The clustering and visualisation of high-dimensional data is a ubiquitous task in modern data science. Popular techniques include nonlinear dimensionality reduction methods like t-SNE or UMAP. These methods face the `scale-problem' of clustering: when dealing with the MNIST dataset, do we want to distinguish different digits or do we want to distinguish different ways of writing the digits? The answer is task dependent and depends on scale. We revisit an idea of Robinson & Pierce-Hoffman that exploits an underlying scaling symmetry in t-SNE to replace 2-dimensional with (2+1)-dimensional embeddings where the additional parameter accounts for scale. This gives rise to the t-SNE tree (short: tree-SNE). We prove that the optimal embedding depends continuously on the scaling parameter for all initial conditions outside a set of measure 0: the tree-SNE tree exists. This idea conceivably extends to other attraction-repulsion methods and is illustrated on several examples.

We revisit the Universal Approximation Theorem(UAT) through the lens of the tropical geometry of neural networks and introduce a constructive, geometry-aware initialization for sigmoidal multi-layer perceptrons (MLPs). Tropical geometry shows that Rectified Linear Unit (ReLU) networks admit decision functions with a combinatorial structure often described as a tropical rational, namely a difference of tropical polynomials. Focusing on planar binary classification, we design purely sigmoidal MLPs that adhere to the finite-sum format of UAT: a finite linear combination of shifted and scaled sigmoids of affine functions. The resulting models yield decision boundaries that already align with prescribed shapes at initialization and can be refined by standard training if desired. This provides a practical bridge between the tropical perspective and smooth MLPs, enabling interpretable, shape-driven initialization without resorting to ReLU architectures. We focus on the construction and empirical demonstrations in two dimensions; theoretical analysis and higher-dimensional extensions are left for future work.

Data attribution for generative models seeks to quantify the influence of individual training examples on model outputs. Existing methods for diffusion models typically require access to model gradients or retraining, limiting their applicability in proprietary or large-scale settings. We propose a nonparametric attribution method that operates entirely on data, measuring influence via patch-level similarity between generated and training images. Our approach is grounded in the analytical form of the optimal score function and naturally extends to multiscale representations, while remaining computationally efficient through convolution-based acceleration. In addition to producing spatially interpretable attributions, our framework uncovers patterns that reflect intrinsic relationships between training data and outputs, independent of any specific model. Experiments demonstrate that our method achieves strong attribution performance, closely matching gradient-based approaches and substantially outperforming existing nonparametric baselines. Code is available at https://github.com/sail-sg/NDA.

Selecting the latent dimensions (ranks) in tensor factorization is a central challenge that often relies on heuristic methods. This paper introduces a rigorous approach to determine rank identifiability in probabilistic tensor models, based on prior predictive moment matching. We transform a set of moment matching conditions into a log-linear system of equations in terms of marginal moments, prior hyperparameters, and ranks; establishing an equivalence between rank identifiability and the solvability of such system. We apply this framework to four foundational tensor-models, demonstrating that the linear structure of the PARAFAC/CP model, the chain structure of the Tensor Train model, and the closed-loop structure of the Tensor Ring model yield solvable systems, making their ranks identifiable. In contrast, we prove that the symmetric topology of the Tucker model leads to an underdetermined system, rendering the ranks unidentifiable by this method. For the identifiable models, we derive explicit closed-form rank estimators based on the moments of observed data only. We empirically validate these estimators and evaluate the robustness of the proposal.

We develop a framework for analyzing the training and learning rate dynamics on a variety of high- dimensional optimization problems trained using one-pass stochastic gradient descent (SGD) with data generated from multiple anisotropic classes. We give exact expressions for a large class of functions of the limiting dynamics, including the risk and the overlap with the true signal, in terms of a deterministic solution to a system of ODEs. We extend the existing theory of high-dimensional SGD dynamics to Gaussian-mixture data and a large (growing with the parameter size) number of classes. We then investigate in detail the effect of the anisotropic structure of the covariance of the data in the problems of binary logistic regression and least square loss. We study three cases: isotropic covariances, data covariance matrices with a large fraction of zero eigenvalues (denoted as the zero-one model), and covariance matrices with spectra following a power-law distribution. We show that there exists a structural phase transition. In particular, we demonstrate that, for the zero-one model and the power-law model with sufficiently large power, SGD tends to align more closely with values of the class mean that are projected onto the "clean directions" (i.e., directions of smaller variance). This is supported by both numerical simulations and analytical studies, which show the exact asymptotic behavior of the loss in the high-dimensional limit.

In this paper we develop a deforestation detection pipeline that incorporates optical and Synthetic Aperture Radar (SAR) data. A crucial component of the pipeline is the construction of anomaly maps of the optical data, which is done using the residual space of a discrete Karhunen-Loève (KL) expansion. Anomalies are quantified using a concentration bound on the distribution of the residual components for the nominal state of the forest. This bound does not require prior knowledge on the distribution of the data. This is in contrast to statistical parametric methods that assume knowledge of the data distribution, an impractical assumption that is especially infeasible for high dimensional data such as ours. Once the optical anomaly maps are computed they are combined with SAR data, and the state of the forest is classified by using a Hidden Markov Model (HMM). We test our approach with Sentinel-1 (SAR) and Sentinel-2 (Optical) data on a $92.19\,km \times 91.80\,km$ region in the Amazon forest. The results show that both the hybrid optical-radar and optical only methods achieve high accuracy that is superior to the recent state-of-the-art hybrid method. Moreover, the hybrid method is significantly more robust in the case of sparse optical data that are common in highly cloudy regions.

Consider two sets of entities and their members' mutual affinity values, say drug-target affinities (DTA). Drugs and targets are said to interact in their effects on DTAs if drug's effect on it depends on the target. Presence of interaction implies that assigning a drug to a target and another drug to another target does not provide the same aggregate DTA as the reversed assignment would provide. Accordingly, correctly capturing interactions enables better decision-making, for example, in allocation of limited numbers of drug doses to their best matching targets. Learning to predict DTAs is popularly done from either solely from known DTAs or together with side information on the entities, such as chemical structures of drugs and targets. In this paper, we introduce interaction directions' prediction performance estimator we call interaction concordance index (IC-index), for both fixed predictors and machine learning algorithms aimed for inferring them. IC-index complements the popularly used DTA prediction performance estimators by evaluating the ratio of correctly predicted directions of interaction effects in data. First, we show the invariance of IC-index on predictors unable to capture interactions. Secondly, we show that learning algorithm's permutation equivariance regarding drug and target identities implies its inability to capture interactions when either drug, target or both are unseen during training. In practical applications, this equivariance is remedied via incorporation of appropriate side information on drugs and targets. We make a comprehensive empirical evaluation over several biomedical interaction data sets with various state-of-the-art machine learning algorithms. The experiments demonstrate how different types of affinity strength prediction methods perform in terms of IC-index complementing existing prediction performance estimators.

We introduce a novel geometric framework for optimal experimental design (OED). Traditional OED approaches, such as those based on mutual information, rely explicitly on probability densities, leading to restrictive invariance properties. To address these limitations, we propose the mutual transport dependence (MTD), a measure of statistical dependence grounded in optimal transport theory which provides a geometric objective for optimizing designs. Unlike conventional approaches, the MTD can be tailored to specific downstream estimation problems by choosing appropriate geometries on the underlying spaces. We demonstrate that our framework produces high-quality designs while offering a flexible alternative to standard information-theoretic techniques.

Diffusion and flow-based generative models have achieved remarkable success in domains such as image synthesis, video generation, and natural language modeling. In this work, we extend these advances to weight space learning by leveraging recent techniques to incorporate structural priors derived from optimization dynamics. Central to our approach is modeling the trajectory induced by gradient descent as a trajectory inference problem. We unify several trajectory inference techniques towards matching a gradient flow, providing a theoretical framework for treating optimization paths as inductive bias. We further explore architectural and algorithmic choices, including reward fine-tuning by adjoint matching, the use of autoencoders for latent weight representation, conditioning on task-specific context data, and adopting informative source distributions such as Kaiming uniform. Experiments demonstrate that our method matches or surpasses baselines in generating in-distribution weights, improves initialization for downstream training, and supports fine-tuning to enhance performance. Finally, we illustrate a practical application in safety-critical systems: detecting harmful covariate shifts, where our method outperforms the closest comparable baseline.

Differential Privacy (DP) provides a rigorous framework for privacy, ensuring the outputs of data-driven algorithms remain statistically indistinguishable across datasets that differ in a single entry. While guaranteeing DP generally requires explicitly injecting noise either to the algorithm itself or to its outputs, the intrinsic randomness of existing algorithms presents an opportunity to achieve DP ``for free''. In this work, we explore the role of regularization in achieving DP across three different decision-making problems: multi-armed bandits, linear contextual bandits, and reinforcement learning from human feedback (RLHF), in offline data settings. We show that adding KL-regularization to the learning objective (a common approach in optimization algorithms) makes the action sampled from the resulting stochastic policy itself differentially private. This offers a new route to privacy guarantees without additional noise injection, while also preserving the inherent advantage of regularization in enhancing performance.