arXiv subjects

Anthea Monod

Publications and source records attributed to Anthea Monod.

At least 19 recordsLinked to original sources

arXiv2026

Density-Robust Spherical Coordinates from Persistent Cohomology

Persistent cohomology provides a principled framework for constructing nonlinear coordinates that reflect the topology of data. However, these topological coordinates can be severely distorted by non-uniform sampling density, limiting their applicability to real-world data. While density-robust circular coordinates have recently been developed, the extension to spherical coordinates remains an open challenge: unlike the circular case, spherical coordinates are obtained through a nonlinear variational problem for sphere-valued maps, to which existing density-correction mechanisms are not directly applicable. In this paper, we introduce the first density-robust construction of spherical coordinates from persistent cohomology. Rather than modifying the coordinate optimization itself, we extend a subsampling-and-alignment framework for circular coordinates to $S^2$, which first computes spherical coordinates on approximately uniform subsamples obtained by rejection sampling and then combines them into a global consensus map. The principal mathematical difficulty is the alignment of independently computed sphere-valued coordinates. We formulate this challenge as a spherical Procrustes problem and establish approximation guarantees for a computationally tractable Euclidean relaxation. Our resulting construction is robust to non-uniform sampling and retains the accuracy of classical spherical coordinates under uniform sampling. Moreover, by computing persistent cohomology only on fixed-size subsamples, our approach avoids the quartic memory bottleneck of the classical spherical coordinate pipeline and scales to substantially larger datasets. We conduct experiments on synthetic data and demonstrate accurate coordinate recovery under severe sampling bias and scalability to datasets of 10,000 points.

math.AT↗

arXiv2026

Tracking Representation Dynamics in Large Language Models with Persistent Homology

Large language models are commonly aligned through supervised fine-tuning, yet little is known about how their internal representations evolve during this process. We study alignment dynamics using persistent homology by tracking the topology of activation spaces throughout fine-tuning. Across four transformer language models ranging from 1B to 7B parameters and three alignment objectives corresponding to helpful, harmless, and mixed training data, we find that the majority of topological reorganization occurs during the earliest stages of training. A dense checkpoint analysis reveals a transient peak in topological activity followed by rapid stabilization. We further show that different alignment objectives induce distinguishable topological trajectories, while instruction-tuned and pretrained models exhibit qualitatively different patterns of evolution. Our results suggest that persistent homology provides a complementary perspective on alignment, revealing representation-level changes that are not apparent from behavioral metrics alone.

cs.LG↗

arXiv2026

Generative Modeling on Metric Graphs via Neural Optimal Transport

We introduce, to our knowledge, the first deep generative modeling framework for probability distributions continuously supported on compact metric graphs. Given source and target measures on a metric graph, our method embeds the graph into a smooth ambient space, solves an entropic Kantorovich problem via a neural semidual parameterization, and projects generated samples back onto the original graph. We study two embedded geometries: an extrinsic Euclidean realization and the intrinsic tropical Abel--Jacobi embedding into the Jacobian torus. In both cases, the resulting generator is graph-supported by construction. We prove that, in the joint limit of increasing neural expressivity, the learned generator converges weakly to a valid transport coupling between the original graph measures. Empirically, across a range of geometrically distinct graphs, our method matches or improves upon heuristic transport baselines based on discrete graph OT, while scaling more favorably. Finally, we demonstrate scalability on real-world urban mobility data by training our model on one million Uber pickup locations in Manhattan, New York City.

stat.ML↗

arXiv2026

Non-Archimedean Polydisc Spaces and Applications to Optimisation

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.

math.OC↗

arXiv2026

Topological Signatures of Grokking

We study the grokking phenomenon through the lens of topology. Using persistent homology on point clouds derived from the embedding matrices of a range of models trained on modular arithmetic with varying primes, we identify a clear and consistent topological signature of grokking: a sharp increase in both the maximum and total persistence of first homology ($H_1$). Persistence diagrams reveal the emergence of a dominant long-lived topological feature together with increasingly structured secondary features, reflecting the underlying cyclic structure of the task. Compared to existing spectral and geometric diagnostics -- specifically, Fourier analysis and local intrinsic dimension -- persistent homology provides a unified geometric and topological characterization of representation learning, capturing both local and global multi-scale structure. Ablations across data regimes and control settings show that these topological transitions are tied to generalization rather than memorization. Our results suggest that persistent homology offers a principled and interpretable framework for analyzing how neural networks internalize latent structure during training.

cs.LG↗

arXiv2026

Feature Starvation as Geometric Instability in Sparse Autoencoders

Sparse autoencoders (SAEs) are used to disentangle the dense, polysemantic internal representations of large language models (LLMs) into interpretable, monosemantic concepts. However, standard $\ell_1$-regularized SAEs suffer from feature starvation (dead neurons) and shrinkage bias, often requiring computationally expensive heuristic resampling and nondifferentiable hard-masking methods to bypass these challenges. We argue that feature starvation is not merely an empirical artifact of poor data diversity, but a fundamental optimization-geometric pathology of overcomplete dictionaries: the $\ell_1$-induced sparse coding map is unstable and fundamentally misaligned with shallow, amortized encoders. To address this structural instability, we introduce adaptive elastic net SAEs (AEN-SAEs), a fully differentiable architecture grounded in classical sparse regression. AEN-SAEs combine an $\ell_2$ structural term that enforces strong convexity and Lipschitz stability with adaptive $\ell_1$ reweighting that eliminates shrinkage bias and suppresses spurious features, thereby jointly controlling the curvature and interaction structure of the induced polyhedral geometry. Theoretically, we show that AEN-SAEs yield a Lipschitz-continuous sparse coding map and recover the global feature support under mild assumptions. Empirically, across synthetic settings and LLMs (Pythia 70M, Llama 3.1 8B), AEN-SAEs mitigate feature starvation without auxiliary heuristics while maintaining competitive reconstruction abilities.

cs.LG↗

arXiv2026

Entropic Riemannian Neural Optimal Transport

Many machine learning problems involve data supported on curved spaces such as spheres, rotation groups, hyperbolic spaces, and general Riemannian manifolds, where Euclidean geometry can distort distances, averages, and the resulting optimal transport (OT) problem. Existing manifold OT methods have pursued amortized out-of-sample maps, while entropic regularization has made discrete OT more scalable, but these advantages have remained largely disjoint. We propose Entropic Riemannian Neural Optimal Transport (Entropic RNOT), a unified framework that combines intrinsic entropic OT with amortized out-of-sample evaluation on Riemannian manifolds. Our method learns a single target-side Schr\"odinger potential through a neural pullback parameterization, recovers the induced Gibbs coupling, and uses the resulting conditional laws to construct intrinsic transport surrogates. These include barycentric projections on Cartan-Hadamard manifolds and heat-smoothed conditional surrogates on stochastically complete manifolds, the latter turning possibly atomic target laws into absolutely continuous ones. For fixed regularization $\varepsilon>0$, we prove that the proposed hypothesis class recovers the entropic optimal coupling in strong probabilistic metrics. As consequences, barycentric surrogates converge in $L^2$, while heat-smoothed surrogates are stable at fixed heat time and asymptotically unbiased as the heat time vanishes. The guarantees hold for compactly supported data on possibly noncompact manifolds. Empirically, our method matches or improves over Euclidean, tangent-space, and log-Euclidean baselines on benchmarks over $\mathbb{S}^2$, $\mathrm{SO}(3)$, $\mathrm{SPD}(3)$, $\mathrm{SE}(3)$, and $\mathbb{H}^2$, scales favorably relative to discrete manifold Sinkhorn, and in a protein-ligand docking application, refines poses on $\mathrm{SE}(3)$ without retraining or per-instance optimization.

stat.ML↗

arXiv2026

Trajectory-Restricted Optimization Conditions and Geometry-Aware Linear Convergence

Linear convergence of first-order methods is typically characterized by global optimization conditions whose constants reflect worst-case geometry of the ambient space. In high-dimensional or structured problems, these global constants can be arbitrarily conservative and fail to capture the geometry actually encountered by optimization trajectories. In this paper, we develop a trajectory-restricted framework for linear convergence based on localized geometric regularity. We introduce restricted variants of the Polyak--{\L}ojasiewicz inequality, error bound, and quadratic growth conditions that are required to hold only on subsets of the domain. We show that classical convergence guarantees extend under these localized conditions, and in key cases, we develop new arguments that yield explicit relationships between the corresponding constants. The resulting rates are governed by geometric quantities associated with the regions traversed by the algorithm. For polyhedral composite problems, we prove that convergence is controlled by restricted Hoffman constants corresponding to the active polyhedral faces visited along the trajectory. Once the iterates enter a well-conditioned face, the effective condition number improves accordingly. Our work provides a geometric quantification for fast local convergence after active-set or manifold identification and more broadly suggests that linear convergence is fundamentally governed by the geometry of the subsets explored by the algorithm, rather than by worst-case global conditioning.

math.OC↗

arXiv2026

On the Uniqueness of Fr\'echet Means for Polytope Norms

Fr\'echet means are a popular type of average for non-Euclidean datasets, defined as those points which minimise the average squared distance to a set of data points. We consider the behaviour of sample Fr\'echet means on normed spaces whose unit ball is a polytope; this setting is rarely covered by existing literature on Fr\'echet means, which focuses on smooth spaces or spaces with bounded curvature. We study the geometry of the set of Fr\'echet means over polytope normed spaces, with a focus on dimension and probabilistic conditions for uniqueness. In particular, we provide a geometric characterisation of the threshold sample size at which Fr\'echet means have a positive probability of being unique, and we prove that this threshold is at most one more than the dimension of our space. We are able to use this geometric characterisation to compute the unique Fr\'echet mean sample threshold in the case of the $\ell_\infty$ and $\ell_1$ norms.

math.PR↗

arXiv2026

Breaking Symmetry Bottlenecks in GNN Readouts

Graph neural networks (GNNs) are widely used for learning on structured data, yet their ability to distinguish non-isomorphic graphs is fundamentally limited. These limitations are usually attributed to message passing; in this work we show that an independent bottleneck arises at the readout stage. Using finite-dimensional representation theory, we prove that all linear permutation-invariant readouts, including sum and mean pooling, factor through the Reynolds (group-averaging) operator and therefore project node embeddings onto the fixed subspace of the permutation action, erasing all non-trivial symmetry-aware components regardless of encoder expressivity. This yields both a new expressivity barrier and an interpretable characterization of what global pooling preserves or destroys. To overcome this collapse, we introduce projector-based invariant readouts that decompose node representations into symmetry-aware channels and summarize them with nonlinear invariant statistics, preserving permutation invariance while retaining information provably invisible to averaging. Empirically, swapping only the readout enables fixed encoders to separate WL-hard graph pairs and improves performance across multiple benchmarks, demonstrating that readout design is a decisive and under-appreciated factor in GNN expressivity.

cs.LG↗

arXiv2026

Riemannian Neural Optimal Transport

Computational optimal transport (OT) offers a principled framework for generative modeling. Neural OT methods, which use neural networks to learn an OT map (or potential) from data in an amortized way, can be evaluated out of sample after training, but existing approaches are tailored to Euclidean geometry. Extending neural OT to high-dimensional Riemannian manifolds remains an open challenge. In this paper, we prove that any method for OT on manifolds that produces discrete approximations of transport maps necessarily suffers from the curse of dimensionality: achieving a fixed accuracy requires a number of parameters that grows exponentially with the manifold dimension. Motivated by this limitation, we introduce Riemannian Neural OT (RNOT) maps, which are continuous neural-network parameterizations of OT maps on manifolds that avoid discretization and incorporate geometric structure by construction. Under mild regularity assumptions, we prove that RNOT maps approximate Riemannian OT maps with sub-exponential complexity in the dimension. Experiments on synthetic and real datasets demonstrate improved scalability and competitive performance relative to discretization-based baselines.

cs.LG↗

arXiv2025

Generalized Persistent Laplacians and their Spectral Properties

Laplacian operators are classical objects that are fundamental in both pure and applied mathematics and are becoming increasingly prominent in modern computational and data science fields such as applied and computational topology and application areas such as machine learning and network science. In this paper, we introduce a unifying operator-theoretic framework of generalized Laplacians as invariants that encompasses and extends all existing constructions, from discrete combinatorial settings to de Rham complexes of smooth manifolds. Within this framework, we introduce and study a generalized notion of persistent Laplacians. While the classical persistent Laplacian fails to satisfy the desirable properties of monotonicity and stability -- both crucial for robustness and interpretability -- our framework allows us to isolate and analyze these properties systematically. We demonstrate that their component maps, the up- and down-persistent Laplacians, satisfy these properties individually. Moreover, we prove that the spectra of these separate components fully determine the spectra of the full Laplacians, making them not only preferable but sufficient for analysis. We study these questions comprehensively, in both the finite and infinite dimensional settings. Our work expands and strengthens the theoretical foundation of generalized Laplacian-based methods in pure, applied, and computational mathematics.

math.AT↗

arXiv2025

The Shape of Adversarial Influence: Characterizing LLM Latent Spaces with Persistent Homology

Existing interpretability methods for Large Language Models (LLMs) predominantly capture linear directions or isolated features. This overlooks the high-dimensional, relational, and nonlinear geometry of model representations. We apply persistent homology (PH) to characterize how adversarial inputs reshape the geometry and topology of internal representation spaces of LLMs. This phenomenon, especially when considered across operationally different attack modes, remains poorly understood. We analyze six models (3.8B to 70B parameters) under two distinct attacks, indirect prompt injection and backdoor fine--tuning, and show that a consistent topological signature persists throughout. Adversarial inputs induce topological compression, where the latent space becomes structurally simpler, collapsing the latent space from varied, compact, small-scale features into fewer, dominant, large-scale ones. This signature is architecture-agnostic, emerges early in the network, and is highly discriminative across layers. By quantifying the shape of activation point clouds and neuron-level information flow, our framework reveals geometric invariants of representational change that complement existing linear interpretability methods.

cs.LG↗

arXiv2025

Metric Graph Kernels via the Tropical Torelli Map

We introduce the first graph kernels for metric graphs via tropical algebraic geometry. In contrast to conventional graph kernels based on graph combinatorics such as nodes, edges, and subgraphs, our metric graph kernels are purely based on the geometry and topology of the underlying metric space. A key characterizing property of our construction is its invariance under edge subdivision, making the kernels intrinsically well-suited for comparing graphs representing different underlying metric spaces. We develop efficient algorithms to compute our kernels and analyze their complexity, which depends primarily on the genus of the input graphs rather than their size. Through experiments on synthetic data and selected real-world datasets, we demonstrate that our kernels capture complementary geometric and topological information overseen by standard combinatorial approaches, particularly in label-free settings. We further showcase their practical utility with an urban road network classification task.

cs.LG↗

arXiv2025

Computing the Tropical Abel--Jacobi Transform and Tropical Distances for Metric Graphs

Metric graphs are important models for capturing the structure of complex data across various domains. While much effort has been devoted to extracting geometric and topological features from graph data, computational aspects of metric graphs as abstract tropical curves remains unexplored. In this paper, we present the first computational and machine learning-driven study of metric graphs from the perspective of tropical algebraic geometry. Specifically, we study the tropical Abel--Jacobi transform, a vectorization of points on a metric graph via the tropical Abel--Jacobi map into its associated flat torus, the tropical Jacobian. We develop algorithms to compute this transform and investigate how the resulting embeddings depend on different combinatorial models of the same metric graph. Once embedded, we compute pairwise distances between points in the tropical Jacobian under two natural metrics: the tropical polarization distance and the Foster--Zhang distance. Computing these distances are generally NP-hard as they turn out to be linked to classical lattice problems in computational complexity, however, we identify a class of metric graphs where fast and explicit computations are feasible. For the general case, we propose practical algorithms for both exact and approximate distance matrix computations using lattice basis reduction and mixed-integer programming solvers. Our work lays the groundwork for future applications of tropical geometry and the tropical Abel--Jacobi transform in machine learning and data analysis.

math.AG↗

arXiv2025

Confidence Bands for Multiparameter Persistence Landscapes

Multiparameter persistent homology is a generalization of classical persistent homology, a central and widely-used methodology from topological data analysis, which takes into account density estimation and is an effective tool for data analysis in the presence of noise. Similar to its classical single-parameter counterpart, however, it is challenging to compute and use in practice due to its complex algebraic construction. In this paper, we study a popular and tractable invariant for multiparameter persistent homology in a statistical setting: the multiparameter persistence landscape. We derive a functional central limit theorem for multiparameter persistence landscapes, from which we compute confidence bands, giving rise to one of the first statistical inference methodologies for multiparameter persistence landscapes. We provide an implementation of confidence bands and demonstrate their application in a machine learning task on synthetic data.

math.ST↗

arXiv2025

Tropical Fr\'echet Means: a polyhedral approach to exact optimization

The Fr\'{e}chet mean is a fundamental notion of central tendency defined as a minimizer of a sum of squared distances in a general metric space. In this paper, we study Fr\'{e}chet means in tropical geometry -- a piecewise linear, combinatorial, and polyhedral variant of algebraic geometry -- by formulating and solving the associated tropical quadratic optimization problem. We give a geometric characterization of the collection of all tropical Fr\'{e}chet means as a bounded set that is simultaneously tropically and classically convex, hence a polytrope. We establish the existence of positivity certificates for maxima of finitely many quadratic polynomials in $\mathbb{R}[x_1,\ldots,x_n]$ whose homogeneous quadratic components are sums of squares, which provides a symbolic framework for exact optimization. Using this structure, we develop algorithms for computing tropical Fr\'{e}chet means and the associated Fr\'{e}chet mean polytrope. We further describe a combinatorial type decomposition of the objective function induced by braid arrangements, yielding a piecewise quadratic representation and a fully symbolic method for exact computation.

math.OC↗

arXiv2024

A Topological Gaussian Mixture Model for Bone Marrow Morphology in Leukaemia

Acute myeloid leukaemia (AML) is a type of blood and bone marrow cancer characterized by the proliferation of abnormal clonal haematopoietic cells in the bone marrow leading to bone marrow failure. Over the course of the disease, angiogenic factors released by leukaemic cells drastically alter the bone marrow vascular niches resulting in observable structural abnormalities. We use a technique from topological data analysis - persistent homology - to quantify the images and infer on the disease through the imaged morphological features. We find that persistent homology uncovers succinct dissimilarities between the control, early, and late stages of AML development. We then integrate persistent homology into stage-dependent Gaussian mixture models for the first time, proposing a new class of models which are applicable to persistent homology summaries and able to both infer patterns in morphological changes between different stages of progression as well as provide a basis for prediction.

stat.AP↗