Real-world training data is often noisy; for example, human annotators assign conflicting class labels to the same instances. Partial-label learning (PLL) is a weakly supervised learning paradigm that allows training classifiers in this context without manual data cleaning. While state-of-the-art methods have good predictive performance, their predictions are sensitive to high noise levels, out-of-distribution data, and adversarial perturbations. We propose a novel PLL method based on subjective logic, which explicitly represents uncertainty by leveraging the magnitudes of the underlying neural network's class activation values. Thereby, we effectively incorporate prior knowledge about the class labels by using a novel label weight re-distribution strategy that we prove to be optimal. We empirically show that our method yields more robust predictions in terms of predictive performance under high PLL noise levels, handling out-of-distribution examples, and handling adversarial perturbations on the test instances.

We analyze the saliency of neural networks and employ interpretable machine learning models to predict whether the Kronecker coefficients of the symmetric group are zero or not. Our models use triples of partitions as input features, as well as b-loadings derived from the principal component of an embedding that captures the differences between partitions. Across all approaches, we achieve an accuracy of approximately 83% and derive explicit formulas for a decision function in terms of b-loadings. Additionally, we develop transformer-based models for prediction, achieving the highest reported accuracy of over 99%.

Many phenomena are naturally characterized by measuring continuous transformations such as shape changes in medicine or articulated systems in robotics. Modeling the variability in such datasets requires performing statistics on Lie groups, that is, manifolds carrying an additional group structure. As the Lie group captures the symmetries in the data, it is essential from a theoretical and practical perspective to ask for statistical methods that respect these symmetries; this way they are insensitive to confounding effects, e.g., due to the choice of reference coordinate systems. In this work, we investigate geodesic regression -- a generalization of linear regression originally derived for Riemannian manifolds. While Lie groups can be endowed with Riemannian metrics, these are generally incompatible with the group structure. We develop a non-metric estimator using an affine connection setting. It captures geodesic relationships respecting the symmetries given by left and right translations. For its computation, we propose an efficient fixed point algorithm requiring simple differential expressions that can be calculated through automatic differentiation. We perform experiments on a synthetic example and evaluate our method on an open-access, clinical dataset studying knee joint configurations under the progression of osteoarthritis.

Building surrogate models with uncertainty quantification capabilities is essential for many engineering applications where randomness, such as variability in material properties, is unavoidable. Polynomial Chaos Expansion (PCE) is widely recognized as a to-go method for constructing stochastic solutions in both intrusive and non-intrusive ways. Its application becomes challenging, however, with complex or high-dimensional processes, as achieving accuracy requires higher-order polynomials, which can increase computational demands and or the risk of overfitting. Furthermore, PCE requires specialized treatments to manage random variables that are not independent, and these treatments may be problem-dependent or may fail with increasing complexity. In this work, we adopt the spectral expansion formalism used in PCE; however, we replace the classical polynomial basis functions with neural network (NN) basis functions to leverage their expressivity. To achieve this, we propose an algorithm that identifies NN-parameterized basis functions in a purely data-driven manner, without any prior assumptions about the joint distribution of the random variables involved, whether independent or dependent. The proposed algorithm identifies each NN-parameterized basis function sequentially, ensuring they are orthogonal with respect to the data distribution. The basis functions are constructed directly on the joint stochastic variables without requiring a tensor product structure. This approach may offer greater flexibility for complex stochastic models, while simplifying implementation compared to the tensor product structures typically used in PCE to handle random vectors. We demonstrate the effectiveness of the proposed scheme through several numerical examples of varying complexity and provide comparisons with classical PCE.

We introduce a simple method for probabilistic predictions on tabular data based on Large Language Models (LLMs) called JoLT (Joint LLM Process for Tabular data). JoLT uses the in-context learning capabilities of LLMs to define joint distributions over tabular data conditioned on user-specified side information about the problem, exploiting the vast repository of latent problem-relevant knowledge encoded in LLMs. JoLT defines joint distributions for multiple target variables with potentially heterogeneous data types without any data conversion, data preprocessing, special handling of missing data, or model training, making it accessible and efficient for practitioners. Our experiments show that JoLT outperforms competitive methods on low-shot single-target and multi-target tabular classification and regression tasks. Furthermore, we show that JoLT can automatically handle missing data and perform data imputation by leveraging textual side information. We argue that due to its simplicity and generality, JoLT is an effective approach for a wide variety of real prediction problems.

The goal in thinning is to summarize a dataset using a small set of representative points. Remarkably, sub-Gaussian thinning algorithms like Kernel Halving and Compress can match the quality of uniform subsampling while substantially reducing the number of summary points. However, existing guarantees cover only a restricted range of distributions and kernel-based quality measures and suffer from pessimistic dimension dependence. To address these deficiencies, we introduce a new low-rank analysis of sub-Gaussian thinning that applies to any distribution and any kernel, guaranteeing high-quality compression whenever the kernel or data matrix is approximately low-rank. To demonstrate the broad applicability of the techniques, we design practical sub-Gaussian thinning approaches that improve upon the best known guarantees for approximating attention in transformers, accelerating stochastic gradient training through reordering, and distinguishing distributions in near-linear time.

Natural data is often organized as a hierarchical composition of features. How many samples do generative models need to learn the composition rules, so as to produce a combinatorial number of novel data? What signal in the data is exploited to learn? We investigate these questions both theoretically and empirically. Theoretically, we consider diffusion models trained on simple probabilistic context-free grammars - tree-like graphical models used to represent the structure of data such as language and images. We demonstrate that diffusion models learn compositional rules with the sample complexity required for clustering features with statistically similar context, a process similar to the word2vec algorithm. However, this clustering emerges hierarchically: higher-level, more abstract features associated with longer contexts require more data to be identified. This mechanism leads to a sample complexity that scales polynomially with the said context size. As a result, diffusion models trained on intermediate dataset size generate data coherent up to a certain scale, but that lacks global coherence. We test these predictions in different domains, and find remarkable agreement: both generated texts and images achieve progressively larger coherence lengths as the training time or dataset size grows. We discuss connections between the hierarchical clustering mechanism we introduce here and the renormalization group in physics.

Integrated Gradients (IG), a widely used axiomatic path-based attribution method, assigns importance scores to input features by integrating model gradients along a straight path from a baseline to the input. While effective in some cases, we show that straight paths can lead to flawed attributions. In this paper, we identify the cause of these misattributions and propose an alternative approach that treats the input space as a Riemannian manifold, computing attributions by integrating gradients along geodesics. We call this method Geodesic Integrated Gradients (GIG). To approximate geodesic paths, we introduce two techniques: a k-Nearest Neighbours-based approach for smaller models and a Stochastic Variational Inference-based method for larger ones. Additionally, we propose a new axiom, Strong Completeness, extending the axioms satisfied by IG. We show that this property is desirable for attribution methods and that GIG is the only method that satisfies it. Through experiments on both synthetic and real-world data, we demonstrate that GIG outperforms existing explainability methods, including IG.

Advances in machine learning and the increasing availability of high-dimensional data have led to the proliferation of social science research that uses the predictions of machine learning models as proxies for measures of human activity or environmental outcomes. However, prediction errors from machine learning models can lead to bias in the estimates of regression coefficients. In this paper, we show how this bias can arise, propose a test for detecting bias, and demonstrate the use of an adversarial machine learning algorithm in order to de-bias predictions. These methods are applicable to any setting where machine-learned predictions are the dependent variable in a regression. We conduct simulations and empirical exercises using ground truth and satellite data on forest cover in Africa. Using the predictions from a naive machine learning model leads to biased parameter estimates, while the predictions from the adversarial model recover the true coefficients.

Nonconvex optimization is central to modern machine learning, but the general framework of nonconvex optimization yields weak convergence guarantees that are too pessimistic compared to practice. On the other hand, while convexity enables efficient optimization, it is of limited applicability to many practical problems. To bridge this gap and better understand the practical success of optimization algorithms in nonconvex settings, we introduce a novel unified parametric assumption. Our assumption is general enough to encompass a broad class of nonconvex functions while also being specific enough to enable the derivation of a unified convergence theorem for gradient-based methods. Notably, by tuning the parameters of our assumption, we demonstrate its versatility in recovering several existing function classes as special cases and in identifying functions amenable to efficient optimization. We derive our convergence theorem for both deterministic and stochastic optimization, and conduct experiments to verify that our assumption can hold practically over optimization trajectories.