This paper studies the role of sinks and diagonal patterns as attention switch and anti-oversmoothing mechanisms. We analyze geometric conditions under which sinks can be represented, showing a necessary alignment between the embedding of the sink and all other embeddings. Next, we refine the current understanding of the role of sinks in oversmoothing prevention: we specify the conditions under which dense attention provably smooths more than sparse attention, and empirically verify that such conditions are often satisfied in practice. We further prove an equivalence between sinks and hard attention switch, in which the output of the attention is identically 0. Finally, we relax the hard attention switch by allowing token self-communication: we provide a quantitative comparison of the costs of representing sinks vs.\ diagonal patterns, showing why sinks are favored in pretrained transformers. The introduction and analysis of diagonal patterns and the generalization of the attention switch close the gap between what oversmoothing prevention requires and what sinks provide, while also establishing when and why attention layers act like MLPs if token communication is not necessary.

We introduce the Sinkhorn treatment effect, an entropic optimal transport measure of divergence between counterfactual distributions. Unlike classical quantities such as the average treatment effect, this measure captures differences across entire distributions. We analyze this divergence as a statistical functional and show it can be written as a smooth transformation of counterfactual mean embeddings with an appropriate kernel. This characterization allows us to establish first-order pathwise differentiability in general, and second-order pathwise differentiability under the null hypothesis of equal counterfactual distributions. Leveraging this smoothness, we construct debiased estimators and use them to obtain asymptotically valid tests for distributional treatment effects with a fixed entropic regularization parameter. Because the power of the test depends on this unknown parameter, we further propose an aggregated test that combines evidence across a grid of regularization choices. Experiments on simulated and image data demonstrate the practical advantages of our estimator and testing procedure.

The Gromov-Wasserstein (GW) problem provides a framework for aligning heterogeneous datasets by matching their intrinsic geometry, but its statistical and computational scaling remains an issue for high-dimensional problems. Slicing techniques offer an appealing route to scalability, but, unlike Wasserstein distances, GW problems do not generally admit closed-form solutions in one-dimension. We resolve this problem for the GW problem with inner product cost (IGW), propose a sliced IGW distance that enjoys a natural rotational invariance property, and comprehensively study its structural and computational properties. Numerical experiments validating our theory are presented, followed by applications to heterogeneous clustering of text data and language model representation comparison.

Independent Component Analysis (ICA) is a foundational tool for unsupervised representation learning, yet its high-dimensional theory remains largely limited to single-component recovery. We develop an asymptotically exact mean-field theory for multi-component online ICA, capturing the coupling induced by simultaneous learning and orthogonalization. In the high-dimensional limit, the joint empirical distribution of learned estimates and ground-truth components converges to a deterministic process, yielding a closed ODE system for the overlap matrix between learned directions and true components. This characterization reveals a genuinely multi-component, initialization-driven phase structure: a decoupled regime, where estimates align with distinct components and evolve nearly independently, and a competition regime, where overlapping initializations induce orthogonality-driven conflicts, slow reorientation, and delayed convergence. Our steady-state analysis gives explicit learnability boundaries and competition conditions linking step size, data moments, and initialization. These conditions show that larger higher-order moments and competition shrink the stable learning-rate window, increase convergence times, and predict a staircase phenomenon in which the number of recoverable components changes discretely with the learning rate. Experiments on synthetic data and hyperspectral remote sensing data validate the predicted trajectories and phase behavior.

Density estimation and reliable prediction regions for outputs are crucial in supervised and unsupervised learning. While conformal prediction effectively generates coverage-guaranteed regions, it struggles with multi-dimensional outputs due to reliance on one-dimensional nonconformity scores. To address this, we introduce CONTRA: CONformal prediction region via normalizing flow TRAnsformation. CONTRA utilizes the latent spaces of normalizing flows to define nonconformity scores based on distances from the center. This allows for the mapping of high-density regions in latent space to sharp prediction regions in the output space, surpassing traditional hyperrectangular or elliptical conformal regions. Further, for scenarios where other predictive models are favored over flow-based models, we extend CONTRA to enhance any such model with a reliable prediction region by training a simple normalizing flow on the residuals. We demonstrate that both CONTRA and its extension maintain guaranteed coverage probability and outperform existing methods in generating accurate prediction regions across various datasets. We conclude that CONTRA is an effective tool for (conditional) density estimation, addressing the under-explored challenge of delivering multi-dimensional prediction regions.

Unlike a standard PINN--which produces an approximate Deep neural networks (DNNs) or multi-layer perceptronssolution by minimizing a PDE-residual loss and thus yields (MLPs) offer various inherent advantages over traditionalonly a point estimate, failing to quantify uncertainty inapproaches of scientific computing and data analysis, suchduced by noisy or limited data, a Bayesian PINN returns a as finite element methods, wavelets and kernel methods, full posterior distribution over solutions by combining the which are often hampered by the irregular and nonlinearuncertain information from the likelihood (data) and the data structures and the high input dimensions. In contrast, DNNs are capable of approximating a rich class of functions prior. Bayesian neural networks, originating in the seminal works of MacKay (MacKay, 1995) and Neal (Neal, 1995), with aforementioned complexities and can also easily en-have been extensively studied over the past three decades codes additional complex physical structures, such as sym- (Lampinen & Vehtari, 2001; Titterington, 2004; Graves, metry and other invariant structures.

We study solving large-scale fixed-point equation x = F(x) with decomposition. Standard strict decomposition assigns each agent a disjoint block and evaluates updates using only owned coordinates. For most operators, however, a block update may depend on variables outside the block. Truncating these dependencies by strict decomposition changes the mean operator and creates structural bias that cannot be removed by more samples, smaller stepsizes, or additional consensus. We therefore propose Core-Halo decomposition, which separates write ownership from read-only evaluation context: each agent updates its own core and reads from an overlapping halo. By aligning the Core-Halo decomposition with the blockdependence structure of F, the original fixed-point problem can be implemented faithfully in a decentralized multi-agent system. We further characterize the fundamental obstruction faced by strict decomposition through a Bellman closure condition and a blockwise bias lower bound, showing that local-only updates can alter the original fixed-point operator. Finally, we conduct extensive experiments across a range of application settings, and demonstrate that Core-Halo achieves near-centralized performance while retaining the parallelism benefits of decentralization.

Unbalanced optimal transport (UOT) extends classical optimal transport to measures with different total masses, but statistical guarantees for Monge-type estimation remain limited. We study unbalanced transport with quadratic cost and Kullback-Leibler marginal penalties and argue that the natural population target is not a map alone, but a transport-growth pair. Consequently, we develop two estimators for the transport-growth pairs under several setups: an optimal transport plan-based estimator for a general case, and a kernel-based estimator for a case with smooth densities. We also show that an error of the estimator achieves the minimax optimal rate by deriving a matching lower bound of the minimax risk. Our main technical contribution is a value-based stability reduction that converts perturbations of the UOT objective into transport and growth risks through a UOT gap condition. These results provide a statistical foundation for Monge-type estimation in unbalanced optimal transport.

Advanced manufacturing technologies allow for the production of intricate parts featuring high shape complexity and spatially-varying material composition. Data fusion of point clouds with chromatic attributes provides 4D point clouds, a compact and informative representation that encodes both shape and material information. In this paper, we present a registration-free framework for Simultaneous Monitoring of shApe and Color (SMAC) via 4D point clouds. The proposed framework leverages Laplace-Beltrami operator spectral properties to capture and monitor geometric features and the relationship between shape and surface color. A combined monitoring scheme is proposed to effectively detect shape deformations and color anomalies, along with a spatially-aware post-signal diagnostic procedure to determine the source of change and localize color anomalies. Importantly, neither component relies on registration or mesh reconstruction, eliminating error-prone and computationally expensive preprocessing steps. A Monte Carlo simulation study and a case study on functionally graded materials demonstrate that SMAC achieves effective detection performance, particularly for subtle defects, while providing diagnostic capabilities to identify the source and location of anomalies.

Mode separation, namely how sharply a distribution fragments into barrier-separated clusters, is a fundamental geometric property of densities, difficult to quantify in high dimensions. It is structurally distinct from dispersion, yet existing tools fall short: differential entropy rises with spread regardless of fragmentation, PCA orders directions by variance regardless of barriers, and mutual information requires a mixture decomposition one usually does not have. We measure mode separation through a single stochastic process intrinsic to the density: a unique reversible diffusion with $f$ as its stationary distribution and constant scalar diffusion coefficient. We extract two readouts from its autocovariance matrix: SSA (Sum of Squared Autocorrelations), a scalar barrier-sensitive measure; and DA (Dominant Autocorrelation directions), linear projections ordered by metastability rather than variance. Under an isotropic-Gaussian null, we derive a closed-form spectrum for the empirical autocovariance that generalizes Marchenko--Pastur, with an analytic upper edge that selects the lag at which DA is read off. Both readouts use only samples and a score function, scaling to high dimensions through pretrained score-based generative models via Tweedie's identity. We apply our framework to three settings: (i) synthetic Gaussian mixtures, where SSA tracks mutual information; (ii) SDXL text-to-image generations, where SSA and DA capture structure that entropy and PCA miss; and (iii) molecular dynamics of alanine dipeptide, where DA recovers the known slow backbone dihedrals from static samples alone.