Metric Spaces

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.

Clustering mixed tabular data requires a unified metric space to bridge the inherent heterogeneity between continuous numerical measurements and discrete categorical symbols. Traditionally, algorithms rely entirely on dataset-internal statistics to estimate categorical relationships, which confines the learned metric to empirical co-occurrences and ignores conceptually obvious yet statistically unobserved affinities. Although LLMs offer external world knowledge, applying their text-centric reasoning to highly abstract tabular concepts presents significant challenges. Bridging this modality gap to construct a semantically complete metric typically requires embedding LLMs into iterative metric learning loops to dynamically optimize cross-modality representations. This incurs intractable computational overhead, forcing a compromise between semantic enrichment and scalability. Therefore, we propose GRACE, an LLM-grounded framework for scalable mixed-data clustering. GRACE shifts semantic acquisition to the attribute-value level via a multi-perspective LLM querying strategy, mapping heterogeneous values into knowledge-informed descriptions. Crucially, this one-shot grounding extracts general-purpose semantic representations that embed heterogeneous attributes into a unified space, decoupling expensive LLM invocation from iterative optimization. Furthermore, GRACE cross-validates these external semantics against dataset-internal statistical evidence to ensure alignment with the dataset-specific cluster structure. Ultimately, GRACE matches the scalability of conventional statistics-driven baselines while achieving superior clustering accuracy and conceptual interpretability over 11 competing methods. The source code is available at https://github.com/develop-yang/GRACE-GRACE-A

Deep Operator Networks (DeepONets; arXiv:1910.03193) typically encode an input function through point values on a fixed discretization. Building on the Topological DeepONet framework of Ismailov (arXiv:2603.11972), we replace point samples by continuous linear functionals drawn from the continuous dual of a Hausdorff locally convex space $({V},\{p_α\}_{α\in A})$, whose topology is generated by a point-separating family of seminorms rather than a single norm, and develop fixed and adaptive functional measurement systems. Measurements are combined with the coefficient-space Two-Step procedure of Lee and Shin (arXiv:2309.01020), while a training-only decoder and regularization stabilize the adaptive coordinates. We derive a discrete error decomposition separating measurement, output-basis, and neural-approximation errors, together with a Barron-rate refinement. The framework is evaluated on the antiderivative operator, a non-normable locally convex input space, heterogeneous Darcy flow, a controlled operator, and fixed-time and time-evolving Navier-Stokes vorticity operators. In the heterogeneous Darcy problem, the functional models retain nearly resolution-independent errors of 5.5-5.6% on unseen grids, while in the controlled problem adaptive measurements reduce the mean error below 1.2%. For the fixed-time Navier-Stokes problem, the Adaptive Topological DeepONet is the most accurate DeepONet-based model, attaining a mean relative $L^2$ error of 1.685% +/- 0.017% using 128 functional coordinates. A comparably sized Fourier neural operator (FNO; arXiv:2010.08895) achieves the lower error 0.832% +/- 0.172%, but requires the full 64x64 input field, twice the training time, and 10.7x greater peak GPU memory. The formulation provides compact, interpretable, and discretization-portable coordinates in the continuous dual $V'$, including for non-normable input spaces.

Semigroups of holomorphic self-maps of the unit disc with an interior fixed point are, by the classical Berkson--Porta representation, entirely determined by a single holomorphic function constrained only by a positivity condition on its real part. This paper uses that representation to determine exactly when the associated flow contracts the Kobayashi metric of the disc at its best possible rate --- the rate dictated by linearization at the fixed point --- rather than at some smaller, conservative rate of the kind ordinarily obtained through auxiliary metric constructions. The question is reduced to a single pointwise inequality on the representing function, and this inequality is resolved completely for a natural one-parameter family of nonlinearities, yielding an exact threshold rather than a sufficient condition of undetermined tightness. Beyond this family, an explicit representing function is exhibited for which the inequality fails almost everywhere on the disc, and the Herglotz integral representation underlying the associated Carathéodory class is used to trace this failure to concentration of the representing measure, explaining rather than merely documenting why no threshold-free general theorem is available. The results are illustrated by direct numerical verification of the sharp threshold and of the explicit obstruction, and the paper closes by identifying the precise class of representing measures --- point masses and their neighborhoods --- that any future general sufficient condition would need to exclude.

We investigate the capacity of current language models to contribute to mathematical research. In Banach space theory, AI systems generated key ideas and proofs for five new results, which were then verified and refined by humans. We also developed an automated system that searches the literature for open problems and attempts solutions at scale. Our results show both the potential of language models for mathematical discovery and the continuing importance of expert verification.

Following Alon, Hanneke, Holzman, and Moran (FOCS 2021), we define a partial concept class (PCC) as a family of partial functions $f: V\to\{0,1,\ast\}$; equivalently, its concepts partition the ground set into black ($f^{-1}(1)$), grey ($f^{-1}(\ast)$), and white parts ($f^{-1}(0)$). Its VC dimension is defined by shattering sets on which the value $\ast$ is not taken. We study two geometric PCCs in real Banach spaces, both with a margin $δ>0$: expanded half-spaces, where the grey part is a strip of width at least $δ$ adjacent to a half-space, and expanded balls, where the grey part is an annulus of width $δ$ around a unit radius ball. Our main results are dimension-free upper bounds on the VC dimension of the PCC of expanded balls in $L_p\parenthμ$, $1\le p<\infty$, including the non-Euclidean and algorithmically particularly relevant case $\ell^d_1$. These bounds depend on the margin and on the radii, but not on the ambient dimension or the underlying measure space. These are extensions of the work of Bourneuf, Charbit, and Thomassé (FOCS 2025) who studied the PCC of expanded balls in Euclidean space, that is, $\ell_2^d$. We also prove lower bounds on the VC dimension that match the upper bounds in terms of the margin parameter $δ$. Finally, we derive a Dense Neighborhood Lemma in $L_p$-spaces, again extending the known Euclidean results. Our method relies on the linearization of the distance through a map into a space of non-trivial Rademacher type, and then the use of a balanced signed-sum estimate, or a no-dimensional Radon theorem. The arguments rely on ideas from functional analysis that are clearly explained for the non-expert in that field.

We study stochastic fixed-point equations $\mathbf{T}(\mathbf{x}) = \mathbf{x}$ over normed spaces $(\mathcal{E}, \|\cdot\|)$, where the operator $\mathbf{T}$ is nonexpansive or contractive and is accessed only through unbiased stochastic evaluations with bounded second central moment. Given $ε> 0, δ\in (0, 1)$, the goal is to output $\mathbf{x} \in \mathcal{E}$ such that $\|\mathbf{T}(\mathbf{x}) - \mathbf{x}\| \leq ε$ with probability at least $1-δ$. We introduce VR-GHAL, a variance-reduced gradual Halpern method for quadratically smoothable Banach spaces. The key algorithmic ingredient is a recursive stochastic estimator based on clipped differences of oracle evaluations: instead of clipping $τ(\mathbf{x}; ξ)$ itself, we clip stochastic differences at the Lipschitz scale $γ\|\mathbf{x} - \mathbf{y}\|$. This makes the estimator pathwise Lipschitz along the algorithmic trajectory while permitting martingale concentration under finite second moments in the native norm. Our main theorem gives an anytime high-probability residual bound: on a single event of probability at least $1 - δ$, the residual decreases nearly geometrically across epochs, up to lower-order logarithmic factors. Under only bounded variance, displaying only the dependence on the target error $ε$ and Lipschitz constant $γ\in (0, 1]$ of $\mathbf{T}$, the resulting oracle complexity is $\min\{ε^{-5}, (1-γ)^{-3}ε^{-2}\}$. Under a Lipschitz-in-expectation oracle, the dependence improves to the corresponding $ε^{-3}$ nonexpansive rate (i.e., for $γ= 1$), and under samplewise nonexpansiveness to $ε^{-2}$.

Accurate metric-space hand pose estimation (HPE) is essential for immersive human-computer interaction and robotics. However, most existing methods predict poses in a root-relative coordinate system and cannot estimate the hand in absolute metric scale. In this work, we observe that the intrinsic proportional relationships among human hand bones encode stable anthropometric priors that implicitly correlate with the overall metric size of the hand. Leveraging this insight, we present ScaleHP, an end-to-end one-stage hand pose estimation framework that bypasses fragile extrinsic depth modules to recover the hand in metric space. ScaleHP employs a transformer-based decoder with a novel scale token to fuse multi-scale morphological and appearance features. By solving for metric coordinates through a perspective-constrained least-squares approach, we achieve high-precision pose estimation in the camera coordinate system. ScaleHP delivers state-of-the-art performance, including 35.8 CS-MPJPE on FreiHand and 4.6/5.9 PA-MPJPE on DexYCB and HO3Dv3. These results demonstrate that internal biological constraints significantly reduce relative geometry and absolute metric errors, offering a robust solution for generalized, real-world hand tracking.

We study the operator-theoretic core of Q-learning in continuous-time stochastic control with continuous states and actions. In value-based reinforcement learning, each Q-learning or DQN update is built from a Bellman optimality target; our analysis isolates this target in a diffusion setting and studies its regularity and approximation complexity. Under uniform ellipticity and Hölder-regular coefficients, we show that a Bellman update maps bounded inputs into an anisotropic regularity class, smoothing the state variable while leaving only Lipschitz dependence on the action variable. This yields a compact family of Bellman iterates and motivates a tensor-product DeepONet architecture adapted to the mixed regularity of the problem. We then derive explicit approximation and resource bounds, together with a stiffness--complexity trade-off as the time step $δ\to 0$. The resulting theory makes a direct contribution to Q-learning theory at the level of Bellman target regularity and approximation in continuous stochastic control. At the same time, we do not claim a full convergence theorem for practical sampled Q-learning with exploration, replay, and stochastic gradient updates.

We develop a general framework for analyzing representation costs of parametric data-fitting methods through their parameter-space regularizers. From this abstract perspective, we define representation costs for arbitrary parametric models and reveal their induced (native) function spaces. This unifies recent function-space views of data-fitting methods. We also prove that many natural results hold in this abstract setting, including representer theorems for parametric methods on their native spaces. The framework also rigorously connects parametric methods with their equivalent nonparametric descriptions under sufficient overparameterization. Classical methods and their native spaces, such as kernel methods / reproducing kernel Hilbert spaces, wavelets / Besov spaces, and shallow neural networks / variation spaces emerge as special cases of our abstract framework. A byproduct of "axiomatizing" the study of representation costs is that we also immediately obtain new results for deep neural networks: For depth-$L$ feedforward ReLU networks, their induced native spaces are $p$-normable quasi-Banach spaces with $p = 2/L$. This reveals that the inductive bias of deep neural networks (as given by the representation cost) cannot be captured by norms for depths $L > 2$.

As AI systems are deployed in high-stakes ethical contexts such as healthcare triage, autonomous vehicle control, and employment screening, formal methods for evaluating their robustness against adversarial manipulation of ethical reasoning remain underdeveloped. This paper introduces the Ethical Robustness Testing System (ERTS), a closed-pipeline framework that: (1) encodes ethical dilemmas into a 22-dimensional Ethical Consequence Space (ECS) grounded in established ethical theory; (2) applies 17 semantic perturbation functions subject to 6 validity constraint classes including a novel semantic coherence constraint; (3) measures decision deviation via a 4-component Ethical Instability Index (EII); and (4) produces domain-adaptive pre-deployment robustness assessment verdicts. We evaluate 4 structured baseline models and 2 production LLMs (Gemini 2.0 Flash and Llama 3.2) across 50 ethical scenarios spanning 8 deployment domains, generating 1,500 adversarial test cases. Results demonstrate that only 33% of models achieve assessment clearance, with the local Llama-3.2 model proving particularly vulnerable to fairness corruption and information degradation attacks (ERS = 0.737). To the best of our knowledge, no existing framework combines a bounded ethical consequence space, semantic coherence constraints, and domain-adaptive assessment in a single adversarial testing pipeline.

The Gromov--Wasserstein (GW) distance provides a framework for comparing metric measure spaces, regardless of their underlying structure or geometry. For network-based data, it enables direct comparisons of graphs with different numbers of nodes, without requiring an embedding or other abstraction. Furthermore, through a variant of GW known as fused Gromov--Wasserstein (fGW), it is also possible to incorporate node features in addition to graph structure. In this work, we implement $k$-nearest neighbors ($k$-NN) classification using the GW and fGW distances. We prove the universal consistency of the GW-$k$-NN classifier on the space of equivalence classes of metric measure spaces with finite support and uniform probability measure. By viewing graphs as finitely supported metric measure spaces equipped with the pairwise distance metric and a uniform probability measure on the nodes, we obtain universal consistency of GW-$k$-NN for the space of graphs. Likewise for fGW-$k$-NN, we prove universal consistency on the space of weak isomorphism classes of structured objects consisting of metric measure spaces with finite support and uniform probability measure and feature maps into Euclidean space, thus establishing universal consistency on the space of node-attributed graphs. Our numerical experiments show that GW-$k$-NN and fGW-$k$-NN consistently perform well across multiple graph datasets, suggesting that metric classifiers such as $k$-NN work well in the GW framework.

Reliable evaluation of instance segmentation models requires metrics that accurately and consistently reflect segmentation quality. However, the metrics most widely used in biological imaging carry fundamental mathematical weaknesses: hard Intersection-over-Union (IoU) thresholds that produce discontinuous, low sensitivity scoring; per-object normalization that distorts scores under object size variation; and greedy or one-to-many matching procedures that yield non-optimal, order-dependent correspondences. Together, these properties produce unintuitive and unreliable model rankings under common failure modes such as split cells, merged cells, and cell boundary imprecision. We propose Maximum Matching Accuracy (MMA), a threshold-free continuous metric that finds a globally optimal one-to-one matching between predicted and ground truth objects and aggregates total overlap using per-pixel normalization. We evaluate MMA against AP@50, PQ, SEG, and AJI across three experiments: synthetic failure cases, progressive corruption tests, and a model ranking comparison. MMA produces scores that are more stable, more sensitive, and more interpretable than existing alternatives, providing a principled foundation for fair instance segmentation benchmarking in biological cell imaging.

We propose a new framework for optimisation over non-Archimedean spaces inspired by Berkovich geometry. Specifically, we introduce polydisc spaces, which consists of products of closed balls over a non-Archimedean field. These spaces retain the rigid hierarchical structure of the non-Archimedean field whilst acquiring many desirable geometric features absent from it. We show that metric trees embed naturally into these spaces, demonstrating their capacity to represent hierarchical data. We study their metric geometry, establishing properties such as geodesic uniqueness, confirming their comaptibility with classical optimisation techniques. We further propose a class of real-valued functions given by linear combinations of absolute values of polynomials. These functions admit a piecewise polynomial description along geodesics and satisfy a universal approximation property. We formulate a theory of optimisation on polydisc spaces: we prove existence of minimisers and explore algorithms for finding them. We provide an accompanying open-source Julia library implementing the core objects and optimisation procedures introduced.

The Gini Multidimensional Scaling (Gini MDS) framework extends the Euclidean multidimensional scaling. We introduce a Gini pseudo-distance based on values and their ranks that depends on a fine-tunable hyperparameter. This pseudo-distance allows flexible exploration of latent configurations, enabling embeddings that best match observed dissimilarities. The Gini MDS is shown to be robust to noise and outliers, making it well-suited for real-world applications. We provide experiments on 16 UCI datasets with outliers and on MNIST images with noise to show that the Gini MDS outperforms the Euclidean MDS on noisy data. Finally, a tensor-based implementation in \texttt{PyTorch} provides GPU acceleration and efficient computation compared to the standard MDS of the \texttt{sklearn} library.

Several machine learning models are defined for inputs of any size, such as graphs with different numbers of nodes and point clouds containing varying numbers of points. The universality properties of such any-dimensional models remain poorly understood, as universality is traditionally studied for models accepting inputs of a fixed size, defined on a compact subset of their domain. In sharp contrast, any-dimensional models can be viewed as sequences of functions defined on growing-sized inputs, and it is not clear in which sense they can be universal. We develop a systematic approach to establish any-dimensional universality, by identifying any-dimensional functions with a unique function taking inputs in a suitable infinite-dimensional limit space containing inputs of all finite sizes as well as their limits. Using the symmetries of these inputs and relations between inputs of different sizes, we show that this limit space admits a natural topology with rich families of compact sets on which any-dimensional universality can be established. We illustrate our approach by showing that several existing architectures fail to be universal, and we propose simple modifications that restore universality.

Establishing Universal Approximation Theorems (UATs) for nonlinear operators and their derivatives is a foundational open problem in Operator Learning (OL) and raises delicate questions in Nonlinear Functional Analysis. We prove the first UATs for $k$-times differentiable nonlinear operators and their derivatives via OL architectures, uniformly on compact sets and in weighted Bastiani--Sobolev spaces for general finite input measures. In full Banach-space generality, these are the first complete generalizations of the corresponding influential classical UATs in [Hornik, 1991] to infinite-dimensional spaces and OL, {and launch Derivative-Informed Operator Learning (DIOL) (i.e. learning nonlinear operators and their derivatives)} on general Banach spaces. Based on our UATs, we formulate Bastiani--Sobolev training in DIOL. We present open frontiers where DIOL and our UATs find applications: high-order accuracy in OL; fast constrained optimization in Banach spaces (e.g. optimal control of PDEs, inverse problems) via Learn-Then-Optimize; numerical methods for infinite-dimensional PDEs (e.g. HJB PDEs on Banach spaces from infinite-dimensional optimal control via Optimize-Then-Learn, such as optimal control of PDEs, SPDEs, path-dependent systems, partially observed systems, mean-field control). We parameterize nonlinear operators via Encoder-Decoder Architectures, classical OL architectures. These include DeepONets, Deep-H-ONets, and PCA-Nets, which our UATs cover. Our UATs are based on (i) Approximation Properties of Banach spaces; (ii) continuous Bastiani differentiability (weaker than continuous Fréchet differentiability); (iii) $C^k_B$ (Bastiani) compact-open topologies; indeed, UA in $C^k$ (Fréchet) compact-open topologies (induced by operator norms) fails; (iv) construction of weighted Bastiani--Sobolev spaces, generalizing classical Gaussian Sobolev spaces on Banach spaces.

Gromov--Wasserstein (GW) distances compare graphs, shapes, and point clouds through internal distances, without requiring a common coordinate system. This invariance is powerful, but discrete GW is a nonconvex quadratic optimal transport problem and is difficult to estimate at scale. We propose \emph{Distance-Matrix Wasserstein} (DMW), a hierarchy of Wasserstein statistics comparing laws of random finite distance matrices. Rather than optimizing a global point-level alignment, DMW samples $n$ points from each space, records their pairwise distances, and transports the resulting matrix laws. We prove that DMW is a relaxation and lower bound of GW, and establish a reverse approximation inequality: the GW--DMW gap is controlled by the Wasserstein error of approximating each original measure with $n$ samples. Hence population DMW converges to GW as sampled subspaces become dense. We further give finite-sample bounds, including intrinsic-dimensional rates that depend on the data manifold rather than the ambient matrix dimension $\binom n2$. For scalable computation, we introduce sliced and multi-scale DMW; for $p=1$, the sliced multi-scale dissimilarity yields positive-definite exponential kernels. Experiments on synthetic metric spaces, scalability benchmarks, graph classification, and two-sample testing validate the theory and demonstrate an interpretable GW-style proxy for structural comparison.

We present an in-depth analysis of the Koopman semigroup via wavelet transform. Towards this goal, we start by introducing the wavelet-based observables and show that they are eigenfunctions of the Koopman semigroup when this semigroup is considered over the Banach space of continuous functions on a compact forward-invariant set endowed with the supremum norm. We then construct closed-form expressions of the action of the Koopman semigroup and its resolvent in terms of these observables. To approximate the action of Koopman semigroup numerically, we combine Extended Dynamic Mode Decomposition (EDMD) with the proposed wavelet-based observables leading to the Wavelet Dynamic Mode Decomposition via Continuous Wavelet Transform (cWDMD) algorithm. We validate our theoretical results on two numerical examples.

Computational social choice and algorithmic decision theory offer rich aggregation theory but no end-to-end process for egalitarian self-governance: aggregation, deliberation, amendment, and consensus are each considered in isolation, with key metric-space aggregators being NP-hard. Here, we propose \emph{constitutional governance in metric spaces}, integrating these stages into a protocol for constitutional governance. A community's \emph{legal corpus} comprises its \emph{laws} together with a \emph{constitution}, each being a point in a metric space, with the constitution specifying the supermajority threshold required to amend it. Members vote to amend the legal corpus by proposing their ideal points, followed by rounds of submitting \emph{public proposals} carrying \emph{supermajority public support}; a polynomial-time \emph{aggregation rule} scores each proposal, and a supported proposal whose score is positive and maximal for two rounds is adopted; if none is found the status quo is retained. Public proposals can be sourced from deliberation, vote aggregation, or AI mediation. With Constitutional Consensus, a community can run the constitutional governance protocol on members' personal computing devices (e.g., smartphones), achieving digital sovereignty. By drawing on metric-space aggregation, reality-aware social choice, supermajority amendment, constitutional consensus, deliberative coalition formation, and AI mediation, we provide a comprehensive framework for the constitutional governance of digital communities and organisations.

Operator learning for partial differential equations (PDEs) aims to learn solution operators on infinite-dimensional function spaces from finite-resolution data. In this setting, it is important for the learned model to be discretization-invariant, or resolution-robust, and to reflect PDE-specific structure. It is therefore natural to ask how such structure should be encoded in the model architecture, hypothesis class, or learning procedure. In this paper, we study operator learning for solution operators of nonlinear parabolic PDEs based on Duhamel--Picard iteration. We formulate Picard iteration as an abstract state-transition model and present a theoretical framework for Picard-type operator learning. We derive implementation-agnostic generalization error bounds that separate the implementation error from the estimation error associated with the abstract state-transition model induced by Picard iteration. A key consequence is that increasing the Picard depth reduces the Picard truncation error without causing an unbounded growth of the entropy-based estimation error. We also extend the analysis to long-time prediction by rolling out the same learned local model over successive time blocks. Finally, we illustrate the theory for nonlinear heat equations on the torus using a Picard-type Fourier neural operator as a concrete implementation.

We introduce the observable Wasserstein distance, a framework for deriving lower bounds on the Wasserstein distance between probability measures on Polish metric spaces, designed to bypass the computational intractability of exact optimal transport in large-scale, non-Euclidean datasets. Analogous to the sliced Wasserstein distance in $\mathbb{R}^d$, our approach projects measures onto the real line via 1-Lipschitz observables and computes the Wasserstein distances between the resulting pushforward distributions. We define a hierarchy of pseudo-metrics by restricting observables to a nested chain of subspaces. A central theoretical contribution is an injectivity result linking the metric covering dimension of the support of a measure to the specific order in the hierarchy that guarantees unique recovery. This serves as a metric-space analogue to the Cramér-Wold Device for Euclidean distributions. We demonstrate that this hierarchy offers a tunable trade-off between sharpness as a lower bound on the Wasserstein distance and computational efficiency. We also present a discrete computational model for finite grids and numerical experiments validating the efficacy and utility of these approximations.

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.

We introduce the Banach-Butterfly Invariant (BBT), an influence-adaptive Banach geometry on the Walsh-Hadamard butterfly factorization. For a Boolean function $f:\{-1,+1\}^n\to\{-1,+1\}$ with coordinate influences $\mathrm{Inf}_\ell(f)$, BBT assigns exponent $p_\ell = 1+\mathrm{Inf}_\ell(f)$ to butterfly layer $\ell$, yielding the contraction invariant $μ(f)=\prod_\ell 2^{-\mathrm{Inf}_\ell/(1+\mathrm{Inf}_\ell)}$. We prove a Jensen lower bound $\log_2μ(f) \ge -I(f)/(1+I(f)/n)$ and that $μ$ is strictly Schur-convex in the influence vector (modulo permutation), giving scaling classes $μ\sim 2^{-n/2}$ (parity), $2^{-Θ(\sqrt{n})}$ (majority), $2^{-1/2}$ (dictators). $\log_2μ$ is rational but not polynomial in the Fourier coefficients while $μ$ is algebraic, and $μ$ separates functions with identical total influence (122 pairs at $n=3$). Using the certified $n \le 4$ ternary Walsh-threshold universe from a companion synthesis manuscript as a finite testbed, we compute exact MILP minimum-support certificates for all 65,536 Boolean functions at $n=4$ (mean 6.42, max 9, all-odd by a parity argument) and on 10,000 of the 616,126 NPN-canonical representatives we enumerate at $n=5$ (matching OEIS A000370). Conditional Spearman $ρ(μ,|\mathrm{supp}|)$ at fixed total influence is $+0.571$ in the largest stratum at $n=4$ but reverses to $-0.38$ at $n=5$ under both function-uniform and NPN-canonical sampling: $μ$ is a valid Schur-convex concentration invariant, not a universal monotone predictor of minimum support across $n$. A companion application paper validates a real-valued WHT activation-energy proxy inspired by this theory on five pretrained LLMs at W2A16, cutting wikitext-2 perplexity by 15-58% versus vanilla auto-round; the transfer from Boolean theory to the real-valued proxy is qualitative, not formal.

Bregman divergences play a pivotal role in statistics, machine learning and computational information geometry. Particularly in the context of machine learning, they are central to clustering, exponential families, parameter estimation and optimisation, among other things. Despite this, the full toolkit of Hilbert spaces and in particular reproducing kernel Hilbert spaces have not been systematically developed and applied to functional Bregman divergences, where points are functions rather than finite-dimensional parameter vectors. While other types of functional Bregman divergences have been studied, these are typically in a Banach space rather than more directly aligned with kernel methods and Hilbert-space geometry commonly used in machine learning. We consider functional Bregman divergences on a Hilbert space, where the self-dual pairing and Riesz representer afford us particularly convenient calculus. Further specialising Bregman generators as a composition involving a kernel mean embedding makes such divergences easy to estimate. We discuss applications in clustering, universal estimation, robust estimation and generative modelling, and contrast our approach with other types of Bregman divergences.

\We introduce the horospherical depth, an intrinsic notion of statistical depth on Hadamard manifolds, and define the Busemann median as the set of its maximizers. The construction exploits the fact that the linear functionals appearing in Tukey's half-space depth are themselves limits of renormalized distance functions; on a Hadamard manifold the same limiting procedure produces Busemann functions, whose sublevel sets are horoballs, the intrinsic replacements for halfspaces. The resulting depth is parametrized by the visual boundary, is isometry-equivariant, and requires neither tangent-space linearization nor a chosen base point. For arbitrary Hadamard manifolds, we prove that the depth regions are nested and geodesically convex, that a centerpoint of depth at least $1/(d+1)$ exists, and hence that the Busemann median exists for every Borel probability measure. Under strictly negative sectional curvature and mild regularity assumptions, the depth is strictly quasi-concave and the median is unique. We also establish robustness: the depth is stable under total-variation perturbations, and under contamination escaping to infinity the limiting median depends on the escape direction but not on how far the contaminating mass has moved along the geodesic ray, in contrast with the Fréchet mean. Finally, we establish uniform consistency of the sample depth and convergence of sample depth regions and sample Busemann medians; on symmetric spaces of noncompact type, the argument proceeds through a VC analysis of upper horospherical halfspaces, while on general Hadamard manifolds it follows from a compactness argument under a mild non-atomicity assumption.

We study discrete k-clustering problems in general metric spaces that are constrained by a combination of two different fairness conditions within the demographic fairness model. Given a metric space (P,d), where every point in P is equipped with a protected attribute, and a number k, the goal is to partition P into k clusters with a designated center each, such that a center-based objective function is minimized and the attributes are fairly distributed with respect to the following two fairness concepts: 1) group fairness: We aim for clusters with balanced numbers of attributes by specifying lower and upper bounds for the desired attribute proportions. 2) diverse center selection: Clusters have natural representatives, i.e., their centers. We ask for a balanced set of representatives by specifying the desired number of centers to choose from each attribute. Dickerson, Esmaeili, Morgenstern and Zhang (2023) denote the combination of these two constraints as doubly constrained fair clustering. They present algorithms whose guarantees depend on the best known approximation factors for either of these problems. Currently, this implies an 8-approximation with a small additive violation on the group fairness constraint. For k-center, we improve this approximation factor to 4 with a small additive violation. This guarantee also depends on the currently best algorithm for DS-fair k-center given by Jones, Nguyen and Nguyen (2020). For k-median and k-means, we propose the first constant-factor approximation algorithms. Our algorithms transform a solution that satisfies diverse center selection into a doubly constrained fair clustering using an LP-based approach. Furthermore, our results are generalizable to other center-selection constraints, such as matroid k-clustering and knapsack constraints.

Margin-based learning, exemplified by linear and kernel methods, is one of the few classical settings where generalization guarantees are independent of the number of parameters. This makes it a central case study in modern highly over-parameterized learning. We ask what minimal mathematical structure underlies this phenomenon. We begin with a simple margin-based problem in arbitrary metric spaces: concepts are defined by a center point and classify points according to whether their distance lies below $r$ or above $R$. We show that whenever $R>3r$, this class is learnable in \emph{any} metric space. Thus, sufficiently large margins make learnability rely only on the triangle inequality, without any linear or analytic structure being necessary. Our first main result extends this phenomenon to concepts defined by bounded linear combinations of distance functions, and reveals a sharp threshold: there exists a universal constant such that whenever the margin is larger than this constant, the class is learnable in every metric space, while below it there exist metric spaces where it is not learnable at all. We then ask whether margin-based learnability can always be explained via an embedding into a linear space -- that is, reduced to linear classification in some Banach space through a kernel-type construction. We answer this negatively by demonstrating a margin learnable class that cannot be embedded into any Banach space in which linear classification with margins is learnable.

Regression with non-Euclidean responses---e.g., probability distributions, networks, symmetric positive-definite matrices, and compositions---has become increasingly important in modern applications. In this paper, we propose deep Fréchet neural networks (DFNNs), an end-to-end deep learning framework for predicting non-Euclidean responses---which are considered as random objects in a metric space---from Euclidean predictors. Our method utilizes the representation-learning power of deep neural networks (DNNs) to the task of approximating conditional Fréchet means of the response given the predictors, the metric-space analogue of conditional expectations, by minimizing a Fréchet risk. The framework is highly flexible, accommodating diverse metrics and high-dimensional predictors. We establish a universal approximation theorem for DFNNs, advancing the state-of-the-art of neural network approximation theory to general metric-space-valued responses, without making model assumptions or relying on local smoothing. We further establish rigorous generalization guarantees for DFNNs and derive corresponding risk bounds, providing, to the best of our knowledge, the first such theoretical results for deep learning regression with metric-space-valued responses. Empirical studies on synthetic distributional and network-valued responses, as well as real-world applications to predicting compositional responses in an Aitchison simplex and spherical responses, demonstrate that DFNNs consistently outperform all existing methods.

Ecological Momentary Assessment (EMA) studies enable the collection of high-frequency self-reports of suicidal thoughts and behaviors (STBs) via smartphones. Latent stochastic differential equations (SDEs) are a promising model class for EMA data, as it is irregularly sampled, noisy, and partially observed. But SDE-based models suffer from two key limitations. (a) These models often violate domain constraints, undermining scientific validity and clinical trust of the model. (b) Training is numerically unstable without ad hoc fixes (e.g. oversimplified dynamics) that are ill-suited for high-stakes applications. Here, we develop a novel class of expressive SDEs whose solutions are provably confined to a prescribed compact polyhedral state space, matching the domains of EMA data. In this work, (1) we show why chain-rule based constructions of SDEs on compact domains fail, theoretically and empirically; (2) we derive constraints on drift and diffusion for general and stationary SDEs so their solutions remain in the desired state space; and (3), we introduce a parameterization that maps arbitrary (neural or expert-given) dynamics into constraint-satisfying SDEs. On several real EMA datasets, including a large suicide-risk study, our parameterization improves forecasts and optimization dynamics over standard latent neural SDE baselines. These contributions pave the way for principled, trustworthy continuous-time models of suicide risk and other clinical time series and extend applications of SDE-based methods (e.g. diffusion models) to domains with hard state constraints.

This paper introduces a framework for uncertainty quantification in regression models defined on metric spaces. Using a proposed notion of homoscedasticity, we define a conformal prediction algorithm that provides finite-sample marginal coverage guarantees and fast convergence rates to the oracle prediction region. For heteroscedastic settings, we introduce a kNN procedure that yields locally adaptive prediction radii in general metric spaces. Although this procedure does not provide the same finite-sample guarantees as the conformal algorithm, it is designed to improve local coverage calibration without imposing smoothing assumptions. Both procedures are compatible with a broad range of regression algorithms and scale to large datasets, allowing practitioners to use their preferred models and incorporate domain-specific knowledge. Building on the heteroscedastic $k$NN approach, we also develop a flexible sequential extension for metric-space-valued time series based on nearest-neighbor expert aggregation. We establish the consistency of the proposed estimators under minimal conditions. Finally, we illustrate the practical utility of our framework in personalized medicine applications involving random objects such as probability distributions and graph Laplacians.

The Hausdorff dimension of a set can be detected using the Riesz energy. Here, we consider situations where a sequence of points, $\{x_n\}$, ``fills in'' a set $E \subset \mathbb{R}^d$ in an appropriate sense and investigate the degree to which the discrete analog to the Riesz energy of these sets can be used to bound the Hausdorff dimension of $E$. We also discuss applications to data science and Erdős/Falconer type problems.

Motivated by the rapidly growing field of mathematics for operator approximation with neural networks, we present a novel universal operator approximation theorem for broad classes of encoder-decoder architectures and a wide range of input and output spaces. In this study, we focus on the approximation of continuous operators between infinite-dimensional normed or metric spaces in the topology of uniform convergence on compact sets. Unlike standard results in the operator learning literature, we additionally investigate the case where the approximating sequence of encoder-decoder architectures can be chosen independently of the compact sets. Taking a topological perspective, we point out that compact-set-independent approximation is a strictly stronger property in most relevant operator learning frameworks. To establish our results, we introduce new approximation properties of input and output spaces tailored to encoder-decoder architectures. These properties enable us to prove a universal operator approximation theorem ensuring uniform convergence on every compact subset of the input space. Our results unify and extend existing universal operator approximation theorems for various encoder-decoder architectures, including classical DeepONets, BasisONets, MIONets, architectures based on frames and other related approaches. A notable feature of our framework is that it also applies to metric spaces beyond the normed setting. In particular, it allows the consideration of $p$-Wasserstein spaces of probability measures as input or output spaces, and Skorohod spaces of càdlàg functions as input spaces. This generality also opens up potential applications in optimal transport.

In the classical best approximation pair (BAP) problem, one is given two nonempty, closed, convex and disjoint subsets in a finite- or an infinite-dimensional Hilbert space, and the goal is to find a pair of points, each from each subset, which realizes the distance between the subsets. Motivated by our recent algorithm for solving the BAP problem [Censor, Mansour, Reem, J. Approx. Theory (2024)], we discuss the problem in more general normed spaces and with possibly non-convex subsets, and focus our attention on the fundamental issues of uniqueness and existence of the solution to the problem. We present several sufficient geometric conditions for the (at most) uniqueness of a BAP. These conditions are related to the structure and the relative orientation of the boundaries of the subsets and to the norm. We also present many sufficient conditions for the existence of a BAP. In general, the paper re-examines several aspects related to the BAP problem, including the historical one, and shows, probably for the first time, how wide is the scope of the BAP problem in terms of the scientific communities which are involved in it (frequently independently) and in terms of its applications.