMA-2/3, Mistral, Qwen3, and Mixtral families demonstrate that LIRQuant achieves state-of-the-art accuracy under the challenging W4A4KV4 setting. Meanwhile, LIRQuant calibrates LLaMA-3-8B in only 25 minutes with 20.3 GB peak GPU memory, achieving the fastest calibration and the lowest memory footprint among recent learnable-rotation methods, and its fused inference kernels deliver up to $2.3\times$ prefill speedup and $3.5\times$ memory saving over FP16.

Accurate evaluation of LLM personalization is hindered by current evaluation paradigms that over- or under-constrain the response space, yielding frequent false rejections and false acceptances. To reduce these false rates, we introduce ConQual-Audit, an audit-based framework that evaluates LLM personalization through a three-stage process: (1) extracting semantic constraints from user profiles and task-grounded reference; (2) qualifying constraints by pruning unnecessary constraints and recovering missing constraints; and (3) verifying candidate responses against the audited constraints via natural language inference. Across PrefEval and LaMP, ConQual-Audit outperforms nine baselines, attaining top AUCs (0.88 and 0.90) and the lowest false rate (0.21) on the leanest constraint sets at an amortizable audit cost. Crucially, this constraint audit is portable: integrating it into ExPerT and PREF helps revise smaller, higher-quality constraint sets to reduce their evaluation false rates.

Many scientific advances come from cross-pollination, carrying an idea from one field over to a problem in another. Yet retrieving papers across fields remains underexplored, with existing benchmarks confined to a narrow range of fields. We study cross-pollination retrieval, the problem of identifying transferable ideas across research fields, and introduce CPR-Dataset, a large-scale dataset constructed by mining cross-topic citations. Its evaluation split CPR-eval has 1,300 queries in 26 fields, and its training split CPR-train has 10,573 pairs. On CPR-eval, the best of eleven existing retrievers reaches a Recall@10 of only 0.26, whereas an LLM reranker reaches 0.81. Standard contrastive fine-tuning on CPR-train raises Recall@10 to as high as 0.70. The gain carries over to domains held out from training and to analogy and reasoning-intensive retrieval. On AnaloBench, it raises nDCG@10 by up to 33% over the untrained retriever, whereas training on three existing retrieval datasets lowers it. On BRIGHT, it raises nDCG@10 by up to 70%, more than twice the gain from those datasets. CPR-Dataset is available at https://huggingface.co/CPR-Dataset, and the code at https://github.com/SeokwonJung-Jay/Cross-Pollination-Retrieval.

Layer-level pruning is attractive for large language model (LLM) deployment as it reduces the model size while preserving a dense architecture that can be efficiently executed on standard hardware. Current methods typically follow a one-shot paradigm, which first removes selected layers to reach a target model size and then performs additional training to recover model capabilities. However, pruning directly to the target size can substantially degrade reasoning capabilities, making subsequent recovery more difficult under a limited training budget. Moreover, this paradigm separates layer selection from recovery, limiting the use of recovery information to guide selection and potentially hindering the recovery of reasoning capabilities. To address these limitations, we propose LOOP, a Layer-level Recovery-Oriented On-Policy Pruning framework that interleaves pruning and recovery across multiple shots. In each shot, LOOP removes a single layer and performs short on-policy distillation, mitigating the abrupt degradation in reasoning capabilities associated with one-shot pruning. To guide layer selection in subsequent shots, LOOP reuses on-policy trajectories and optimization signals from recovery to jointly assess each remaining layer's reasoning contribution and recovery sensitivity, thereby making layer selection recovery-aware. Evaluations on several benchmarks show that LOOP consistently outperforms one-shot baselines in average accuracy across different model scales and pruning ratios.