In finite-width deep neural networks, the empirical kernel $G$ evolves stochastically across layers. We develop a collective kernel effective field theory (EFT) for pre-activation ResNets based on a $G$-only closure hierarchy and diagnose its finite validity window. Exploiting the exact conditional Gaussianity of residual increments, we derive an exact stochastic recursion for $G$. Applying Gaussian approximations systematically yields a continuous-depth ODE system for the mean kernel $K_0$, the kernel covariance $V_4$, and the $1/n$ mean correction $K_{1,\mathrm{EFT}}$, which emerges diagrammatically as a one-loop tadpole correction. Numerically, $K_0$ remains accurate at all depths. However, the $V_4$ equation residual accumulates to an $O(1)$ error at finite time, primarily driven by approximation errors in the $G$-only transport term. Furthermore, $K_{1,\mathrm{EFT}}$ fails due to the breakdown of the source closure, which exhibits a systematic mismatch even at initialization. These findings highlight the limitations of $G$-only state-space reduction and suggest extending the state space to incorporate the sigma-kernel.

We propose an adaptive MCMC method that learns a linear preconditioner which is dense in its off-diagonal elements but sparse in its parametrisation. Due to this sparsity, we achieve a per-iteration computational complexity of $O(m^2d)$ for a user-determined parameter $m$, compared with the $O(d^2)$ complexity of existing adaptive strategies that can capture correlation information from the target. Diagonal preconditioning has an $O(d)$ per-iteration complexity, but is known to fail in the case that the target distribution is highly correlated, see \citet[Section 3.5]{hird2025a}. Our preconditioner is constructed using eigeninformation from the target covariance which we infer using online principal components analysis on the MCMC chain. It is composed of a diagonal matrix and a product of carefully chosen reflection matrices. On various numerical tests we show that it outperforms diagonal preconditioning in terms of absolute performance, and that it outperforms traditional dense preconditioning and multiple diagonal plus low-rank alternatives in terms of time-normalised performance.

Machine learning (ML) models have achieved strikingly high accuracies in spectroscopic classification tasks, often without a clear proof that those models used chemically meaningful features. Existing studies have linked these results to data preprocessing choices, noise sensitivity, and model complexity, but no unifying explanation is available so far. In this work, we show that these phenomena arise naturally from the intrinsic high dimensionality of spectral data. Using a theoretical analysis grounded in the Feldman-Hajek theorem and the concentration of measure, we show that even infinitesimal distributional differences, caused by noise, normalisation, or instrumental artefacts, may become perfectly separable in high-dimensional spaces. Through a series of specific experiments on synthetic and real fluorescence spectra, we illustrate how models can achieve near-perfect accuracy even when chemical distinctions are absent, and why feature-importance maps may highlight spectrally irrelevant regions. We provide a rigorous theoretical framework, confirm the effect experimentally, and conclude with practical recommendations for building and interpreting ML models in spectroscopy.

There remain theoretical gaps in deep neural network estimators for the nonparametric Cox proportional hazards model. In particular, it is unclear how gradient-based optimization error propagates to population risk under partial likelihood, how pointwise bias can be controlled to permit valid inference, and how ensemble-based uncertainty quantification behaves under realistic variance decay regimes. We develop an asymptotic distribution theory for deep Cox estimators that addresses these issues. First, we establish nonasymptotic oracle inequalities for general trained networks that link in-sample optimization error to population risk without requiring the exact empirical risk optimizer. We then construct a structured neural parameterization that achieves infinity-norm approximation rates compatible with the oracle bound, yielding control of the pointwise bias. Under these conditions and using the Hajek--Hoeffding projection, we prove pointwise and multivariate asymptotic normality for subsampled ensemble estimators. We derive a range of subsample sizes that balances bias correction with the requirement that the Hajek--Hoeffding projection remain dominant. This range accommodates decay conditions on the single-overlap covariance, which measures how strongly a single shared observation influences the estimator, and is weaker than those imposed in the subsampling literature. An infinitesimal jackknife representation provides analytic covariance estimation and valid Wald-type inference for relative risk contrasts such as log-hazard ratios. Finally, we illustrate the finite-sample implications of the theory through simulations and a real data application.

Data assimilation algorithms estimate the state of a dynamical system from partial observations, where the successful performance of these algorithms hinges on costly parameter tuning and on employing an accurate model for the dynamics. This paper introduces a framework for jointly learning the state, dynamics, and parameters of filtering algorithms in data assimilation through a process we refer to as auto-differentiable filtering. The framework leverages a theoretically motivated loss function that enables learning from partial, noisy observations via gradient-based optimization using auto-differentiation. We further demonstrate how several well-known data assimilation methods can be learned or tuned within this framework. To underscore the versatility of auto-differentiable filtering, we perform experiments on dynamical systems spanning multiple scientific domains, such as the Clohessy-Wiltshire equations from aerospace engineering, the Lorenz-96 system from atmospheric science, and the generalized Lotka-Volterra equations from systems biology. Finally, we provide guidelines for practitioners to customize our framework according to their observation model, accuracy requirements, and computational budget.

Classical Gaussian processes and Kriging models are commonly based on stationary kernels, whereby correlations between observations depend exclusively on the relative distance between scattered data. While this assumption ensures analytical tractability, it limits the ability of Gaussian processes to represent heterogeneous correlation structures. In this work, we investigate variably scaled kernels as an effective tool for constructing non-stationary Gaussian processes by explicitly modifying the correlation structure of the data. Through a scaling function, variably scaled kernels alter the correlations between data and enable the modeling of targets exhibiting abrupt changes or discontinuities. We analyse the resulting predictive uncertainty via the variably scaled kernel power function and clarify the relationship between variably scaled kernels-based constructions and classical non-stationary kernels. Numerical experiments demonstrate that variably scaled kernels-based Gaussian processes yield improved reconstruction accuracy and provide uncertainty estimates that reflect the underlying structure of the data

Square-root Kalman filters propagate state covariances in Cholesky-factor form for numerical stability, and are a natural target for gradient-based parameter learning in state-space models. Their core operation, triangularization of a matrix $M \in \mathbb{R}^{n \times m}$, is computed via a QR decomposition in practice, but naively differentiating through it causes two problems: the semi-orthogonal factor is non-unique when $m > n$, yielding undefined gradients; and the standard Jacobian formula involves inverses, which diverges when $M$ is rank-deficient. Both are resolved by the observation that all filter outputs relevant to learning depend on the input matrix only through the Gramian $MM^\top$, so the composite loss is smooth in $M$ even where the triangularization is not. We derive a closed-form chain-rule directly from the differential of this Gramian identity, prove it exact for the Kalman log-marginal likelihood and filtered moments, and extend it to rank-deficient inputs via a two-component decomposition: a column-space term based on the Moore--Penrose pseudoinverse, and a null-space correction for perturbations outside the column space of $M$.

Scaling laws for neural networks, in which the loss decays as a power-law in the number of parameters, data, and compute, depend fundamentally on the spectral structure of the data covariance, with power-law eigenvalue decay appearing ubiquitously in vision and language tasks. A central question is whether this spectral structure is preserved or destroyed when data passes through the basic building block of a neural network: a random linear projection followed by a nonlinear activation. We study this question for the random feature model: given data $x \sim N(0,H)\in \mathbb{R}^v$ where $H$ has $α$-power-law spectrum ($λ_j(H ) \asymp j^{-α}$, $α> 1$), a Gaussian sketch matrix $W \in \mathbb{R}^{v\times d}$, and an entrywise monomial $f(y) = y^{p}$, we characterize the eigenvalues of the population random-feature covariance $\mathbb{E}_{x }[\frac{1}{d}f(W^\top x )^{\otimes 2}]$. We prove matching upper and lower bounds: for all $1 \leq j \leq c_1 d \log^{-(p+1)}(d)$, the $j$-th eigenvalue is of order $\left(\log^{p-1}(j+1)/j\right)^α$. For $ c_1 d \log^{-(p+1)}(d)\leq j\leq d$, the $j$-th eigenvalue is of order $j^{-α}$ up to a polylog factor. That is, the power-law exponent $α$ is inherited exactly from the input covariance, modified only by a logarithmic correction that depends on the monomial degree $p$. The proof combines a dyadic head-tail decomposition with Wick chaos expansions for higher-order monomials and random matrix concentration inequalities.

We derive a tighter inequality than Cauchy-Schwarz Inequality, leverage it to refine Pearson's