Reference-Free Metric

Despite their high accuracy on point-wise metrics, machine learning weather forecasting models can exhibit different failure modes such as blurring, periodic irregularities, and other unphysical spatial artifacts. This has motivated a variety of metrics to detect known failure cases. Existing metrics fix a representation or transformation in advance, and that choice limits the artifacts they can detect. We propose to train a discriminator for separating reference data from the model's output, and using its output logit to obtain a divergence-like realism score. The discriminator learns whatever separates the model's fields from real weather, adapting to whichever failure mode that model exhibits. We compare our learned atmospheric critic to existing metrics using various synthetic corruptions applied to ERA5 reanalysis data. Our method successfully identifies the corruptions and ranks their severity, while existing metrics fail on at least one corruption. Additionally, we evaluate forecasts from real weather models, and find that the realism score degrades with longer lead times and the metric generally assigns higher realism to numerical models than to machine learning models.

Training data constrains optimizer geometry through the covectors visible to a declared information channel. We study how such partial information determines a full positive cometric relative to a reference and which degrees of freedom remain unidentified. Our central result resolves full-column-rank positive-definite compression under affine-invariant Riemannian geometry. The compression map is a split-Hadamard metric submetry and admits an explicit unique completion that is the affine-invariant nearest full geometry realizing a visible target and yields exact full-to-visible variational reduction. When the channel moves, the completions form a gauge-invariant rank stratification of the positive-definite cone. Its closed-form pullback pair metric separates visible-metric motion from subspace rotation through a reference-mismatch weight, yields an explicit positive-semidefinite multi-direction Gram matrix, and exposes the precise singularity of reference-valued modes. The mechanism is explained by a metric theorem equating ball submetry, attained fiber distance, and lossless reduction of every monotone radial visible decision problem. A smooth split-Hadamard theorem supplies coherent information sheets, proximal commutation, and solution-wise gradient-flow lifting. The positive-definite realization also gives closed-form prior-data shrinkage. Diagonal and block optimizer families reduce to relative-interior conic image tests with valid facial certificates, while deterministic and finite-sample bounds quantify recovery of the visible geometry and its subspace. Together these results characterize exact reduction, reference-dependent completion, and structured expressivity for the stated finite-dimensional affine-invariant model.

In Yorùbá, pitch alone separates \d{o}k\d{o} (husband, Mid), \d{o}k\d{ò} (vehicle, Low), and \d{o}k\d{ó} (hoe, High) -- the diacritics ARE the tone marks. Yet character error rate (CER), the standard automated metric for text-to-speech (TTS), is in practice computed from ASR output that drops those marks: a synthesizer can ace CER and still say vehicle for husband. We introduce DunDun -- named for the dùndún, the Yorùbá talking drum that speaks through pitch alone -- an automated, reference-free lexical-tone metric that needs no tone-labelled corpus. The gold High/Mid/Low sequence is read from the input text's diacritics (in TTS that text exists by construction, so no reference recording is needed); the prediction comes from the audio's pitch track. We validate three ways. Flattening pitch with PSOLA resynthesis collapses DunDun while CER does not move. Inverting High and Low in the answer key of 300 native recordings drives the two-class readout to 0.14, symmetrically below its 0.35 chance level -- a consistency check on the scoring path, not independent evidence. And three native listeners, over 67 blind A/B trials, pick the tone-correct clip 89.6% of the time (95% CI 80.0-94.8; p < 1e-4); whether DunDun tracks those judgements trial by trial is not resolved at this sample size. Applied to a massively multilingual zero-shot TTS model, DunDun shows what CER cannot: Yorùbá tone sits near the native anchor before any Yorùbá fine-tuning (0.567 +/- 0.02 over five decode seeds vs. 0.596; chance 0.33), despite the 21.4% CER the model's own paper reports; and a few hours of clean audio halve CER (5.6% to 2.7% by 5h, 1.7% by 15h) while tone saturates within the hour. On non-tonal Swahili, CER already captures the gains: the metric a language needs is language-dependent. We release the metric and the complete validation protocol.

Code-level autonomous research loops (ARLs) have recently emerged as a concrete object of study in automated machine learning research. In such loops, an LLM agent proposes modifications to an experimental training pipeline, executes the modified pipeline, and retains edits that improve a verifiable in-loop metric. Although executable metrics may appear to provide a reliable signal of progress, it remains unclear whether repeated metric-driven code editing leads to genuine improvements that generalize beyond the loop. We provide a systematic diagnosis of this question. Across various experiment settings, we identify a robust failure mode that we call \textbf{algorithmic mode collapse}. In this regime, surface-level edit diversity remains stable, but semantic and mechanism-level diversity collapse: the agent continues to edit different lines of code while repeatedly proposing the same kinds of algorithmic changes. This collapse is accompanied by a widening gap between in-loop metric gains and gains measured on independent held-out evaluations. We then propose Diversity-Aware Proposal Sampling (\textsc{DAPS}), a lightweight mitigation that combines category-coverage reweighting, persistent edit memory, and a validation gate. Under a three-tier protocol separating the in-loop metric, the audit metric read by the gate, and a blind metric no loop component ever accesses, \textsc{DAPS} reduces semantic-cluster decay of edits by $69.1\%$ and improves relative faithfulness by $83.7\%$ blind and $81.6\%$ audited, while preserving in-loop optimization speed. We provide the code in Github \href{https://github.com/BokwaiHo/arl-mode-collapse}{repository}.

Large language models can write parametric CAD programs from a natural-language description (text-to-CAD generation), but a single sample is often wrong. Increasing test-time compute by sampling multiple candidates only helps if a good candidate can be identified, yet no ground-truth model is available at generation time. Existing systems often require a separate verifier, such as a vision-language judge, to select among candidates. We investigate whether the candidate pool itself provides enough signal for effective selection and a verifier-free alternative. We introduce 3D CAD consensus selection, hereafter consensus selection: sample $N$ parametric CAD programs, compile them to 3D models, and return the candidate that agrees most with the rest of the pool. The method is training-free and compatible with existing CAD agents. We investigate geometric and topological notions of agreement, each of which improves its corresponding evaluation metric. On the exact candidate pools of a state-of-the-art CAD generation method, geometric consensus improves all three geometric metrics over the method's verifier, while topological consensus matches it on topology. Across every tested LLM and prompt variant, geometric consensus also improves geometric accuracy over random selection from the same pool, reducing Chamfer distance by $1-10\%$.

Probabilistic Circuits (PCs) are tractable generative models whose internal nodes encode a hierarchy of probabilistic sum- maries over different variable scopes. Existing PC-based out- of-distribution (OOD) detection methods ignore this hierar- chy, reducing the entire circuit to the scalar likelihood (or its uncertainty) computed at the root. We introduce Hierar- chical Likelihood Vector (HLV), a representation whose en- tries are the likelihoods associated with selected PC nodes and define the Hierarchical Likelihood Distance (HLD), a PC-induced pseudo-metric that compares the probability dis- tributions through the expectations of their HLVs. We show that HLD is an integral probability metric over a function class naturally induced by the PC and develop a principled goodness-of-fit hypothesis test for unsupervised OOD detec- tion. Unlike existing approaches, the trained PC alone serves as the representation of the in-distribution: no held-out in- distribution data are required at deployment. We further show that the quantities required by the hypothesis test can be com- puted exactly, directly from the trained circuit, yielding an ap- proximate analytic decision threshold. Experiments on tabular and MNIST datasets demonstrate that exploiting the hierarchi- cal probabilistic summaries encoded through the PC improve OOD detection over root-likelihood, uncertainty-, typicality- and kernel-based baselines, while naturally localizing distri- bution shifts to the PC nodes responsible for the shift.

Although large language models can translate some historical languages surprisingly well, their usefulness in digital humanities workflows is limited by the lack of reliable evaluation. We investigate whether existing automatic evaluation metrics developed for modern languages are reliable in this setting, using translation from Classical Chinese to English as a test case. We introduce a diagnostic framework based on minimal pairs capturing error types salient in scholarly use, probing both reference-based and reference-free metrics for error sensitivity and tolerance to valid variation. We find that all metrics exhibit blind spots, however MetricX24 performs best overall. Our findings highlight the need for more robust and interpretable metrics for historically and culturally distinct translation settings.

Computing distances between sets is essential in robotic motion planning and control, where differentiable gradients enable real-time optimization. The Euclidean Signed Distance Function (SDF), however, is not differentiable everywhere, and existing alternatives often sacrifice differentiability, sign information, or computational efficiency. In this letter, we introduce a novel differentiable signed distance between convex polyhedra. To this end, we first propose differentiable versions of the minimum and maximum operators, termed the Hölder minimum and Hölder maximum. We then replace the original min-max operators in the classical SDF formulation, yielding the Hölder signed distance. Unlike prior differentiable distance formulations that rely on iterative algorithms, our approach is computed in closed form, eliminating convergence issues while remaining naturally amenable to GPU parallelization. We validate the practical advantages and computational performance of the proposed distance through runtime comparisons with existing approaches. We also present a robotic manipulator experiment, demonstrating its suitability for applications in control.

We introduce a continuous metric field framework trained by a single causal contrastive loss. The framework encodes a scene into coefficients of a fixed symmetric matrix basis, assembles them into a Lie algebra element, and exponentiates the result to a Riemannian or Lorentzian metric. Across dimensions, this field discovers the full spectrum of geometric structures: from obstacle-avoiding geodesics in robot navigation across planar and manipulator configuration spaces, to event horizons of black holes in Lorentzian spacetime. Extensive zero-shot generalization studies demonstrate that the field captures transferable geometric structure rather than memorizing specific configurations. In the black hole setting, the causal loss spontaneously evolves genuine black-hole-like structures with the correct Lorentzian signature. The same loss, the same architecture, and the same training protocol produce the full range of geometric phenomena across dimensions. The field knows geometry, and geometry knows physics.

Adaptive rounding methods such as GPTQ, or equivalently Babai's nearest plane algorithm, round a real matrix to integers under a quadratic metric. They process the entries in a fixed order, one at a time, propagating each rounding error to the entries not yet processed through a triangular feedback matrix. We study the two-sided version of this task, in which fixed nonsingular basis matrices act on both the left and the right of the residual; the familiar one-sided case is the special case of an identity right basis. Vectorizing the matrix turns the two-sided objective into a quadratic metric whose Gram matrix is a Kronecker product, so the one-dimensional algorithm applies verbatim, but takes quartic time in the matrix dimension. We present GPTQ-2D, which produces the identical rounded matrix in cubic time. It rounds the entries anti-diagonal by anti-diagonal; entries on the same anti-diagonal are independent and are rounded in parallel.

Monocular foundation models provide dense geometry but usually lack a stable metric scale. This paper presents VIDAR, a visual-inertial dense reconstruction framework that couples SVO+IMU odometry with Depth Anything 3. VIDAR uses the visual-inertial front end as a metric anchor: it provides camera poses, scale, and a consistent world frame for aligning dense foundation-model predictions across time. The foundation model then contributes detailed local geometry that is fused into a global reconstruction. We study both pose-conditioned DA3 and a decoupled alignment strategy. On EuRoC, pose injection reduces scale error to about 1% and reaches 0.463 mean F@0.10; the decoupled hybrid improves this to 0.676 without ground-truth poses. Results on EuRoC and TUM RGB-D show that VIDAR is a practical route to metric dense monocular reconstruction.

We present FoundationGeo, a two-stage framework that explicitly bridges relative and metric prediction via spatial calibration and principled data design. Stage 1 learns a high-fidelity, affine-invariant geometry model by initializing with DINOv3 and training on a curated 10.2M-sample multi-domain corpus with complementary local-detail supervision, yielding sharp boundaries and strong cross-domain generalization. Stage 2 moves beyond global scaling by introducing lightweight pixel-wise calibration fields for metric estimation: a scale field for spatially varying metric alignment and a ray-direction correction field that mitigates directional bias in point-map geometry, together producing metrically consistent 3D point maps. Beyond model design, we identify camera intrinsic coverage, especially focal length distribution mismatch between training and test data, as a key bottleneck for zero-shot metric generalization: performance drops sharply when test intrinsics fall outside the training distribution. To address this, we synthesize additional training data across diverse focal lengths using a Blender-based data engine, repairing under-covered focal regimes and improving robustness under intrinsic shift. Extensive zero-shot evaluations across seven benchmarks show that FoundationGeo significantly strengthens cross-domain robustness, staying near the top across diverse domains while avoiding the sharp cross-domain performance drops observed in other methods. This consistency translates into the best overall performance, surpassing heavier baselines by over 5.2% on average.

LLM agent benchmarks measure task completion, reliability, and inference cost, but not the persistent data an agent run leaves on disk, including logs, context snapshots, checkpoints, and debug traces. We introduce AgentFootprint, a cross-framework benchmark of post-run agent storage footprint. Its serialization-aware metric suite measures total retention, channel composition, duplication, growth, compressibility, and conversation-history reconstructability. It addresses a measurement trap: naive byte-level measurement understates duplication by an order of magnitude because database paging and JSON escaping obscure repeated content. A fixed-trace control separates agent-generated logical volume from persistence-layer amplification: replaying the same trajectory through seven persisting frameworks yields a 6.7x spread. Under identical models, tools, and tasks, configurations with 100% accuracy differ by 15.7x in retained bytes, although their defaults support different recovery and audit capabilities. Three full-history configurations grow superlinearly on a repeated-observation stress task. Exported trajectories from 108 instance-normalized SWE-bench Verified submissions span three orders of magnitude per instance, with no detectable correlation with resolve rate. A content-addressed store reduces retention by 4.8x-32.7x while preserving every reconstructability score. These results establish persistent storage as a resource metric to report jointly with accuracy and reconstructability.

Vision-centric autonomous driving requires accurate metric geometry and ego-motion estimation from synchronized multi-camera observations. Recent visual geometry models show strong performance in pose estimation, depth prediction, and 3D reconstruction, but are not tailored to rigid multi-camera driving systems. They often encode camera poses as entangled representations, in which time-varying ego-motion and static camera-rig geometry are jointly modeled, limiting the utilization of vehicle-side geometric priors. We propose Trajectory-Rig Decoupled Metric Geometry Learning (TRIG), a geometry perception framework for autonomous driving. TRIG factorizes camera poses into ego-trajectory and camera-rig components, enabling separate modeling of ego-motion and static multi-camera topology. We introduce decoupled pose encoding and supervision, which separately constrain trajectory evolution and rig geometry for metric-consistent learning. Moreover, sparse Temporal--Spatial attention separates cross-camera interaction from temporal aggregation, reducing global attention cost while preserving geometric reasoning. Experiments on five autonomous driving benchmarks show that TRIG achieves state-of-the-art performance in pose estimation, metric depth prediction, and 3D reconstruction.

The AInstein architecture introduced an unsupervised neural method for solving the Riemannian Einstein equations on arbitrary manifolds. This Physics Informed Neural Network approach (PINN) is extended here to Lorentzian signature, validated by recovering the maximally extended Schwarzschild geometry, and tested as novel search method for arbitrary black hole solutions. The topology is built into the architecture by treating $S^{2}$ globally through its standard embedding, such that the network learns an ambient metric on the manifold $\mathbb{R}^{2} \times \mathbb{R}^{3}$, where Penrose coordinates are chosen for $\mathbb{R}^2$ and the metric on $S^{2}$ is obtained by pullback. The architecture is first trained with the objective of recovering the Schwarzschild metric via losses encoding the vacuum Einstein equation, a quadratic Weyl scalar constraint, and the $SO(3)$ symmetry of the resultant metric; directly motivated by the Birkhoff--Jebsen theorem. Following this, the objective is generalised to use the Petrov speciality index, a horizon curvature anchor, and a trapped-surface constraint, to allow search for algebraically general Petrov type I solutions, finding potentially novel general-type Lorentzian Einstein metrics with a genuinely trapped interior.

The standard way to compare two text embeddings is cosine similarity. Scattered studies report that a different metric does better, but never pin down the geometric condition that decides when, or why. We settle both with a comprehensive empirical study: nineteen parameter-free similarity metrics on nineteen encoders, from compact sentence transformers up to seven-billion-parameter large language models, across seven datasets. The answer is geometric. When an encoder spreads its variance evenly across directions, cosine is the best parameter-free choice and no other metric helps by a usable margin. When the variance concentrates into a few dominant directions, a property known as anisotropy, rank-based and L1-type metrics beat cosine by a clear margin. The absolute gain is modest, but because cosine starts low on these encoders it is a sizable relative improvement, around twenty percent on average and largest where cosine is weakest. What decides this is the geometry of the embedding space, not how the model was trained: where the two disagree, the metric follows the geometry. One number, the fraction of variance held by the single most dominant dimension, predicts how much the alternatives help across all nineteen encoders, with a rank correlation of 0.86 and a linear correlation of 0.95. To test this as the cause rather than a correlate, we project out the dominant directions: cosine recovers and the advantage of the other metrics nearly vanishes, but only on the encoders that were anisotropic to begin with. The effect is directional, not magnitude based, since it survives normalizing every vector to unit length. Among parameter-free metrics, then, cosine is the right tool wherever an encoder is well spread, which includes the fine-tuned embedders commonly deployed for retrieval, and we give a one-number diagnostic for when it is not.

Graph-based narrative extraction relies on a coherence function to score transitions between events, but the coherence metrics in current use are defined operationally and lack an information-theoretic foundation. We study the composite metric $C=\sqrt{A\cdot T}$, where $A$ is the angular similarity of document embeddings and $T=1-d_{\mathrm{JS}}$ is a topic proximity from the Jensen-Shannon distance of soft memberships, and give it an information-geometric reading together with an axiomatic characterization of the geometric-mean combinator. On the product manifold $\mathbb{S}^{d-1}\timesΔ^{K-1}$, the negative log-coherence decomposes additively into an angular and a topic cost. Because the Riemannian metric tensor induced by the Jensen-Shannon distance on the simplex is proportional to the Fisher information matrix, the topic component is locally consistent with the Fisher-Rao metric singled out by Chentsov's theorem. Within the compensability spectrum of combinators, the geometric mean is the unique one consistent with four natural axioms (a boundary/veto condition, symmetry, log-additivity, normalization), and the construction motivates a proper product metric $d_\times$. Experiments on four corpora, three embedding families, and three topic models are consistent with the framework: the Fisher identity holds ($R\ge0.99$), the geometric mean tracks $d_\times$ closely ($ρ=0.999$), and a downstream LLM-as-judge check finds it is not dominated by any alternative combinator or single-channel baseline. Sweeping the spectrum, the bottleneck-coherence gap between extracted and random storylines splits into a symmetric component, maximized at the geometric mean across five corpora, and a displacement term; a cross-modal image-narrative case study reproduces the effect. These results justify the composite coherence metric and articulate when the geometric mean is the natural choice.

High-dimensional optimal transport is seldom available in closed form. The one-dimensional case is exceptional because the order of the real line is compatible with convex transport costs, making monotone rearrangement optimal. This paper studies when an analogous Monge structure can be recovered in higher dimensions from a partial order. We introduce a cone-compatible Monge geometry: a closed convex cone (K) induces the order (x\preceq_K y) whenever (y-x\in K), and is compatible with a cost if ordered pairs satisfy a Monge exchange inequality. For squared Mahalanobis costs (c_M(x,y)=(x-y)^\top M(x-y)), we prove a sharp characterization: compatibility holds exactly when (K) is acute under the (M)-inner product, namely (u^\top Mv\ge0) for all (u,v\in K), equivalently (K\subseteq K_M^*). Under this condition, measures supported on cone chains admit a quantile-type closed-form optimal coupling, yielding exact transport under the original ground cost rather than after projection or metric replacement. We distinguish the resulting cone-chain Wasserstein metric on canonically ordered chain distributions from an extended directed cone transport cost on general measures, and develop feasibility, duality, stability, approximation, Gaussian recovery, statistical, and computational results. The theory is complementary to sliced and tree Wasserstein distances: it is not a universal fast surrogate, but a way to obtain interpretable, direction-valid, original-space monotone transport for ordered high-dimensional data.

Many real-world classification tasks require predicting multiple labels per instance, necessitating the optimization of complex evaluation metrics such as the $F$-measure and Jaccard index. While the Empirical Utility Maximization (EUM) framework is natural for these population-level metrics, existing theoretical results are largely limited to asymptotic Bayes-consistency. In this paper, we develop principled learning algorithms for optimizing a broad class of generalized metrics within the EUM framework, grounded in the stronger notion of $H$-consistency. Our key contribution is the design of novel surrogate loss functions for multi-label learning that admit provable $H$-consistency bounds, enabling optimization with non-asymptotic guarantees tailored to the hypothesis class and finite samples. Crucially, we prove these combinatorially formulated surrogates decompose exactly, operating in strictly $O(l)$ time without approximations. Building on this foundation, we introduce MMO (Multi-Label Metric Optimization), a new family of algorithms for optimizing generalized linear-fractional metrics. We validate our approach through extensive experiments, demonstrating robust scalability and superior performance over state-of-the-art continuous baselines on large-scale datasets (MS-COCO, Reuters-21578) in high-sparsity, deep learning regimes. Our results offer both theoretical rigor and practical effectiveness for general multi-label metric optimization.

Model merging offers a promising avenue for knowledge integration and parallel development without retraining. Yet, existing methods either ignore the geometry of the loss landscape or rely on intractable full-space Hessian approximations. We propose EpiMer, a framework that casts model merging as solving the Fréchet mean on a Riemannian manifold and restricts the computation to a low-rank subspace spanned by the task vectors. With the expected Hessian as the metric, we reveal a connection between local curvature and epistemic uncertainty of the parameters. Our theoretical analysis decomposes the merging error bound into the subspace Fréchet variance and the residual energy, and provides a closed-form characterization of when curvature-aware merging provably outperforms flat-geometry methods. In addition, our framework unifies both curvature-aware methods and recent spectral methods as special cases of the subspace Fréchet mean with different geometric metrics. Merging fine-tuned CLIP-ViT models on eight image classification tasks, Epistemic Merging strictly outperforms the baselines on all three CLIP-ViT backbones at matched rank, improving the across-task average accuracy and worst-task accuracy on every backbone.

Geometric deep learning organises neural architectures around the symmetries of their data domain, with the choice of symmetry group serving as a geometric prior that determines what representations can be learned. Metric-Aware Principal Component Analysis (MAPCA) parameterises principal component analysis by a positive-definite metric matrix, with a canonical subfamily interpolating between standard PCA and output whitening and a diagonal-metric point recovering Invariant PCA (IPCA). This paper positions MAPCA within the geometric deep learning framework. The metric is read as the geometric prior; the orthogonal group preserving it is the symmetry group it induces; MAPCA solutions are equivariant under this group with the resulting spectrum invariant; and MAPCA's defining constraint is the linear analogue of the Schur-type weight constraints used in equivariant networks. Across six axes - domain, symmetry group, equivariance, invariance, architectural primitive, and geometric prior - we construct a precise dictionary between MAPCA and geometric deep learning. The technical anchor is a uniqueness theorem characterising IPCA as the unique linear data-derived metric in the MAPCA family that is equivariant under arbitrary diagonal rescaling and projects onto the fixed-point set of the action, equivalent under normalisation to the variance-maximisation criterion in its precise form. The paper closes with three bridges: kernel PCA as the nonlinear extension, spectral graph methods as MAPCA on graphs, and a deep MAPCA construction extending the positioning into deep equivariant networks

Traditional robotic hand metrics focus on static properties such as workspace, manipulability, and grasp stability. However, these metrics do not directly measure dexterity under the standard definition in robotic manipulation: the ability to continuously change an object's pose within the hand while maintaining contact from an initial grasp. We introduce Kinematic Rolling Manipulation Ability (KaRMA), a kinematic-only metric for fine manipulation that quantifies reachable in-hand translation and reorientation of a spherical test object within a two-finger precision pinch through feasible rolling motions. KaRMA enforces joint limits, collision constraints, rolling contact, and antipodal force feasibility, then investigates reachable in-hand object poses via breadth-first search over translation and rotation primitives. KaRMA reports three scores: translational coverage (KaRMA-T), rotational coverage (KaRMA-R), and sensitivity to the initial grasp (KaRMA-S). We evaluate KaRMA on 16 widely used robotic hands and compare against static baselines, showing that KaRMA separates hands that rank identically under static proxies, reveals translation-rotation tradeoffs invisible to existing baselines, and is qualitatively consistent with selected published task benchmarks where Jacobian-based metrics can be misleading.

In many real-world computer vision applications, including medical imaging and industrial inspection, binary classification tasks are characterized by a severe scarcity of positive samples. A widely adopted solution is to generate synthetic positive data using image-to-image transformations applied to negative samples. However, a fundamental challenge remains: how can we reliably assess whether such synthetic data will improve downstream model performance? In this work, we propose a geometry-driven metric that predicts the utility of synthetic data without requiring model training. Our approach operates in the embedding space of a pre-trained foundation model and represents the dataset through difference vectors between samples. We evaluate whether the weight vector of a linear classifier can be expressed within the subspace spanned by these variations by measuring the relative projection error. Intuitively, if the variations induced by synthetic data capture task-relevant directions, their span can approximate the classifier, resulting in low projection error. Conversely, poor synthetic data fails to span these directions, leading to higher error. Across multiple datasets and architectures, we show that this metric exhibits strong correlation with downstream classification performance of CNNs trained on mixtures of real negative and synthetic positive data. These findings suggest that the proposed metric serves as a practical and informative tool for evaluating synthetic data quality in data-scarce settings.

Mean-pooled cosine similarity is the default metric for comparing neural representations across languages, modalities, and tasks. We establish that this metric is not length-invariant: under the anisotropy that characterizes modern transformer representations, mean-pooled cosine grows monotonically in sequence length, independent of representational content. Empirically, on HumanEvalPack across four code LLMs, the length ratio alone explains $R^2 = 0.52$--$0.75$ of cross-language "Python proximity," while AST depth and shared-token fraction add less than 3% of explained variance beyond length. Substituting Centered Kernel Alignment (CKA) reduces explained variance by 83% and reverses the sign of the length coefficient ($β_{\mathrm{len}}: +0.86 \to -0.37$). The same pattern holds in Mistral-7B on parallel WMT pairs ($R^2 = 0.23$ EN-FR, $R^2 = 0.33$ EN-DE for cosine; $R^2 < 0.01$ for CKA). In CLIP ViT-B/32, mean-pooling reduces the length effect relative to EOS-pooling ($R^2: 0.21 \to {<}0.01$), as predicted by the theory's dependence on anisotropy. We argue that length-invariant metrics such as CKA should be the default for cross-representation comparisons, and that recent claims of cross-lingual representational convergence built on mean-pooled cosine warrant re-examination.

Evaluating image captions without references remains challenging because global embedding similarity often misses fine-grained mismatches such as hallucinated objects, missing attributes, or incorrect relations. We propose MSD-Score, a reference-free metric that models image patch and text token embeddings as von Mises-Fisher mixtures on the unit hypersphere. Instead of treating each modality as a single point, MSD-Score formulates image-text matching as a multi-scale distributional scoring problem. Semantic discrepancies are quantified via a weighted bi-directional KL divergence and combined with global similarity in a multi-scale framework for both single- and multi-candidate evaluations. Extensive experiments show that MSD-Score achieves state-of-the-art correlation with human judgments among reference-free metrics. Beyond accuracy, its probabilistic formulation yields transparent and decomposable diagnostics of local grounding errors, providing a deterministic complementary signal to holistic similarity metrics and judge-based evaluators.

Optimal transport (OT) is a central framework for modeling distribution shifts. Because OT compares distributions directly in input space, a well-designed ground metric between observations is essential to ensure that the optimizer does not violate the true geometry of change. We propose Displacement-Reshaped Optimal Transport (ReshapeOT), a method that reshapes the ground metric by integrating observed sample displacements as an additional source of knowledge. Technically, ReshapeOT replaces the Euclidean metric with a Mahalanobis distance estimated from displacement second moments. This effectively carves expressways through the input space, inviting transport solutions that better align with observed displacements. Our method is computationally lightweight, integrates seamlessly into any OT solver that operates on a cost matrix, and can be kernelized for further flexibility. Experiments on synthetic and real-world data show that ReshapeOT achieves substantial gains in transport reliability. We further demonstrate our method's usefulness in two practical use cases.

We investigate \emph{magnitude} as a new unary and strictly Pareto-compliant quality indicator for finite approximation sets to the Pareto front in multiobjective optimization. Magnitude originates in enriched category theory and metric geometry, where it is a notion of size or point content for compact metric spaces and a generalization of cardinality. For dominated regions in the $\ell_1$ box setting, magnitude is close to hypervolume but not identical: it contains the top-dimensional hypervolume term together with positive lower-dimensional projection and boundary contributions. This paper gives a first theoretical study of magnitude as an indicator. We consider multiobjective maximization with a common anchor point. For dominated sets generated by finite approximation sets, we derive an all-dimensional projection formula, prove weak and strict set monotonicity on finite unions of anchored boxes, and thereby obtain weak and strict Pareto compliance. Unlike hypervolume, magnitude assigns positive value to boundary points sharing one or more coordinates with the anchor point, even when their top-dimensional hypervolume contribution vanishes. We then formulate projected set-gradient methods and compare hypervolume and magnitude on biobjective and three-dimensional simplex examples. Numerically, magnitude favors boundary-including populations and, for suitable cardinalities, complete Das--Dennis grids, whereas hypervolume prefers more interior-filling configurations. Computationally, magnitude reduces to hypervolume on coordinate projections; for fixed dimension this yields the same asymptotic complexity up to a factor $2^d-1$, and in dimensions two and three $Θ(n\log n)$ time. These results identify magnitude as a mathematically natural and computationally viable alternative to hypervolume for finite Pareto front approximations.

Expert specialization is fundamental to Mixture-of-Experts (MoE) model success, yet existing metrics (cosine similarity, routing entropy) lack theoretical grounding and yield inconsistent conclusions under reparameterization. We present an information-geometric framework providing the first rigorous characterization of MoE specialization dynamics. Our key insight is that expert routing distributions evolve on the probability simplex equipped with the Fisher information metric, enabling formal analysis via Riemannian geometry. We prove that standard heuristic metrics violate parameterization invariance (Theorem 1), establish that specialization corresponds to geodesic flow with quantified approximation bounds (Theorem 2), and derive a failure predictor with theoretical threshold justification (Theorem 3). The framework introduces two principled metrics: Fisher Specialization Index (FSI) achieving r=0.91+/-0.02 correlation with downstream performance, and Fisher Heterogeneity Score (FHS) predicting training failure at 10% completion with AUC=0.89+/-0.03 -- outperforming validation-loss-based early stopping by 23% while requiring 40x fewer compute cycles. We validate intervention protocols achieving 87% recovery rate when FHS>1 is detected. Comprehensive experiments across language modeling (WikiText-103, C4), vision MoE (ImageNet), and scaling studies (8-64 experts, 125M-2.7B parameters) validate our theoretical predictions.

Artificial intelligence is increasingly embedded in human decision-making, yet distinguishing systems that genuinely amplify human cognition from those promoting excessive dependence remains underdefined. This paper introduces a framework to distinguish cognitive amplification (improving hybrid performance without degrading human capability) from cognitive delegation (outsourcing reasoning to the AI). We define four metrics: the Cognitive Amplification Index (CAI*), Dependency Ratio (D), Human Reliance Index (HRI), and Human Cognitive Drift Rate (HCDR). We test this framework in an agent-based NetLogo simulation across three reliance regimes and multiple dependency-atrophy configurations, performing constrained optimizations and parameter sweeps to determine if positive collaborative gain is recoverable. Finally, we introduce an extension with an explicit human-AI interaction term. Our metrics effectively distinguish degenerate AI-dominated delegation, capability-preserving but weakly competitive interaction, and structurally dependent boundary regimes. Across all baseline configurations, no regime achieves positive collaborative gain relative to the best standalone baseline, even when reducing capability atrophy to zero. This limitation proves structural rather than merely parametric. Positive collaborative gain (CAI* > 0) becomes attainable only after introducing an explicit interaction term allowing retained human capability to contribute directly to the assisted output. This framework provides a basis for evaluating whether human-AI systems remain cognitively sustainable. The results suggest that preventing capability erosion alone is insufficient for genuine amplification if the architecture remains delegation-oriented. Amplification requires both preserved human capability and a coupling mechanism through which it contributes productively to the hybrid outcome.

Geometric properties of Transformer weights, particularly the unembedding matrix, have been widely useful in language model interpretability research. Yet, their utility for estimating downstream performance remains unclear. In this work, we systematically investigate the relationship between model performance and the unembedding matrix geometry, particularly its effective rank. Our experiments, involving a suite of 108 OLMo-style language models trained under controlled variation, reveal several key findings. While the best-performing models often exhibit a high effective rank, this trend is not universal across tasks and training setups. Contrary to prior work, we find that low effective rank does not cause late-stage performance degradation in small models, but instead co-occurs with it; we find adversarial cases where low-rank models do not exhibit saturation. Moreover, we show that effective rank is strongly influenced by pre-training hyperparameters, such as batch size and weight decay, which in-turn affect the model's performance. Lastly, extending our analysis to other geometric metrics and final-layer representation, we find that these metrics are largely aligned, but none can reliably predict downstream performance. Overall, our findings suggest that the model's geometry, as captured by existing metrics, primarily reflects training choices rather than performance.

Representational similarity analysis and related methods compare the internal geometries of neural networks, but they measure only alignment between spaces, leaving a blind spot -- whether a representation's structure is reliably recoverable, not merely similar. We introduce geometric stability, a distinct axis, and \textit{Shesha}, a metric that quantifies it from a single representation by correlating dissimilarity matrices built from complementary random halves of the feature dimensions. Unlike CKA and Procrustes distance, Shesha is provably non-invariant to orthogonal rotations of the feature basis. This is by design: the basis is privileged for learned models, since probes, patching, and steering act on coordinates, and a rotation-invariant metric cannot see whether the targeted structure survives them. A double dissociation isolates the mechanism -- removing the top principal component collapses CKA while Shesha holds, whereas rotating a representation into its eigenbasis, which preserves the spectrum and CKA exactly, collapses Shesha. Across 2,463 encoder configurations in seven domains, the metrics are redundant under geometry-preserving transforms and anti-correlate under compression ($ρ=-0.47$). Across 170 vision models spanning 6 clean and 38 corruption-shifted datasets, DINOv2 ranks first or second in transferability on three of six clean datasets yet bottom-quartile in stability on five, an isolated dissociation rather than a trade-off.