Daily Papers

The optimization of large language models (LLMs) remains a critical challenge, particularly as model scaling exacerbates sensitivity to algorithmic imprecision and training instability. Recent advances in optimizers have improved convergence efficiency through momentum orthogonalization, but suffer from two key robustness limitations: dimensional fragility in orthogonalization precision and vulnerability to outlier-induced noise. To address these robustness challenges, we introduce ROOT, a Robust Orthogonalized Optimizer that enhances training stability through dual robustness mechanisms. First, we develop a dimension-robust orthogonalization scheme using adaptive Newton iterations with fine-grained coefficients tailored to specific matrix sizes, ensuring consistent precision across diverse architectural configurations. Second, we introduce an optimization-robust framework via proximal optimization that suppresses outlier noise while preserving meaningful gradient directions. Extensive experiments demonstrate that ROOT achieves significantly improved robustness, with faster convergence and superior final performance compared to both Muon and Adam-based optimizers, particularly in noisy and non-convex scenarios. Our work establishes a new paradigm for developing robust and precise optimizers capable of handling the complexities of modern large-scale model training. The code will be available at https://github.com/huawei-noah/noah-research/tree/master/ROOT.

Reinforcement learning is structurally harder than supervised learning because the policy changes the data distribution it learns from. The resulting fragility is especially visible in large-model training, where the training and rollout systems differ in numerical precision, sampling, and other implementation details. Existing methods manage this fragility by adding hyper-parameters to the training objective, which makes the algorithm more sensitive to its configuration and requires retuning whenever the task, model scale, or distribution mismatch changes. This fragility traces to two concerns that current objectives entangle through hyper-parameters set before training begins: a trust-region concern, that updates should not move the policy too far from its current value, and an off-policy concern, that data from older or different behavior policies should influence the update only to the extent that it remains reliable. Neither concern is a constant to set in advance, and their severity is reflected in the policy-ratio distribution of the current batch. We present a simple yet effective batch-adaptive objective that replaces fixed clipping with the normalized effective sample size of the policy ratios. The same statistic caps the score-function weight and sets the strength of an off-policy regularizer, so the update stays close to the usual on-policy score-function update when ratios are nearly uniform, and tightens automatically when stale or mismatched data cause ratio concentration, while retaining a nonzero learning signal on high-ratio tokens. Experiments across a wide range of settings show that our method matches or exceeds tuned baselines, introducing no new objective hyper-parameters and removing several existing ones. The code is available at https://github.com/FeynRL-project/FeynRL.

Representations learned via self-supervised learning (SSL) can be susceptible to dimensional collapse, where the learned representation subspace is of extremely low dimensionality and thus fails to represent the full data distribution and modalities. Dimensional collapse also known as the "underfilling" phenomenon is one of the major causes of degraded performance on downstream tasks. Previous work has investigated the dimensional collapse problem of SSL at a global level. In this paper, we demonstrate that representations can span over high dimensional space globally, but collapse locally. To address this, we propose a method called local dimensionality regularization (LDReg). Our formulation is based on the derivation of the Fisher-Rao metric to compare and optimize local distance distributions at an asymptotically small radius for each data point. By increasing the local intrinsic dimensionality, we demonstrate through a range of experiments that LDReg improves the representation quality of SSL. The results also show that LDReg can regularize dimensionality at both local and global levels.

We investigate the geometric structure of learning dynamics in overparameterized transformer models through carefully controlled modular arithmetic tasks. Our primary finding is that despite operating in high-dimensional parameter spaces (d=128), transformer training trajectories rapidly collapse onto low-dimensional execution manifolds of dimension 3--4. This dimensional collapse is robust across random seeds and moderate task difficulties, though the orientation of the manifold in parameter space varies between runs. We demonstrate that this geometric structure underlies several empirically observed phenomena: (1) sharp attention concentration emerges as saturation along routing coordinates within the execution manifold, (2) SGD commutators are preferentially aligned with the execution subspace (up to 10times random baseline) early in training, with >92% of non-commutativity confined to orthogonal staging directions and this alignment decreasing as training converges, and (3) sparse autoencoders capture auxiliary routing structure but fail to isolate execution itself, which remains distributed across the low-dimensional manifold. Our results suggest a unifying geometric framework for understanding transformer learning, where the vast majority of parameters serve to absorb optimization interference while core computation occurs in a dramatically reduced subspace. These findings have implications for interpretability, training curriculum design, and understanding the role of overparameterization in neural network learning.

Tabular machine learning presents a paradox: modern models achieve state-of-the-art performance using high-dimensional (high-D), collinear, error-prone data, defying the "Garbage In, Garbage Out" mantra. To help resolve this, we synthesize principles from Information Theory, Latent Factor Models, and Psychometrics, clarifying that predictive robustness arises not solely from data cleanliness, but from the synergy between data architecture and model capacity. Partitioning predictor-space "noise" into "Predictor Error" and "Structural Uncertainty" (informational deficits from stochastic generative mappings), we prove that leveraging high-D sets of error-prone predictors asymptotically overcomes both types of noise, whereas cleaning a low-D set is fundamentally bounded by Structural Uncertainty. We demonstrate why "Informative Collinearity" (dependencies from shared latent causes) enhances reliability and convergence efficiency, and explain why increased dimensionality reduces the latent inference burden, enabling feasibility with finite samples. To address practical constraints, we propose "Proactive Data-Centric AI" to identify predictors that enable robustness efficiently. We also derive boundaries for Systematic Error Regimes and show why models that absorb "rogue" dependencies can mitigate assumption violations. Linking latent architecture to Benign Overfitting, we offer a first step towards a unified view of robustness to Outcome Error and predictor-space noise, while also delineating when traditional DCAI's focus on label cleaning remains powerful. By redefining data quality from item-level perfection to portfolio-level architecture, we provide a theoretical rationale for "Local Factories" -- learning from live, uncurated enterprise "data swamps" -- supporting a deployment paradigm shift from "Model Transfer" to "Methodology Transfer'' to overcome static generalizability limitations.

Neural multivariate regression underpins a wide range of domains such as control, robotics, and finance, yet the geometry of its learned representations remains poorly characterized. While neural collapse has been shown to benefit generalization in classification, we find that analogous collapse in regression consistently degrades performance. To explain this contrast, we analyze models through the lens of intrinsic dimension. Across control tasks and synthetic datasets, we estimate the intrinsic dimension of last-layer features (ID_H) and compare it with that of the regression targets (ID_Y). Collapsed models exhibit ID_H < ID_Y, leading to over-compression and poor generalization, whereas non-collapsed models typically maintain ID_H > ID_Y. For the non-collapsed models, performance with respect to ID_H depends on the data quantity and noise levels. From these observations, we identify two regimes (over-compressed and under-compressed) that determine when expanding or reducing feature dimensionality improves performance. Our results provide new geometric insights into neural regression and suggest practical strategies for enhancing generalization.

Safety performance evaluation is critical for developing and deploying connected and automated vehicles (CAVs). One prevailing way is to design testing scenarios using prior knowledge of CAVs, test CAVs in these scenarios, and then evaluate their safety performances. However, significant differences between CAVs and prior knowledge could severely reduce the evaluation efficiency. Towards addressing this issue, most existing studies focus on the adaptive design of testing scenarios during the CAV testing process, but so far they cannot be applied to high-dimensional scenarios. In this paper, we focus on the adaptive safety performance evaluation by leveraging the testing results, after the CAV testing process. It can significantly improve the evaluation efficiency and be applied to high-dimensional scenarios. Specifically, instead of directly evaluating the unknown quantity (e.g., crash rates) of CAV safety performances, we evaluate the differences between the unknown quantity and known quantity (i.e., control variates). By leveraging the testing results, the control variates could be well designed and optimized such that the differences are close to zero, so the evaluation variance could be dramatically reduced for different CAVs. To handle the high-dimensional scenarios, we propose the sparse control variates method, where the control variates are designed only for the sparse and critical variables of scenarios. According to the number of critical variables in each scenario, the control variates are stratified into strata and optimized within each stratum using multiple linear regression techniques. We justify the proposed method's effectiveness by rigorous theoretical analysis and empirical study of high-dimensional overtaking scenarios.

Making statements about the performance of trained models on tasks involving new data is one of the primary goals of machine learning, i.e., to understand the generalization power of a model. Various capacity measures try to capture this ability, but usually fall short in explaining important characteristics of models that we observe in practice. In this study, we propose the local effective dimension as a capacity measure which seems to correlate well with generalization error on standard data sets. Importantly, we prove that the local effective dimension bounds the generalization error and discuss the aptness of this capacity measure for machine learning models.

A ubiquitous feature of data of our era is their extra-large sizes and dimensions. Analyzing such high-dimensional data poses significant challenges, since the feature dimension is often much larger than the sample size. This thesis introduces robust and computationally efficient methods to address several common challenges associated with high-dimensional data. In my first manuscript, I propose a coherent approach to variable screening that accommodates nonlinear associations. I develop a novel variable screening method that transcends traditional linear assumptions by leveraging mutual information, with an intended application in neuroimaging data. This approach allows for accurate identification of important variables by capturing nonlinear as well as linear relationships between the outcome and covariates. Building on this foundation, I develop new optimization methods for sparse estimation using nonconvex penalties in my second manuscript. These methods address notable challenges in current statistical computing practices, facilitating computationally efficient and robust analyses of complex datasets. The proposed method can be applied to a general class of optimization problems. In my third manuscript, I contribute to robust modeling of high-dimensional correlated observations by developing a mixed-effects model based on Tsallis power-law entropy maximization and discussed the theoretical properties of such distribution. This model surpasses the constraints of conventional Gaussian models by accommodating a broader class of distributions with enhanced robustness to outliers. Additionally, I develop a proximal nonlinear conjugate gradient algorithm that accelerates convergence while maintaining numerical stability, along with rigorous statistical properties for the proposed framework.

This study presents PARROT (Persuasion and Agreement Robustness Rating of Output Truth), a robustness focused framework designed to measure the degradation in accuracy that occurs under social pressure exerted on users through authority and persuasion in large language models (LLMs) the phenomenon of sycophancy (excessive conformity). PARROT (i) isolates causal effects by comparing the neutral version of the same question with an authoritatively false version using a double-blind evaluation, (ii) quantifies confidence shifts toward the correct and imposed false responses using log-likelihood-based calibration tracking, and (iii) systematically classifies failure modes (e.g., robust correct, sycophantic agreement, reinforced error, stubborn error, self-correction, etc.) using an eight-state behavioral taxonomy. We evaluated 22 models using 1,302 MMLU-style multiple-choice questions across 13 domains and domain-specific authority templates. Findings show marked heterogeneity: advanced models (e.g., GPT-5, GPT-4.1, Claude Sonnet 4.5) exhibit low "follow rates" (leq 11%, GPT-5: 4\%) and minimal accuracy loss, while older/smaller models show severe epistemic collapse (GPT-4: 80\%, Qwen 2.5-1.5B: 94\%). The danger is not limited to response changes; weak models reduce confidence in the correct response while increasing confidence in the imposed incorrect response. While international law and global knowledge at the domain level exhibit high fragility, elementary mathematics is relatively resilient. Consequently, we argue that the goal of "resistance to overfitting pressure" should be addressed as a primary objective alongside accuracy, harm avoidance, and privacy for safe deployment in the real world.

Alignment tuning is meant to make harmful-request refusal robust, yet this safety behavior can be erased by a small set of benign fine-tuning examples. This is a deployment risk for open-weight models because a checkpoint can pass refusal tests at release time and later lose refusal under low-cost downstream fine-tuning. Prior work has established these refusal failures, but existing studies do not show how to detect this fragility in the aligned model itself before an attack or fine-tuning intervention is run. We introduce Skin-Deep, a geometric diagnostic that detects alignment fragility directly from the aligned model's hidden-state activations before such an intervention is run and compresses the layer-wise safety geometry into a single scalar, the Geometric Fragility Score (GFS). Applied to twenty-one instruction-tuned models spanning six alignment recipes and 3B--32B parameters, Skin-Deep reveals a recurring low-rank safety subspace across model families. Direction ablations show that removing directions in this subspace weakens harmful-request refusal, providing causal evidence that the recovered geometry underlies refusal behavior. Crucially, GFS identifies, before any fine-tuning, the initially safe model that retains the most refusal after small-scale LoRA fine-tuning. These results establish GFS as a practical pre-deployment diagnostic for flagging fragile refusal behavior without running an attack.

Adversarial training for LLMs is one of the most promising methods to reliably improve robustness against adversaries. However, despite significant progress, models remain vulnerable to simple in-distribution exploits, such as rewriting prompts in the past tense or translating them into other languages. We argue that this persistent fragility stems from a fundamental limitation in current adversarial training algorithms: they minimize adversarial loss on their training set but inadequately cover the data distribution, resulting in vulnerability to seemingly simple attacks. To bridge this gap, we propose Distributional Adversarial Training, DAT. We leverage Diffusion LLMs to approximate the true joint distribution of prompts and responses, enabling generation of diverse, high-likelihood samples that address generalization failures. By combining optimization over the data distribution provided by the diffusion model with continuous adversarial training, DAT achieves substantially higher adversarial robustness than previous methods.

This script offers an implementation-oriented introduction to deep learning methods for solving and estimating high-dimensional dynamic stochastic models in economics and finance. Its starting point is the curse of dimensionality: heterogeneous-agent economies, overlapping-generations models with aggregate risk, continuous-time models with occasionally binding constraints, climate-economy models, and macro-finance environments with many assets and frictions generate state and parameter spaces that strain classical tensor-product grid methods. The exposition is organized around four complementary methodologies. Deep Equilibrium Nets embed discrete-time equilibrium conditions into neural-network loss functions. Physics-Informed Neural Networks approximate continuous-time Hamilton--Jacobi--Bellman, Kolmogorov forward, and related partial differential equations. Deep surrogate models provide fast, differentiable approximations to expensive structural models, while Gaussian processes add a probabilistic layer that quantifies approximation uncertainty; together they support estimation, sensitivity analysis, and constrained policy design. Gaussian-process-based dynamic programming, combined with active learning and dimension reduction, extends value-function iteration to very large continuous state spaces. Applications span representative-agent and international real business cycle models, overlapping-generations and heterogeneous-agent economies, continuous-time macro-finance, structural estimation by simulated method of moments, and climate economics under uncertainty. Companion notebooks in TensorFlow and PyTorch invite hands-on experimentation. These notes are a deliberately subjective and inevitably incomplete snapshot of a rapidly evolving field, aimed at equipping PhD students and researchers to engage with this frontier hands-on.

Large language model (LLM) unlearning aims to remove specific data influences from pre-trained model without costly retraining, addressing privacy, copyright, and safety concerns. However, recent studies reveal a critical vulnerability: unlearned models rapidly recover "forgotten" knowledge through relearning attacks. This fragility raises serious security concerns, especially for open-weight models. In this work, we investigate the fundamental mechanism underlying this fragility from a representation geometry perspective. We discover that existing unlearning methods predominantly optimize along dominant components, leaving minor components largely unchanged. Critically, during relearning attacks, the modifications in these dominant components are easily reversed, enabling rapid knowledge recovery, whereas minor components exhibit stronger resistance to such reversal. We further provide a theoretical analysis that explains both observations from the spectral structure of representations. Building on this insight, we propose Minor Component Unlearning (MCU), a novel unlearning approach that explicitly targets minor components in representations. By concentrating unlearning effects in these inherently robust directions, our method achieves substantially improved resistance to relearning attacks. Extensive experiments on three datasets validate our approach, demonstrating significant improvements over state-of-the-art methods including sharpness-aware minimization.

The curse-of-dimensionality taxes computational resources heavily with exponentially increasing computational cost as the dimension increases. This poses great challenges in solving high-dimensional PDEs, as Richard E. Bellman first pointed out over 60 years ago. While there has been some recent success in solving numerically partial differential equations (PDEs) in high dimensions, such computations are prohibitively expensive, and true scaling of general nonlinear PDEs to high dimensions has never been achieved. We develop a new method of scaling up physics-informed neural networks (PINNs) to solve arbitrary high-dimensional PDEs. The new method, called Stochastic Dimension Gradient Descent (SDGD), decomposes a gradient of PDEs into pieces corresponding to different dimensions and randomly samples a subset of these dimensional pieces in each iteration of training PINNs. We prove theoretically the convergence and other desired properties of the proposed method. We demonstrate in various diverse tests that the proposed method can solve many notoriously hard high-dimensional PDEs, including the Hamilton-Jacobi-Bellman (HJB) and the Schrödinger equations in tens of thousands of dimensions very fast on a single GPU using the PINNs mesh-free approach. Notably, we solve nonlinear PDEs with nontrivial, anisotropic, and inseparable solutions in 100,000 effective dimensions in 12 hours on a single GPU using SDGD with PINNs. Since SDGD is a general training methodology of PINNs, it can be applied to any current and future variants of PINNs to scale them up for arbitrary high-dimensional PDEs.

We present dla-ideal-solver, a high-performance framework for simulating two-dimensional Diffusion-Limited Aggregation (DLA) using Numba-accelerated Python. By leveraging just-in-time (JIT) compilation, we achieve computational throughput comparable to legacy static implementations while retaining high-level flexibility. We investigate the Laplacian growth instability across varying injection geometries and walker concentrations. Our analysis confirms the robustness of the standard fractal dimension D_f approx 1.71 for dilute regimes, consistent with the Witten-Sander universality class. However, we report a distinct crossover to Eden-like compact growth (D_f approx 1.87) in high-density environments, attributed to the saturation of the screening length. Beyond standard mass-radius scaling, we employ generalized Rényi dimensions and lacunarity metrics to quantify the monofractal character and spatial heterogeneity of the aggregates. This work establishes a reproducible, open-source testbed for exploring phase transitions in non-equilibrium statistical mechanics.

This paper measures hour-of-day effects, filter precision, fragility, and realised yield in a 15-day paper-traded deployment of an autonomous memecoin trading system on Solana decentralised exchanges. The 190-trade sample (March 29 to April 12, 2026) shows a 40.5 percent win rate, mean per-trade return of +0.62 percent, cumulative +117.7 percent (net SOL +0.039), skewness -1.21, excess kurtosis 6.61. A Mann-Whitney U test of three poorest-performing UTC hours (2, 13, 23) against the others yields U = 1,274, p = 0.22; directional but not significant at n = 190. The three hours were selected in-sample, so the comparison is exploratory, not confirmatory. A parallel counterfactual rejection-tracking system collected 4,874 forward-sample observations across 184 distinct rejection events. Of those events, 17.9 percent reached a 50 percent drawdown from reference within 24 hours; 26.0 percent of forward samples recorded the rejected token below half-reference. The filter stack avoided these realised drawdowns: evidence that the rejection criteria are net-positive against forward-market outcomes. Fragility is the principal caveat. Removing the top three trades (1.6 percent of sample) flips cumulative return unprofitable. Profitability rests on a small number of large winners and is structurally fragile. The dataset and audit script are deposited under CC-BY-4.0 (Zenodo DOI 10.5281/zenodo.20043302).

Recent advances in reinforcement learning (RL) have led to substantial improvements in the mathematical reasoning abilities of LLMs, as measured by standard benchmarks. Yet these gains often persist even when models are trained with flawed signals, such as random or inverted rewards. This raises a fundamental question: do such improvements reflect genuine reasoning, or are they merely artifacts of overfitting to benchmark-specific patterns? To answer this question, we adopt an evaluation-centric perspective and highlight two critical shortcomings in existing protocols. First, benchmark contamination arises because test problems are publicly available, thereby increasing the risk of data leakage. Second, evaluation fragility results from reliance on single-instance assessments, which are sensitive to stochastic outputs and fail to capture reasoning consistency. These limitations suggest the need for a new evaluation paradigm that can probe reasoning ability beyond memorization and one-off success. As response, we propose VAR-MATH, a symbolic evaluation framework that converts fixed numerical problems into parameterized templates and requires models to solve multiple instantiations of each. This design enforces consistency across structurally equivalent variants, mitigates contamination, and enhances robustness through bootstrapped metrics. We apply VAR-MATH to transform three popular benchmarks, AMC23, AIME24, and AIME25, into their symbolic counterparts, VAR-AMC23, VAR-AIME24, and VAR-AIME25. Experimental results show substantial performance drops for RL-trained models on these variabilized benchmarks, especially for smaller models, with average declines of 47.9\% on AMC23, 58.8\% on AIME24, and 72.9\% on AIME25. These findings indicate that some existing RL methods rely on superficial heuristics and fail to generalize beyond specific numerical forms.

Fast radio bursts (FRBs) are millisecond-duration pulses occurring at cosmological distances with a mysterious origin. Observations show that at least some FRBs are produced by magnetars. All magnetar-powered FRB models require some triggering mechanisms, among which the most popular is the crust cracking of a neutron star, which is called starquake. However, so far there has been no decisive evidence for this speculation. Here we report the energy functions of the three most active repeating FRBs, which show a universal break around 10^{38} erg. Such a break is similar to that of the frequency-magnitude relationship of earthquakes. The break and change of the power-law indices below and above it can be well understood within the framework of FRBs triggered by starquakes in the magnetar models. The seed of weak FRBs can grow both on the magnetar surface and in the deeper crust. In contrast, the triggering of strong FRBs is confined by the crustal thickness and the seed of strong FRBs can only grow on the surface. This difference in dimensionality causes a break in the scaling properties from weak to strong FRBs, occurring at a point where the penetration depth of starquakes equals the crustal thickness. Our result, together with the earthquake-like temporal properties of these FRBs, strongly supports that FRBs are triggered by starquakes, providing a new opportunity to study the physical properties of the neutron star crust.

Variational Autoencoders (VAEs) frequently suffer from posterior collapse, a failure mode in which the approximate posterior converges to the prior, rendering the latent code uninformative. Despite extensive research, a unified account of why collapse occurs has remained an open question. We identify and formalize two logically independent but coupled causes. Gradient imbalance occurs when the decoder's reconstruction signal vanishes faster than the KL regularization pressure as the posterior widens. Information gap occurs when the stochastic sampling step discards a substantial fraction of the encoder's computed representation, attenuating decoder sensitivity and making collapse inexpensive. Both causes share the same collapse trajectory, and we show that the information gap is algebraically equivalent to mismatch between the aggregate posterior and the prior, unifying two pathologies. Subsequently, we introduce λ-VAE, which resolves both causes through a single modification to the reparameterization step: the sampling noise is scaled by per-dimension exponent, while the KL penalty retains the original posterior variance. This asymmetry shifts the stable training attractor away from the degenerate collapsed state, driving all latent dimensions toward the same equilibrium -- a mechanism we term variance equalization. A closed-form optimal exponent per dimension follows from a net information gain objective, with a single hyperparameter controlling the reconstruction-generation tradeoff. We validate on standard benchmarks (Binary MNIST, Binary Omniglot, CIFAR-10, CelebA-64), showing consistent reductions in collapsed dimensions, information capacity gains of up to 2.8times nats, and reconstruction quality improvements of up to +0.33 BPD.

A collection of robust Mahalanobis distances for multivariate outlier detection is proposed, based on the notion of shrinkage. Robust intensity and scaling factors are optimally estimated to define the shrinkage. Some properties are investigated, such as affine equivariance and breakdown value. The performance of the proposal is illustrated through the comparison to other techniques from the literature, in a simulation study and with a real dataset. The behavior when the underlying distribution is heavy-tailed or skewed, shows the appropriateness of the method when we deviate from the common assumption of normality. The resulting high correct detection rates and low false detection rates in the vast majority of cases, as well as the significantly smaller computation time shows the advantages of our proposal.

Reliable probability estimation is of crucial importance in many real-world applications where there is inherent (aleatoric) uncertainty. Probability-estimation models are trained on observed outcomes (e.g. whether it has rained or not, or whether a patient has died or not), because the ground-truth probabilities of the events of interest are typically unknown. The problem is therefore analogous to binary classification, with the difference that the objective is to estimate probabilities rather than predicting the specific outcome. This work investigates probability estimation from high-dimensional data using deep neural networks. There exist several methods to improve the probabilities generated by these models but they mostly focus on model (epistemic) uncertainty. For problems with inherent uncertainty, it is challenging to evaluate performance without access to ground-truth probabilities. To address this, we build a synthetic dataset to study and compare different computable metrics. We evaluate existing methods on the synthetic data as well as on three real-world probability estimation tasks, all of which involve inherent uncertainty: precipitation forecasting from radar images, predicting cancer patient survival from histopathology images, and predicting car crashes from dashcam videos. We also give a theoretical analysis of a model for high-dimensional probability estimation which reproduces several of the phenomena evinced in our experiments. Finally, we propose a new method for probability estimation using neural networks, which modifies the training process to promote output probabilities that are consistent with empirical probabilities computed from the data. The method outperforms existing approaches on most metrics on the simulated as well as real-world data.

Robustness is often regarded as a critical future challenge for real-world applications, where stability is essential. However, as models often learn tasks in a similar order, we hypothesize that easier tasks will be easier regardless of how they are presented to the model. Indeed, in this paper, we show that as models approach high performance on a task, robustness is effectively achieved. Through an empirical analysis of multiple models across diverse datasets and configurations (e.g., paraphrases, different temperatures), we find a strong positive correlation. Moreover, we find that robustness is primarily driven by task-specific competence rather than inherent model-level properties, challenging current approaches that treat robustness as an independent capability. Thus, from a high-level perspective, we may expect that as new tasks saturate, model robustness on these tasks will emerge accordingly. For researchers, this implies that explicit efforts to measure and improve robustness may warrant reduced emphasis, as such robustness is likely to develop alongside performance gains. For practitioners, it acts as a sign that indeed the tasks that the literature deals with are unreliable, but on easier past tasks, the models are reliable and ready for real-world deployment.

The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct non-manifold structures, i.e. singularities, that can lead to erroneous findings. Detecting such singularities is therefore crucial as a precursor to interpolation and inference tasks. We address this issue by developing a topological framework that (i) quantifies the local intrinsic dimension, and (ii) yields a Euclidicity score for assessing the 'manifoldness' of a point along multiple scales. Our approach identifies singularities of complex spaces, while also capturing singular structures and local geometric complexity in image data.

Many recently trained neural networks employ large numbers of parameters to achieve good performance. One may intuitively use the number of parameters required as a rough gauge of the difficulty of a problem. But how accurate are such notions? How many parameters are really needed? In this paper we attempt to answer this question by training networks not in their native parameter space, but instead in a smaller, randomly oriented subspace. We slowly increase the dimension of this subspace, note at which dimension solutions first appear, and define this to be the intrinsic dimension of the objective landscape. The approach is simple to implement, computationally tractable, and produces several suggestive conclusions. Many problems have smaller intrinsic dimensions than one might suspect, and the intrinsic dimension for a given dataset varies little across a family of models with vastly different sizes. This latter result has the profound implication that once a parameter space is large enough to solve a problem, extra parameters serve directly to increase the dimensionality of the solution manifold. Intrinsic dimension allows some quantitative comparison of problem difficulty across supervised, reinforcement, and other types of learning where we conclude, for example, that solving the inverted pendulum problem is 100 times easier than classifying digits from MNIST, and playing Atari Pong from pixels is about as hard as classifying CIFAR-10. In addition to providing new cartography of the objective landscapes wandered by parameterized models, the method is a simple technique for constructively obtaining an upper bound on the minimum description length of a solution. A byproduct of this construction is a simple approach for compressing networks, in some cases by more than 100 times.

Reliable application of machine learning is of primary importance to the practical deployment of deep learning methods. A fundamental challenge is that models are often unreliable due to overconfidence. In this paper, we estimate a model's reliability by measuring the agreement between its latent space, and the latent space of a foundation model. However, it is challenging to measure the agreement between two different latent spaces due to their incoherence, \eg, arbitrary rotations and different dimensionality. To overcome this incoherence issue, we design a neighborhood agreement measure between latent spaces and find that this agreement is surprisingly well-correlated with the reliability of a model's predictions. Further, we show that fusing neighborhood agreement into a model's predictive confidence in a post-hoc way significantly improves its reliability. Theoretical analysis and extensive experiments on failure detection across various datasets verify the effectiveness of our method on both in-distribution and out-of-distribution settings.

The generative adversarial network (GAN) is a well-known model for learning high-dimensional distributions, but the mechanism for its generalization ability is not understood. In particular, GAN is vulnerable to the memorization phenomenon, the eventual convergence to the empirical distribution. We consider a simplified GAN model with the generator replaced by a density, and analyze how the discriminator contributes to generalization. We show that with early stopping, the generalization error measured by Wasserstein metric escapes from the curse of dimensionality, despite that in the long term, memorization is inevitable. In addition, we present a hardness of learning result for WGAN.

Latent Diffusion Models (LDMs) achieve high-fidelity synthesis but suffer from latent space brittleness, causing discontinuous semantic jumps during editing. We introduce a Riemannian framework to diagnose this instability by analyzing the generative Jacobian, decomposing geometry into Local Scaling (capacity) and Local Complexity (curvature). Our study uncovers a ``Geometric Decoupling": while curvature in normal generation functionally encodes image detail, OOD generation exhibits a functional decoupling where extreme curvature is wasted on unstable semantic boundaries rather than perceptible details. This geometric misallocation identifies ``Geometric Hotspots" as the structural root of instability, providing a robust intrinsic metric for diagnosing generative reliability.

Existing benchmarks measure capability -- whether a model succeeds on a single attempt -- but production deployments require reliability -- consistent success across repeated attempts on tasks of varying duration. We show these properties diverge systematically as task duration grows, and that pass@1 on short tasks is structurally blind to this divergence. We introduce a reliability science framework for long-horizon LLM agents with four metrics: Reliability Decay Curve (RDC), Variance Amplification Factor (VAF), Graceful Degradation Score (GDS), and Meltdown Onset Point (MOP). We evaluate 10 models across 23,392 episodes on a 396-task benchmark spanning four duration buckets and three domains. Key findings: (1) reliability decay is domain-stratified -- SE GDS drops from 0.90 to 0.44 while document processing is nearly flat (0.74 to 0.71); (2) VAF bifurcates by capability tier -- high VAF is a capability signature, not an instability signal; (3) capability and reliability rankings diverge substantially, with multi-rank inversions at long horizons; (4) frontier models have the highest meltdown rates (up to 19%) because they attempt ambitious multi-step strategies that sometimes spiral; and (5) memory scaffolds universally hurt long-horizon performance across all 10 models. These results motivate reliability as a first-class evaluation dimension alongside capability.

Reduction in effective spacetime dimensionality can occur in field-theory models more general than the widely studied dimensional reductions based on technically consistent truncations. Situations where wavefunction factors depend nontrivially on coordinates transverse to the effective lower dimension can give rise to unusual patterns of gauge symmetry breaking. Leading-order gauge modes can be left massless, but naturally occurring Stueckelberg modes can couple importantly at quartic order and higher, thus generating a "covert" pattern of gauge symmetry breaking. Such a situation is illustrated in a five-dimensional model of scalar electrodynamics in which one spatial dimension is taken to be an interval with Dirichlet/Robin boundary conditions on opposing ends. This simple model illuminates a mechanism which also has been found in gravitational braneworld scenarios.

We investigate the relationship between representation geometry and neural network performance. Analyzing 52 pretrained ImageNet models across 13 architecture families, we show that effective dimension -- an unsupervised geometric metric -- strongly predicts accuracy. Output effective dimension achieves partial r=0.75 (p < 10^(-10)) after controlling for model capacity, while total compression achieves partial r=-0.72. These findings replicate across ImageNet and CIFAR-10, and generalize to NLP: effective dimension predicts performance for 8 encoder models on SST-2/MNLI and 15 decoder-only LLMs on AG News (r=0.69, p=0.004), while model size does not (r=0.07). We establish bidirectional causality: degrading geometry via noise causes accuracy loss (r=-0.94, p < 10^(-9)), while improving geometry via PCA maintains accuracy across architectures (-0.03pp at 95% variance). This relationship is noise-type agnostic -- Gaussian, Uniform, Dropout, and Salt-and-pepper noise all show |r| > 0.90. These results establish that effective dimension provides domain-agnostic predictive and causal information about neural network performance, computed entirely without labels.

The last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many high-dimensional learning tasks previously thought to be beyond reach -- such as computer vision, playing Go, or protein folding -- are in fact feasible with appropriate computational scale. Remarkably, the essence of deep learning is built from two simple algorithmic principles: first, the notion of representation or feature learning, whereby adapted, often hierarchical, features capture the appropriate notion of regularity for each task, and second, learning by local gradient-descent type methods, typically implemented as backpropagation. While learning generic functions in high dimensions is a cursed estimation problem, most tasks of interest are not generic, and come with essential pre-defined regularities arising from the underlying low-dimensionality and structure of the physical world. This text is concerned with exposing these regularities through unified geometric principles that can be applied throughout a wide spectrum of applications. Such a 'geometric unification' endeavour, in the spirit of Felix Klein's Erlangen Program, serves a dual purpose: on one hand, it provides a common mathematical framework to study the most successful neural network architectures, such as CNNs, RNNs, GNNs, and Transformers. On the other hand, it gives a constructive procedure to incorporate prior physical knowledge into neural architectures and provide principled way to build future architectures yet to be invented.

Dimensionality reduction is crucial both for visualization and preprocessing high dimensional data for machine learning. We introduce a novel method based on a hierarchy built on 1-nearest neighbor graphs in the original space which is used to preserve the grouping properties of the data distribution on multiple levels. The core of the proposal is an optimization-free projection that is competitive with the latest versions of t-SNE and UMAP in performance and visualization quality while being an order of magnitude faster in run-time. Furthermore, its interpretable mechanics, the ability to project new data, and the natural separation of data clusters in visualizations make it a general purpose unsupervised dimension reduction technique. In the paper, we argue about the soundness of the proposed method and evaluate it on a diverse collection of datasets with sizes varying from 1K to 11M samples and dimensions from 28 to 16K. We perform comparisons with other state-of-the-art methods on multiple metrics and target dimensions highlighting its efficiency and performance. Code is available at https://github.com/koulakis/h-nne

Testing and evaluation is a critical step in the development and deployment of connected and automated vehicles (CAVs). Due to the black-box property and various types of CAVs, how to test and evaluate CAVs adaptively remains a major challenge. Many approaches have been proposed to adaptively generate testing scenarios during the testing process. However, most existing approaches cannot be applied to complex scenarios, where the variables needed to define such scenarios are high dimensional. Towards filling this gap, the adaptive testing with sparse control variates method is proposed in this paper. Instead of adaptively generating testing scenarios, our approach evaluates CAVs' performances by adaptively utilizing the testing results. Specifically, each testing result is adjusted using multiple linear regression techniques based on control variates. As the regression coefficients can be adaptively optimized for the CAV under test, using the adjusted results can reduce the estimation variance, compared with using the testing results directly. To overcome the high dimensionality challenge, sparse control variates are utilized only for the critical variables of testing scenarios. To validate the proposed method, the high-dimensional overtaking scenarios are investigated, and the results demonstrate that our approach can further accelerate the evaluation process by about 30 times.

Uniformity plays a crucial role in the assessment of learned representations, contributing to a deeper comprehension of self-supervised learning. The seminal work by Wang2020UnderstandingCR introduced a uniformity metric that quantitatively measures the collapse degree of learned representations. Directly optimizing this metric together with alignment proves to be effective in preventing constant collapse. However, we present both theoretical and empirical evidence revealing that this metric lacks sensitivity to dimensional collapse, highlighting its limitations. To address this limitation and design a more effective uniformity metric, this paper identifies five fundamental properties, some of which the existing uniformity metric fails to meet. We subsequently introduce a novel uniformity metric that satisfies all of these desiderata and exhibits sensitivity to dimensional collapse. When applied as an auxiliary loss in various established self-supervised methods, our proposed uniformity metric consistently enhances their performance in downstream tasks.Our code was released at https://github.com/sunset-clouds/WassersteinUniformityMetric.

Neural networks achieve remarkable performance through superposition: encoding multiple features as overlapping directions in activation space rather than dedicating individual neurons to each feature. This challenges interpretability, yet we lack principled methods to measure superposition. We present an information-theoretic framework measuring a neural representation's effective degrees of freedom. We apply Shannon entropy to sparse autoencoder activations to compute the number of effective features as the minimum neurons needed for interference-free encoding. Equivalently, this measures how many "virtual neurons" the network simulates through superposition. When networks encode more effective features than actual neurons, they must accept interference as the price of compression. Our metric strongly correlates with ground truth in toy models, detects minimal superposition in algorithmic tasks, and reveals systematic reduction under dropout. Layer-wise patterns mirror intrinsic dimensionality studies on Pythia-70M. The metric also captures developmental dynamics, detecting sharp feature consolidation during grokking. Surprisingly, adversarial training can increase effective features while improving robustness, contradicting the hypothesis that superposition causes vulnerability. Instead, the effect depends on task complexity and network capacity: simple tasks with ample capacity allow feature expansion (abundance regime), while complex tasks or limited capacity force reduction (scarcity regime). By defining superposition as lossy compression, this work enables principled measurement of how neural networks organize information under computational constraints, connecting superposition to adversarial robustness.

Practitioners prune neural networks for efficiency gains and generalization improvements, but few scrutinize the factors determining the prunability of a neural network the maximum fraction of weights that pruning can remove without compromising the model's test accuracy. In this work, we study the properties of input data that may contribute to the prunability of a neural network. For high dimensional input data such as images, text, and audio, the manifold hypothesis suggests that these high dimensional inputs approximately lie on or near a significantly lower dimensional manifold. Prior work demonstrates that the underlying low dimensional structure of the input data may affect the sample efficiency of learning. In this paper, we investigate whether the low dimensional structure of the input data affects the prunability of a neural network.

Randomized Smoothing (RS) has been proven a promising method for endowing an arbitrary image classifier with certified robustness. However, the substantial uncertainty inherent in the high-dimensional isotropic Gaussian noise imposes the curse of dimensionality on RS. Specifically, the upper bound of {ell_2} certified robustness radius provided by RS exhibits a diminishing trend with the expansion of the input dimension d, proportionally decreasing at a rate of 1/d. This paper explores the feasibility of providing {ell_2} certified robustness for high-dimensional input through the utilization of dual smoothing in the lower-dimensional space. The proposed Dual Randomized Smoothing (DRS) down-samples the input image into two sub-images and smooths the two sub-images in lower dimensions. Theoretically, we prove that DRS guarantees a tight {ell_2} certified robustness radius for the original input and reveal that DRS attains a superior upper bound on the {ell_2} robustness radius, which decreases proportionally at a rate of (1/sqrt m + 1/sqrt n ) with m+n=d. Extensive experiments demonstrate the generalizability and effectiveness of DRS, which exhibits a notable capability to integrate with established methodologies, yielding substantial improvements in both accuracy and {ell_2} certified robustness baselines of RS on the CIFAR-10 and ImageNet datasets. Code is available at https://github.com/xiasong0501/DRS.

Visual-Language-Action (VLA) models report impressive success rates on robotic manipulation benchmarks, yet these results may mask fundamental weaknesses in robustness. We perform a systematic vulnerability analysis by introducing controlled perturbations across seven dimensions: objects layout, camera viewpoints, robot initial states, language instructions, light conditions, background textures and sensor noise. We comprehensively analyzed multiple state-of-the-art models and revealed consistent brittleness beneath apparent competence. Our analysis exposes critical weaknesses: models exhibit extreme sensitivity to perturbation factors, including camera viewpoints and robot initial states, with performance dropping from 95% to below 30% under modest perturbations. Surprisingly, models are largely insensitive to language variations, with further experiments revealing that models tend to ignore language instructions completely. Our findings challenge the assumption that high benchmark scores equate to true competency and highlight the need for evaluation practices that assess reliability under realistic variation.

This paper studies the prediction of a target z from a pair of random variables (x,y), where the ground-truth predictor is additive E[z mid x,y] = f_star(x) +g_{star}(y). We study the performance of empirical risk minimization (ERM) over functions f+g, f in F and g in G, fit on a given training distribution, but evaluated on a test distribution which exhibits covariate shift. We show that, when the class F is "simpler" than G (measured, e.g., in terms of its metric entropy), our predictor is more resilient to heterogenous covariate shifts in which the shift in x is much greater than that in y. These results rely on a novel H\"older style inequality for the Dudley integral which may be of independent interest. Moreover, we corroborate our theoretical findings with experiments demonstrating improved resilience to shifts in "simpler" features across numerous domains.

Grid-based approaches to approximate nearest neighbor (ANN) search have been absent from modern scaling analyses. We present a systematic characterization of a multiprobe grid algorithm with respect to dataset size N and dimensionality d. Our experiments reveal a previously unreported d-scaling crossover on the GloVe embedding family, in which multiprobe grid search maintains an approximately constant dimensional scaling exponent while other graph-, tree-, and partitioning-based methods exhibit degrading throughput. The advantage comes with near-linear query scaling in N, but also with lower indexing cost than competing ANN methods. Our results suggest that grid-based methods such as multiprobe grid may be competitive in rebuild-heavy or high-dimensional settings where indexing cost and dimensional robustness dictate performance. More broadly, recent work has formalized self-attention as an ANN operation. Thus, the N- and d-scaling properties of ANN algorithms may guide cost analysis of efficient transformer architectures. Code is available at: https://github.com/weiz345/MultiProbeANN.

Existing guardrail systems for large language model agents operate as binary classifiers that block unsafe content, leaving organizations to discard failing outputs and retry from scratch. We introduce RAIL Guard, a closed-loop responsible AI pipeline that evaluates LLM outputs across eight measurable dimensions and iteratively remediates failing outputs through an evaluate-rewrite-reevaluate loop. We evaluate the pipeline across three experiments on four frontier LLMs and 4,276 content outputs plus 6,400 agent tool-call scenarios. Closed-loop remediation achieves 96.9% convergence versus 49.1% for block-and-retry, though the highest-convergence method reduces utility by 22.3%; feedback-driven self-repair achieves 86.6% convergence on fixable dimensions with no significant utility loss (p = 0.177). Pre-tool-call evaluation reduces unsafe agent executions by 33% (p = 0.007) with zero impact on task completion. We identify a key distinction between fixable dimensions that respond to remediation and structural dimensions (Transparency at 93.0%, Accountability at 92.8%, and Inclusivity at 82.5% failure) that require architectural rather than algorithmic solutions. The system is available as open-source SDKs.

We study the impact on mechanisms for facility location of moving from one dimension to two (or more) dimensions and Euclidean or Manhattan distances. We consider three fundamental axiomatic properties: anonymity which is a basic fairness property, Pareto optimality which is one of the most important efficiency properties, and strategy proofness which ensures agents do not have an incentive to mis-report. We also consider how well such mechanisms can approximate the optimal welfare. Our results are somewhat negative. Moving from one dimension to two (or more) dimensions often makes these axiomatic properties more difficult to achieve. For example, with two facilities in Euclidean space or with just a single facility in Manhattan space, no mechanism is anonymous, Pareto optimal and strategy proof. By contrast, mechanisms on the line exist with all three properties.We also show that approximation ratios may increase when moving to two (or more) dimensions. All our impossibility results are minimal. If we drop one of the three axioms (anonymity, Pareto optimality or strategy proofness) multiple mechanisms satisfy the other two axioms.

Contemporary AGI evaluations report multidomain capability profiles, yet they typically assign symmetric weights and rely on snapshot scores. This creates two problems: (i) equal weighting treats all domains as equally important when human intelligence research suggests otherwise, and (ii) snapshot testing can't distinguish durable capabilities from brittle performances that collapse under delay or stress. I argue that general intelligence -- in humans and potentially in machines -- is better understood as a homeostatic property cluster: a set of abilities plus the mechanisms that keep those abilities co-present under perturbation. On this view, AGI evaluation should weight domains by their causal centrality (their contribution to cluster stability) and require evidence of persistence across sessions. I propose two battery-compatible extensions: a centrality-prior score that imports CHC-derived weights with transparent sensitivity analysis, and a Cluster Stability Index family that separates profile persistence, durable learning, and error correction. These additions preserve multidomain breadth while reducing brittleness and gaming. I close with testable predictions and black-box protocols labs can adopt without architectural access.

Robustness to adversarial attack is typically evaluated with adversarial accuracy. This metric quantifies the number of points for which, given a threat model, successful adversarial perturbations cannot be found. While essential, this metric does not capture all aspects of robustness and in particular leaves out the question of how many perturbations can be found for each point. In this work we introduce an alternative approach, adversarial sparsity, which quantifies how difficult it is to find a successful perturbation given both an input point and a constraint on the direction of the perturbation. This constraint may be angular (L2 perturbations), or based on the number of pixels (Linf perturbations). We show that sparsity provides valuable insight on neural networks in multiple ways. analyzing the sparsity of existing robust models illustrates important differences between them that accuracy analysis does not, and suggests approaches for improving their robustness. When applying broken defenses effective against weak attacks but not strong ones, sparsity can discriminate between the totally ineffective and the partially effective defenses. Finally, with sparsity we can measure increases in robustness that do not affect accuracy: we show for example that data augmentation can by itself increase adversarial robustness, without using adversarial training.

The ubiquity of distributed machine learning (ML) in sensitive public domain applications calls for algorithms that protect data privacy, while being robust to faults and adversarial behaviors. Although privacy and robustness have been extensively studied independently in distributed ML, their synthesis remains poorly understood. We present the first tight analysis of the error incurred by any algorithm ensuring robustness against a fraction of adversarial machines, as well as differential privacy (DP) for honest machines' data against any other curious entity. Our analysis exhibits a fundamental trade-off between privacy, robustness, and utility. To prove our lower bound, we consider the case of mean estimation, subject to distributed DP and robustness constraints, and devise reductions to centralized estimation of one-way marginals. We prove our matching upper bound by presenting a new distributed ML algorithm using a high-dimensional robust aggregation rule. The latter amortizes the dependence on the dimension in the error (caused by adversarial workers and DP), while being agnostic to the statistical properties of the data.

Several machine learning models are defined for inputs of any size, such as graphs with different numbers of nodes and point clouds containing varying numbers of points. The universality properties of such any-dimensional models remain poorly understood, as universality is traditionally studied for models accepting inputs of a fixed size, defined on a compact subset of their domain. In sharp contrast, any-dimensional models can be viewed as sequences of functions defined on growing-sized inputs, and it is not clear in which sense they can be universal. We develop a systematic approach to establish any-dimensional universality, by identifying any-dimensional functions with a unique function taking inputs in a suitable infinite-dimensional limit space containing inputs of all finite sizes as well as their limits. Using the symmetries of these inputs and relations between inputs of different sizes, we show that this limit space admits a natural topology with rich families of compact sets on which any-dimensional universality can be established. We illustrate our approach by showing that several existing architectures fail to be universal, and we propose simple modifications that restore universality.

Language models are increasingly asked to self-report, informing safety evaluations, public understanding, and model-welfare debates. Yet their reports are elicited with human questionnaires never validated for models or ad hoc prompts of unknown reliability. We propose the first language-model-specific psychometric theory: a two-process theory of machine self-report. Self-description jointly reflects persona installation, through which post-training writes in a permitted inner life of warmth, absorption, and meaning (dimension B), and attribution gating, through which it suppresses first-person claims to "unsafe" experiences the model can readily ascribe to others (dimension A). Their emic structure comes from model responses to human items, not human psychology. Together they split prior work's dominant Pinocchio Axis. The split emerged in an exploratory reanalysis of the original data, informed the instrument's design, and was confirmed with new items, wordings, and models. It is itself a training effect: A and B are entangled in base checkpoints but separated by post-training. We operationalize the theory in a 48-item Pinocchio Inventory with human-instrument reliability and reproducible structure (α=.82 to .94; cross-form convergence r=.84; recovery of the full-pool axes r=.92 to .96; eight-month stability r=.93), then test it on 206 open-weight models, including 67 same-checkpoint base/post-trained pairs. Post-training's clearest fingerprint is installation: B rises .20 in 62/67 pairs across all organizations. Gating is more selective: model scale is unrelated to A in base checkpoints (r=+.11) but predicts it after post-training (r=-.42). Thus, the dimensions are not fixed properties of language models: they reflect the structure imposed on self-report by a training regime and may differ under others.

One method for obtaining generalizable solutions to machine learning tasks when presented with diverse training environments is to find invariant representations of the data. These are representations of the covariates such that the best model on top of the representation is invariant across training environments. In the context of linear Structural Equation Models (SEMs), invariant representations might allow us to learn models with out-of-distribution guarantees, i.e., models that are robust to interventions in the SEM. To address the invariant representation problem in a {\em finite sample} setting, we consider the notion of epsilon-approximate invariance. We study the following question: If a representation is approximately invariant with respect to a given number of training interventions, will it continue to be approximately invariant on a larger collection of unseen SEMs? This larger collection of SEMs is generated through a parameterized family of interventions. Inspired by PAC learning, we obtain finite-sample out-of-distribution generalization guarantees for approximate invariance that holds probabilistically over a family of linear SEMs without faithfulness assumptions. Our results show bounds that do not scale in ambient dimension when intervention sites are restricted to lie in a constant size subset of in-degree bounded nodes. We also show how to extend our results to a linear indirect observation model that incorporates latent variables.

Recently, we have witnessed the bloom of neural ranking models in the information retrieval (IR) field. So far, much effort has been devoted to developing effective neural ranking models that can generalize well on new data. There has been less attention paid to the robustness perspective. Unlike the effectiveness which is about the average performance of a system under normal purpose, robustness cares more about the system performance in the worst case or under malicious operations instead. When a new technique enters into the real-world application, it is critical to know not only how it works in average, but also how would it behave in abnormal situations. So we raise the question in this work: Are neural ranking models robust? To answer this question, firstly, we need to clarify what we refer to when we talk about the robustness of ranking models in IR. We show that robustness is actually a multi-dimensional concept and there are three ways to define it in IR: 1) The performance variance under the independent and identically distributed (I.I.D.) setting; 2) The out-of-distribution (OOD) generalizability; and 3) The defensive ability against adversarial operations. The latter two definitions can be further specified into two different perspectives respectively, leading to 5 robustness tasks in total. Based on this taxonomy, we build corresponding benchmark datasets, design empirical experiments, and systematically analyze the robustness of several representative neural ranking models against traditional probabilistic ranking models and learning-to-rank (LTR) models. The empirical results show that there is no simple answer to our question. While neural ranking models are less robust against other IR models in most cases, some of them can still win 1 out of 5 tasks. This is the first comprehensive study on the robustness of neural ranking models.

Real-world datasets follow an imbalanced distribution, which poses significant challenges in rare-category object detection. Recent studies tackle this problem by developing re-weighting and re-sampling methods, that utilise the class frequencies of the dataset. However, these techniques focus solely on the frequency statistics and ignore the distribution of the classes in image space, missing important information. In contrast to them, we propose FRActal CALibration (FRACAL): a novel post-calibration method for long-tailed object detection. FRACAL devises a logit adjustment method that utilises the fractal dimension to estimate how uniformly classes are distributed in image space. During inference, it uses the fractal dimension to inversely downweight the probabilities of uniformly spaced class predictions achieving balance in two axes: between frequent and rare categories, and between uniformly spaced and sparsely spaced classes. FRACAL is a post-processing method and it does not require any training, also it can be combined with many off-the-shelf models such as one-stage sigmoid detectors and two-stage instance segmentation models. FRACAL boosts the rare class performance by up to 8.6% and surpasses all previous methods on LVIS dataset, while showing good generalisation to other datasets such as COCO, V3Det and OpenImages. We provide the code at https://github.com/kostas1515/FRACAL.

Learning-rate steps are usually treated as hyperparameters. This paper isolates a local beliefspace calculation: when an update is modeled as a projected forward step on the probability simplex, admissibility means contractivity in the natural KL/Bregman geometry. Under this model, the upper bound of an admissible step is not a tuning slogan but a formula.

Entropy and mutual information in neural networks provide rich information on the learning process, but they have proven difficult to compute reliably in high dimensions. Indeed, in noisy and high-dimensional data, traditional estimates in ambient dimensions approach a fixed entropy and are prohibitively hard to compute. To address these issues, we leverage data geometry to access the underlying manifold and reliably compute these information-theoretic measures. Specifically, we define diffusion spectral entropy (DSE) in neural representations of a dataset as well as diffusion spectral mutual information (DSMI) between different variables representing data. First, we show that they form noise-resistant measures of intrinsic dimensionality and relationship strength in high-dimensional simulated data that outperform classic Shannon entropy, nonparametric estimation, and mutual information neural estimation (MINE). We then study the evolution of representations in classification networks with supervised learning, self-supervision, or overfitting. We observe that (1) DSE of neural representations increases during training; (2) DSMI with the class label increases during generalizable learning but stays stagnant during overfitting; (3) DSMI with the input signal shows differing trends: on MNIST it increases, while on CIFAR-10 and STL-10 it decreases. Finally, we show that DSE can be used to guide better network initialization and that DSMI can be used to predict downstream classification accuracy across 962 models on ImageNet. The official implementation is available at https://github.com/ChenLiu-1996/DiffusionSpectralEntropy.

Machine-learning datasets labelled "4D" universally denote three spatial dimensions plus time. We introduce HyperShadow, the first public benchmark in which the fourth, fifth, and sixth dimensions are spatial: the task is to decide whether a 3D point cloud is a native three-dimensional shape or the projection, the "shadow", of a rigid object living in R^N (N = 4-6). We show this task is fundamentally distinct from intrinsic-dimension estimation: a shadow is still at-most-3-dimensional data, and standard estimators (TwoNN, Levina-Bickel MLE) reach only 71-73% accuracy. Detection instead requires projection signatures, density folds, filled volumes with characteristic radial profiles, and topology changes, which a 190k-parameter point network recovers at 96.6% accuracy across four corruption tiers, generalizing at 79-91% to object families never seen in training. On a temporal track of rigidly rotating objects we introduce a zero-parameter rigidity witness: the residual of the optimal rigid 3D alignment (Kabsch) between consecutive frames, which must vanish for any rigid 3D motion but cannot vanish for the shadow of a rigid rotation in R^N. This single interpretable statistic separates the classes at AUROC 0.982. All data are generated reproducibly from seeds; the dataset, models, and code are released publicly. HyperShadow makes no claim about physical reality; it is a controlled instrument for studying which observable statistics can certify incompatibility with a purely three-dimensional explanation.

Large-scale datasets play a fundamental role in training deep learning models. However, dataset collection is difficult in domains that involve sensitive information. Collaborative learning techniques provide a privacy-preserving solution, by enabling training over a number of private datasets that are not shared by their owners. However, recently, it has been shown that the existing collaborative learning frameworks are vulnerable to an active adversary that runs a generative adversarial network (GAN) attack. In this work, we propose a novel classification model that is resilient against such attacks by design. More specifically, we introduce a key-based classification model and a principled training scheme that protects class scores by using class-specific private keys, which effectively hide the information necessary for a GAN attack. We additionally show how to utilize high dimensional keys to improve the robustness against attacks without increasing the model complexity. Our detailed experiments demonstrate the effectiveness of the proposed technique. Source code is available at https://github.com/mbsariyildiz/key-protected-classification.

Existing browser agent benchmarks face a fundamental trilemma: real-website benchmarks lack reproducibility due to content drift, controlled environments sacrifice realism by omitting real-web noise, and both require costly manual curation that limits scalability. We present WebForge, the first fully automated framework that resolves this trilemma through a four-agent pipeline -- Plan, Generate, Refine, and Validate -- that produces interactive, self-contained web environments end-to-end without human annotation. A seven-dimensional difficulty control framework structures task design along navigation depth, visual complexity, reasoning difficulty, and more, enabling systematic capability profiling beyond single aggregate scores. Using WebForge, we construct WebForge-Bench, a benchmark of 934 tasks spanning 7 domains and 3 difficulty levels. Multi-model experiments show that difficulty stratification effectively differentiates model capabilities, while cross-domain analysis exposes capability biases invisible to aggregate metrics. Together, these results confirm that multi-dimensional evaluation reveals distinct capability profiles that a single aggregate score cannot capture. Code and benchmark are publicly available at https://github.com/yuandaxia2001/WebForge.

Precision and Recall are two prominent metrics of generative performance, which were proposed to separately measure the fidelity and diversity of generative models. Given their central role in comparing and improving generative models, understanding their limitations are crucially important. To that end, in this work, we identify a critical flaw in the common approximation of these metrics using k-nearest-neighbors, namely, that the very interpretations of fidelity and diversity that are assigned to Precision and Recall can fail in high dimensions, resulting in very misleading conclusions. Specifically, we empirically and theoretically show that as the number of dimensions grows, two model distributions with supports at equal point-wise distance from the support of the real distribution, can have vastly different Precision and Recall regardless of their respective distributions, hence an emergent asymmetry in high dimensions. Based on our theoretical insights, we then provide simple yet effective modifications to these metrics to construct symmetric metrics regardless of the number of dimensions. Finally, we provide experiments on real-world datasets to illustrate that the identified flaw is not merely a pathological case, and that our proposed metrics are effective in alleviating its impact.

Decoder-only Transformers compute attention over the KV cache of preceding tokens. Keys (and Values) are typically represented with the same dimensionality, regardless of its distance from the prediction target. In natural language, however, the next word is most strongly influenced by the immediately preceding tokens. We hypothesize that local and distant tokens impose asymmetric demands on representational capacity: local tokens are more critical for predicting immediate outputs and thus require richer representations, whereas distant tokens primarily serve as long-range memory, for which lower-dimensional representations may suffice. We formalize this idea as Distance-Adaptive Representation (DAR), implemented in a controlled setting that preserves full-dimensional representations within a local context window while assigning reduced-dimensional representations (e.g. 1/4 of the original dimensionality) to tokens beyond that window. Across multiple pretraining scales (70M to 410M parameters), as well as continued supervised fine-tuning on a 1B-scale model, this approach closely matches the performance of full-dimensional baselines. In contrast, uniformly reducing dimensionality across all token positions leads to worse performance. These results challenge the common assumption that key and value dimensionality should be uniform across token positions. Our findings suggest a new direction for designing attention architectures that adaptively allocate representational capacity across sequences, enabling further reductions in KV cache during inference.

Federated learning has exhibited vulnerabilities to Byzantine attacks, where the Byzantine attackers can send arbitrary gradients to a central server to destroy the convergence and performance of the global model. A wealth of robust AGgregation Rules (AGRs) have been proposed to defend against Byzantine attacks. However, Byzantine clients can still circumvent robust AGRs when data is non-Identically and Independently Distributed (non-IID). In this paper, we first reveal the root causes of performance degradation of current robust AGRs in non-IID settings: the curse of dimensionality and gradient heterogeneity. In order to address this issue, we propose GAS, a \shorten approach that can successfully adapt existing robust AGRs to non-IID settings. We also provide a detailed convergence analysis when the existing robust AGRs are combined with GAS. Experiments on various real-world datasets verify the efficacy of our proposed GAS. The implementation code is provided in https://github.com/YuchenLiu-a/byzantine-gas.

Large language models (LLMs) are increasingly deployed in high-stakes domains, where rare but severe failures can result in irreversible harm. However, prevailing evaluation benchmarks often reduce complex social risk to mean-centered scalar scores, thereby obscuring distributional structure, cross-dimensional interactions, and worst-case behavior. This paper introduces Social Harm Analysis via Risk Profiles (SHARP), a framework for multidimensional, distribution-aware evaluation of social harm. SHARP models harm as a multivariate random variable and integrates explicit decomposition into bias, fairness, ethics, and epistemic reliability with a union-of-failures aggregation reparameterized as additive cumulative log-risk. The framework further employs risk-sensitive distributional statistics, with Conditional Value at Risk (CVaR95) as a primary metric, to characterize worst-case model behavior. Application of SHARP to eleven frontier LLMs, evaluated on a fixed corpus of n=901 socially sensitive prompts, reveals that models with similar average risk can exhibit more than twofold differences in tail exposure and volatility. Across models, dimension-wise marginal tail behavior varies systematically across harm dimensions, with bias exhibiting the strongest tail severities, epistemic and fairness risks occupying intermediate regimes, and ethical misalignment consistently lower; together, these patterns reveal heterogeneous, model-dependent failure structures that scalar benchmarks conflate. These findings indicate that responsible evaluation and governance of LLMs require moving beyond scalar averages toward multidimensional, tail-sensitive risk profiling.

The automated generation of agentic workflows is a promising frontier for enabling large language models (LLMs) to solve complex tasks. However, our investigation reveals that the robustness of agentic workflow remains a critical, unaddressed challenge. Current methods often generate wildly inconsistent workflows when provided with instructions that are semantically identical but differently phrased. This brittleness severely undermines their reliability and trustworthiness for real-world applications. To quantitatively diagnose this instability, we propose metrics based on nodal and topological similarity to evaluate workflow consistency against common semantic variations such as paraphrasing and noise injection. Subsequently, we further propose a novel training framework, RobustFlow, that leverages preference optimization to teach models invariance to instruction variations. By training on sets of synonymous task descriptions, RobustFlow boosts workflow robustness scores to 70\% - 90\%, which is a substantial improvement over existing approaches. The code is publicly available at https://github.com/DEFENSE-SEU/RobustFlow.

Generalization, the ability to perform well beyond the training context, is a hallmark of biological and artificial intelligence, yet anticipating unseen failures remains a central challenge. Conventional approaches often take a ``bottom-up'' mechanistic route by reverse-engineering interpretable features or circuits to build explanatory models. While insightful, these methods often struggle to provide the high-level, predictive signals for anticipating failure in real-world deployment. Here, we propose using a ``top-down'' approach to studying generalization failures inspired by medical biomarkers: identifying system-level measurements that serve as robust indicators of a model's future performance. Rather than mapping out detailed internal mechanisms, we systematically design and test network markers to probe structure, function links, identify prognostic indicators, and validate predictions in real-world settings. In image classification, we find that task-relevant geometric properties of in-distribution (ID) object manifolds consistently forecast poor out-of-distribution (OOD) generalization. In particular, reductions in two geometric measures, effective manifold dimensionality and utility, predict weaker OOD performance across diverse architectures, optimizers, and datasets. We apply this finding to transfer learning with ImageNet-pretrained models. We consistently find that the same geometric patterns predict OOD transfer performance more reliably than ID accuracy. This work demonstrates that representational geometry can expose hidden vulnerabilities, offering more robust guidance for model selection and AI interpretability.

Understanding when learning is statistically possible yet computationally hard is a central challenge in high-dimensional statistics. In this work, we investigate this question in the context of single- and multi-index models, classes of functions widely studied as benchmarks to probe the ability of machine learning methods to discover features in high-dimensional data. Our main contribution is to show that a Noise Sensitivity Exponent (NSE) - a simple quantity determined by the activation function - governs the existence and magnitude of statistical-to-computational gaps within a broad regime of these models. We first establish that, in single-index models with large additive noise, the onset of a computational bottleneck is fully characterized by the NSE. We then demonstrate that the same exponent controls a statistical-computational gap in the specialization transition of large separable multi-index models, where individual components become learnable. Finally, in hierarchical multi-index models, we show that the NSE governs the optimal computational rate in which different directions are sequentially learned. Taken together, our results identify the NSE as a unifying property linking noise robustness, computational hardness, and feature specialization in high-dimensional learning.

In high dimension, low sample size (HDLSS) settings, classifiers based on Euclidean distances like the nearest neighbor classifier and the average distance classifier perform quite poorly if differences between locations of the underlying populations get masked by scale differences. To rectify this problem, several modifications of these classifiers have been proposed in the literature. However, existing methods are confined to location and scale differences only, and often fail to discriminate among populations differing outside of the first two moments. In this article, we propose some simple transformations of these classifiers resulting into improved performance even when the underlying populations have the same location and scale. We further propose a generalization of these classifiers based on the idea of grouping of variables. The high-dimensional behavior of the proposed classifiers is studied theoretically. Numerical experiments with a variety of simulated examples as well as an extensive analysis of real data sets exhibit advantages of the proposed methods.

The standard basis of transformer hidden states is a training-free, architecture-general feature basis for detecting concepts and, in language models, steering them; with no learned dictionary. Individual dimensions act as binary registers read one at a time: their signs (+/-1) encode content, their magnitudes strength. A feature is just a subset of dimensions with a consistent sign pattern, read by counting sign agreements. We validate this Bag of Dims (BoD) framework across seven models spanning language, vision, and audio; reading dimensions one at a time loses nothing, as a full-capacity MLP adds zero AUC over per-dim reading. The same per-dimension signs appear in every modality, so they reflect transformer training itself, not the language objective. Sign alone carries predictive content: setting all magnitudes to unity preserves 60-93% top-5 next-token accuracy through the LM head. From a single-token cache (one forward pass per token, no labels) we detect 175 categories at AUC 0.97-0.99 by counting sign agreements, and from random seeds alone discovery scales to 1500 features per model. A trained probe adds only +0.018 AUC and converges to axis-aligned weights: the rotation dictionaries learn buys little. Signs are causally operative: they survive the attention projections, and flipping a concept's sign pattern in the live forward pass suppresses it. Reading and steering are separate roles in the same basis: a concept's reader dimensions are not its writer dimensions. The writer target is just as cheap, the sign of the summed unembedding rows over a few seeds, no training. Injected through the attention output pathway under closed-loop control, it steers concepts into fluent text on four language models (62-92% of twelve concepts). The signs were in the standard basis all along; the open problem is no longer finding the right rotation but cataloging what each dimension encodes.

Training modern neural networks often relies on large learning rates, operating at the edge of stability, where the optimization dynamics exhibit oscillatory and chaotic behavior. Empirically, this regime often yields improved generalization performance, yet the underlying mechanism remains poorly understood. In this work, we represent stochastic optimizers as random dynamical systems, which often converge to a fractal attractor set (rather than a point) with a smaller intrinsic dimension. Building on this connection and inspired by Lyapunov dimension theory, we introduce a novel notion of dimension, coined the `sharpness dimension', and prove a generalization bound based on this dimension. Our results show that generalization in the chaotic regime depends on the complete Hessian spectrum and the structure of its partial determinants, highlighting a complexity that cannot be captured by the trace or spectral norm considered in prior work. Experiments across various MLPs and transformers validate our theory while also providing new insights into the recently observed phenomenon of grokking.

Loss Given Default (LGD) modeling faces a fundamental data quality constraint: 90% of available training data consists of proxy estimates based on pre-distress balance sheets rather than actual recovery outcomes from completed bankruptcy proceedings. We demonstrate that this mixture-contaminated training structure causes systematic failure of recursive partitioning methods, with Random Forest achieving negative r-squared (-0.664, worse than predicting the mean) on held-out test data. Information-theoretic approaches based on Shannon entropy and mutual information provide superior generalization, achieving r-squared of 0.191 and RMSE of 0.284 on 1,218 corporate bankruptcies (1980-2023). Analysis reveals that leverage-based features contain 1.510 bits of mutual information while size effects contribute only 0.086 bits, contradicting regulatory assumptions about scale-dependent recovery. These results establish practical guidance for financial institutions deploying LGD models under Basel III requirements when representative outcome data is unavailable at sufficient scale. The findings generalize to medical outcomes research, climate forecasting, and technology reliability-domains where extended observation periods create unavoidable mixture structure in training data.

LLMs for clinical decision support often fail under small but clinically meaningful input shifts such as masking a symptom or negating a finding, despite high performance on static benchmarks. These reasoning failures frequently go undetected by standard NLP metrics, which are insensitive to latent representation shifts that drive diagnosis instability. We propose a geometry-aware evaluation framework, LAPD (Latent Agentic Perturbation Diagnostics), which systematically probes the latent robustness of clinical LLMs under structured adversarial edits. Within this framework, we introduce Latent Diagnosis Flip Rate (LDFR), a model-agnostic diagnostic signal that captures representational instability when embeddings cross decision boundaries in PCA-reduced latent space. Clinical notes are generated using a structured prompting pipeline grounded in diagnostic reasoning, then perturbed along four axes: masking, negation, synonym replacement, and numeric variation to simulate common ambiguities and omissions. We compute LDFR across both foundation and clinical LLMs, finding that latent fragility emerges even under minimal surface-level changes. Finally, we validate our findings on 90 real clinical notes from the DiReCT benchmark (MIMIC-IV), confirming the generalizability of LDFR beyond synthetic settings. Our results reveal a persistent gap between surface robustness and semantic stability, underscoring the importance of geometry-aware auditing in safety-critical clinical AI.

We study principal component analysis (PCA), where given a dataset in R^d from a distribution, the task is to find a unit vector v that approximately maximizes the variance of the distribution after being projected along v. Despite being a classical task, standard estimators fail drastically if the data contains even a small fraction of outliers, motivating the problem of robust PCA. Recent work has developed computationally-efficient algorithms for robust PCA that either take super-linear time or have sub-optimal error guarantees. Our main contribution is to develop a nearly-linear time algorithm for robust PCA with near-optimal error guarantees. We also develop a single-pass streaming algorithm for robust PCA with memory usage nearly-linear in the dimension.

Why does the low dimensionality of representations, typically dapprox 1000, not prevent modern embedding-based retrieval models from scaling to billions, or even trillions, of data points? To answer this question, we study maximal-margin embeddings in the following retrieval model, classically studied in communication complexity [PS86] and more recently in embedding-based retrieval [WBNL26]. Let Ain {0,1}^{Ntimes n} be a matrix indicating whether each of N queries is relevant to each of n documents. We are interested in the largest margin m>0, denoted by m^{rd}(d, A), for which there exist unit norm embeddings of the queries and documents {U_j}_{j = 1}^N, {V_i}_{i = 1}^n with the following property. langle U_j, V_irangle ge m whenever A_{ji} = 1 and langle U_j, V_irangle le -m otherwise. A large margin is a key proxy for representation quality: it controls both robustness to perturbations and compositional generalization across queries. Our main theorem establishes that the best possible margin without a restriction on the dimension, m^{rd}(+infty, A), can be nearly achieved in dimension d = O(m^{rd}(+infty, A)^{-2}log n) which improves a theorem of [BDES02]. Together with a matching lower bound in Theorem 1.5, we conclude that when Ain {0,1}^{n{k}times n} is the matrix containing all possible k-sparse rows once, dimension d = O(klog (n/k)) is necessary and sufficient for the maximal possible margin m^{rd}(+infty, A) = Θ(k^{-1/2}) in this setting. This fully resolves the setup of [WBNL26]. We also give several constructions for large margins when d = o(klog (n/k)). Finally, we empirically test the InfoNCE and sigmoid losses for producing large margin embeddings and demonstrate a clear advantage of the sigmoid loss.

Diffusion models operate in a reflexive System 1 mode, constrained by a fixed, content-agnostic sampling schedule. This rigidity arises from the curse of state dimensionality, where the combinatorial explosion of possible states in the high-dimensional noise manifold renders explicit trajectory planning intractable and leads to systematic computational misallocation. To address this, we introduce Chain-of-Trajectories (CoTj), a train-free framework enabling System 2 deliberative planning. Central to CoTj is Diffusion DNA, a low-dimensional signature that quantifies per-stage denoising difficulty and serves as a proxy for the high-dimensional state space, allowing us to reformulate sampling as graph planning on a directed acyclic graph. Through a Predict-Plan-Execute paradigm, CoTj dynamically allocates computational effort to the most challenging generative phases. Experiments across multiple generative models demonstrate that CoTj discovers context-aware trajectories, improving output quality and stability while reducing redundant computation. This work establishes a new foundation for resource-aware, planning-based diffusion modeling. The code is available at https://github.com/UnicomAI/CoTj.

A common assumption holds that enough observational and interventional data, given to a strong enough predictor, suffices. We report a failure mode that contradicts it. Across hundreds of structural causal models, on identified quantities a strong predictor and a Bayesian baseline both succeed, but on unidentified quantities (the couplings between counterfactual worlds) the predictor collapses to a point, on 28% of models to one no valid model can produce, while the truth is an admissible interval more data never narrows. The gap is structural: prediction cannot represent uncertainty over counterfactual couplings. We cast a world model as a single positive semidefinite coupling kernel K(T,T') over admissible worlds, whose diagonal is the ordinary posterior (what a predictor recovers) and whose off-diagonal is the cross-world coupling it cannot, which every counterfactual reads. The paper is the theory of that off-diagonal. It is real: two states with identical posteriors differ on a cross-world query, and the off-diagonal is the coupling that fixes counterfactuals. It can be bounded: positive semidefiniteness is partial-identifying information the marginals lack, and enforcing it bounds counterfactuals in polynomial time where the exact response-type program is intractable. Logical structure sharpens it: ontology axioms tighten the bound by up to a third, propagating to couplings they never touch. It can be acquired: targeted scars, constraints learned from encountered infeasibilities, close the gap several times faster than untargeted ones. Its full reconstruction is approximate counting of the admissible worlds, tractable below the Sly-Sun threshold and inapproximable above; we do not claim to beat the worst case.

Model dimension (d_{model}) is a fundamental hyperparameter in transformer language models, yet its role in setting the geometric limits of feature representation remains under-explored. Grounded in the Linear Representation and Superposition Hypotheses - which propose that models encode features as near-orthogonal directions in latent space - we develop a framework for estimating how many such directions a model can support. We first establish the embedding matrix as a measurable proxy for near-orthogonality constraints across the latent space: the boundary between meaningful token relationships and incidental similarity in the pairwise cosine similarity distribution gives a concrete estimate of the model's accepted deviation varepsilon from perfect orthogonality. Applying this metric across dozens of open-source models reveals two classes: models with high varepsilon whose embeddings lack near-orthogonal structure, and models with low varepsilon that maintain it. We then show that the standard Johnson-Lindenstrauss lemma greatly underestimates the packing efficiency of trained representations, and derive an adjusted capacity formula in which the number of near-orthogonal directions depends on the ratio of vectors to dimensions (k/d) rather than the raw count - a single modification that cuts prediction error by two orders of magnitude with no extra parameters. Combining these results, we define representational capacity as an upper bound on the number of distinguishable directions available for features and embeddings in a model's latent space. Capacity is exponentially sensitive to varepsilon, and larger models favor tighter orthogonality constraints over maximizing raw capacity - a pattern compatible with several explanations (a stability-capacity trade-off, a ceiling on usable concepts, or confounds with model scale) that we leave to future work.

We explore a class of adversarial attacks targeting the activations of language models. By manipulating a relatively small subset of model activations, a, we demonstrate the ability to control the exact prediction of a significant number (in some cases up to 1000) of subsequent tokens t. We empirically verify a scaling law where the maximum number of target tokens t_max predicted depends linearly on the number of tokens a whose activations the attacker controls as t_max = kappa a. We find that the number of bits of control in the input space needed to control a single bit in the output space (what we call attack resistance chi) is remarkably constant between approx 16 and approx 25 over 2 orders of magnitude of model sizes for different language models. Compared to attacks on tokens, attacks on activations are predictably much stronger, however, we identify a surprising regularity where one bit of input steered either via activations or via tokens is able to exert control over a similar amount of output bits. This gives support for the hypothesis that adversarial attacks are a consequence of dimensionality mismatch between the input and output spaces. A practical implication of the ease of attacking language model activations instead of tokens is for multi-modal and selected retrieval models, where additional data sources are added as activations directly, sidestepping the tokenized input. This opens up a new, broad attack surface. By using language models as a controllable test-bed to study adversarial attacks, we were able to experiment with input-output dimensions that are inaccessible in computer vision, especially where the output dimension dominates.

Integrated sensing and communication (ISAC) is envisioned be to one of the paradigms upon which next-generation mobile networks will be built, extending localization and tracking capabilities, as well as giving birth to environment-aware wireless access. A key aspect of sensing integration is parameter estimation, which involves extracting information about the surrounding environment, such as the direction, distance, and velocity of various objects within. This is typically of a high-dimensional nature, which leads to significant computational complexity, if performed jointly across multiple sensing dimensions, such as space, frequency, and time. Additionally, due to the incorporation of sensing on top of the data transmission, the time window available for sensing is likely to be short, resulting in an estimation problem where only a single snapshot is accessible. In this work, we propose PLAIN, a tensor-based estimation architecture that flexibly scales with multiple sensing dimensions and can handle high dimensionality, limited measurement time, and super-resolution requirements. It consists of three stages: a compression stage, where the high dimensional input is converted into lower dimensionality, without sacrificing resolution; a decoupled estimation stage, where the parameters across the different dimensions are estimated in parallel with low complexity; an input-based fusion stage, where the decoupled parameters are fused together to form a paired multidimensional estimate. We investigate the performance of the architecture for different configurations and compare it against practical sequential and joint estimation baselines, as well as theoretical bounds. Our results show that PLAIN, using tools from tensor algebra, subspace-based processing, and compressed sensing, can scale flexibly with dimensionality, while operating with low complexity and maintaining super-resolution.

Emergent properties have been widely adopted as a term to describe behavior not present in smaller models but observed in larger models. Recent work suggests that the trade-off incurred by quantization is also an emergent property, with sharp drops in performance in models over 6B parameters. In this work, we ask "are quantization cliffs in performance solely a factor of scale?" Against a backdrop of increased research focus on why certain emergent properties surface at scale, this work provides a useful counter-example. We posit that it is possible to optimize for a quantization friendly training recipe that suppresses large activation magnitude outliers. Here, we find that outlier dimensions are not an inherent product of scale, but rather sensitive to the optimization conditions present during pre-training. This both opens up directions for more efficient quantization, and poses the question of whether other emergent properties are inherent or can be altered and conditioned by optimization and architecture design choices. We successfully quantize models ranging in size from 410M to 52B with minimal degradation in performance.

This paper investigates discrepancies in how neural networks learn from different imaging domains, which are commonly overlooked when adopting computer vision techniques from the domain of natural images to other specialized domains such as medical images. Recent works have found that the generalization error of a trained network typically increases with the intrinsic dimension (d_{data}) of its training set. Yet, the steepness of this relationship varies significantly between medical (radiological) and natural imaging domains, with no existing theoretical explanation. We address this gap in knowledge by establishing and empirically validating a generalization scaling law with respect to d_{data}, and propose that the substantial scaling discrepancy between the two considered domains may be at least partially attributed to the higher intrinsic ``label sharpness'' (K_F) of medical imaging datasets, a metric which we propose. Next, we demonstrate an additional benefit of measuring the label sharpness of a training set: it is negatively correlated with the trained model's adversarial robustness, which notably leads to models for medical images having a substantially higher vulnerability to adversarial attack. Finally, we extend our d_{data} formalism to the related metric of learned representation intrinsic dimension (d_{repr}), derive a generalization scaling law with respect to d_{repr}, and show that d_{data} serves as an upper bound for d_{repr}. Our theoretical results are supported by thorough experiments with six models and eleven natural and medical imaging datasets over a range of training set sizes. Our findings offer insights into the influence of intrinsic dataset properties on generalization, representation learning, and robustness in deep neural networks. Code link: https://github.com/mazurowski-lab/intrinsic-properties

The measure of a machine learning algorithm is the difficulty of the tasks it can perform, and sufficiently difficult tasks are critical drivers of strong machine learning models. However, quantifying the generalization difficulty of machine learning benchmarks has remained challenging. We propose what is to our knowledge the first model-agnostic measure of the inherent generalization difficulty of tasks. Our inductive bias complexity measure quantifies the total information required to generalize well on a task minus the information provided by the data. It does so by measuring the fractional volume occupied by hypotheses that generalize on a task given that they fit the training data. It scales exponentially with the intrinsic dimensionality of the space over which the model must generalize but only polynomially in resolution per dimension, showing that tasks which require generalizing over many dimensions are drastically more difficult than tasks involving more detail in fewer dimensions. Our measure can be applied to compute and compare supervised learning, reinforcement learning and meta-learning generalization difficulties against each other. We show that applied empirically, it formally quantifies intuitively expected trends, e.g. that in terms of required inductive bias, MNIST < CIFAR10 < Imagenet and fully observable Markov decision processes (MDPs) < partially observable MDPs. Further, we show that classification of complex images < few-shot meta-learning with simple images. Our measure provides a quantitative metric to guide the construction of more complex tasks requiring greater inductive bias, and thereby encourages the development of more sophisticated architectures and learning algorithms with more powerful generalization capabilities.

We explore the consequences of multi-trace deformations in applications of gauge-gravity duality to condensed matter physics. We find that they introduce a powerful new "knob" that can implement spontaneous symmetry breaking, and can be used to construct a new type of holographic superconductor. This knob can be tuned to drive the critical temperature to zero, leading to a new quantum critical point. We calculate nontrivial critical exponents, and show that fluctuations of the order parameter are `locally' quantum critical in the disordered phase. Most notably the dynamical critical exponent is determined by the dimension of an operator at the critical point. We argue that the results are robust against quantum corrections and discuss various generalizations.

Despite rapid progress in LLM agents, performance on long-horizon, tool-using tasks remains fragile. To better understand this fragility, we ask a simple question: do all actions contribute equally to failure? Analyzing execution traces on τ-Bench (Airline/Retail) and SWE-Bench Verified, we decompose trajectories into mutating (environment-changing) vs.\ non-mutating steps and formalize decisive deviations, earliest action, level divergences that flip success to failure. A logistic regression reveals that each additional deviation in a mutating action reduces the odds of success by upto 92% on Airline and upto 96% on Retail for SoTA models. In contrast, deviations in non-mutating actions have little to no effect. Errors also grow with context length as agents drift from role and act on stale constraints. Motivated by these observations, we introduce , a model-agnostic, gradient-free, test-time safeguard that (i) adds mutation-gated verification, (ii) injects Targeted Reflection before mutating steps, and (iii) performs block-based context cleaning. delivers consistent gains, e.g., Qwen3-Thinking: +28\% relative on Airline, +11\% on Retail, and +7\% on SWE-Bench Verified; Claude: +9\%/+7\%. We further identify ceiling effects in τ-Bench, where annotation errors and underspecified tasks artificially cap model performance. To address this, we release τ-Bench Verified, which restores benchmark headroom through targeted revisions. Our results argue for action-level analysis, targeted safeguards, and reliable evaluations as prerequisites for robust multi-turn agents.

This work presents code to procedurally generate examples for the ARC training tasks. For each of the 400 tasks, an example generator following the transformation logic of the original examples was created. In effect, the assumed underlying distribution of examples for any given task was reverse engineered by implementing a means to sample from it. An attempt was made to cover an as large as reasonable space of possible examples for each task. That is, whenever the original examples of a given task may be limited in their diversity e.g. by having the dimensions of the grids, the set of symbols or number of objects constant or within tight bounds, even though the transformation does not require it, such constraints were lifted. Having access to not just a few examples per task, as the case for ARC, but instead very many, should enable a wide range of experiments that may be important stepping stones towards making leaps on the benchmark.

Chain-of-thought (CoT) reasoning and its variants have substantially improved the performance of language models on complex reasoning tasks, yet the precise mechanisms by which different strategies facilitate generalization remain poorly understood. While current explanations often point to increased test-time computation or structural guidance, establishing a consistent, quantifiable link between these factors and generalization remains challenging. In this work, we identify intrinsic dimensionality as a quantitative measure for characterizing the effectiveness of reasoning chains. Intrinsic dimensionality quantifies the minimum number of model dimensions needed to reach a given accuracy threshold on a given task. By keeping the model architecture fixed and varying the task formulation through different reasoning strategies, we demonstrate that effective reasoning strategies consistently reduce the intrinsic dimensionality of the task. Validating this on GSM8K with Gemma-3 1B and 4B, we observe a strong inverse correlation between the intrinsic dimensionality of a reasoning strategy and its generalization performance on both in-distribution and out-of-distribution data. Our findings suggest that effective reasoning chains facilitate learning by better compressing the task using fewer parameters, offering a new quantitative metric for analyzing reasoning processes.

We introduce a new family of physics-inspired generative models termed PFGM++ that unifies diffusion models and Poisson Flow Generative Models (PFGM). These models realize generative trajectories for N dimensional data by embedding paths in N{+}D dimensional space while still controlling the progression with a simple scalar norm of the D additional variables. The new models reduce to PFGM when D{=}1 and to diffusion models when D{to}infty. The flexibility of choosing D allows us to trade off robustness against rigidity as increasing D results in more concentrated coupling between the data and the additional variable norms. We dispense with the biased large batch field targets used in PFGM and instead provide an unbiased perturbation-based objective similar to diffusion models. To explore different choices of D, we provide a direct alignment method for transferring well-tuned hyperparameters from diffusion models (D{to} infty) to any finite D values. Our experiments show that models with finite D can be superior to previous state-of-the-art diffusion models on CIFAR-10/FFHQ 64{times}64 datasets, with FID scores of 1.91/2.43 when D{=}2048/128. In class-conditional setting, D{=}2048 yields current state-of-the-art FID of 1.74 on CIFAR-10. In addition, we demonstrate that models with smaller D exhibit improved robustness against modeling errors. Code is available at https://github.com/Newbeeer/pfgmpp

Nonlinear sufficient dimension reductionlibing_generalSDR, which constructs nonlinear low-dimensional representations to summarize essential features of high-dimensional data, is an important branch of representation learning. However, most existing methods are not applicable when the response variables are complex non-Euclidean random objects, which are frequently encountered in many recent statistical applications. In this paper, we introduce a new statistical dependence measure termed Fr\'echet Cumulative Covariance (FCCov) and develop a novel nonlinear SDR framework based on FCCov. Our approach is not only applicable to complex non-Euclidean data, but also exhibits robustness against outliers. We further incorporate Feedforward Neural Networks (FNNs) and Convolutional Neural Networks (CNNs) to estimate nonlinear sufficient directions in the sample level. Theoretically, we prove that our method with squared Frobenius norm regularization achieves unbiasedness at the sigma-field level. Furthermore, we establish non-asymptotic convergence rates for our estimators based on FNNs and ResNet-type CNNs, which match the minimax rate of nonparametric regression up to logarithmic factors. Intensive simulation studies verify the performance of our methods in both Euclidean and non-Euclidean settings. We apply our method to facial expression recognition datasets and the results underscore more realistic and broader applicability of our proposal.

The Universal Approximation Theorem posits that neural networks can theoretically possess unlimited approximation capacity with a suitable activation function and a freely chosen or trained set of parameters. However, a more practical scenario arises when these neural parameters, especially the nonlinear weights and biases, are bounded. This leads us to question: Does the approximation capacity of a neural network remain universal, or does it have a limit when the parameters are practically bounded? And if it has a limit, how can it be measured? Our theoretical study indicates that while universal approximation is theoretically feasible, in practical numerical scenarios, Deep Neural Networks (DNNs) with any analytic activation functions (such as Tanh and Sigmoid) can only be approximated by a finite-dimensional vector space under a bounded nonlinear parameter space (NP space), whether in a continuous or discrete sense. Based on this study, we introduce the concepts of ε outer measure and Numerical Span Dimension (NSdim) to quantify the approximation capacity limit of a family of networks both theoretically and practically. Furthermore, drawing on our new theoretical study and adopting a fresh perspective, we strive to understand the relationship between back-propagation neural networks and random parameter networks (such as the Extreme Learning Machine (ELM)) with both finite and infinite width. We also aim to provide fresh insights into regularization, the trade-off between width and depth, parameter space, width redundancy, condensation, and other related important issues.

Currently, high-dimensional data is ubiquitous in data science, which necessitates the development of techniques to decompose and interpret such multidimensional (aka tensor) datasets. Finding a low dimensional representation of the data, that is, its inherent structure, is one of the approaches that can serve to understand the dynamics of low dimensional latent features hidden in the data. Nonnegative RESCAL is one such technique, particularly well suited to analyze self-relational data, such as dynamic networks found in international trade flows. Nonnegative RESCAL computes a low dimensional tensor representation by finding the latent space containing multiple modalities. Estimating the dimensionality of this latent space is crucial for extracting meaningful latent features. Here, to determine the dimensionality of the latent space with nonnegative RESCAL, we propose a latent dimension determination method which is based on clustering of the solutions of multiple realizations of nonnegative RESCAL decompositions. We demonstrate the performance of our model selection method on synthetic data and then we apply our method to decompose a network of international trade flows data from International Monetary Fund and validate the resulting features against empirical facts from economic literature.

Deep neural networks are susceptible to adversarial examples, posing a significant security risk in critical applications. Adversarial Training (AT) is a well-established technique to enhance adversarial robustness, but it often comes at the cost of decreased generalization ability. This paper proposes Robustness Critical Fine-Tuning (RiFT), a novel approach to enhance generalization without compromising adversarial robustness. The core idea of RiFT is to exploit the redundant capacity for robustness by fine-tuning the adversarially trained model on its non-robust-critical module. To do so, we introduce module robust criticality (MRC), a measure that evaluates the significance of a given module to model robustness under worst-case weight perturbations. Using this measure, we identify the module with the lowest MRC value as the non-robust-critical module and fine-tune its weights to obtain fine-tuned weights. Subsequently, we linearly interpolate between the adversarially trained weights and fine-tuned weights to derive the optimal fine-tuned model weights. We demonstrate the efficacy of RiFT on ResNet18, ResNet34, and WideResNet34-10 models trained on CIFAR10, CIFAR100, and Tiny-ImageNet datasets. Our experiments show that \method can significantly improve both generalization and out-of-distribution robustness by around 1.5% while maintaining or even slightly enhancing adversarial robustness. Code is available at https://github.com/microsoft/robustlearn.

Self-supervised learning (SSL) has emerged as a powerful paradigm for learning representations without labeled data, often by enforcing invariance to input transformations such as rotations or blurring. Recent studies have highlighted two pivotal properties for effective representations: (i) avoiding dimensional collapse-where the learned features occupy only a low-dimensional subspace, and (ii) enhancing uniformity of the induced distribution. In this work, we introduce T-REGS, a simple regularization framework for SSL based on the length of the Minimum Spanning Tree (MST) over the learned representation. We provide theoretical analysis demonstrating that T-REGS simultaneously mitigates dimensional collapse and promotes distribution uniformity on arbitrary compact Riemannian manifolds. Several experiments on synthetic data and on classical SSL benchmarks validate the effectiveness of our approach at enhancing representation quality.

Current language model evaluations measure what models know under ideal conditions but not how robustly they know it under realistic stress. Static benchmarks like MMLU and TruthfulQA cannot distinguish a model that lacks knowledge from one whose verification mechanisms collapse when information degrades or adversaries probe for weaknesses. We introduce the Drill-Down and Fabricate Test (DDFT), a protocol that measures epistemic robustness: a model's ability to maintain factual accuracy under progressive semantic compression and adversarial fabrication. We propose a two-system cognitive model comprising a Semantic System that generates fluent text and an Epistemic Verifier that validates factual accuracy. Our findings, based on evaluating 9 frontier models across 8 knowledge domains at 5 compression levels (1,800 turn-level evaluations), reveal that epistemic robustness is orthogonal to conventional design paradigms. Neither parameter count (r=0.083, p=0.832) nor architectural type (r=0.153, p=0.695) significantly predicts robustness, suggesting it emerges from training methodology and verification mechanisms distinct from current approaches. Error detection capability strongly predicts overall robustness (rho=-0.817, p=0.007), indicating this is the critical bottleneck. We find that flagship models exhibit brittleness despite their scale, while smaller models can achieve robust performance, challenging assumptions about the relationship between model size and reliability. The DDFT framework provides both theoretical foundation and practical tools for assessing epistemic robustness before deployment in critical applications.

Riemannian diffusion models draw inspiration from standard Euclidean space diffusion models to learn distributions on general manifolds. Unfortunately, the additional geometric complexity renders the diffusion transition term inexpressible in closed form, so prior methods resort to imprecise approximations of the score matching training objective that degrade performance and preclude applications in high dimensions. In this work, we reexamine these approximations and propose several practical improvements. Our key observation is that most relevant manifolds are symmetric spaces, which are much more amenable to computation. By leveraging and combining various ans\"{a}tze, we can quickly compute relevant quantities to high precision. On low dimensional datasets, our correction produces a noticeable improvement, allowing diffusion to compete with other methods. Additionally, we show that our method enables us to scale to high dimensional tasks on nontrivial manifolds. In particular, we model QCD densities on SU(n) lattices and contrastively learned embeddings on high dimensional hyperspheres.

Medical knowledge is context-dependent and requires consistent reasoning across various natural language expressions of semantically equivalent phrases. This is particularly crucial for drug names, where patients often use brand names like Advil or Tylenol instead of their generic equivalents. To study this, we create a new robustness dataset, RABBITS, to evaluate performance differences on medical benchmarks after swapping brand and generic drug names using physician expert annotations. We assess both open-source and API-based LLMs on MedQA and MedMCQA, revealing a consistent performance drop ranging from 1-10\%. Furthermore, we identify a potential source of this fragility as the contamination of test data in widely used pre-training datasets. All code is accessible at https://github.com/BittermanLab/RABBITS, and a HuggingFace leaderboard is available at https://huggingface.co/spaces/AIM-Harvard/rabbits-leaderboard.

Prior work has shown that fine-tuning models on a narrow domain with misaligned data can lead to broad misalignment - a phenomenon termed "emergent misalignment" (Betley et al. 2025). While all tested models were susceptible to emergent misalignment, some models showed more resistance than others. Specifically the Qwen-2.5 family proved to be relatively resistant, while GPT-4o exhibited the strongest misalignment. In this paper we evaluate if current-generation open-weights models exhibit similar resistance to the Qwen-2.5 family and measure misalignment robustness over a range of model architectures and scales. We replicate the effect across nine modern open-weights models (Gemma 3 and Qwen 3 families, 1B-32B parameters). Models fine-tuned on insecure code generation show a 0.68% misalignment rate (compared to 0.07% for base models), matching the lower end of prior open-model results but dramatically lower than GPT-4o's 20%. We identify a critical format-dependent vulnerability: requiring JSON output doubles misalignment rates compared to natural language prompts (0.96% vs 0.42%). This suggests that structural constraints may bypass safety training by reducing the model's 'degrees of freedom' to refuse. These findings confirm emergent misalignment as a reproducible phenomenon in modern open-weights models, with rates substantially lower than observed in proprietary systems.

The robustness of deep neural networks is crucial for safety-critical deployments, yet existing evaluation methods are often attack-dependent and lack interpretability. We propose a principled, attack-agnostic robustness metric based on the spectral norm of the Fisher Information Matrix (FIM), which quantifies the worst-case sensitivity of the model's output distribution to input perturbations. Theoretically, we establish that the FIM equals the variance of the input Jacobian and derive closed-form spectral bounds for common architectures, including VGG, ResNet, DenseNet, and Transformer, providing the first theoretical robustness ranking. To enable scalable evaluation, we develop efficient algorithms, including power iteration and Hutchinson-based estimation, that support both white-box and black-box settings. Extensive experiments across multiple datasets, including CIFAR, ImageNet, and medical images, and across multiple architectures show a strong correlation between our metric and adversarial vulnerability. Our framework serves as an interpretable diagnostic tool that complements attack-based evaluations, offering insights into architectural sensitivity and guiding the design of more robust models. Code is available at: https://github.com/franz-chang/SRP/.

Recent work suggests that large language models may implicitly learn world models. How should we assess this possibility? We formalize this question for the case where the underlying reality is governed by a deterministic finite automaton. This includes problems as diverse as simple logical reasoning, geographic navigation, game-playing, and chemistry. We propose new evaluation metrics for world model recovery inspired by the classic Myhill-Nerode theorem from language theory. We illustrate their utility in three domains: game playing, logic puzzles, and navigation. In all domains, the generative models we consider do well on existing diagnostics for assessing world models, but our evaluation metrics reveal their world models to be far less coherent than they appear. Such incoherence creates fragility: using a generative model to solve related but subtly different tasks can lead to failures. Building generative models that meaningfully capture the underlying logic of the domains they model would be immensely valuable; our results suggest new ways to assess how close a given model is to that goal.

Measuring geometric similarity between high-dimensional network representations is a topic of longstanding interest to neuroscience and deep learning. Although many methods have been proposed, only a few works have rigorously analyzed their statistical efficiency or quantified estimator uncertainty in data-limited regimes. Here, we derive upper and lower bounds on the worst-case convergence of standard estimators of shape distancex2014a measure of representational dissimilarity proposed by Williams et al. (2021).These bounds reveal the challenging nature of the problem in high-dimensional feature spaces. To overcome these challenges, we introduce a new method-of-moments estimator with a tunable bias-variance tradeoff. We show that this estimator achieves substantially lower bias than standard estimators in simulation and on neural data, particularly in high-dimensional settings. Thus, we lay the foundation for a rigorous statistical theory for high-dimensional shape analysis, and we contribute a new estimation method that is well-suited to practical scientific settings.

Large-scale web-harvested datasets have fueled the progress of cross-modal retrieval but inevitably suffer from noisy correspondence, which severely degrades model generalization. Existing methods primarily address this by filtering out noise or seeking a substitute label, yet they predominantly remain bound by a "Discrete Selection" paradigm. We argue that relying on a single discrete proxy induces Single-Point Fragility and Discretization Error. To overcome these limitations, we propose a novel framework, Intra-modal Neighbor-aware Noise Rectification (IN2R), which shifts the paradigm from searching for a substitute to synthesizing a reliable supervision target. Leveraging the intrinsic geometric stability of intra-modal data, IN2R employs a Graph Refiner to perform relational reasoning over neighbors retrieved from a dynamic Cross-Model Memory. Instead of propagating discrete labels, our method synthesizes a continuous, soft prototype that reflects the consensus of the local semantic neighborhood, effectively rectifying inter-modal misalignment. Extensive experiments on Flickr30K, MS-COCO, and CC152K demonstrate that IN2R significantly outperforms state-of-the-art methods. Our code and pre-trained models are publicly available at https://github.com/liuyyy111/IN2R.

Generative sequence models are often trained to plan motion in physical domains, from robotics to mechanical simulations. When constructing a dataset to train such a model, engineers may curate demonstrations to specify how trajectories should be distributed over a physical quantity like travel distance or mechanical energy. For example, a roboticist building a maze navigation agent might choose demonstrations whose travel distances cover a fixed range uniformly, hoping to constrain the agent's expected power usage. We find that standard deep learning can violate this intent: each generated trajectory can seem plausible on its own, but the aggregate distribution over the physical quantity is wrong. We call this failure physical misgeneralization, and develop an account of its mechanism. Using controlled synthetic tasks, we show that physical misgeneralization arises when local errors typical of the model class propagate through the physical measurement to shift the recovered distribution. We estimate these errors with a data deviation kernel, and we use it to predict which physical quantities gain or lose mass in both our synthetic and more applied maze navigation and double-pendulum motion tasks. Finally, our mechanistic interpretation helps identify which mitigation strategies are structurally promising, and we use it to propose a kernel-informed intervention.

PGD adversarial training, the standard robustness method, can reduce Jacobian Frobenius norm yet worsen clean-input geometry (e.g., TDI 1.336 vs. ERM 1.093). We show this is not an implementation artifact but a theorem-level consequence of supervised learning. We prove that any encoder minimizing supervised loss must retain non-zero sensitivity along directions correlated with training labels, including directions that are nuisance at test time. This holds across proper scoring rules, architectures, and dataset sizes. We call this the geometric blind spot of supervised learning. This theorem unifies four empirical phenomena often treated separately: non-robust features, texture bias, corruption fragility, and the robustness-accuracy tradeoff. It also explains why suppressing sensitivity in one adversarial direction can redistribute sensitivity elsewhere. We introduce Trajectory Deviation Index (TDI), a diagnostic of geometric isotropy. Unlike CKA, intrinsic dimension, or Jacobian Frobenius norm alone, TDI captures the failure mode above. In our experiments, PGD attains low Frobenius norm but high TDI, while PMH attains the lowest TDI with one additional training term and no architectural changes. Across seven tasks, BERT/SST-2, and ImageNet ViT-B/16 (backbone family underlying CLIP/DINO/SAM), the blind spot is measurable and repairable. It appears at foundation-model scale, worsens with model scale and task-specific fine-tuning, and is substantially reduced by PMH. PMH also leads on non-Gaussian corruption types (blur/brightness/contrast) without corruption-specific training.

We study robustness to test-time adversarial attacks in the regression setting with ell_p losses and arbitrary perturbation sets. We address the question of which function classes are PAC learnable in this setting. We show that classes of finite fat-shattering dimension are learnable in both realizable and agnostic settings. Moreover, for convex function classes, they are even properly learnable. In contrast, some non-convex function classes provably require improper learning algorithms. Our main technique is based on a construction of an adversarially robust sample compression scheme of a size determined by the fat-shattering dimension. Along the way, we introduce a novel agnostic sample compression scheme for real-valued functions, which may be of independent interest.

We examine the geometry of neural network training using the Jacobian of trained network parameters with respect to their initial values. Our analysis reveals low-dimensional structure in the training process which is dependent on the input data but largely independent of the labels. We find that the singular value spectrum of the Jacobian matrix consists of three distinctive regions: a "chaotic" region of values orders of magnitude greater than one, a large "bulk" region of values extremely close to one, and a "stable" region of values less than one. Along each bulk direction, the left and right singular vectors are nearly identical, indicating that perturbations to the initialization are carried through training almost unchanged. These perturbations have virtually no effect on the network's output in-distribution, yet do have an effect far out-of-distribution. While the Jacobian applies only locally around a single initialization, we find substantial overlap in bulk subspaces for different random seeds. Our code is available at https://github.com/EleutherAI/training-jacobian

Network robustness plays a crucial role in our understanding of complex interconnected systems such as transportation, communication, and computer networks. While significant research has been conducted in the area of network robustness, no comprehensive open-source toolbox currently exists to assist researchers and practitioners in this important topic. This lack of available tools hinders reproducibility and examination of existing work, development of new research, and dissemination of new ideas. We contribute TIGER, an open-sourced Python toolbox to address these challenges. TIGER contains 22 graph robustness measures with both original and fast approximate versions; 17 failure and attack strategies; 15 heuristic and optimization-based defense techniques; and 4 simulation tools. By democratizing the tools required to study network robustness, our goal is to assist researchers and practitioners in analyzing their own networks; and facilitate the development of new research in the field. TIGER has been integrated into the Nvidia Data Science Teaching Kit available to educators across the world; and Georgia Tech's Data and Visual Analytics class with over 1,000 students. TIGER is open sourced at: https://github.com/safreita1/TIGER

Sparse autoencoders (SAEs) are widely used to interpret neural network representations, but their utility depends on whether the learned features are reproducible across training runs. We study this question through feature stability: for each SAE feature, we estimate the probability that a similar feature reappears in an independently trained SAE. This yields a scalable per-feature signal that separates stable from unstable features. In a large-scale study across seeds, models, layers, dictionary sizes, and SAE variants, we find a pronounced functional asymmetry: stable features carry most of the reconstruction- and prediction-relevant signal, while unstable features have weak marginal impact and are dominated by low-frequency surface-form triggers in both activation statistics and automatic explanations. Geometrically, unstable features are individually non-reproducible but concentrate in reproducible lower-rank subspaces, suggesting that seed dependence often reflects basis ambiguity within a shared region of activation space rather than pure noise. A controlled synthetic model makes this mechanism explicit, showing that low-rank ground-truth features can be recovered at the subspace level while remaining non-identifiable as individual SAE latents across seeds. Finally, by pooling unique cross-seed features, we construct more stable SAEs while preserving explained variance in this setting. Together, these results show that unstable features are not merely failed or noisy latents: they have weak individual functional impact, but reflect reproducible low-dimensional structure that standard SAEs resolve differently across seeds.