The method introduces structure-aware optimal transport diagnostics to assess if empirical time-series dynamics align with a prescribed predictable-drift class. It translates stochastic path properties into a finite-dimensional projection problem over empirical transition laws, using row-wise moment constraints for admissibility. Deviations are quantified by a Kullback-Leibler projection gap, $G_{\lambda,k}$. This yields a diagnostic and a scalable Bregman-Dykstra algorithm that incorporates structural constraints into Sinkhorn-type methods. The technique can distinguish between process classes that share identical second-order statistics, acting as a general tool for sequential data model validation.
ELPAC is a Bayesian framework designed to jointly infer latent aging stages and progression subtypes from multimodal data. It integrates subtype trajectories by working backward from endpoint groups using biologically structured velocity components. The method further factorizes residual multimodal variation into sparse coordination axes that reveal subtype-specific patterns. ELPAC employs a scalable variational inference scheme for joint posterior inference at the cohort level. Testing on Alzheimer's data shows ELPAC accurately recovers latent structures and provides biologically interpretable multimodal coordination programs.
Kernel-based greedy algorithms, often p-greedy, select approximation nodes by minimizing point-wise error indicators. In noisy data, the separation of approximation error and noise amplification is crucial, which is governed by the power function and a Lebesgue-type term. This motivates using the Stein's unbiased risk estimator (SURE) as a proxy for the mean squared error. SURE provides a principled stopping criterion for selecting the number of nodes in the p-greedy procedure. Numerical experiments confirm that this method effectively balances deterministic accuracy and statistical noise.
Blind Noise Equalization for Large Data Matrices under General Variance Profiles introduces BRINE, a data-driven method to equalize noise variance across rows and columns in noisy matrices. This procedure aims to resolve the distortion caused by heteroskedasticity, which obscures weak signals in large datasets. The core technique involves equating the diagonal entries of the data resolvent as a proxy for equalizing unknown noise variances. The method restores the Marchenko-Pastur law for the noise spectrum and can amplify signal components without prior knowledge of the signal structure. The approach proves convergence to the true equalizer with geometric rates, providing estimates directly from noisy data.
The research proposes an Onsager-corrected, target-policy-weighted criterion for high-dimensional off-policy evaluation without sample splitting. This method estimates the full-data reward model's out-of-sample risk using only its training observations. It enables early stopping during gradient-descent without requiring data sample splitting. The criterion is shown to provide uniform consistency and near-oracle guarantees for neural networks under specific conditions. Experiments indicate that this correction reduces risk estimation error and improves reward model selection compared to held-out baselines.
The lower-star filtration (LSF) is limited because it can fill topological features immediately, resulting in incomplete persistence diagrams. Three alternative filtration methods are proposed to address this limitation: $\varepsilon$-lower-star filtration (ELSF), $L_p$ filtration ($L_p F$), and softmax filtration (SMF). These alternatives introduce dimension-dependent delays or use aggregation functions based on all vertex values instead of just the maximum. These modifications allow topological features to persist over longer intervals, potentially revealing more information about data structures. Experimental results show that these methods are computationally efficient and offer greater sensitivity to data differences than LSF.
Score-Repellent Monte Carlo improves MCMC sampling by tilting the target density using the score function. Geometry-Aware SRMC generalizes this by incorporating a symmetric positive-definite matrix $M$ for anisotropic landscapes. The method identifies the optimal direction by relating the Fisher information to the repulsion-side matrix. This approach yields a bound-optimal repulsion-side matrix, providing theoretical guarantees for online implementation. Experimental results show GA-SRMC significantly outperforms SRMC in terms of ESS for various targets.
The research explores kernel-based learning of analytic operators from finite samples, emphasizing how regularity affects kernel selection. It establishes embeddings of analytic operators into vector-valued reproducing kernel Hilbert spaces. The study provides finite-sample, high-probability guarantees for uncertainty based on sampling, showing exponential rates for analytic kernels and polynomial rates for Sobolev-type kernels. The theory is applied to the solution operator of parametric elliptic PDEs, separating sampling and discretization errors. Ultimately, the work prescribes how mesh size should relate to sample size for accurate approximations.
Language model systems can select prompts and parameter adapters repeatedly, with earlier choices affecting later contexts. The study analyzes selection among candidates by comparing cumulative loss against the best initial configuration. Excess loss is broken down into regret within the family, the gap from the family, and the impact of the deployment history. Sublinear expected regret is achieved when switching limits the scope, and fewer switches can either improve or worsen deployed loss. Experiments suggest that text-only selection has lower regret but more failures than joint selection.
Machine learning is redefined as successfully computing a function with a proper derivation in a formal system, extending Gödel's theorem to empirical targets. Learning true probability requires both a Truth Condition (convergence to the true probability) and an Assurance Condition (the machine deriving its own convergence). Learning therefore extends beyond statistical consistency to necessitate a derivation of convergence from the observed record. Modern AI is hybrid, combining probabilistic tasks like prediction with deductive tasks like proving and coding. This framework explains the hybrid nature of modern AI within a single formal system.