Persistent entropy is the Shannon entropy of a persistence-based probability measure defined on a persistence diagram. However, its cross-entropy version is not naturally defined because two persistence diagrams generally have different event spaces. To bridge these event spaces, we combine a similarity function with persistence weighting to define an induced probability. The induced probability reflects information from one diagram on the event space of the other diagram and assigns unexplained probability mass to the unexplained event. Using the induced probability, we extend cross entropy to persistence diagrams, called persistent cross entropy (PCE). We establish the main properties of both the induced probability and PCE and prove stability theorems for both. Through three numerical studies, we show that PCE distinguishes diagrams with the same persistent entropy, separates causal directions in dynamical systems without constructing a joint persistent diagram, and can be used as a directional topology loss for knowledge distillation.
Modern AI models such as tabular foundation models and gradient-boosted ensembles can outpredict classical methods, but provide little basis for reasoning about their predictions. High-stakes decisions call for models that are both accurate and interpretable as built. Local linear modeling offers a path forward: a smooth regression function is locally well approximated by a linear one, allowing a linear fit near each query point to achieve high accuracy without sacrificing transparency. The challenges lie in learning what is "local" and developing statistical tools for interpretation. Here, we propose local distillation, in which a black-box "teacher" guides a regularized linear "student" model at each query point. The teacher (1) defines locality by upweighting training observations with similar predicted outcomes, and (2) anchors the fit with its prediction at the query point, included as a pseudo-observation whose weight is estimated from the data. For interpretation, we add a small amount of Gaussian randomization to the local objective and use refits to assess stability: selection frequencies identify reliable features at a query point, and clustering the randomized fits identifies stable subgroups across the data. Under the lasso penalty, we prove that this randomization yields feature-selection probabilities that are stable under small perturbations of the training responses. Across 17 benchmark datasets, local distillation nearly matches its AI teacher's accuracy while producing a sparse linear model at each test point. In a high-dimensional cancer gene expression example, the framework identifies patient subgroups whose local models use different genes; this heterogeneity is invisible to a global linear model, and difficult to surface in a black-box model.
Retail demand forecasts are reused across replenishment, capacity, labor, and transportation planning cycles. Point-error objectives do not constrain abrupt movement between adjacent forecasts, while post-hoc smoothing acts only after model fitting. We ask whether a training-time penalty on consecutive within-series movement can improve horizontal forecast-path stability without materially changing point accuracy. The penalty is evaluated in a temporal-structured pipeline combining recent-demand embeddings with calendar, price, hierarchy, item, and store features. On selected M5 demand series at 1000, 3000, and 4000-series scales, the stability-aware hybrid model improves Forecast Stability Score over XGBoost by 6.91%, 6.66%, and 7.68%, respectively, while RMSE changes remain within 0.72% across three random seeds. Post-hoc exponential smoothing attains lower raw movement but incurs a larger RMSE cost; training-time regularization preserves more point accuracy and performs favorably under normalized stability. These findings extend forecast evaluation from point-error minimization toward an accuracy-stability trade-off perspective for operational retail forecasting.
As black-box models become foundational to modern research, ensuring their stability is paramount for the realization of trustworthy artificial intelligence. The inherent diversity of inputs - ranging from structured Gaussian distributions to complex data with unknown structures - poses a significant challenge: how to stabilize black-box outputs while effectively leveraging available prior information. This paper introduces a task-oriented randomization methodology that adaptively tailors its strategy to the underlying generative mechanisms of the input data, specifically addressing unstructured complexities. A comprehensive suite of stability guarantees is proposed. Beyond establishing rigorous theoretical foundations for stability, the research provides a detailed analysis of the intrinsic trade-off between stability and exploration. Motivated by the architecture of Large Language Models, the framework is further extended to top-k ranking problems. The validity and effectiveness of the proposal are demonstrated through extensive numerical simulations and applications to the real-world dataset.
Optimal transport (OT) provides a principled framework for mapping between probability distributions. Despite extensive progress, applying OT to large-scale data remains computationally demanding, and the resulting pointwise transport plans are often difficult to interpret. We introduce Optimal Mixture Transport (OMT), a scalable framework that shifts the transport paradigm from individual samples to mixtures of subpopulations, reformulating the transport problem as a strictly biconvex optimization with a unique global minimizer. We further establish theoretical guarantees on the stability of the OMT map, showing that bounded perturbations of the underlying distributions lead to bounded changes in the transport plan. By formulating subpopulations as exponential-family distributions, OMT decouples computational complexity from the sample size, scaling solely with the number of mixture components. We demonstrate the effectiveness and practicality of OMT on a wide range of synthetic benchmarks and real-world datasets, including image data and large-scale single-cell RNA sequencing measurements.
This paper analyzes identifiability and stability for the drifting field underlying distributional matching in the Generative Drifting framework of Deng et al. First, we introduce the class of companion-elliptic kernels, which includes the Laplace kernel and is characterized by a second-order elliptic coupling between each kernel $κ$ in this class and its companion function $η$. For each kernel in this class and each pair of Borel probability measures, we prove that the drifting field vanishes if and only if the two probability measures are equal. We further show that this class consists precisely of Gaussian kernels and Matérn kernels with $ν\ge 1/2$. Second, by constructing counterexamples, we exhibit sequences for which mass escapes to infinity while the field tends to zero; in particular, control of the field norm alone does not guarantee weak convergence. Nevertheless, we prove that the only possible mode of failure is confined to the one-dimensional ray $\{c\,p:0\le c\le 1\}$. Consequently, weak convergence can be restored by imposing an asymptotic lower bound on the intrinsic overlap scalar, a linear observable defined by the kernel and the target measure.