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Evaluations of LLM personas via psychometric questionnaires typically rely on aggregate scores, discarding within-instance correlation structure. We test whether this geometric structure is intrinsic or frame-dependent. Constructing within-instance correlation matrices from IPIP-50 responses, we analyze geometry on SPD manifolds under manipulated question orderings in GPT-4o simulating American and Chinese-American personas. We find that persona expression comprises two dissociable components: aggregated features (Big Five scores) degrade under randomization (21% drop) but are frame-robust; geometric features (SPD manifold) collapse under frame misalignment (42% drop) but recover substantially (to 84%) under shared frames, surpassing aggregated features (76%). This collapse-recovery pattern reveals that persona geometry is not intrinsic but a frame-dependent coordination pattern encoding information invisible to aggregation. Our findings establish a dual-nature framework for LLM personas, frame-dependent geometry versus frame-robust aggregates, necessitating frame-aware evaluation and challenging static trait conceptions.

Bogomolny, Bohigas and Schmit (BBS) found that the spectrum of the pairwise distance matrix on N points sampled from a smooth d-dimensional manifold encodes a signature of the underlying geometry. We develop I-BBS (Inference-BBS), a coordinate-free method that identifies a low-dimensional latent sub-manifold embedded in a high-dimensional ambient distance matrix alone, without accessing an ambient high-dimensional vector space. It therefore applies even when that space is only partly observable or undefined. We model the ambient embedding by two classes of generative noise, model-based and model-free. The noise mixes the latent signal with off-manifold components, so the eigenvalues reorganise collectively and the latent geometry cannot be read off eigenvalue by eigenvalue. We recover it instead from two integer-stable signatures that survive the noise: the multiplicity of the top non-Perron multiplet, which fixes $d$, and a parameter-free law for how the multiplet positions shrink as the noise grows. On synthetic spheres $S^1$, $S^2$ and $S^3$ these integer signatures are far more stable under noise than the continuous spectral slope, and a blind test recovers both the manifold and the noise model from a single distance matrix. Applications to neural-network representations and to the dynamic training regime are developed in two companion papers.

The common factor analytic model is related to Helmholtz and Boltzmann machines, can be conceived as a linear autoencoder, or can be thought of as a single-hidden-layer generative neural network. We thus consider it a basal generative representation learner that can be used as a minimal model for studying the foundational characteristics of (deep) generative model architectures. We focus on the fundamental problem of indeterminacy in latent factor projections. This indeterminacy implies that, even when the intrinsic dimension of the latent vector is known, regularity conditions are met, and rotational indeterminacy is resolved, an inherent indefiniteness in the retrieval of causative latent sources remains: they will be uncertain, distributionally deviant, and non-unique. This can have major implications for data representation but remains an elusive issue, even to practitioners and theorists well-versed in the factor model. Moreover, this classic psychometric problem is intricately related to the modern issue of latent variable collapse in the variational autoencoder framework for deep generative modeling. Here, we assess this indeterminacy from various perspectives and show how these are mathematically and conceptually related and we discuss subsequent implications for the Psychometrics, Statistics, and Artificial Intelligence communities. We show that one has latent factor determinacy across all its facets when the feature-dimension grows to infinity. This feeds into an essentially distribution-free estimation approach in the sample case when the number of features grows very large. We conclude, as these are emergent properties at scale, that the factor model is suited for representation learning of very-high-dimensional data.

This paper examines how metric adjustments to Multidimensional Scaling (MDS) can enhance its effectiveness as a visual tool for pattern recognition. The distance under consideration, referred to as Max-D-SW, is an adjustment of the Max-Sliced Wasserstein distance. In contrast to the original formulation, which optimizes over single unit directions, Max-D-SW aggregates contributions over orthonormal bases. This modification provides a clear numerical advantage in MDS outcomes, particularly when applied to heavy-tailed distributions. We also establish sample-complexity bounds showing that Max-D-SW remains statistically tractable, with rates comparable to those of its max-sliced counterpart. Moreover, we show that a better sample complexity for a metric does not necessarily translate into better performance when the metric is used as an input for MDS.

Polynomial Stein discrepancies (PSD) provide a scalable alternative to kernel Stein methods for measuring sample quality and goodness-of-fit testing, but their statistical properties remain poorly understood. We show that increasing polynomial degree primarily amplifies signal without adequately controlling variance, rather than directly optimising the signal-to-noise ratio (SNR). Under suitable assumptions, this might lead to a failure mode in which the $\text{SNR}^2$ can provably decay exponentially with polynomial degree. Motivated by this observation, we reformulate Stein discrepancy construction as an explicit $\text{SNR}^2$ maximisation problem, yielding a Rayleigh quotient over Stein features. This perspective motivates $λ$-PSD, an approximate scalable covariance-aware reweighting scheme defined in a low-dimensional subspace. Under Gaussian settings, we show that $λ$-PSD avoids the exponential $\text{SNR}^2$ collapse and achieves a stable $\text{SNR}^2$. Empirically, $λ$-PSD substantially improves test power while retaining linear-time complexity in the number of samples, highlighting the importance of SNR-aware design for scalable Stein discrepancies.

Textual data are often collected alongside structured response variables, but prediction and interpretation are commonly treated as separate tasks. This paper studies rating prediction as an initial case of interpretable text-response modelling, where the aim is to learn textual representations that are both semantically meaningful and aligned with an external response. We propose a joint non-negative matrix factorisation and binomial regression model, in which the document-topic representation is learned from both text reconstruction and rating prediction. Simulation experiments and a real-world review dataset show that the model can recover stable response-relevant textual signals and achieve competitive performance against linear and ridge regression baselines. The framework provides a practical step towards interpretable modelling of text-linked outcomes, with potential extensions to other response types beyond bounded ratings.

Recent works have highlighted optimization difficulties faced by gradient descent in training the first and last layers of transformer-based language models, which are overcome by optimizers such as Adam. These works suggest that the difficulty is linked to the heavy-tailed distribution of words in text data, where the frequency of the kth most frequent word πk is proportional to 1/k, following Zipf's law. To better understand the impact of the data distribution on training performance, we study a linear bigram model for next-token prediction when the tokens follow a power law πk 1/kα parameterized by the exponent α > 0. We derive optimization scaling laws for deterministic gradient descent and sign descent as a proxy for Adam as a function of the exponent α. Existing theoretical investigations in scaling laws assume that the eigenvalues of the data decay as a power law with exponent α > 1. This assumption effectively makes the problem "finite dimensional" as most of the loss comes from a few of the largest eigencomponents. In comparison, we show that the problem is more difficult when the data have heavier tails. The case α = 1 as found in language is "worst-case" for gradient descent, in that the number of iterations required to reach a small relative error scales almost linearly with dimension. While the performance of sign descent also depends on the dimension, for Zipf-distributed data the number of iterations scales only with the square-root of the dimension, leading to a large improvement for large vocabularies.

Neural combinatorial optimization (NCO) has achieved remarkable performance, yet its learned model representations and decision rationale remain a black box. This impedes both academic research and practical deployment, since researchers and stakeholders require deeper insights into NCO models. In this paper, we take the first critical step towards interpreting NCO models by investigating their representations through various probing tasks. Moreover, we introduce a novel probing tool named Coefficient Significance Probing (CS-Probing) to enable deeper analysis of NCO representations by examining the coefficients and statistical significance during probing. Extensive experiments and analysis reveal that NCO models encode low-level information essential for solution construction, while capturing high-level knowledge to facilitate better decisions. Using CS-Probing, we find that prevalent NCO models impose varying inductive biases on their learned representations, uncover direct evidence related to model generalization, and identify key embedding dimensions associated with specific knowledge. These insights can be potentially translated into practice, for example, with minor code modifications, we improve the generalization of the analyzed model. Our work represents a first systematic attempt to interpret black-box NCO models, showcasing probing as a promising tool for analyzing their internal mechanisms and revealing insights for the NCO community. The source code is publicly available 2.

Long-separated research has been conducted on two highly correlated tracks: traffic and incidents. Traffic track witnesses complicating deep learning models, e.g., to push the prediction a few percent more accurate, and the incident track only studies the incidents alone, e.g., to infer the incident risk. We, for the first time, spatiotemporally aligned the two tracks in a large-scale region (16,972 traffic nodes) from year 2022 to 2024: our TraffiDent dataset includes traffic, i.e., time-series indexes on traffic flow, lane occupancy, and average vehicle speed, and incident, whose records are spatiotemporally aligned with traffic data, with seven different incident classes. Additionally, each node includes detailed physical and policylevel meta-attributes of lanes. Previous datasets typically contain only traffic or incident data in isolation, limiting research to general forecasting tasks.