Daily Papers

Evaluating LLM agents for scientific tasks has focused on token costs while ignoring tool-use costs like simulation time and experimental resources. As a result, metrics like pass@k become impractical under realistic budget constraints. To address this gap, we introduce SimulCost, the first benchmark targeting cost-sensitive parameter tuning in physics simulations. SimulCost compares LLM tuning cost-sensitive parameters against traditional scanning approach in both accuracy and computational cost, spanning 2,916 single-round (initial guess) and 1,900 multi-round (adjustment by trial-and-error) tasks across 12 simulators from fluid dynamics, solid mechanics, and plasma physics. Each simulator's cost is analytically defined and platform-independent. Frontier LLMs achieve 46--64% success rates in single-round mode, dropping to 35--54% under high accuracy requirements, rendering their initial guesses unreliable especially for high accuracy tasks. Multi-round mode improves rates to 71--80%, but LLMs are 1.5--2.5x slower than traditional scanning, making them uneconomical choices. We also investigate parameter group correlations for knowledge transfer potential, and the impact of in-context examples and reasoning effort, providing practical implications for deployment and fine-tuning. We open-source SimulCost as a static benchmark and extensible toolkit to facilitate research on improving cost-aware agentic designs for physics simulations, and for expanding new simulation environments. Code and data are available at https://github.com/Rose-STL-Lab/SimulCost-Bench.

As neural networks grow in scale, their training becomes both computationally demanding and rich in dynamics. Amidst the flourishing interest in these training dynamics, we present a novel observation: Parameters during training exhibit intrinsic correlations over time. Capitalizing on this, we introduce Correlation Mode Decomposition (CMD). This algorithm clusters the parameter space into groups, termed modes, that display synchronized behavior across epochs. This enables CMD to efficiently represent the training dynamics of complex networks, like ResNets and Transformers, using only a few modes. Moreover, test set generalization is enhanced. We introduce an efficient CMD variant, designed to run concurrently with training. Our experiments indicate that CMD surpasses the state-of-the-art method for compactly modeled dynamics on image classification. Our modeling can improve training efficiency and lower communication overhead, as shown by our preliminary experiments in the context of federated learning.

Linear structural equation models are multivariate statistical models encoded by mixed graphs. In particular, the set of covariance matrices for distributions belonging to a linear structural equation model for a fixed mixed graph G=(V, D,B) is parameterized by a rational function with parameters for each vertex and edge in G. This rational parametrization naturally allows for the study of these models from an algebraic and combinatorial point of view. Indeed, this point of view has led to a collection of results in the literature, mainly focusing on questions related to identifiability and determining relationships between covariances (i.e., finding polynomials in the Gaussian vanishing ideal). So far, a large proportion of these results has focused on the case when D, the directed part of the mixed graph G, is acyclic. This is due to the fact that in the acyclic case, the parametrization becomes polynomial and there is a description of the entries of the covariance matrices in terms of a finite sum. We move beyond the acyclic case and give a closed form expression for the entries of the covariance matrices in terms of the one-connections in a graph obtained from D through some small operations. This closed form expression then allows us to show that if G is simple, then the parametrization map is generically finite-to-one. Finally, having a closed form expression for the covariance matrices allows for the development of an algorithm for systematically exploring possible polynomials in the Gaussian vanishing ideal.

Partial correlations quantify linear association between two variables adjusting for the influence of the remaining variables. They form the backbone for graphical models and are readily obtained from the inverse of the covariance matrix. For compositional data, the covariance structure is specified from log ratios of variables, so unless we try to "open" the data via a normalization, this implies changes in the definition and interpretation of partial correlations. In the present work, we elucidate how results derived by Aitchison (1986) lead to a natural definition of partial correlation that has a number of advantages over current measures of association. For this, we show that the residuals of log-ratios between a variable with a reference, when adjusting for all remaining variables including the reference, are reference-independent. Since the reference itself can be controlled for, correlations between residuals are defined for the variables directly without the necessity to recur to ratios except when specifying which variables are partialled out. Thus, perhaps surprisingly, partial correlations do not have the problems commonly found with measures of pairwise association on compositional data. They are well-defined between two variables, are properly scaled, and allow for negative association. By design, they are subcompositionally incoherent, but they share this property with conventional partial correlations (where results change when adjusting for the influence of fewer variables). We discuss the equivalence with normalization-based approaches whenever the normalizing variables are controlled for. We also discuss the partial variances and correlations we obtain from a previously studied data set of Roman glass cups.

Invariance and equivariance to geometrical transformations have proven to be very useful inductive biases when training (convolutional) neural network models, especially in the low-data regime. Much work has focused on the case where the symmetry group employed is compact or abelian, or both. Recent work has explored enlarging the class of transformations used to the case of Lie groups, principally through the use of their Lie algebra, as well as the group exponential and logarithm maps. The applicability of such methods to larger transformation groups is limited by the fact that depending on the group of interest G, the exponential map may not be surjective. Further limitations are encountered when G is neither compact nor abelian. Using the structure and geometry of Lie groups and their homogeneous spaces, we present a framework by which it is possible to work with such groups primarily focusing on the Lie groups G = GL^{+}(n, R) and G = SL(n, R), as well as their representation as affine transformations R^{n} rtimes G. Invariant integration as well as a global parametrization is realized by decomposing the `larger` groups into subgroups and submanifolds which can be handled individually. Under this framework, we show how convolution kernels can be parametrized to build models equivariant with respect to affine transformations. We evaluate the robustness and out-of-distribution generalisation capability of our model on the standard affine-invariant benchmark classification task, where we outperform all previous equivariant models as well as all Capsule Network proposals.

Understanding the performance of machine learning (ML) models across diverse data distributions is critically important for reliable applications. Despite recent empirical studies positing a near-perfect linear correlation between in-distribution (ID) and out-of-distribution (OOD) accuracies, we empirically demonstrate that this correlation is more nuanced under subpopulation shifts. Through rigorous experimentation and analysis across a variety of datasets, models, and training epochs, we demonstrate that OOD performance often has a nonlinear correlation with ID performance in subpopulation shifts. Our findings, which contrast previous studies that have posited a linear correlation in model performance during distribution shifts, reveal a "moon shape" correlation (parabolic uptrend curve) between the test performance on the majority subpopulation and the minority subpopulation. This non-trivial nonlinear correlation holds across model architectures, hyperparameters, training durations, and the imbalance between subpopulations. Furthermore, we found that the nonlinearity of this "moon shape" is causally influenced by the degree of spurious correlations in the training data. Our controlled experiments show that stronger spurious correlation in the training data creates more nonlinear performance correlation. We provide complementary experimental and theoretical analyses for this phenomenon, and discuss its implications for ML reliability and fairness. Our work highlights the importance of understanding the nonlinear effects of model improvement on performance in different subpopulations, and has the potential to inform the development of more equitable and responsible machine learning models.

This paper introduces a structural equation formulation that gives rise to a new family of quasi-periodic Gaussian processes, useful to process a broad class of natural and physiological signals. The proposed formulation simplifies generation and forecasting, and provides hyperparameter estimates, which we exploit in a convergent and consistent iterative estimation algorithm. A bootstrap approach for standard error estimation and confidence intervals is also provided. We demonstrate the computational and scaling benefits of the proposed approach on a broad class of problems, including water level tidal analysis, CO_{2} emission data, and sunspot numbers data. By leveraging the structural equations, our method reduces the cost of likelihood evaluations and predictions from O(k^2 p^2) to O(p^2), significantly improving scalability.

A model involving Gaussian processes (GPs) is introduced to simultaneously handle multi-task learning, clustering, and prediction for multiple functional data. This procedure acts as a model-based clustering method for functional data as well as a learning step for subsequent predictions for new tasks. The model is instantiated as a mixture of multi-task GPs with common mean processes. A variational EM algorithm is derived for dealing with the optimisation of the hyper-parameters along with the hyper-posteriors' estimation of latent variables and processes. We establish explicit formulas for integrating the mean processes and the latent clustering variables within a predictive distribution, accounting for uncertainty on both aspects. This distribution is defined as a mixture of cluster-specific GP predictions, which enhances the performances when dealing with group-structured data. The model handles irregular grid of observations and offers different hypotheses on the covariance structure for sharing additional information across tasks. The performances on both clustering and prediction tasks are assessed through various simulated scenarios and real datasets. The overall algorithm, called MagmaClust, is publicly available as an R package.

Mechanistic interpretability aims to understand neural networks by identifying which learned features mediate specific behaviors. Attribution graphs reveal these feature pathways, but interpreting them requires extensive manual analysis -- a single prompt can take approximately 2 hours for an experienced circuit tracer. We present probe prompting, an automated pipeline that transforms attribution graphs into compact, interpretable subgraphs built from concept-aligned supernodes. Starting from a seed prompt and target logit, we select high-influence features, generate concept-targeted yet context-varying probes, and group features by cross-prompt activation signatures into Semantic, Relationship, and Say-X categories using transparent decision rules. Across five prompts including classic "capitals" circuits, probe-prompted subgraphs preserve high explanatory coverage while compressing complexity (Completeness 0.83, mean across circuits; Replacement 0.54). Compared to geometric clustering baselines, concept-aligned groups exhibit higher behavioral coherence: 2.3x higher peak-token consistency (0.425 vs 0.183) and 5.8x higher activation-pattern similarity (0.762 vs 0.130), despite lower geometric compactness. Entity-swap tests reveal a layerwise hierarchy: early-layer features transfer robustly (64% transfer rate, mean layer 6.3), while late-layer Say-X features specialize for output promotion (mean layer 16.4), supporting a backbone-and-specialization view of transformer computation. We release code (https://github.com/peppinob-ol/attribution-graph-probing), an interactive demo (https://huggingface.co/spaces/Peppinob/attribution-graph-probing), and minimal artifacts enabling immediate reproduction and community adoption.

State-of-the-art space science missions increasingly rely on automation due to spacecraft complexity and the costs of human oversight. The high volume of data, including scientific and telemetry data, makes manual inspection challenging. Machine learning offers significant potential to meet these demands. The Euclid space telescope, in its survey phase since February 2024, exemplifies this shift. Euclid's success depends on accurate monitoring and interpretation of housekeeping telemetry and science-derived data. Thousands of telemetry parameters, monitored as time series, may or may not impact the quality of scientific data. These parameters have complex interdependencies, often due to physical relationships (e.g., proximity of temperature sensors). Optimising science operations requires careful anomaly detection and identification of hidden parameter states. Moreover, understanding the interactions between known anomalies and physical quantities is crucial yet complex, as related parameters may display anomalies with varied timing and intensity. We address these challenges by analysing temperature anomalies in Euclid's telemetry from February to August 2024, focusing on eleven temperature parameters and 35 covariates. We use a predictive XGBoost model to forecast temperatures based on historical values, detecting anomalies as deviations from predictions. A second XGBoost model predicts anomalies from covariates, capturing their relationships to temperature anomalies. We identify the top three anomalies per parameter and analyse their interactions with covariates using SHAP (Shapley Additive Explanations), enabling rapid, automated analysis of complex parameter relationships. Our method demonstrates how machine learning can enhance telemetry monitoring, offering scalable solutions for other missions with similar data challenges.

We used a dataset of daily Bloomberg Financial Market Summaries from 2010 to 2023, reposted on large financial media, to determine how global news headlines may affect stock market movements using ChatGPT and a two-stage prompt approach. We document a statistically significant positive correlation between the sentiment score and future equity market returns over short to medium term, which reverts to a negative correlation over longer horizons. Validation of this correlation pattern across multiple equity markets indicates its robustness across equity regions and resilience to non-linearity, evidenced by comparison of Pearson and Spearman correlations. Finally, we provide an estimate of the optimal horizon that strikes a balance between reactivity to new information and correlation.

Graph measures that express closeness or distance between nodes can be employed for graph nodes clustering using metric clustering algorithms. There are numerous measures applicable to this task, and which one performs better is an open question. We study the performance of 25 graph measures on generated graphs with different parameters. While usually measure comparisons are limited to general measure ranking on a particular dataset, we aim to explore the performance of various measures depending on graph features. Using an LFR graph generator, we create a dataset of 11780 graphs covering the whole LFR parameter space. For each graph, we assess the quality of clustering with k-means algorithm for each considered measure. Based on this, we determine the best measure for each area of the parameter space. We find that the parameter space consists of distinct zones where one particular measure is the best. We analyze the geometry of the resulting zones and describe it with simple criteria. Given particular graph parameters, this allows us to recommend a particular measure to use for clustering.

Predicting the dependencies between observations from multiple time series is critical for applications such as anomaly detection, financial risk management, causal analysis, or demand forecasting. However, the computational and numerical difficulties of estimating time-varying and high-dimensional covariance matrices often limits existing methods to handling at most a few hundred dimensions or requires making strong assumptions on the dependence between series. We propose to combine an RNN-based time series model with a Gaussian copula process output model with a low-rank covariance structure to reduce the computational complexity and handle non-Gaussian marginal distributions. This permits to drastically reduce the number of parameters and consequently allows the modeling of time-varying correlations of thousands of time series. We show on several real-world datasets that our method provides significant accuracy improvements over state-of-the-art baselines and perform an ablation study analyzing the contributions of the different components of our model.

Researchers are usually interested in examining the impact of covariates when separating heterogeneous samples into latent classes that are more homogeneous. The majority of theoretical and empirical studies with such aims have focused on identifying covariates as predictors of class membership in the structural equation modeling framework. In other words, the covariates only indirectly affect the sample heterogeneity. However, the covariates' influence on between-individual differences can also be direct. This article presents a mixture model that investigates covariates to explain within-cluster and between-cluster heterogeneity simultaneously, known as a mixture-of-experts (MoE) model. This study aims to extend the MoE framework to investigate heterogeneity in nonlinear trajectories: to identify latent classes, covariates as predictors to clusters, and covariates that explain within-cluster differences in change patterns over time. Our simulation studies demonstrate that the proposed model generally estimates the parameters unbiasedly, precisely and exhibits appropriate empirical coverage for a nominal 95% confidence interval. This study also proposes implementing structural equation model forests to shrink the covariate space of the proposed mixture model. We illustrate how to select covariates and construct the proposed model with longitudinal mathematics achievement data. Additionally, we demonstrate that the proposed mixture model can be further extended in the structural equation modeling framework by allowing the covariates that have direct effects to be time-varying.

Simulation-based inference with conditional neural density estimators is a powerful approach to solving inverse problems in science. However, these methods typically treat the underlying forward model as a black box, with no way to exploit geometric properties such as equivariances. Equivariances are common in scientific models, however integrating them directly into expressive inference networks (such as normalizing flows) is not straightforward. We here describe an alternative method to incorporate equivariances under joint transformations of parameters and data. Our method -- called group equivariant neural posterior estimation (GNPE) -- is based on self-consistently standardizing the "pose" of the data while estimating the posterior over parameters. It is architecture-independent, and applies both to exact and approximate equivariances. As a real-world application, we use GNPE for amortized inference of astrophysical binary black hole systems from gravitational-wave observations. We show that GNPE achieves state-of-the-art accuracy while reducing inference times by three orders of magnitude.

Learning causal structure from observational data often assumes that we observe independent and identically distributed (i.\,i.\,d) data. The traditional approach aims to find a graphical representation that encodes the same set of conditional independence relationships as those present in the observed distribution. It is known that under i.\,i.\,d assumption, even with infinite data, there is a limit to how fine-grained a causal structure we can identify. To overcome this limitation, recent work has explored using data originating from different, related environments to learn richer causal structure. These approaches implicitly rely on the independent causal mechanisms (ICM) principle, which postulates that the mechanism giving rise to an effect given its causes and the mechanism which generates the causes do not inform or influence each other. Thus, components of the causal model can independently change from environment to environment. Despite its wide application in machine learning and causal inference, there is a lack of statistical formalization of the ICM principle and how it enables identification of richer causal structures from grouped data. Here we present new causal de Finetti theorems which offer a first statistical formalization of ICM principle and show how causal structure identification is possible from exchangeable data. Our work provides theoretical justification for a broad range of techniques leveraging multi-environment data to learn causal structure.

Multi-subject fMRI studies are challenging due to the high variability of both brain anatomy and functional brain topographies across participants. An effective way of aggregating multi-subject fMRI data is to extract a shared representation that filters out unwanted variability among subjects. Some recent work has implemented probabilistic models to extract a shared representation in task fMRI. In the present work, we improve upon these models by incorporating temporal information in the common latent structures. We introduce a new model, Shared Gaussian Process Factor Analysis (S-GPFA), that discovers shared latent trajectories and subject-specific functional topographies, while modelling temporal correlation in fMRI data. We demonstrate the efficacy of our model in revealing ground truth latent structures using simulated data, and replicate experimental performance of time-segment matching and inter-subject similarity on the publicly available Raider and Sherlock datasets. We further test the utility of our model by analyzing its learned model parameters in the large multi-site SPINS dataset, on a social cognition task from participants with and without schizophrenia.

Time series clustering promises to uncover hidden structural patterns in data with applications across healthcare, finance, industrial systems, and other critical domains. However, without validated ground truth information, researchers cannot objectively assess clustering quality or determine whether poor results stem from absent structures in the data, algorithmic limitations, or inappropriate validation methods, raising the question whether clustering is "more art than science" (Guyon et al., 2009). To address these challenges, we introduce CSTS (Correlation Structures in Time Series), a synthetic benchmark for evaluating the discovery of correlation structures in multivariate time series data. CSTS provides a clean benchmark that enables researchers to isolate and identify specific causes of clustering failures by differentiating between correlation structure deterioration and limitations of clustering algorithms and validation methods. Our contributions are: (1) a comprehensive benchmark for correlation structure discovery with distinct correlation structures, systematically varied data conditions, established performance thresholds, and recommended evaluation protocols; (2) empirical validation of correlation structure preservation showing moderate distortion from downsampling and minimal effects from distribution shifts and sparsification; and (3) an extensible data generation framework enabling structure-first clustering evaluation. A case study demonstrates CSTS's practical utility by identifying an algorithm's previously undocumented sensitivity to non-normal distributions, illustrating how the benchmark enables precise diagnosis of methodological limitations. CSTS advances rigorous evaluation standards for correlation-based time series clustering.

When rows of an n times d matrix A are given in a stream, we study algorithms for approximating the top eigenvector of the matrix {A}^TA (equivalently, the top right singular vector of A). We consider worst case inputs A but assume that the rows are presented to the streaming algorithm in a uniformly random order. We show that when the gap parameter R = σ_1(A)^2/σ_2(A)^2 = Ω(1), then there is a randomized algorithm that uses O(h cdot d cdot polylog(d)) bits of space and outputs a unit vector v that has a correlation 1 - O(1/R) with the top eigenvector v_1. Here h denotes the number of heavy rows in the matrix, defined as the rows with Euclidean norm at least |{A}|_F/d cdot operatorname{polylog(d)}. We also provide a lower bound showing that any algorithm using O(hd/R) bits of space can obtain at most 1 - Ω(1/R^2) correlation with the top eigenvector. Thus, parameterizing the space complexity in terms of the number of heavy rows is necessary for high accuracy solutions. Our results improve upon the R = Ω(log n cdot log d) requirement in a recent work of Price and Xun (FOCS 2024). We note that the algorithm of Price and Xun works for arbitrary order streams whereas our algorithm requires a stronger assumption that the rows are presented in a uniformly random order. We additionally show that the gap requirements in their analysis can be brought down to R = Ω(log^2 d) for arbitrary order streams and R = Ω(log d) for random order streams. The requirement of R = Ω(log d) for random order streams is nearly tight for their analysis as we obtain a simple instance with R = Ω(log d/loglog d) for which their algorithm, with any fixed learning rate, cannot output a vector approximating the top eigenvector v_1.

This paper considers the problem of variable selection allowing for parameter instability. It distinguishes between signal and pseudo-signal variables that are correlated with the target variable, and noise variables that are not, and investigate the asymptotic properties of the One Covariate at a Time Multiple Testing (OCMT) method proposed by Chudik et al. (2018) under parameter insatiability. It is established that OCMT continues to asymptotically select an approximating model that includes all the signals and none of the noise variables. Properties of post selection regressions are also investigated, and in-sample fit of the selected regression is shown to have the oracle property. The theoretical results support the use of unweighted observations at the selection stage of OCMT, whilst applying down-weighting of observations only at the forecasting stage. Monte Carlo and empirical applications show that OCMT without down-weighting at the selection stage yields smaller mean squared forecast errors compared to Lasso, Adaptive Lasso, and boosting.

We propose Equivariance by Contrast (EbC) to learn equivariant embeddings from observation pairs (y, g cdot y), where g is drawn from a finite group acting on the data. Our method jointly learns a latent space and a group representation in which group actions correspond to invertible linear maps -- without relying on group-specific inductive biases. We validate our approach on the infinite dSprites dataset with structured transformations defined by the finite group G:= (R_m times Z_n times Z_n), combining discrete rotations and periodic translations. The resulting embeddings exhibit high-fidelity equivariance, with group operations faithfully reproduced in latent space. On synthetic data, we further validate the approach on the non-abelian orthogonal group O(n) and the general linear group GL(n). We also provide a theoretical proof for identifiability. While broad evaluation across diverse group types on real-world data remains future work, our results constitute the first successful demonstration of general-purpose encoder-only equivariant learning from group action observations alone, including non-trivial non-abelian groups and a product group motivated by modeling affine equivariances in computer vision.

We study the problem of detecting the correlation between two Gaussian databases XinR^{ntimes d} and Y^{ntimes d}, each composed of n users with d features. This problem is relevant in the analysis of social media, computational biology, etc. We formulate this as a hypothesis testing problem: under the null hypothesis, these two databases are statistically independent. Under the alternative, however, there exists an unknown permutation sigma over the set of n users (or, row permutation), such that X is rho-correlated with Y^sigma, a permuted version of Y. We determine sharp thresholds at which optimal testing exhibits a phase transition, depending on the asymptotic regime of n and d. Specifically, we prove that if rho^2dto0, as dtoinfty, then weak detection (performing slightly better than random guessing) is statistically impossible, irrespectively of the value of n. This compliments the performance of a simple test that thresholds the sum all entries of X^TY. Furthermore, when d is fixed, we prove that strong detection (vanishing error probability) is impossible for any rho<rho^star, where rho^star is an explicit function of d, while weak detection is again impossible as long as rho^2dto0. These results close significant gaps in current recent related studies.

This article studies the change in the prediction accuracy of a response variable when the number of predictors increases, and all variables follow a multivariate normal distribution. Assuming that the correlations between variables are independently drawn, I show that adding variables leads to globally increasing returns to scale when the mean of the correlation distribution is zero. The speed of learning depends positively on the variance of the correlation distribution. I use simulations to study the more complex case of correlation distributions with a non-zero mean and find a pattern of decreasing returns followed by increasing returns to scale - as long as the variance of correlations is not degenerate, in which case globally decreasing returns emerge. I train a collaborative filtering algorithm using the MovieLens 1M dataset to analyze returns from adding variables in a more realistic setting and find globally increasing returns to scale across 2,000 variables. The results suggest significant scale advantages from additional variables in prediction tasks.

Correlation does not imply causation, but patterns of statistical association between variables can be exploited to infer a causal structure (even with purely observational data) with the burgeoning field of causal discovery. As a purely observational science, astrophysics has much to gain by exploiting these new methods. The supermassive black hole (SMBH)--galaxy interaction has long been constrained by observed scaling relations, that is low-scatter correlations between variables such as SMBH mass and the central velocity dispersion of stars in a host galaxy's bulge. This study, using advanced causal discovery techniques and an up-to-date dataset, reveals a causal link between galaxy properties and dynamically-measured SMBH masses. We apply a score-based Bayesian framework to compute the exact conditional probabilities of every causal structure that could possibly describe our galaxy sample. With the exact posterior distribution, we determine the most likely causal structures and notice a probable causal reversal when separating galaxies by morphology. In elliptical galaxies, bulge properties (built from major mergers) tend to influence SMBH growth, while in spiral galaxies, SMBHs are seen to affect host galaxy properties, potentially through feedback in gas-rich environments. For spiral galaxies, SMBHs progressively quench star formation, whereas in elliptical galaxies, quenching is complete, and the causal connection has reversed. Our findings support theoretical models of hierarchical assembly of galaxies and active galactic nuclei feedback regulating galaxy evolution. Our study suggests the potentiality for further exploration of causal links in astrophysical and cosmological scaling relations, as well as any other observational science.

Based on the theory of homogeneous spaces we derive geometrically optimal edge attributes to be used within the flexible message-passing framework. We formalize the notion of weight sharing in convolutional networks as the sharing of message functions over point-pairs that should be treated equally. We define equivalence classes of point-pairs that are identical up to a transformation in the group and derive attributes that uniquely identify these classes. Weight sharing is then obtained by conditioning message functions on these attributes. As an application of the theory, we develop an efficient equivariant group convolutional network for processing 3D point clouds. The theory of homogeneous spaces tells us how to do group convolutions with feature maps over the homogeneous space of positions R^3, position and orientations R^3 {times} S^2, and the group SE(3) itself. Among these, R^3 {times} S^2 is an optimal choice due to the ability to represent directional information, which R^3 methods cannot, and it significantly enhances computational efficiency compared to indexing features on the full SE(3) group. We support this claim with state-of-the-art results -- in accuracy and speed -- on five different benchmarks in 2D and 3D, including interatomic potential energy prediction, trajectory forecasting in N-body systems, and generating molecules via equivariant diffusion models.

Despite the widespread application of latent factor analysis, existing methods suffer from the following weaknesses: requiring the number of factors to be known, lack of theoretical guarantees for learning the model structure, and nonidentifiability of the parameters due to rotation invariance properties of the likelihood. We address these concerns by proposing a fast correlation thresholding (CT) algorithm that simultaneously learns the number of latent factors and a rotationally identifiable model structure. Our novel approach translates this structure learning problem into the search for so-called independent maximal cliques in a thresholded correlation graph that can be easily constructed from the observed data. Our clique analysis technique scales well up to thousands of variables, while competing methods are not applicable in a reasonable amount of running time. We establish a finite-sample error bound and high-dimensional consistency for the structure learning of our method. Through a series of simulation studies and a real data example, we show that the CT algorithm is an accurate method for learning the structure of factor analysis models and is robust to violations of its assumptions.

We analyse the robustness of the DESI 2024 cosmological inference from fits to the full shape of the galaxy power spectrum to uncertainties in the Halo Occupation Distribution (HOD) model of the galaxy-halo connection and the choice of priors on nuisance parameters. We assess variations in the recovered cosmological parameters across a range of mocks populated with different HOD models and find that shifts are often greater than 20% of the expected statistical uncertainties from the DESI data. We encapsulate the effect of such shifts in terms of a systematic covariance term, C_{rm HOD}, and an additional diagonal contribution quantifying the impact of our choice of nuisance parameter priors on the ability of the effective field theory (EFT) model to correctly recover the cosmological parameters of the simulations. These two covariance contributions are designed to be added to the usual covariance term, C_{rm stat}, describing the statistical uncertainty in the power spectrum measurement, in order to fairly represent these sources of systematic uncertainty. This approach is more general and robust to choices of model free parameters or additional external datasets used in cosmological fits than the alternative approach of adding systematic uncertainties at the level of the recovered marginalised parameter posteriors. We compare the approaches within the context of a fixed LambdaCDM model and demonstrate that our method gives conservative estimates of the systematic uncertainty that nevertheless have little impact on the final posteriors obtained from DESI data.

We reanalyze the spectral lag data for GRB 160625B using frequentist inference in order to constrain the energy scale (E_{QG}) of Lorentz Invariance Violation (LIV). For this purpose, we use profile likelihood to deal with the astrophysical nuisance parameters. This is in contrast to Bayesian inference implemented in previous works, where marginalization was carried out over the nuisance parameters. We show that with profile likelihood, we do not find a global minimum for chi^2 as a function of E_{QG} below the Planck scale for both linear and quadratic models of LIV, whereas bounded credible intervals were previously obtained using Bayesian inference. Therefore, we can set one-sided lower limits in a straightforward manner. We find that E_{QG} geq 2.55 times 10^{16} GeV and E_{QG} geq 1.85 times 10^7 GeV at 95\% c.l., for linear and quadratic LIV, respectively. Therefore, this is the first proof-of-principles application of profile likelihood method to the analysis of GRB spectral lag data to constrain LIV.

Training large language models (LLMs) poses challenges due to their massive scale and heterogeneous architectures. While adaptive optimizers like AdamW help address gradient variations, they still struggle with efficient and effective parameter-wise learning rate estimation, resulting in training instability, slow convergence, and poor compatibility with parameter-efficient fine-tuning (PEFT) techniques. This work introduces Scaling with Gradient Grouping (SGG), an optimizer wrapper that improves adaptive learning rate estimation by dynamic grouping and group-specific scaling. SGG first groups gradient statistics in each layer into clusters and then applies cluster-specific scaling to calibrate learning rates for each parameter, thus imposing collective group-wise constraints while maintaining precise per-parameter adaptation. Experiments on diverse (M)LLM benchmarks show that SGG integrates seamlessly with existing optimizers, and offers consistent gains and faster convergence over baselines, with various model sizes. Its stability across varying batch sizes and learning rates establishes SGG as a robust choice for LLM optimization.

The singlet-doublet vector-like fermion dark matter model has been extensively studied in the literature over the past decade. An important parameter in this model is the singlet-doublet mixing angle (sintheta). All the previous studies have primarily focused on annihilation and co-annihilation processes for obtaining the correct dark matter relic density, assuming that the singlet and doublet components decouple at the same epoch. In this work, we demonstrate that this assumption holds only for larger mixing angles with a dependency on the mass of the dark matter. However, it badly fails for the mixing angle sintheta<0.05. We present a systematic study of the parameter space of the singlet-doublet dark matter relic, incorporating annihilation, co-annihilation, and, for the first time, co-scattering processes. Additionally, the freeze-in parameter space is also explored. We found that due to the inclusion of co-scattering processes, the correct relic density parameter space is shifted towards the detection sensitivity range of the LHC and MATHUSLA via displaced vertex signatures.

In this work, we aim to estimate the stellar parameters of the primary (Aa) by performing asteroseismic analysis on its period-spacing pattern. We use the C-3PO neural network to perform asteroseismic modelling of the g-mode period-spacing pattern of Aa, discussing the interplay of this information with external constraints from spectroscopy (T_{rm eff} and log(g)) and eclipse modelling (R). To estimate the level of uncertainty due to different frequency extraction and pattern identification processes, we consider four different variations on the period-spacing patterns. To better understand the correlations between and the uncertainty structure of our parameter estimates, we also employed a classical, parameter-based MCMC grid search on four different stellar grids. The best-fitting, externally constrained model to the period-spacing pattern arrives at estimates of the stellar properties for Aa of: M=1.51 pm 0.05 M_odot, X_c =0.43 pm 0.04, R=1.66 pm 0.1 R_odot, f_{rm ov}=0.010, Omega_c=1.58 pm 0.01 d^{-1} with rigid rotation to within the measurement errors, log(T_{rm eff})=3.856 pm 0.008 dex, log(g)=4.18 pm 0.04 dex, and log(L)=0.809 pm 0.005 dex, which agree well with previous measurements from eclipse modelling, spectroscopy, and the Gaia DR3 luminosity. We find that the near-core properties of the best-fitting asteroseismic models are consistent with external constraints from eclipse modelling and spectroscopy. Aa appears to be a typical example of a gamma Dor star, fitting well within existing populations. We find that Aa is quasi-rigidly rotating to within the uncertainties, and note that the asteroseismic age estimate for Aa (1100 pm 100 Myr) is considerably older than the young (35 Myr) age implied by previous isochrone fits to the B binary in the literature. Our MCMC parameter-based grid-search agrees well with our pattern-modelling approach.

Small sample sizes are common in many disciplines, which necessitates pooling roughly similar datasets across multiple institutions to study weak but relevant associations between images and disease outcomes. Such data often manifest shift/imbalance in covariates (i.e., secondary non-imaging data). Controlling for such nuisance variables is common within standard statistical analysis, but the ideas do not directly apply to overparameterized models. Consequently, recent work has shown how strategies from invariant representation learning provides a meaningful starting point, but the current repertoire of methods is limited to accounting for shifts/imbalances in just a couple of covariates at a time. In this paper, we show how viewing this problem from the perspective of Category theory provides a simple and effective solution that completely avoids elaborate multi-stage training pipelines that would otherwise be needed. We show the effectiveness of this approach via extensive experiments on real datasets. Further, we discuss how this style of formulation offers a unified perspective on at least 5+ distinct problem settings, from self-supervised learning to matching problems in 3D reconstruction.

Epidemiologists often wish to estimate quantities that are easy to communicate and correspond to the results of realistic public health scenarios. Methods from causal inference can answer these questions. We adopt the language of potential outcomes under Rubin's original Bayesian framework and show that the parametric g-formula is easily amenable to a Bayesian approach. We show that the frequentist properties of the Bayesian g-formula suggest it improves the accuracy of estimates of causal effects in small samples or when data may be sparse. We demonstrate our approach to estimate the effect of environmental tobacco smoke on body mass index z-scores among children aged 4-9 years who were enrolled in a longitudinal birth cohort in New York, USA. We give a general algorithm and supply SAS and Stan code that can be adopted to implement our computational approach in both time-fixed and longitudinal data.

We introduce a general, flexible, parametric survival modelling framework which encompasses key shapes of hazard function (constant, increasing, decreasing, up-then-down, down-then-up), various common survival distributions (log-logistic, Burr type XII, Weibull, Gompertz), and includes defective distributions (i.e., cure models). This generality is achieved using four basic distributional parameters: two scale-type parameters and two shape parameters. Generalising to covariate dependence, the scale-type regression components correspond to accelerated failure time (AFT) and proportional hazards (PH) models. Therefore, this general formulation unifies the most popular survival models which allows us to consider the practical value of possible modelling choices for survival data. Furthermore, in line with our proposed flexible baseline distribution, we advocate the use of multi-parameter regression in which more than one distributional parameter depends on covariates - rather than the usual convention of having a single covariate-dependent (scale) parameter. While many choices are available, we suggest introducing covariates through just one or other of the two scale parameters, which covers AFT and PH models, in combination with a `power' shape parameter, which allows for more complex non-AFT/non-PH effects, while the other shape parameter remains covariate-independent, and handles automatic selection of the baseline distribution. We explore inferential issues in simulations, both with and without a covariate, with particular focus on evidence concerning the need, or otherwise, to include both AFT and PH parameters. We illustrate the efficacy of our modelling framework by investigating differences between treatment groups using data from a lung cancer study and a melanoma study. Censoring is accommodated throughout.

Quantifying the similarity between two mathematical structures or datasets constitutes a particularly interesting and useful operation in several theoretical and applied problems. Aimed at this specific objective, the Jaccard index has been extensively used in the most diverse types of problems, also motivating some respective generalizations. The present work addresses further generalizations of this index, including its modification into a coincidence index capable of accounting also for the level of relative interiority between the two compared entities, as well as respective extensions for sets in continuous vector spaces, the generalization to multiset addition, densities and generic scalar fields, as well as a means to quantify the joint interdependence between two random variables. The also interesting possibility to take into account more than two sets has also been addressed, including the description of an index capable of quantifying the level of chaining between three structures. Several of the described and suggested eneralizations have been illustrated with respect to numeric case examples. It is also posited that these indices can play an important role while analyzing and integrating datasets in modeling approaches and pattern recognition activities, including as a measurement of clusters similarity or separation and as a resource for representing and analyzing complex networks.

Objective: Electroencephalography signals are recorded as a multidimensional dataset. We propose a new framework based on the augmented covariance extracted from an autoregressive model to improve motor imagery classification. Methods: From the autoregressive model can be derived the Yule-Walker equations, which show the emergence of a symmetric positive definite matrix: the augmented covariance matrix. The state-of the art for classifying covariance matrices is based on Riemannian Geometry. A fairly natural idea is therefore to extend the standard approach using these augmented covariance matrices. The methodology for creating the augmented covariance matrix shows a natural connection with the delay embedding theorem proposed by Takens for dynamical systems. Such an embedding method is based on the knowledge of two parameters: the delay and the embedding dimension, respectively related to the lag and the order of the autoregressive model. This approach provides new methods to compute the hyper-parameters in addition to standard grid search. Results: The augmented covariance matrix performed noticeably better than any state-of-the-art methods. We will test our approach on several datasets and several subjects using the MOABB framework, using both within-session and cross-session evaluation. Conclusion: The improvement in results is due to the fact that the augmented covariance matrix incorporates not only spatial but also temporal information, incorporating nonlinear components of the signal through an embedding procedure, which allows the leveraging of dynamical systems algorithms. Significance: These results extend the concepts and the results of the Riemannian distance based classification algorithm.

We introduce fast algorithms for correlation clustering with respect to the Min Max objective that provide constant factor approximations on complete graphs. Our algorithms are the first purely combinatorial approximation algorithms for this problem. We construct a novel semi-metric on the set of vertices, which we call the correlation metric, that indicates to our clustering algorithms whether pairs of nodes should be in the same cluster. The paper demonstrates empirically that, compared to prior work, our algorithms sacrifice little in the objective quality to obtain significantly better run-time. Moreover, our algorithms scale to larger networks that are effectively intractable for known algorithms.

Mixture-of-Experts (MoE) layers have emerged as an important tool in scaling up modern neural networks by decoupling total trainable parameters from activated parameters in the forward pass for each token. However, sparse MoEs add complexity to training due to (i) new trainable parameters (router weights) that, like all other parameter groups, require hyperparameter (HP) tuning; (ii) new architecture scale dimensions (number of and size of experts) that must be chosen and potentially taken large. To make HP selection cheap and reliable, we propose a new parameterization for transformer models with MoE layers when scaling model width, depth, number of experts, and expert (hidden) size. Our parameterization is justified by a novel dynamical mean-field theory (DMFT) analysis. When varying different model dimensions trained at a fixed token budget, we find empirically that our parameterization enables reliable HP transfer across models from 51M to over 2B total parameters. We further take HPs identified from sweeping small models on a short token horizon to train larger models on longer horizons and report performant model behaviors.

We provide a concise framework for generalized ensemble theory through a matrix-based approach. By introducing an observation matrix, any discrete probability distribution, including those for non-equilibrium steady states, can be expressed as a generalized Boltzmann distribution, with observables and conjugate variables as the basis and coordinates in a linear space. In this framework, we identify the minimal sufficient statistics required for inferring the Boltzmann distribution. Furthermore, we show that the Hadamard and Vandermonde matrices are suitable observation matrices for spin systems and random walks. In master equation systems, the probability flux observation matrix facilitates the identification of detailed balance violations. Our findings provide a new approach to developing generalized ensemble theory for non-equilibrium steady-state systems.

Heterogeneity and comorbidity are two interwoven challenges associated with various healthcare problems that greatly hampered research on developing effective treatment and understanding of the underlying neurobiological mechanism. Very few studies have been conducted to investigate heterogeneous causal effects (HCEs) in graphical contexts due to the lack of statistical methods. To characterize this heterogeneity, we first conceptualize heterogeneous causal graphs (HCGs) by generalizing the causal graphical model with confounder-based interactions and multiple mediators. Such confounders with an interaction with the treatment are known as moderators. This allows us to flexibly produce HCGs given different moderators and explicitly characterize HCEs from the treatment or potential mediators on the outcome. We establish the theoretical forms of HCEs and derive their properties at the individual level in both linear and nonlinear models. An interactive structural learning is developed to estimate the complex HCGs and HCEs with confidence intervals provided. Our method is empirically justified by extensive simulations and its practical usefulness is illustrated by exploring causality among psychiatric disorders for trauma survivors.

Spurious correlations in the data, where multiple cues are predictive of the target labels, often lead to a phenomenon known as simplicity bias, where a model relies on erroneous, easy-to-learn cues while ignoring reliable ones. In this work, we propose an ensemble diversification framework exploiting Diffusion Probabilistic Models (DPMs) for shortcut bias mitigation. We show that at particular training intervals, DPMs can generate images with novel feature combinations, even when trained on images displaying correlated input features. We leverage this crucial property to generate synthetic counterfactuals to increase model diversity via ensemble disagreement. We show that DPM-guided diversification is sufficient to remove dependence on primary shortcut cues, without a need for additional supervised signals. We further empirically quantify its efficacy on several diversification objectives, and finally show improved generalization and diversification performance on par with prior work that relies on auxiliary data collection.

Demystifying the interactions among multiple agents from their past trajectories is fundamental to precise and interpretable trajectory prediction. However, previous works mainly consider static, pair-wise interactions with limited relational reasoning. To promote more comprehensive interaction modeling and relational reasoning, we propose DynGroupNet, a dynamic-group-aware network, which can i) model time-varying interactions in highly dynamic scenes; ii) capture both pair-wise and group-wise interactions; and iii) reason both interaction strength and category without direct supervision. Based on DynGroupNet, we further design a prediction system to forecast socially plausible trajectories with dynamic relational reasoning. The proposed prediction system leverages the Gaussian mixture model, multiple sampling and prediction refinement to promote prediction diversity, training stability and trajectory smoothness, respectively. Extensive experiments show that: 1)DynGroupNet can capture time-varying group behaviors, infer time-varying interaction category and interaction strength during trajectory prediction without any relation supervision on physical simulation datasets; 2)DynGroupNet outperforms the state-of-the-art trajectory prediction methods by a significant improvement of 22.6%/28.0%, 26.9%/34.9%, 5.1%/13.0% in ADE/FDE on the NBA, NFL Football and SDD datasets and achieve the state-of-the-art performance on the ETH-UCY dataset.

We study the scalar curvature of the Fisher information metric on the microscopic coupling-parameter manifold of lattice spin models at criticality. For a d-dimensional lattice with periodic boundary conditions and n = L^d sites, the Fisher manifold has m = d cdot n dimensions (one per bond), and we find |R(J_c)| sim n^{d_R} with d_R = (dν+ 2η)/(dν+ η), where ν and η are the correlation-length and anomalous-dimension critical exponents. For 2D Ising (ν= 1, η= 1/4), this predicts d_R = 10/9, confirmed by exact transfer-matrix computations (L = 6--9: d_R = 1.1115 pm 0.0002) and multi-seed MCMC through L = 24. For 3D Ising (ν= 0.630, η= 0.0363), the prediction d_R = 1.019 is consistent with MCMC on L^3 tori up to L = 10 (power-law fit: d_R = 1.040). For 2D Potts q = 3 (predicted 33/29 approx 1.138), FFT-MCMC through L = 40 shows d_eff oscillating non-monotonically around sim 1.20, consistent with O(1/(ln L)^2) logarithmic corrections. For q = 4 (predicted 22/19), effective exponents oscillate with strong logarithmic corrections. The Ricci decomposition identity R_3 = -R_1/2, R_4 = -R_2/2 holds to 5--6 digits for all models. This exponent is distinct from Ruppeiner thermodynamic curvature and reflects the collective geometry of the growing Fisher manifold. We provide falsification criteria and predictions for additional universality classes.

SO(3)-equivariant networks are the dominant models for machine learning interatomic potentials (MLIPs). The key operation of such networks is the Clebsch-Gordan (CG) tensor product, which is computationally expensive. To accelerate the computation, we develop tensor decomposition networks (TDNs) as a class of approximately equivariant networks in which CG tensor products are replaced by low-rank tensor decompositions, such as the CANDECOMP/PARAFAC (CP) decomposition. With the CP decomposition, we prove (i) a uniform bound on the induced error of SO(3)-equivariance, and (ii) the universality of approximating any equivariant bilinear map. To further reduce the number of parameters, we propose path-weight sharing that ties all multiplicity-space weights across the O(L^3) CG paths into a single shared parameter set without compromising equivariance, where L is the maximum angular degree. The resulting layer acts as a plug-and-play replacement for tensor products in existing networks, and the computational complexity of tensor products is reduced from O(L^6) to O(L^4). We evaluate TDNs on PubChemQCR, a newly curated molecular relaxation dataset containing 105 million DFT-calculated snapshots. We also use existing datasets, including OC20, and OC22. Results show that TDNs achieve competitive performance with dramatic speedup in computations. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS/tree/main/OpenMol/TDN{https://github.com/divelab/AIRS/}).

Starting from the Mellin-Barnes integral representation of a Feynman integral depending on set of kinematic variables z_i, we derive a system of partial differential equations w.r.t.\ new variables x_j, which parameterize the differentiable constraints z_i=y_i(x_j). In our algorithm, the powers of propagators can be considered as arbitrary parameters. Our algorithm can also be used for the reduction of multiple hypergeometric sums to sums of lower dimension, finding special values and reduction equations of hypergeometric functions in a singular locus of continuous variables, or finding systems of partial differential equations for master integrals with arbitrary powers of propagators. As an illustration, we produce a differential equation of fourth order in one variable for the one-loop two-point Feynman diagram with two different masses and arbitrary propagator powers.

The use of kernel functions is a common technique to extract important features from data sets. A quantum computer can be used to estimate kernel entries as transition amplitudes of unitary circuits. Quantum kernels exist that, subject to computational hardness assumptions, cannot be computed classically. It is an important challenge to find quantum kernels that provide an advantage in the classification of real-world data. We introduce a class of quantum kernels that can be used for data with a group structure. The kernel is defined in terms of a unitary representation of the group and a fiducial state that can be optimized using a technique called kernel alignment. We apply this method to a learning problem on a coset-space that embodies the structure of many essential learning problems on groups. We implement the learning algorithm with 27 qubits on a superconducting processor.

A variety of recent papers discuss the application of Shapley values, a concept for explaining coalitional games, for feature attribution in machine learning. However, the correct way to connect a machine learning model to a coalitional game has been a source of controversy. The two main approaches that have been proposed differ in the way that they condition on known features, using either (1) an interventional or (2) an observational conditional expectation. While previous work has argued that one of the two approaches is preferable in general, we argue that the choice is application dependent. Furthermore, we argue that the choice comes down to whether it is desirable to be true to the model or true to the data. We use linear models to investigate this choice. After deriving an efficient method for calculating observational conditional expectation Shapley values for linear models, we investigate how correlation in simulated data impacts the convergence of observational conditional expectation Shapley values. Finally, we present two real data examples that we consider to be representative of possible use cases for feature attribution -- (1) credit risk modeling and (2) biological discovery. We show how a different choice of value function performs better in each scenario, and how possible attributions are impacted by modeling choices.

The paper shows that the application of the fixed-effect multiple linear regression model to an overparameterized dataset is equivalent to fitting the data with a hyper-curve parameterized by a single scalar parameter. This equivalence allows for a predictor-focused approach, where each predictor is described by a function of the chosen parameter. It is proven that a linear model will produce exact predictions even in the presence of nonlinear dependencies that violate the model assumptions. Parameterization in terms of the dependent variable and the monomial basis in the predictor function space are applied here to both synthetic and experimental data. The hyper-curve approach is especially suited for the regularization of problems with noise in predictor variables and can be used to remove noisy and "improper" predictors from the model.

Practical identifiability is a critical concern in data-driven modeling of mathematical systems. In this paper, we propose a novel framework for practical identifiability analysis to evaluate parameter identifiability in mathematical models of biological systems. Starting with a rigorous mathematical definition of practical identifiability, we demonstrate its equivalence to the invertibility of the Fisher Information Matrix. Our framework establishes the relationship between practical identifiability and coordinate identifiability, introducing a novel metric that simplifies and accelerates the evaluation of parameter identifiability compared to the profile likelihood method. Additionally, we introduce new regularization terms to address non-identifiable parameters, enabling uncertainty quantification and improving model reliability. To guide experimental design, we present an optimal data collection algorithm that ensures all model parameters are practically identifiable. Applications to Hill functions, neural networks, and dynamic biological models demonstrate the feasibility and efficiency of the proposed computational framework in uncovering critical biological processes and identifying key observable variables.

We present a general framework for symmetrizing an arbitrary neural-network architecture and making it equivariant with respect to a given group. We build upon the proposals of Kim et al. (2023); Kaba et al. (2023) for symmetrization, and improve them by replacing their conversion of neural features into group representations, with an optimization whose loss intuitively measures the distance between group orbits. This change makes our approach applicable to a broader range of matrix groups, such as the Lorentz group O(1, 3), than these two proposals. We experimentally show our method's competitiveness on the SO(2) image classification task, and also its increased generality on the task with O(1, 3). Our implementation will be made accessible at https://github.com/tiendatnguyen-vision/Orbit-symmetrize.

In this manuscript we consider the problem of generalized linear estimation on Gaussian mixture data with labels given by a single-index model. Our first result is a sharp asymptotic expression for the test and training errors in the high-dimensional regime. Motivated by the recent stream of results on the Gaussian universality of the test and training errors in generalized linear estimation, we ask ourselves the question: "when is a single Gaussian enough to characterize the error?". Our formula allow us to give sharp answers to this question, both in the positive and negative directions. More precisely, we show that the sufficient conditions for Gaussian universality (or lack of thereof) crucially depend on the alignment between the target weights and the means and covariances of the mixture clusters, which we precisely quantify. In the particular case of least-squares interpolation, we prove a strong universality property of the training error, and show it follows a simple, closed-form expression. Finally, we apply our results to real datasets, clarifying some recent discussion in the literature about Gaussian universality of the errors in this context.

Studies show that the model-independent, fully non-perturbative covariant cosmographic approach is suitable for analyzing the local Universe (zlesssim 0.1). However, accurately characterizing large and inhomogeneous mass distributions requires the fourth-order term in the redshift expansion of the covariant luminosity distance d_L(z,n). We calculate the covariant snap parameter S and its spherical harmonic multipole moments using the matter expansion tensor and the evolution equations for lightray bundles. The fourth-order term adds 36 degrees of freedom, since the highest independent multipole of the snap is the 32-pole (dotriacontapole) (ell=5). Including this term helps to de-bias estimations of the covariant deceleration parameter. Given that observations suggest axially symmetric anisotropies in the Hubble diagram for z lesssim 0.1 and theory shows that only a subset of multipoles contributes to the signal, we demonstrate that only 12 degrees of freedom are needed for a model-independent description of the local universe. We use an analytical axisymmetric model of the local Universe, with data that matches the Zwicky Transient Facility survey, in order to provide a numerical example of the amplitude of the snap multipoles and to forecast precision.

Graph Neural Networks (GNNs) show promising results for graph tasks. However, existing GNNs' generalization ability will degrade when there exist distribution shifts between testing and training graph data. The cardinal impetus underlying the severe degeneration is that the GNNs are architected predicated upon the I.I.D assumptions. In such a setting, GNNs are inclined to leverage imperceptible statistical correlations subsisting in the training set to predict, albeit it is a spurious correlation. In this paper, we study the problem of the generalization ability of GNNs in Out-Of-Distribution (OOD) settings. To solve this problem, we propose the Learning to Reweight for Generalizable Graph Neural Network (L2R-GNN) to enhance the generalization ability for achieving satisfactory performance on unseen testing graphs that have different distributions with training graphs. We propose a novel nonlinear graph decorrelation method, which can substantially improve the out-of-distribution generalization ability and compares favorably to previous methods in restraining the over-reduced sample size. The variables of the graph representation are clustered based on the stability of the correlation, and the graph decorrelation method learns weights to remove correlations between the variables of different clusters rather than any two variables. Besides, we interpose an efficacious stochastic algorithm upon bi-level optimization for the L2R-GNN framework, which facilitates simultaneously learning the optimal weights and GNN parameters, and avoids the overfitting problem. Experimental results show that L2R-GNN greatly outperforms baselines on various graph prediction benchmarks under distribution shifts.

Studies of density matrices for random quantum states lead naturally to the fixed trace Laguerre ensemble in random matrix theory. Previous studies have uncovered explicit rational function formulas for moments of purity statistic (trace of the squared density matrix), and also a third order linear differential equation satisfied by the eigenvalue density. We further probe the origin of these results from the viewpoint of integrability, which is taken here to mean wider classes of recursions and differential equations, and give extensions. Prominent in our study are first order linear matrix differential equations. One application given is to the derivation of the third order scalar equation for the density. Another is to obtain the explicit rational function formula for the variance of the purity statistic in the β generalised fixed trace Laguerre ensemble. In the original case (β= 2), the purity cumulants are expressed in terms of the large argument expansion of a particular σ-Painlevé IV transcendent. In a different but related direction, the exact computation of the two-point correlation for the fixed determinant circular unitary ensemble SU(N) is given the Appendix.

Overparameterized neural networks can be highly accurate on average on an i.i.d. test set yet consistently fail on atypical groups of the data (e.g., by learning spurious correlations that hold on average but not in such groups). Distributionally robust optimization (DRO) allows us to learn models that instead minimize the worst-case training loss over a set of pre-defined groups. However, we find that naively applying group DRO to overparameterized neural networks fails: these models can perfectly fit the training data, and any model with vanishing average training loss also already has vanishing worst-case training loss. Instead, the poor worst-case performance arises from poor generalization on some groups. By coupling group DRO models with increased regularization---a stronger-than-typical L2 penalty or early stopping---we achieve substantially higher worst-group accuracies, with 10-40 percentage point improvements on a natural language inference task and two image tasks, while maintaining high average accuracies. Our results suggest that regularization is important for worst-group generalization in the overparameterized regime, even if it is not needed for average generalization. Finally, we introduce a stochastic optimization algorithm, with convergence guarantees, to efficiently train group DRO models.

Pre-trained models have revolutionized natural language understanding. However, researchers have found they can encode artifacts undesired in many applications, such as professions correlating with one gender more than another. We explore such gendered correlations as a case study for how to address unintended correlations in pre-trained models. We define metrics and reveal that it is possible for models with similar accuracy to encode correlations at very different rates. We show how measured correlations can be reduced with general-purpose techniques, and highlight the trade offs different strategies have. With these results, we make recommendations for training robust models: (1) carefully evaluate unintended correlations, (2) be mindful of seemingly innocuous configuration differences, and (3) focus on general mitigations.

We study short-horizon forecasting of weekly terrorism incident counts using the Global Terrorism Database (GTD, 1970--2016). We build a reproducible pipeline with fixed time-based splits and evaluate a Bidirectional LSTM (BiLSTM) against strong classical anchors (seasonal-naive, linear/ARIMA) and a deep LSTM-Attention baseline. On the held-out test set, the BiLSTM attains RMSE 6.38, outperforming LSTM-Attention (9.19; +30.6\%) and a linear lag-regression baseline (+35.4\% RMSE gain), with parallel improvements in MAE and MAPE. Ablations varying temporal memory, training-history length, spatial grain, lookback size, and feature groups show that models trained on long historical data generalize best; a moderate lookback (20--30 weeks) provides strong context; and bidirectional encoding is critical for capturing both build-up and aftermath patterns within the window. Feature-group analysis indicates that short-horizon structure (lagged counts and rolling statistics) contributes most, with geographic and casualty features adding incremental lift. We release code, configs, and compact result tables, and provide a data/ethics statement documenting GTD licensing and research-only use. Overall, the study offers a transparent, baseline-beating reference for GTD incident forecasting.

We find candidate macroscopic gravity duals for scale-invariant but non-Lorentz invariant fixed points, which do not have particle number as a conserved quantity. We compute two-point correlation functions which exhibit novel behavior relative to their AdS counterparts, and find holographic renormalization group flows to conformal field theories. Our theories are characterized by a dynamical critical exponent z, which governs the anisotropy between spatial and temporal scaling t to lambda^z t, x to lambda x; we focus on the case with z=2. Such theories describe multicritical points in certain magnetic materials and liquid crystals, and have been shown to arise at quantum critical points in toy models of the cuprate superconductors. This work can be considered a small step towards making useful dual descriptions of such critical points.

Using the example of configurations generated with the worm algorithm for the two-dimensional Ising model, we propose renormalization group (RG) transformations, inspired by the tensor RG, that can be applied to sets of images. We relate criticality to the logarithmic divergence of the largest principal component. We discuss the changes in link occupation under the RG transformation, suggest ways to obtain data collapse, and compare with the two state tensor RG approximation near the fixed point.

The parameter space for any fixed architecture of feedforward ReLU neural networks serves as a proxy during training for the associated class of functions - but how faithful is this representation? It is known that many different parameter settings can determine the same function. Moreover, the degree of this redundancy is inhomogeneous: for some networks, the only symmetries are permutation of neurons in a layer and positive scaling of parameters at a neuron, while other networks admit additional hidden symmetries. In this work, we prove that, for any network architecture where no layer is narrower than the input, there exist parameter settings with no hidden symmetries. We also describe a number of mechanisms through which hidden symmetries can arise, and empirically approximate the functional dimension of different network architectures at initialization. These experiments indicate that the probability that a network has no hidden symmetries decreases towards 0 as depth increases, while increasing towards 1 as width and input dimension increase.

Energy-energy correlators are constructed by averaging the number of charged particle pairs within jets, weighted by the product of their transverse momenta, as a function of the angular separation of the particles within a pair. They are sensitive to a multitude of perturbative and nonperturbative quantum chromodynamics phenomena in high-energy particle collisions. Using lead-lead data recorded with the CMS detector, energy-energy correlators inside high transverse momentum jets are measured in heavy ion collisions for the first time. The data are obtained at a nucleon-nucleon center-of-mass energy of 5.02 TeV and correspond to an integrated luminosity of 1.70 nb^{-1}. A similar analysis is done for proton-proton collisions at the same center-of-mass energy to establish a reference. The ratio of lead-lead to proton-proton energy-energy correlators reveals significant jet substructure modifications in the quark-gluon plasma. The results are compared to different models that incorporate either color coherence or medium response effects, where the two effects predict similar substructure modifications.

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an n-sample in a space M can be considered as an element of the quotient space of M^n modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces. We fully describe the orbifold and path-metric structure of the sample space when M is a manifold or path-metric space, respectively. These results are non-trivial even when M is Euclidean. We show that the infinite sample space exists in a Gromov-Hausdorff type sense and coincides with the Wasserstein space of probability distributions on M. We exhibit Fr\'echet means and k-means as metric projections onto 1-skeleta or k-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.

A key step in reverse engineering neural networks is to decompose them into simpler parts that can be studied in relative isolation. Linear parameter decomposition -- a framework that has been proposed to resolve several issues with current decomposition methods -- decomposes neural network parameters into a sum of sparsely used vectors in parameter space. However, the current main method in this framework, Attribution-based Parameter Decomposition (APD), is impractical on account of its computational cost and sensitivity to hyperparameters. In this work, we introduce Stochastic Parameter Decomposition (SPD), a method that is more scalable and robust to hyperparameters than APD, which we demonstrate by decomposing models that are slightly larger and more complex than was possible to decompose with APD. We also show that SPD avoids other issues, such as shrinkage of the learned parameters, and better identifies ground truth mechanisms in toy models. By bridging causal mediation analysis and network decomposition methods, this demonstration opens up new research possibilities in mechanistic interpretability by removing barriers to scaling linear parameter decomposition methods to larger models. We release a library for running SPD and reproducing our experiments at https://github.com/goodfire-ai/spd.

In this article, we study the consistency of the template estimation with the Fr\'echet mean in quotient spaces. The Fr\'echet mean in quotient spaces is often used when the observations are deformed or transformed by a group action. We show that in most cases this estimator is actually inconsistent. We exhibit a sufficient condition for this inconsistency, which amounts to the folding of the distribution of the noisy template when it is projected to the quotient space. This condition appears to be fulfilled as soon as the support of the noise is large enough. To quantify this inconsistency we provide lower and upper bounds of the bias as a function of the variability (the noise level). This shows that the consistency bias cannot be neglected when the variability increases.

Redundancies and correlations in the responses of sensory neurons seem to waste neural resources but can carry cues about structured stimuli and may help the brain to correct for response errors. To assess how the retina negotiates this tradeoff, we measured simultaneous responses from populations of ganglion cells presented with natural and artificial stimuli that varied greatly in correlation structure. We found that pairwise correlations in the retinal output remained similar across stimuli with widely different spatio-temporal correlations including white noise and natural movies. Meanwhile, purely spatial correlations tended to increase correlations in the retinal response. Responding to more correlated stimuli, ganglion cells had faster temporal kernels and tended to have stronger surrounds. These properties of individual cells, along with gain changes that opposed changes in effective contrast at the ganglion cell input, largely explained the similarity of pairwise correlations across stimuli where receptive field measurements were possible.

To enhance group robustness to spurious correlations, prior work often relies on auxiliary annotations for groups or spurious features and assumes identical sets of groups across source and target domains. These two requirements are both unnatural and impractical in real-world settings. To overcome these limitations, we propose a method that leverages the semantic structure inherent in class labels--specifically, superclass information--to naturally reduce reliance on spurious features. Our model employs gradient-based attention guided by a pre-trained vision-language model to disentangle superclass-relevant and irrelevant features. Then, by promoting the use of all superclass-relevant features for prediction, our approach achieves robustness to more complex spurious correlations without the need to annotate any source samples. Experiments across diverse datasets demonstrate that our method significantly outperforms baselines in domain generalization tasks, with clear improvements in both quantitative metrics and qualitative visualizations.

In this paper, we study the flow of signals through linear paths with the nonlinear condition that a node emits a signal when it receives external stimuli or when two incoming signals from other nodes arrive coincidentally with a combined amplitude above a fixed threshold. Sets of such nodes form a polychrony group and can sometimes lead to cascades. In the context of this work, cascades are polychrony groups in which the number of nodes activated as a consequence of other nodes is greater than the number of externally activated nodes. The difference between these two numbers is the so-called profit. Given the initial conditions, we predict the conditions for a vertex to activate at a prescribed time and provide an algorithm to efficiently reconstruct a cascade. We develop a dictionary between polychrony groups and graph theory. We call the graph corresponding to a cascade a chinampa. This link leads to a topological classification of chinampas. We enumerate the chinampas of profits zero and one and the description of a family of chinampas isomorphic to a family of partially ordered sets, which implies that the enumeration problem of this family is equivalent to computing the Stanley-order polynomials of those partially ordered sets.

There exist a number of well known multiplicative generating functions for series of Schur functions. Amongst these are some related to the dual Cauchy identity whose expansion coefficients are rather simple, and in some cases periodic in parameters specifying the Schur functions. More recently similar identities have been found involving expansions in terms of characters of the symplectic group. Here these results are extended and generalised to all classical Lie groups. This is done through the derivation of explicit recurrence relations for the expansion coefficients based on the action of the Weyl groups of both the symplectic and orthogonal groups. Copious results are tabulated in the form of explicit values of the expansion coefficients as functions of highest weight parameters. An alternative approach is then based on dual pairs of symplectic and/or orthogonal groups. A byproduct of this approach is that expansions in terms of spin orthogonal group characters can always be recovered from non-spin cases.

Manifold-valued measurements exist in numerous applications within computer vision and machine learning. Recent studies have extended Deep Neural Networks (DNNs) to manifolds, and concomitantly, normalization techniques have also been adapted to several manifolds, referred to as Riemannian normalization. Nonetheless, most of the existing Riemannian normalization methods have been derived in an ad hoc manner and only apply to specific manifolds. This paper establishes a unified framework for Riemannian Batch Normalization (RBN) techniques on Lie groups. Our framework offers the theoretical guarantee of controlling both the Riemannian mean and variance. Empirically, we focus on Symmetric Positive Definite (SPD) manifolds, which possess three distinct types of Lie group structures. Using the deformation concept, we generalize the existing Lie groups on SPD manifolds into three families of parameterized Lie groups. Specific normalization layers induced by these Lie groups are then proposed for SPD neural networks. We demonstrate the effectiveness of our approach through three sets of experiments: radar recognition, human action recognition, and electroencephalography (EEG) classification. The code is available at https://github.com/GitZH-Chen/LieBN.git.

Moment restrictions and their conditional counterparts emerge in many areas of machine learning and statistics ranging from causal inference to reinforcement learning. Estimators for these tasks, generally called methods of moments, include the prominent generalized method of moments (GMM) which has recently gained attention in causal inference. GMM is a special case of the broader family of empirical likelihood estimators which are based on approximating a population distribution by means of minimizing a varphi-divergence to an empirical distribution. However, the use of varphi-divergences effectively limits the candidate distributions to reweightings of the data samples. We lift this long-standing limitation and provide a method of moments that goes beyond data reweighting. This is achieved by defining an empirical likelihood estimator based on maximum mean discrepancy which we term the kernel method of moments (KMM). We provide a variant of our estimator for conditional moment restrictions and show that it is asymptotically first-order optimal for such problems. Finally, we show that our method achieves competitive performance on several conditional moment restriction tasks.

Clustering is a widely deployed unsupervised learning tool. Model-based clustering is a flexible framework to tackle data heterogeneity when the clusters have different shapes. Likelihood-based inference for mixture distributions often involves non-convex and high-dimensional objective functions, imposing difficult computational and statistical challenges. The classic expectation-maximization (EM) algorithm is a computationally thrifty iterative method that maximizes a surrogate function minorizing the log-likelihood of observed data in each iteration, which however suffers from bad local maxima even in the special case of the standard Gaussian mixture model with common isotropic covariance matrices. On the other hand, recent studies reveal that the unique global solution of a semidefinite programming (SDP) relaxed K-means achieves the information-theoretically sharp threshold for perfectly recovering the cluster labels under the standard Gaussian mixture model. In this paper, we extend the SDP approach to a general setting by integrating cluster labels as model parameters and propose an iterative likelihood adjusted SDP (iLA-SDP) method that directly maximizes the exact observed likelihood in the presence of data heterogeneity. By lifting the cluster assignment to group-specific membership matrices, iLA-SDP avoids centroids estimation -- a key feature that allows exact recovery under well-separateness of centroids without being trapped by their adversarial configurations. Thus iLA-SDP is less sensitive than EM to initialization and more stable on high-dimensional data. Our numeric experiments demonstrate that iLA-SDP can achieve lower mis-clustering errors over several widely used clustering methods including K-means, SDP and EM algorithms.

With an increasing number of replication studies performed in psychological science, the question of how to evaluate the outcome of a replication attempt deserves careful consideration. Bayesian approaches allow to incorporate uncertainty and prior information into the analysis of the replication attempt by their design. The Replication Bayes Factor, introduced by Verhagen & Wagenmakers (2014), provides quantitative, relative evidence in favor or against a successful replication. In previous work by Verhagen & Wagenmakers (2014) it was limited to the case of t-tests. In this paper, the Replication Bayes Factor is extended to F-tests in multi-group, fixed-effect ANOVA designs. Simulations and examples are presented to facilitate the understanding and to demonstrate the usefulness of this approach. Finally, the Replication Bayes Factor is compared to other Bayesian and frequentist approaches and discussed in the context of replication attempts. R code to calculate Replication Bayes factors and to reproduce the examples in the paper is available at https://osf.io/jv39h/.

The probabilistic diffusion model has become highly effective across various domains. Typically, sampling from a diffusion model involves using a denoising distribution characterized by a Gaussian with a learned mean and either fixed or learned covariances. In this paper, we leverage the recently proposed covariance moment matching technique and introduce a novel method for learning the diagonal covariance. Unlike traditional data-driven diagonal covariance approximation approaches, our method involves directly regressing the optimal diagonal analytic covariance using a new, unbiased objective named Optimal Covariance Matching (OCM). This approach can significantly reduce the approximation error in covariance prediction. We demonstrate how our method can substantially enhance the sampling efficiency, recall rate and likelihood of commonly used diffusion models.

Time series forecasting has attracted significant attention in recent decades. Previous studies have demonstrated that the Channel-Independent (CI) strategy improves forecasting performance by treating different channels individually, while it leads to poor generalization on unseen instances and ignores potentially necessary interactions between channels. Conversely, the Channel-Dependent (CD) strategy mixes all channels with even irrelevant and indiscriminate information, which, however, results in oversmoothing issues and limits forecasting accuracy. There is a lack of channel strategy that effectively balances individual channel treatment for improved forecasting performance without overlooking essential interactions between channels. Motivated by our observation of a correlation between the time series model's performance boost against channel mixing and the intrinsic similarity on a pair of channels, we developed a novel and adaptable Channel Clustering Module (CCM). CCM dynamically groups channels characterized by intrinsic similarities and leverages cluster information instead of individual channel identities, combining the best of CD and CI worlds. Extensive experiments on real-world datasets demonstrate that CCM can (1) boost the performance of CI and CD models by an average margin of 2.4% and 7.2% on long-term and short-term forecasting, respectively; (2) enable zero-shot forecasting with mainstream time series forecasting models; (3) uncover intrinsic time series patterns among channels and improve interpretability of complex time series models.

We describe nonparametric deconvolution models (NDMs), a family of Bayesian nonparametric models for collections of data in which each observation is the average over the features from heterogeneous particles. For example, these types of data are found in elections, where we observe precinct-level vote tallies (observations) of individual citizens' votes (particles) across each of the candidates or ballot measures (features), where each voter is part of a specific voter cohort or demographic (factor). Like the hierarchical Dirichlet process, NDMs rely on two tiers of Dirichlet processes to explain the data with an unknown number of latent factors; each observation is modeled as a weighted average of these latent factors. Unlike existing models, NDMs recover how factor distributions vary locally for each observation. This uniquely allows NDMs both to deconvolve each observation into its constituent factors, and also to describe how the factor distributions specific to each observation vary across observations and deviate from the corresponding global factors. We present variational inference techniques for this family of models and study its performance on simulated data and voting data from California. We show that including local factors improves estimates of global factors and provides a novel scaffold for exploring data.

We discuss methods for modeling eclipsing binary stars using the "physical", "simplified" and "phenomenological" models. There are few realizations of the "physical" Wilson-Devinney (1971) code and its improvements, e.g. Binary Maker, Phoebe. A parameter search using the Monte-Carlo method was realized by Zola et al. (2010), which is efficient in expense of too many evaluations of the test function. We compare existing algorithms of minimization of multi-parametric functions and propose to use a "combined" algorithm, depending on if the Hessian matrix is positively determined. To study methods, a simply fast-computed function resembling the "complete" test function for the physical model. Also we adopt a simplified model of an eclipsing binary at a circular orbit assuming spherical components with an uniform brightness distribution. This model resembles more advanced models in a sense of correlated parameter estimates due to a similar topology of the test function. Such a model may be applied to detached Algol-type systems, where the tidal distortion of components is negligible.

We consider the problem of graph matching, or learning vertex correspondence, between two correlated stochastic block models (SBMs). The graph matching problem arises in various fields, including computer vision, natural language processing and bioinformatics, and in particular, matching graphs with inherent community structure has significance related to de-anonymization of correlated social networks. Compared to the correlated Erdos-Renyi (ER) model, where various efficient algorithms have been developed, among which a few algorithms have been proven to achieve the exact matching with constant edge correlation, no low-order polynomial algorithm has been known to achieve exact matching for the correlated SBMs with constant correlation. In this work, we propose an efficient algorithm for matching graphs with community structure, based on the comparison between partition trees rooted from each vertex, by extending the idea of Mao et al. (2021) to graphs with communities. The partition tree divides the large neighborhoods of each vertex into disjoint subsets using their edge statistics to different communities. Our algorithm is the first low-order polynomial-time algorithm achieving exact matching between two correlated SBMs with high probability in dense graphs.

We introduce ensembles within score-based sampling methods to develop gradient-free approximate sampling techniques that leverage the collective dynamics of particle ensembles to compute approximate reverse diffusion drifts. We introduce the underlying methodology, emphasizing its relationship with generative diffusion models and the previously introduced F\"ollmer sampler. We demonstrate the efficacy of ensemble strategies through various examples, ranging from low- to medium-dimensionality sampling problems, including multi-modal and highly non-Gaussian probability distributions, and provide comparisons to traditional methods like NUTS. Our findings highlight the potential of ensemble strategies for modeling complex probability distributions in situations where gradients are unavailable. Finally, we showcase its application in the context of Bayesian inversion problems within the geophysical sciences.

We develop a novel way to probe subgalactic-scale matter distribution with diffractive lensing on gravitational waves. Five-year observations from Einstein Telescope and DECIGO are expected to probe k= 10^5sim 10^8 ,{rm Mpc}^{-1} down to P(k) = 10^{-16} sim 10^{-14} ,{rm Mpc}^3 level. These results can be interpreted in terms of primordial black holes in the range M_{rm PBH} gtrsim 10^{-3}M_odot down to f_{rm PBH} = 10^{-6} level, or QCD axion minihalos in the range m_a = 10^{-3} sim 10^{-12} ,{rm eV}. A key result of the paper is the approximate relation between the scale k and the gravitational wave frequency f, derived in an ensemble of `multi-lensing' events. This relation enables direct measurement of the power spectrum at specific scales, with sensitivities characterized by model-independent kernels delta P(k). Additionally, we delineate the statistical properties of `multi-lensing' based on the `Fresnel number' N_F. When N_F cal O(1), the statistical significance can be approximately calculated by Variance of lensing effects, which is directly related to the power spectrum among other moments of matter distribution.

Spurious correlations allow flexible models to predict well during training but poorly on related test distributions. Recent work has shown that models that satisfy particular independencies involving correlation-inducing nuisance variables have guarantees on their test performance. Enforcing such independencies requires nuisances to be observed during training. However, nuisances, such as demographics or image background labels, are often missing. Enforcing independence on just the observed data does not imply independence on the entire population. Here we derive mmd estimators used for invariance objectives under missing nuisances. On simulations and clinical data, optimizing through these estimates achieves test performance similar to using estimators that make use of the full data.

The recent trend towards Personalized Federated Learning (PFL) has garnered significant attention as it allows for the training of models that are tailored to each client while maintaining data privacy. However, current PFL techniques primarily focus on modeling the conditional distribution heterogeneity (i.e. concept shift), which can result in suboptimal performance when the distribution of input data across clients diverges (i.e. covariate shift). Additionally, these techniques often lack the ability to adapt to unseen data, further limiting their effectiveness in real-world scenarios. To address these limitations, we propose a novel approach, FedGMM, which utilizes Gaussian mixture models (GMM) to effectively fit the input data distributions across diverse clients. The model parameters are estimated by maximum likelihood estimation utilizing a federated Expectation-Maximization algorithm, which is solved in closed form and does not assume gradient similarity. Furthermore, FedGMM possesses an additional advantage of adapting to new clients with minimal overhead, and it also enables uncertainty quantification. Empirical evaluations on synthetic and benchmark datasets demonstrate the superior performance of our method in both PFL classification and novel sample detection.

Covariates provide valuable information on external factors that influence time series and are critical in many real-world time series forecasting tasks. For example, in retail, covariates may indicate promotions or peak dates such as holiday seasons that heavily influence demand forecasts. Recent advances in pretraining large language model architectures for time series forecasting have led to highly accurate forecasters. However, the majority of these models do not readily use covariates as they are often specific to a certain task or domain. This paper introduces a new method to incorporate covariates into pretrained time series forecasting models. Our proposed approach incorporates covariate information into pretrained forecasting models through modular blocks that inject past and future covariate information, without necessarily modifying the pretrained model in consideration. In order to evaluate our approach, we introduce a benchmark composed of 32 different synthetic datasets with varying dynamics to evaluate the effectivity of forecasting models with covariates. Extensive evaluations on both synthetic and real datasets show that our approach effectively incorporates covariate information into pretrained models, outperforming existing baselines.

Using data from the Galaxy and Mass Assembly (GAMA) survey, we investigate how galaxy properties correlate with the local environment, focusing on the two southern regions of the survey (G02 and G23) that have not previously been examined in this context. We employ two-point and marked correlation functions to quantify the environmental dependence of galaxy color, stellar mass, luminosity across the u, g, r, J, and K bands, as well as star formation rate (SFR) and specific star formation rate (sSFR). We also assess the impact of redshift incompleteness and cosmic variance on these clustering measurements. Our results show that u-r and g-r colors are most strongly correlated with local overdensity, followed by stellar mass. The sSFR exhibits a clear inverse relationship with density of the environment, consistent with the trend observed for u-band luminosity, which traces young stellar populations. In contrast, galaxies brighter in the g, J, and K bands preferentially inhabit denser regions. By comparing our measurements from the southern regions with those from the equatorial regions of GAMA, we find that cosmic variance does not significantly influence our conclusions. However, redshift incompleteness affects the clustering measurements, as revealed through comparisons of subsets within the G02 region. The measured correlations provide key constraints for models of galaxy assembly across mass and environment, while the environmental trends in color and near-infrared luminosity offer a means to trace stellar mass growth and quenching with redshift.

Dirichlet Process mixture models (DPMM) in combination with Gaussian kernels have been an important modeling tool for numerous data domains arising from biological, physical, and social sciences. However, this versatility in applications does not extend to strong theoretical guarantees for the underlying parameter estimates, for which only a logarithmic rate is achieved. In this work, we (re)introduce and investigate a metric, named Orlicz-Wasserstein distance, in the study of the Bayesian contraction behavior for the parameters. We show that despite the overall slow convergence guarantees for all the parameters, posterior contraction for parameters happens at almost polynomial rates in outlier regions of the parameter space. Our theoretical results provide new insight in understanding the convergence behavior of parameters arising from various settings of hierarchical Bayesian nonparametric models. In addition, we provide an algorithm to compute the metric by leveraging Sinkhorn divergences and validate our findings through a simulation study.

This study presents the modeling of the gravitational wave (GW) bias parameter by bridging a connection between simulated GW sources and galaxies in low redshift galaxy surveys 2MPZ and WISExSCOS (WISC). We study this connection by creating a mock GW catalog, populating galaxy surveys with binary black holes (BBHs) for different scenarios of the GW host-galaxy probability as a function of the galaxy stellar mass. We probe the observable consequences of this connection by exploring the spatial clustering of the GW sources in terms of the GW bias parameter. We consider a phenomenological broken power law model for the host-galaxy probability function, with a potential turnover M_{K} at high stellar mass (10^{11} M_{odot} in the fiducial model) where the star formation efficiency begins to drop. We vary the parameters of the GW host-galaxy probability function and find that generically the GW bias increases as M_{K} increases (and gets suppressed as M_{K} decreases). The change in the GW bias parameter shows a maximum change of about 30% for different scenarios explored in this work in comparison to the galaxy bias. Future measurements of the GW bias can help constrain M_{K} and the slopes of the host-galaxy probability function and thus offer insights into the underlying astrophysical processes.

We study counterfactual identifiability in causal models with bijective generation mechanisms (BGM), a class that generalizes several widely-used causal models in the literature. We establish their counterfactual identifiability for three common causal structures with unobserved confounding, and propose a practical learning method that casts learning a BGM as structured generative modeling. Learned BGMs enable efficient counterfactual estimation and can be obtained using a variety of deep conditional generative models. We evaluate our techniques in a visual task and demonstrate its application in a real-world video streaming simulation task.

We present a neural network emulator to constrain the thermal parameters of the intergalactic medium (IGM) at 5.4z6.0 using the Lyman-displaystylealpha (Lydisplaystylealpha) forest flux auto-correlation function. Our auto-differentiable JAX-based framework accelerates the surrogate model generation process using approximately 100 sparsely sampled Nyx hydrodynamical simulations with varying combinations of thermal parameters, i.e., the temperature at mean density T_{{0}}, the slope of the temperaturedisplaystyle-density relation displaystylegamma, and the mean transmission flux langle{F}{rangle}. We show that this emulator has a typical accuracy of 1.0% across the specified redshift range. Bayesian inference of the IGM thermal parameters, incorporating emulator uncertainty propagation, is further expedited using NumPyro Hamiltonian Monte Carlo. We compare both the inference results and computational cost of our framework with the traditional nearest-neighbor interpolation approach applied to the same set of mock Lyalpha flux. By examining the credibility contours of the marginalized posteriors for T_{{0}},gamma,and{langle}{F}{rangle} obtained using the emulator, the statistical reliability of measurements is established through inference on 100 realistic mock data sets of the auto-correlation function.

Observational studies of causal effects require adjustment for confounding factors. In the tabular setting, where these factors are well-defined, separate random variables, the effect of confounding is well understood. However, in public policy, ecology, and in medicine, decisions are often made in non-tabular settings, informed by patterns or objects detected in images (e.g., maps, satellite or tomography imagery). Using such imagery for causal inference presents an opportunity because objects in the image may be related to the treatment and outcome of interest. In these cases, we rely on the images to adjust for confounding but observed data do not directly label the existence of the important objects. Motivated by real-world applications, we formalize this challenge, how it can be handled, and what conditions are sufficient to identify and estimate causal effects. We analyze finite-sample performance using simulation experiments, estimating effects using a propensity adjustment algorithm that employs a machine learning model to estimate the image confounding. Our experiments also examine sensitivity to misspecification of the image pattern mechanism. Finally, we use our methodology to estimate the effects of policy interventions on poverty in African communities from satellite imagery.

A number of methods have been proposed for causal effect estimation, yet few have demonstrated efficacy in handling data with complex structures, such as images. To fill this gap, we propose Causal Multi-task Deep Ensemble (CMDE), a novel framework that learns both shared and group-specific information from the study population. We provide proofs demonstrating equivalency of CDME to a multi-task Gaussian process (GP) with a coregionalization kernel a priori. Compared to multi-task GP, CMDE efficiently handles high-dimensional and multi-modal covariates and provides pointwise uncertainty estimates of causal effects. We evaluate our method across various types of datasets and tasks and find that CMDE outperforms state-of-the-art methods on a majority of these tasks.

This article presents Individual Conditional Expectation (ICE) plots, a tool for visualizing the model estimated by any supervised learning algorithm. Classical partial dependence plots (PDPs) help visualize the average partial relationship between the predicted response and one or more features. In the presence of substantial interaction effects, the partial response relationship can be heterogeneous. Thus, an average curve, such as the PDP, can obfuscate the complexity of the modeled relationship. Accordingly, ICE plots refine the partial dependence plot by graphing the functional relationship between the predicted response and the feature for individual observations. Specifically, ICE plots highlight the variation in the fitted values across the range of a covariate, suggesting where and to what extent heterogeneities might exist. In addition to providing a plotting suite for exploratory analysis, we include a visual test for additive structure in the data generating model. Through simulated examples and real data sets, we demonstrate how ICE plots can shed light on estimated models in ways PDPs cannot. Procedures outlined are available in the R package ICEbox.

In observational studies, balancing covariates in different treatment groups is essential to estimate treatment effects. One of the most commonly used methods for such purposes is weighting. The performance of this class of methods usually depends on strong regularity conditions for the underlying model, which might not hold in practice. In this paper, we investigate weighting methods from a functional estimation perspective and argue that the weights needed for covariate balancing could differ from those needed for treatment effects estimation under low regularity conditions. Motivated by this observation, we introduce a new framework of weighting that directly targets the treatment effects estimation. Unlike existing methods, the resulting estimator for a treatment effect under this new framework is a simple kernel-based U-statistic after applying a data-driven transformation to the observed covariates. We characterize the theoretical properties of the new estimators of treatment effects under a nonparametric setting and show that they are able to work robustly under low regularity conditions. The new framework is also applied to several numerical examples to demonstrate its practical merits.

Causal representation learning aims to unveil latent high-level causal representations from observed low-level data. One of its primary tasks is to provide reliable assurance of identifying these latent causal models, known as identifiability. A recent breakthrough explores identifiability by leveraging the change of causal influences among latent causal variables across multiple environments liu2022identifying. However, this progress rests on the assumption that the causal relationships among latent causal variables adhere strictly to linear Gaussian models. In this paper, we extend the scope of latent causal models to involve nonlinear causal relationships, represented by polynomial models, and general noise distributions conforming to the exponential family. Additionally, we investigate the necessity of imposing changes on all causal parameters and present partial identifiability results when part of them remains unchanged. Further, we propose a novel empirical estimation method, grounded in our theoretical finding, that enables learning consistent latent causal representations. Our experimental results, obtained from both synthetic and real-world data, validate our theoretical contributions concerning identifiability and consistency.

The last decade has seen a marked shift in how the internal structure of hadrons is understood. Modern experimental facilities, new theoretical techniques for the continuum bound-state problem and progress with lattice-regularised QCD have provided strong indications that soft quark+quark (diquark) correlations play a crucial role in hadron physics. For example, theory indicates that the appearance of such correlations is a necessary consequence of dynamical chiral symmetry breaking, viz. a corollary of emergent hadronic mass that is responsible for almost all visible mass in the universe; experiment has uncovered signals for such correlations in the flavour-separation of the proton's electromagnetic form factors; and phenomenology suggests that diquark correlations might be critical to the formation of exotic tetra- and penta-quark hadrons. A broad spectrum of such information is evaluated herein, with a view to consolidating the facts and therefrom moving toward a coherent, unified picture of hadron structure and the role that diquark correlations might play.

The k-means++ seeding algorithm (Arthur & Vassilvitskii, 2007) is widely used in practice for the k-means clustering problem where the goal is to cluster a dataset X subset R ^d into k clusters. The popularity of this algorithm is due to its simplicity and provable guarantee of being O(log k) competitive with the optimal solution in expectation. However, its running time is O(|X|kd), making it expensive for large datasets. In this work, we present a simple and effective rejection sampling based approach for speeding up k-means++. Our first method runs in time O(nnz (X) + beta k^2d) while still being O(log k ) competitive in expectation. Here, beta is a parameter which is the ratio of the variance of the dataset to the optimal k-means cost in expectation and O hides logarithmic factors in k and |X|. Our second method presents a new trade-off between computational cost and solution quality. It incurs an additional scale-invariant factor of k^{-Omega( m/beta)} Var (X) in addition to the O(log k) guarantee of k-means++ improving upon a result of (Bachem et al, 2016a) who get an additional factor of m^{-1}Var(X) while still running in time O(nnz(X) + mk^2d). We perform extensive empirical evaluations to validate our theoretical results and to show the effectiveness of our approach on real datasets.

We show that PLDR-LLMs pretrained at self-organized criticality exhibit reasoning at inference time. The characteristics of PLDR-LLM deductive outputs at criticality is similar to second-order phase transitions. At criticality, the correlation length diverges, and the deductive outputs attain a metastable steady state. The steady state behaviour suggests that deductive outputs learn representations equivalent to scaling functions, universality classes and renormalization groups from the training dataset, leading to generalization and reasoning capabilities in the process. We can then define an order parameter from the global statistics of the model's deductive output parameters at inference. The reasoning capabilities of a PLDR-LLM is better when its order parameter is close to zero at criticality. This observation is supported by the benchmark scores of the models trained at near-criticality and sub-criticality. Our results provide a self-contained explanation on how reasoning manifests in large language models, and the ability to reason can be quantified solely from global model parameter values of the deductive outputs at steady state, without any need for evaluation of curated benchmark datasets through inductive output for reasoning and comprehension.

A key to knowledge graph embedding (KGE) is to choose a proper representation space, e.g., point-wise Euclidean space and complex vector space. In this paper, we propose a unified perspective of embedding and introduce uncertainty into KGE from the view of group theory. Our model can incorporate existing models (i.e., generality), ensure the computation is tractable (i.e., efficiency) and enjoy the expressive power of complex random variables (i.e., expressiveness). The core idea is that we embed entities/relations as elements of a symmetric group, i.e., permutations of a set. Permutations of different sets can reflect different properties of embedding. And the group operation of symmetric groups is easy to compute. In specific, we show that the embedding of many existing models, point vectors, can be seen as elements of a symmetric group. To reflect uncertainty, we first embed entities/relations as permutations of a set of random variables. A permutation can transform a simple random variable into a complex random variable for greater expressiveness, called a normalizing flow. We then define scoring functions by measuring the similarity of two normalizing flows, namely NFE. We construct several instantiating models and prove that they are able to learn logical rules. Experimental results demonstrate the effectiveness of introducing uncertainty and our model. The code is available at https://github.com/changyi7231/NFE.

In this work we take a Category Theoretic perspective on the relationship between probabilistic modeling and function approximation. We begin by defining two extensions of function composition to stochastic process subordination: one based on the co-Kleisli category under the comonad (Omega x -) and one based on the parameterization of a category with a Lawvere theory. We show how these extensions relate to the category Stoch and other Markov Categories. Next, we apply the Para construction to extend stochastic processes to parameterized statistical models and we define a way to compose the likelihood functions of these models. We conclude with a demonstration of how the Maximum Likelihood Estimation procedure defines an identity-on-objects functor from the category of statistical models to the category of Learners. Code to accompany this paper can be found at https://github.com/dshieble/Categorical_Stochastic_Processes_and_Likelihood

We use tools from geometric statistics to analyze the usual estimation procedure of a template shape. This applies to shapes from landmarks, curves, surfaces, images etc. We demonstrate the asymptotic bias of the template shape estimation using the stratified geometry of the shape space. We give a Taylor expansion of the bias with respect to a parameter sigma describing the measurement error on the data. We propose two bootstrap procedures that quantify the bias and correct it, if needed. They are applicable for any type of shape data. We give a rule of thumb to provide intuition on whether the bias has to be corrected. This exhibits the parameters that control the bias' magnitude. We illustrate our results on simulated and real shape data.

Several studies have compared the in-distribution (ID) and out-of-distribution (OOD) performance of models in computer vision and NLP. They report a frequent positive correlation and some surprisingly never even observe an inverse correlation indicative of a necessary trade-off. The possibility of inverse patterns is important to determine whether ID performance can serve as a proxy for OOD generalization capabilities. This paper shows with multiple datasets that inverse correlations between ID and OOD performance do happen in real-world data - not only in theoretical worst-case settings. We also explain theoretically how these cases can arise even in a minimal linear setting, and why past studies could miss such cases due to a biased selection of models. Our observations lead to recommendations that contradict those found in much of the current literature. - High OOD performance sometimes requires trading off ID performance. - Focusing on ID performance alone may not lead to optimal OOD performance. It may produce diminishing (eventually negative) returns in OOD performance. - In these cases, studies on OOD generalization that use ID performance for model selection (a common recommended practice) will necessarily miss the best-performing models, making these studies blind to a whole range of phenomena.

Entropy and its various generalizations are important in many fields, including mathematical statistics, communication theory, physics and computer science, for characterizing the amount of information associated with a probability distribution. In this paper we propose goodness-of-fit statistics for the multivariate Student and multivariate Pearson type II distributions, based on the maximum entropy principle and a class of estimators for Renyi entropy based on nearest neighbour distances. We prove the L2-consistency of these statistics using results on the subadditivity of Euclidean functionals on nearest neighbour graphs, and investigate their rate of convergence and asymptotic distribution using Monte Carlo methods.

The striking fractal geometry of strange attractors underscores the generative nature of chaos: like probability distributions, chaotic systems can be repeatedly measured to produce arbitrarily-detailed information about the underlying attractor. Chaotic systems thus pose a unique challenge to modern statistical learning techniques, while retaining quantifiable mathematical properties that make them controllable and interpretable as benchmarks. Here, we present a growing database currently comprising 131 known chaotic dynamical systems spanning fields such as astrophysics, climatology, and biochemistry. Each system is paired with precomputed multivariate and univariate time series. Our dataset has comparable scale to existing static time series databases; however, our systems can be re-integrated to produce additional datasets of arbitrary length and granularity. Our dataset is annotated with known mathematical properties of each system, and we perform feature analysis to broadly categorize the diverse dynamics present across the collection. Chaotic systems inherently challenge forecasting models, and across extensive benchmarks we correlate forecasting performance with the degree of chaos present. We also exploit the unique generative properties of our dataset in several proof-of-concept experiments: surrogate transfer learning to improve time series classification, importance sampling to accelerate model training, and benchmarking symbolic regression algorithms.

We study the problem of model selection in causal inference, specifically for the case of conditional average treatment effect (CATE) estimation under binary treatments. Unlike model selection in machine learning, there is no perfect analogue of cross-validation as we do not observe the counterfactual potential outcome for any data point. Towards this, there have been a variety of proxy metrics proposed in the literature, that depend on auxiliary nuisance models estimated from the observed data (propensity score model, outcome regression model). However, the effectiveness of these metrics has only been studied on synthetic datasets as we can access the counterfactual data for them. We conduct an extensive empirical analysis to judge the performance of these metrics introduced in the literature, and novel ones introduced in this work, where we utilize the latest advances in generative modeling to incorporate multiple realistic datasets. Our analysis suggests novel model selection strategies based on careful hyperparameter tuning of CATE estimators and causal ensembling.

With the rapid growth in language processing applications, fairness has emerged as an important consideration in data-driven solutions. Although various fairness definitions have been explored in the recent literature, there is lack of consensus on which metrics most accurately reflect the fairness of a system. In this work, we propose a new formulation : ACCUMULATED PREDICTION SENSITIVITY, which measures fairness in machine learning models based on the model's prediction sensitivity to perturbations in input features. The metric attempts to quantify the extent to which a single prediction depends on a protected attribute, where the protected attribute encodes the membership status of an individual in a protected group. We show that the metric can be theoretically linked with a specific notion of group fairness (statistical parity) and individual fairness. It also correlates well with humans' perception of fairness. We conduct experiments on two text classification datasets : JIGSAW TOXICITY, and BIAS IN BIOS, and evaluate the correlations between metrics and manual annotations on whether the model produced a fair outcome. We observe that the proposed fairness metric based on prediction sensitivity is statistically significantly more correlated with human annotation than the existing counterfactual fairness metric.

We explore a generative machine learning-based approach for estimating multi-dimensional probability density functions (PDFs) in a target sample using a statistically independent but related control sample - a common challenge in particle physics data analysis. The generative model must accurately reproduce individual observable distributions while preserving the correlations between them, based on the input multidimensional distribution from the control sample. Here we present a conditional normalizing flow model (CNF) based on a chain of bijectors which learns to transform unpaired simulation events to data events. We assess the performance of the CNF model in the context of LHC Higgs to diphoton analysis, where we use the CNF model to convert a Monte Carlo diphoton sample to one that models data. We show that the CNF model can accurately model complex data distributions and correlations. We also leverage the recently popularized Modified Differential Multiplier Method (MDMM) to improve the convergence of our model and assign physical meaning to usually arbitrary loss-function parameters.

In regression and causal inference, controlled subgroup selection aims to identify, with inferential guarantees, a subgroup (defined as a subset of the covariate space) on which the average response or treatment effect is above a given threshold. E.g., in a clinical trial, it may be of interest to find a subgroup with a positive average treatment effect. However, existing methods either lack inferential guarantees, heavily restrict the search for the subgroup, or sacrifice efficiency by naive data splitting. We propose a novel framework called chiseling that allows the analyst to interactively refine and test a candidate subgroup by iteratively shrinking it. The sole restriction is that the shrinkage direction only depends on the points outside the current subgroup, but otherwise the analyst may leverage any prior information or machine learning algorithm. Despite this flexibility, chiseling controls the probability that the discovered subgroup is null (e.g., has a non-positive average treatment effect) under minimal assumptions: for example, in randomized experiments, this inferential validity guarantee holds under only bounded moment conditions. When applied to a variety of simulated datasets and a real survey experiment, chiseling identifies substantially better subgroups than existing methods with inferential guarantees.

In many neural networks, different values of the parameters may result in the same loss value. Parameter space symmetries are loss-invariant transformations that change the model parameters. Teleportation applies such transformations to accelerate optimization. However, the exact mechanism behind this algorithm's success is not well understood. In this paper, we show that teleportation not only speeds up optimization in the short-term, but gives overall faster time to convergence. Additionally, teleporting to minima with different curvatures improves generalization, which suggests a connection between the curvature of the minimum and generalization ability. Finally, we show that integrating teleportation into a wide range of optimization algorithms and optimization-based meta-learning improves convergence. Our results showcase the versatility of teleportation and demonstrate the potential of incorporating symmetry in optimization.

Group equivariant neural networks are used as building blocks of group invariant neural networks, which have been shown to improve generalisation performance and data efficiency through principled parameter sharing. Such works have mostly focused on group equivariant convolutions, building on the result that group equivariant linear maps are necessarily convolutions. In this work, we extend the scope of the literature to self-attention, that is emerging as a prominent building block of deep learning models. We propose the LieTransformer, an architecture composed of LieSelfAttention layers that are equivariant to arbitrary Lie groups and their discrete subgroups. We demonstrate the generality of our approach by showing experimental results that are competitive to baseline methods on a wide range of tasks: shape counting on point clouds, molecular property regression and modelling particle trajectories under Hamiltonian dynamics.

In critical decision-making scenarios, optimizing accuracy can lead to a biased classifier, hence past work recommends enforcing group-based fairness metrics in addition to maximizing accuracy. However, doing so exposes the classifier to another kind of bias called infra-marginality. This refers to individual-level bias where some individuals/subgroups can be worse off than under simply optimizing for accuracy. For instance, a classifier implementing race-based parity may significantly disadvantage women of the advantaged race. To quantify this bias, we propose a general notion of eta-infra-marginality that can be used to evaluate the extent of this bias. We prove theoretically that, unlike other fairness metrics, infra-marginality does not have a trade-off with accuracy: high accuracy directly leads to low infra-marginality. This observation is confirmed through empirical analysis on multiple simulated and real-world datasets. Further, we find that maximizing group fairness often increases infra-marginality, suggesting the consideration of both group-level fairness and individual-level infra-marginality. However, measuring infra-marginality requires knowledge of the true distribution of individual-level outcomes correctly and explicitly. We propose a practical method to measure infra-marginality, and a simple algorithm to maximize group-wise accuracy and avoid infra-marginality.

Understanding the gradient variance of black-box variational inference (BBVI) is a crucial step for establishing its convergence and developing algorithmic improvements. However, existing studies have yet to show that the gradient variance of BBVI satisfies the conditions used to study the convergence of stochastic gradient descent (SGD), the workhorse of BBVI. In this work, we show that BBVI satisfies a matching bound corresponding to the ABC condition used in the SGD literature when applied to smooth and quadratically-growing log-likelihoods. Our results generalize to nonlinear covariance parameterizations widely used in the practice of BBVI. Furthermore, we show that the variance of the mean-field parameterization has provably superior dimensional dependence.

In the artificial neuron, I replace the dot product with the weighted Lehmer mean, which may emulate different cases of a generalized mean. The single neuron instance is replaced by a multiplet of neurons which have the same averaging weights. A group of outputs feed forward, in lieu of the single scalar. The generalization parameter is typically set to a different value for each neuron in the multiplet. I further extend the concept to a multiplet taken from the Gini mean. Derivatives with respect to the weight parameters and with respect to the two generalization parameters are given. Some properties of the network are investigated, showing the capacity to emulate the classical exclusive-or problem organically in two layers and perform some multiplication and division. The network can instantiate truncated power series and variants, which can be used to approximate different functions, provided that parameters are constrained. Moreover, a mean case slope score is derived that can facilitate a learning-rate novelty based on homogeneity of the selected elements. The multiplet neuron equation provides a way to segment regularization timeframes and approaches.

We introduce Bayesian Poisson Tucker decomposition (BPTD) for modeling country--country interaction event data. These data consist of interaction events of the form "country i took action a toward country j at time t." BPTD discovers overlapping country--community memberships, including the number of latent communities. In addition, it discovers directed community--community interaction networks that are specific to "topics" of action types and temporal "regimes." We show that BPTD yields an efficient MCMC inference algorithm and achieves better predictive performance than related models. We also demonstrate that it discovers interpretable latent structure that agrees with our knowledge of international relations.

We study sampling algorithms for β-ensembles with time complexity less than cubic in the cardinality of the ensemble. Following Dumitriu & Edelman (2002), we see the ensemble as the eigenvalues of a random tridiagonal matrix, namely a random Jacobi matrix. First, we provide a unifying and elementary treatment of the tridiagonal models associated to the three classical Hermite, Laguerre and Jacobi ensembles. For this purpose, we use simple changes of variables between successive reparametrizations of the coefficients defining the tridiagonal matrix. Second, we derive an approximate sampler for the simulation of β-ensembles, and illustrate how fast it can be for polynomial potentials. This method combines a Gibbs sampler on Jacobi matrices and the diagonalization of these matrices. In practice, even for large ensembles, only a few Gibbs passes suffice for the marginal distribution of the eigenvalues to fit the expected theoretical distribution. When the conditionals in the Gibbs sampler can be simulated exactly, the same fast empirical convergence is observed for the fluctuations of the largest eigenvalue. Our experimental results support a conjecture by Krishnapur et al. (2016), that the Gibbs chain on Jacobi matrices of size N mixes in O(log(N)).

Standard methods for detecting discontinuities in conditional means are not applicable to outcomes that are complex, non-Euclidean objects like distributions, networks, or covariance matrices. This article develops a nonparametric test for jumps in conditional means when outcomes lie in a non-Euclidean metric space. Using local Fr\'echet regressionx2014which generalizes standard regression to metric-space valued datax2014the method estimates a mean path on either side of a candidate cutoff, extending existing k-sample tests to a flexible regression setting. Key theoretical contributions include a central limit theorem for the local estimator of the conditional Fr\'echet variance and the asymptotic validity and consistency of the proposed test. Simulations confirm nominal size control and robust power in finite samples. Two applications demonstrate the method's value by revealing effects invisible to scalar-based tests. First, I detect a sharp change in work-from-home compositions at Washington State's income threshold for non-compete enforceability during COVID-19, highlighting remote work's role as a bargaining margin. Second, I find that countries restructure their input-output networks after losing preferential US trade access. These findings underscore that analyzing regression functions within their native metric spaces can reveal structural discontinuities that scalar summaries would miss.

We investigate spectral features of bottomonium at high temperature, in particular the thermal mass shift and width of ground state S-wave and P-wave state. We employ and compare a range of methods for determining these features from lattice NRQCD correlators, including direct correlator analyses (multi-exponential fits and moments of spectral functions), linear methods (Backus-Gilbert, Tikhonov and HLT methods), and Bayesian methods for spectral function reconstruction (MEM and BR). We comment on the reliability and limitations of the various methods.

We propose algorithms for approximate filtering and smoothing in high-dimensional Factorial hidden Markov models. The approximation involves discarding, in a principled way, likelihood factors according to a notion of locality in a factor graph associated with the emission distribution. This allows the exponential-in-dimension cost of exact filtering and smoothing to be avoided. We prove that the approximation accuracy, measured in a local total variation norm, is "dimension-free" in the sense that as the overall dimension of the model increases the error bounds we derive do not necessarily degrade. A key step in the analysis is to quantify the error introduced by localizing the likelihood function in a Bayes' rule update. The factorial structure of the likelihood function which we exploit arises naturally when data have known spatial or network structure. We demonstrate the new algorithms on synthetic examples and a London Underground passenger flow problem, where the factor graph is effectively given by the train network.

Model attributions are important in deep neural networks as they aid practitioners in understanding the models, but recent studies reveal that attributions can be easily perturbed by adding imperceptible noise to the input. The non-differentiable Kendall's rank correlation is a key performance index for attribution protection. In this paper, we first show that the expected Kendall's rank correlation is positively correlated to cosine similarity and then indicate that the direction of attribution is the key to attribution robustness. Based on these findings, we explore the vector space of attribution to explain the shortcomings of attribution defense methods using ell_p norm and propose integrated gradient regularizer (IGR), which maximizes the cosine similarity between natural and perturbed attributions. Our analysis further exposes that IGR encourages neurons with the same activation states for natural samples and the corresponding perturbed samples, which is shown to induce robustness to gradient-based attribution methods. Our experiments on different models and datasets confirm our analysis on attribution protection and demonstrate a decent improvement in adversarial robustness.

The assumption that response and predictor belong to the same statistical unit may be violated in practice. Unbiased estimation and recovery of true label ordering based on unlabeled data are challenging tasks and have attracted increasing attentions in the recent literature. In this paper, we present a relatively complete analysis of label permutation problem for the generalized linear model with multivariate responses. The theory is established under different scenarios, with knowledge of true parameters, with partial knowledge of underlying label permutation matrix and without any knowledge. Our results remove the stringent conditions required by the current literature and are further extended to the missing observation setting which has never been considered in the field of label permutation problem. On computational side, we propose two methods, "maximum likelihood estimation" algorithm and "two-step estimation" algorithm, to accommodate for different settings. When the proportion of permuted labels is moderate, both methods work effectively. Multiple numerical experiments are provided and corroborate our theoretical findings.

Searches for stochastic gravitational-wave backgrounds using pulsar timing arrays look for correlations in the timing residuals induced by the background across the pulsars in the array. The correlation signature of an isotropic, unpolarized gravitational-wave background predicted by general relativity follows the so-called Hellings and Downs curve, which is a relatively simple function of the angle between a pair of Earth-pulsar baselines. In this paper, we give a pedagogical discussion of the Hellings and Downs curve for pulsar timing arrays, considering simpler analogous scenarios involving sound and electromagnetic waves. We calculate Hellings-and-Downs-type functions for these two scenarios and develop a framework suitable for doing more general correlation calculations.

Straight line equation y=mx with slope m, when singularly perturbed as ay^3+y=mx with a positive parameter a, results in S-shaped curves or S-curves on a real plane. As arightarrow 0, we get back y=mx which is a cumulative distribution function of a continuous uniform distribution that describes the occurrence of every event in an interval to be equally probable. As arightarrowinfty, the derivative of y has finite support only at y=0 resembling a degenerate distribution. Based on these arguments, in this work, we propose that these S-curves can represent maximum entropy uniform distribution to a zero entropy single value. We also argue that these S-curves are superposable as they are only parametrically nonlinear but fundamentally linear. So far, the superposed forms have been used to capture the patterns of natural systems such as nonlinear dynamics of biological growth and kinetics of enzyme reactions. Here, we attempt to use the S-curve and its superposed form as statistical models. We fit the models on a classical dataset containing flower measurements of iris plants and analyze their usefulness in pattern recognition. Based on these models, we claim that any non-uniform pattern can be represented as a singular perturbation to uniform distribution. However, our parametric estimation procedure have some limitations such as sensitivity to initial conditions depending on the data at hand.

We introduce a new class of algorithms, Stochastic Generalized Method of Moments (SGMM), for estimation and inference on (overidentified) moment restriction models. Our SGMM is a novel stochastic approximation alternative to the popular Hansen (1982) (offline) GMM, and offers fast and scalable implementation with the ability to handle streaming datasets in real time. We establish the almost sure convergence, and the (functional) central limit theorem for the inefficient online 2SLS and the efficient SGMM. Moreover, we propose online versions of the Durbin-Wu-Hausman and Sargan-Hansen tests that can be seamlessly integrated within the SGMM framework. Extensive Monte Carlo simulations show that as the sample size increases, the SGMM matches the standard (offline) GMM in terms of estimation accuracy and gains over computational efficiency, indicating its practical value for both large-scale and online datasets. We demonstrate the efficacy of our approach by a proof of concept using two well known empirical examples with large sample sizes.