A Closed-Form Formula for Consistent Lipschitz Regression on Metric Spaces with Sparse Neural Network Realizations

Several classical machine-learning methods, such as KRRs and SVRs, are both computationally and analytically tractable since their estimators either admit closed-form expressions or are obtained by minimizing convex training objectives; neither feature is generally available for deep neural networks. We address this by introducing a simple closed-form ``two-stage'' compositional formula $\hat{f}$ for reconstructing an unknown Lipschitz function $f:\mathcal{X}\to \mathbb{R}$ on a metric space $(\mathcal X,ρ)$ from $N$ i.i.d. noisy observations. Our main result is a high-probability uniform ($L^{\infty}$) recovery guarantee that jointly controls approximation and statistical errors while enjoying an optimization error of zero; in particular, we do not assume oracle access to an approximate ERM. Our secondary main results establish the optimality of our formula in three complementary senses. 1) Function space: On Ahlfors-regular metric spaces, the hypothesis class parameterized by our formula attains the optimal fat-shattering dimension. 2) Parameter space: Its dependence on the parameters is maximally numerically stable, in the sense that a smaller approximation error cannot be achieved with a smaller Lipschitz dependence on the model parameters. 3) Forward pass: Its dependence on the input is maximally regular, matching the Lipschitz constant of the target function $f$. When $\mathcal X=[0,1]^d$ is equipped with the $\ell^\infty$ norm, $\hat{f}$ admits algorithmic ReLU-MLP and exact ReLU-multi-head transformer realizations of depth $\mathcal{O}(\log(N))$ with $\mathcal{O}(N)$ nonzero parameters.

Common Geodesics Do Not Guarantee Fisher Consistency of the Structured SVM: Minimal Counterexamples and a Tree-Metric Classification

A known necessary condition for Fisher consistency of the structured support vector machine requires the task loss to be a metric for which every output triple has a common geodesic point. We show that this condition is not sufficient for the canonical coordinate-wise argmax decoder. A four-output unit star admits an exactly optimal score vector whose maximizers are all strictly non-Bayes, and four outputs are minimal among metrics satisfying the condition. We then completely classify positively weighted tree metrics whose vertex set is the output space: argmax consistency holds if and only if the tree is a path. The failure on branching trees is confined to boundary distributions; every tree retains the argmax property at every full-support distribution. Among metrics satisfying the common-geodesic condition, five outputs are necessary and sufficient for a full-support counterexample; $K_{2,3}$ is the smallest member of an infinite $K_{m,n}$ family. We additionally give a full-support counterexample for the three-dimensional Hamming cube. All optimality claims have exact primal-dual certificates. The counterexamples expose a concrete decoder gap: in this polyhedral setting, an embedding can guarantee the existence of a calibrated link without validating a prescribed argmax link on every surrogate-risk minimizer.

Diversity-Based Active Learning: An Evaluation of Metric Spaces for Active Learning Selection

With rapid advancement over the last few years, many different methods are now widely used for classification. However, training these models requires substantial labeled data. Active Learning is a potential solution to this problem. Pool-based active learning minimizes costs by querying only the most informative samples from an unlabeled dataset. Diversity-based approaches, on the other hand, attempt to select a representative subset of the data. There are many different objectives for determining the selection process, including exact K-center, exact K-median, and Greedy K-center. In this paper, we will focus on evaluating the performance of Greedy K-center across a variety of metric spaces: the raw feature space, a Linear Discriminant Analysis (LDA) space, and a model-derived probability space (with and without entropy-based weighting). Using Random Forest classifiers as a baseline evaluator, our empirical results on synthetic and real-world datasets demonstrate that mapping unlabeled instances into a predictive probability space and weighting the result by entropy often dominates the other options for active learning selection with Greedy K-center.

Distance Profile Embedding for Independence and Conditional Independence Testing of Random Objects

Testing independence or conditional independence is fundamental to statistical inference, yet existing methods for non-Euclidean random objects often face a difficult trade-off between geometric flexibility and theoretical tractability. We introduce the Distance Profile Embedding (DPE), a novel representation that maps random objects from general metric spaces into a Hilbert space of square-integrable functions. We prove that this mapping is injective and preserves full distributional information without requiring isometric Hilbert embeddings or one-to-one correspondence conditions. Leveraging the DPE, we develop a unified framework for marginal and conditional independence testing of random objects that enjoys a rigorous asymptotic theory for both size and power. Notably, our framework is the first in the literature to accommodate object-valued conditioning variables when testing conditional independence, overcoming the Euclidean or Hilbertian constraints of existing methodologies. We facilitate the calculation of analytic $p$-values using closed-form asymptotic null distributions, which avoids the computational burden of permutation tests common in existing metric-based methods. The numerical properties of our methods are demonstrated through both simulations and two real-world applications involving gut microbiome compositions and global human mortality distributions, respectively.

Random-Effects Algorithm for Random Objects in Metric Spaces

Across many scientific disciplines, multiple observations are collected from the same experimental units, and in modern datasets these observations often arise as non-Euclidean random objects. In such settings, the incorporation of random effects is a critical modeling step for efficient estimation and personalized prediction. Although mixed-effects models are well established for scalar outcomes and, more recently, for functional data in Hilbert spaces, general random-effects frameworks for objects in metric spaces remain underdeveloped. In this paper, we propose a nonlinear Fréchet-based algorithm for random-effects modeling of arbitrary random objects defined on a metric space. Using M-estimation theory, we establish conditions under which the proposed metric-space prediction target is consistently estimated under a working random-effects formulation. We then evaluate the empirical performance of the proposed method using both synthetic data and digital health datasets that require practical tools for analyzing random objects in metric spaces, such as multivariate probability distributions and random graphs. We show that, although our method is developed beyond Hilbert spaces, it can outperform existing Hilbert space-based methods.