Modern LLM workflows move coordinate-indexed objects across checkpoints: steering vectors, sparse autoencoders, top-$k$ neuron sets, attribution lists, and merge alignments. This is only well posed after fixing the model's residual-stream gauge, which we show is architecture-dependent: LayerNorm residual charts have permutation gauge $S_d$ (up to a global sign flip), while RMSNorm charts with generic per-channel gain have signed-permutation gauge $B_d = S_d \ltimes \{\pm 1\}^d$. Permutation-only alignment is therefore symmetry-incomplete for RMSNorm models. We introduce sign-marginalized Hungarian matching and prove a sharp failure mode: with decorrelated coordinates, raw signed-correlation matching has a structural permutation-accuracy ceiling at the positive-sign fraction of the true gauge, which sign-marginalization removes. We then make coordinate-preserving transport, not function-level merging, the primary object: composing saved-checkpoint local $B_d$ gauges along same-base fine-tuning trajectories recovers 91.1% of cross-run coordinates at 1500 steps versus 60.3% for endpoint matching, and the gain is not explained by merely routing through the base. The recovered gauge transfers tools that permutation-only alignment breaks: TinyLlama SAE reconstruction has NMSE 0.004 under $B_d$ versus 1.08 under $S_d$; Qwen sentiment steering preserves 95.8% of its effect versus 17.2%; refusal steering reverses sign under $S_d$; coordinate-preserving merges behave the same way. The same covariance governs stateful training: signed transport of AdamW state preserves the resumed trajectory, while permutation-only state follows a different one from a functionally identical checkpoint. Finally, gauge-sweep audits show index-level interpretability claims are reproducible only relative to an explicit gauge.

A deep network's loss is invariant to continuous symmetries of its parameters: the logit shift, the ReLU rescaling, the LayerNorm scale, the per-head attention rotation. Adam's per-coordinate preconditioner drifts along each symmetry orbit, which pulls the trajectory off the symmetry quotient where the optimization lives and blurs the singular-learning rate the quotient makes readable. We build DDC, a Dead-Direction Conditioner that lifts a base optimizer into a $G$-equivariant one: it conditions the optimizer's state in the orbit decomposition of a $G$-invariant metric, so the trajectory stays a preconditioned gradient flow on the quotient $\barΘ= Θ/G$. The construction carries four architectural gauges (cross-entropy shift, ReLU and SwiGLU rescaling, LayerNorm and RMSNorm scale, and a per-head $O(d_{\rm head})$ attention rotation matched to RoPE), proves exactly equivariant on an Adam base, and composes with a Muon base through a gauge-equivariant orthogonaliser. Respecting the symmetry changes both the minimum the optimizer reaches and what it leaves measurable there. On a language model trained past the point of fit, DDCAdam resists the over-training collapse AdamW falls into, holding a validation-train loss gap of 0.67 against 5.88, and reads the dead-direction rate in 32 of 65 layer-by-observable cells where AdamW reads it in 7. A vision transformer trained from scratch reaches lower validation loss (1.71 against 2.12) while compressing spare feed-forward capacity a matched AdamW leaves intact. On a Muon base, where the rotation gauge composes exactly, DDCMuon groks ten of eleven seeds at depth 24 that a plain Muon never reaches. Built into the optimizer, a network's gauge symmetry sharpens the minimum it finds and turns that minimum's geometry into something the trajectory can measure.

High dimensional representation drift is commonly quantified using Euclidean or cosine distances, which presuppose fixed coordinates when comparing representations across time, training or preprocessing stages. While effective in many settings, these measures entangle intrinsic changes in the data with variations induced by arbitrary parametrizations. We introduce a projective geometric view of representation drift grounded in the Fubini Study metric, which identifies representations that differ only by gauge transformations such as global rescalings or sign flips. Applying this framework to empirical high dimensional datasets, we explicitly construct representation trajectories and track their evolution through cumulative geometric drift. Comparing Euclidean, cosine and Fubini Study distances along these trajectories reveals that conventional metrics systematically overestimate change whenever representations carry genuine projective ambiguity. By contrast, the Fubini Study metric isolates intrinsic evolution by remaining invariant under gauge-induced fluctuations. We further show that the difference between cosine and Fubini Study drift defines a computable, monotone quantity that directly captures representation churn attributable to gauge freedom. This separation provides a diagnostic for distinguishing meaningful structural evolution from parametrization artifacts, without introducing model-specific assumptions. Overall, we establish a geometric criterion for assessing representation stability in high-dimensional systems and clarify the limits of angular distances. Embedding representation dynamics in projective space connects data analysis with established geometric programs and yields observables that are directly testable in empirical workflows.

Autonomous inspection in hazardous environments requires AI agents that can interpret high-level goals and execute precise control. A key capability for such agents is spatial grounding, for example when a drone must center a detected object in its camera view to enable reliable inspection. While large language models provide a natural interface for specifying goals, using them directly for visual control achieves only 58\% success in this task. We envision that equipping agents with a world model as a tool would allow them to roll out candidate actions and perform better in spatially grounded settings, but conventional world models are data and compute intensive. To address this, we propose a task-specific latent dynamics model that learns state-specific action-induced shifts in a shared latent space using only goal-state supervision. The model leverages global action embeddings and complementary training losses to stabilize learning. In experiments, our approach achieves 71\% success and generalizes to unseen images and instructions, highlighting the potential of compact, domain-specific latent dynamics models for spatial alignment in autonomous inspection.

Both provide lower bounds for the partition function by utilizing the so-called gauge transformation which modifies factors of GM while keeping the partition function invariant. Moreover, we prove that both G-MF and G-BP are exact for GMs with a single loop of a special structure, even though the bare MF and BP perform badly in this case.

Mappings from biological sequences (DNA, RNA, protein) to quantitative measures of sequence functionality play an important role in contemporary biology. We are interested in the related tasks of (i) inferring predictive sequence-to-function maps and (ii) decomposing sequence-function maps to elucidate the contributions of individual subsequences. Because each sequence-function map can be written as a weighted sum over subsequences in multiple ways, meaningfully interpreting these weights requires "gauge-fixing," i.e., defining a unique representation for each map. Recent work has established that most existing gauge-fixed representations arise as the unique solutions to $L_2$-regularized regression in an overparameterized "weight space" where the choice of regularizer defines the gauge. Here, we establish the relationship between regularized regression in overparameterized weight space and Gaussian process approaches that operate in "function space," i.e. the space of all real-valued functions on a finite set of sequences. We disentangle how weight space regularizers both impose an implicit prior on the learned function and restrict the optimal weights to a particular gauge. We also show how to construct regularizers that correspond to arbitrary explicit Gaussian process priors combined with a wide variety of gauges. Next, we derive the distribution of gauge-fixed weights implied by the Gaussian process posterior and demonstrate that even for long sequences this distribution can be efficiently computed for product-kernel priors using a kernel trick. Finally, we characterize the implicit function space priors associated with the most common weight space regularizers. Overall, our framework unifies and extends our ability to infer and interpret sequence-function relationships.

Maximum-entropy methods, rooted in the inverse Ising/Potts problem from statistical mechanics, have become indispensable tools for modeling pairwise interactions in disciplines such as bioinformatics, ecology, and neuroscience. Despite their remarkable success, these methods often overlook high-order interactions that may be crucial in complex systems. Conversely, while modern machine learning approaches can capture such interactions, existing interpretable frameworks are computationally expensive, making it impractical to assess the relevance of high-order interactions in real-world scenarios. Restricted Boltzmann Machines (RBMs) offer a computationally efficient alternative by encoding statistical correlations via hidden nodes in a bipartite neural network. Here, we present a method that maps RBMs exactly onto generalized Potts models with interactions of arbitrary high order. This approach leverages large-$N$ approximations, facilitated by the simple architecture of the RBM, to enable the efficient extraction of effective many-body couplings with minimal computational cost. This mapping also enables the development of a general formal framework for the extraction of effective higher-order interactions in arbitrarily complex probabilistic models. Additionally, we introduce a robust formalism for gauge fixing within the generalized Potts model. We validate our method by accurately recovering two- and three-body interactions from synthetic datasets. Additionally, applying our framework to protein sequence data demonstrates its effectiveness in reconstructing protein contact maps, achieving performance comparable to state-of-the-art inverse Potts models. These results position RBMs as a powerful and efficient tool for investigating high-order interactions in complex systems.

Several recent works seek to develop foundation models specifically for medical applications, adapting general-purpose large language models (LLMs) and vision-language models (VLMs) via continued pretraining on publicly available biomedical corpora. These works typically claim that such domain-adaptive pretraining (DAPT) improves performance on downstream medical tasks, such as answering medical licensing exam questions. In this paper, we compare seven public "medical" LLMs and two VLMs against their corresponding base models, arriving at a different conclusion: all medical VLMs and nearly all medical LLMs fail to consistently improve over their base models in the zero-/few-shot prompting regime for medical question-answering (QA) tasks. For instance, across the tasks and model pairs we consider in the 3-shot setting, medical LLMs only outperform their base models in 12.1% of cases, reach a (statistical) tie in 49.8% of cases, and are significantly worse than their base models in the remaining 38.2% of cases. Our conclusions are based on (i) comparing each medical model head-to-head, directly against the corresponding base model; (ii) optimizing the prompts for each model separately; and (iii) accounting for statistical uncertainty in comparisons. While these basic practices are not consistently adopted in the literature, our ablations show that they substantially impact conclusions. Our findings suggest that state-of-the-art general-domain models may already exhibit strong medical knowledge and reasoning capabilities, and offer recommendations to strengthen the conclusions of future studies.

The safety of train operations is largely dependent on the health of rail tracks, necessitating regular and meticulous inspection and maintenance. A significant part of such inspections involves geometric measurements of the tracks to detect any potential problems. Traditional methods for track geometry measurements, while proven to be accurate, require track closures during inspections, and consume a considerable amount of time as the inspection area grows, causing significant disruptions to regular operations. To address this challenge, this paper proposes a track geometry measurement system (TGMS) that utilizes an unmanned aerial vehicle (UAV) platform equipped with a light detection and ranging (LiDAR) sensor. Integrated with a state-of-the-art machine-learning-based computer vision algorithm, and a simultaneous localization and mapping (SLAM) algorithm, this platform can conduct track geometry inspections seamlessly over a larger area without interrupting rail operations. In particular, this semi-or fully automated measurement is found capable of measuring critical track geometry irregularities in gauge, curvature, and profile with subinch accuracy. Cross-level and warp are not measured due to the absence of gravity data. By eliminating operational interruptions, our system offers a more streamlined, cost-effective, and safer solution for inspecting and maintaining rail infrastructure.

Gauge fixing is applied in several contexts within lattice field theory calculations, for example to give meaning to gauge-variant observables used in RI-MOM renormalization schemes [1], as a computational trick to replace gauge-invariant operators with cheaper gauge-variant operators [2], or as inputs for comparison to phenomenological models [3, 4]. Recently, gauge-variant operators have also been used for contour deformations to reduce statistical noise [5-7]. In these contexts, the choice of gauge-fixing scheme can affect the efficiency of the calculation, and it may be desirable to systematically explore options for the scheme. Two kinds of gauge-fixing schemes are commonly used: gauge fixing by functional minimization (e.g. Landau and Coulomb gauge) or gauge fixing a maximal tree of links to the identity. Our work makes several contributions in this context. First, we parameterize a continuous family of gauge-fixing schemes that include the former as special cases. Second, we derive the gradients with respect to these parameters of an arbitrary loss function computed from gauge-fixed configurations, which can be used for gradient-based optimization within the family. Finally, we discuss the restriction of this method to a subfamily consisting of maximal trees alone, addressing the discrete nature of this space by introducing a temperature regulator, and demonstrate the effectiveness of this approach in solving two regression problems.