arXiv subjects

Darrick Lee

Publications and source records attributed to Darrick Lee.

At least 19 recordsLinked to original sources

arXiv2026

Interpreting Latent Protein Language Model Features with Geometric Annotations

Protein language models (pLMs) encode information about protein sequences which enable downstream tasks such as structure prediction, but their internal representations are not well understood. Sparse autoencoders (SAEs) provide a promising tool to disentangle latent pLM representations into interpretable features, but existing annotation pipelines largely rely on protein-level annotations derived from database labels and LLM annotations of top activating sequences. Such annotations can overlook the localized residue-level and geometric patterns encoded by sparse features. We introduce an automated and scalable method for interpreting SAE features in ESM-2 by using geometrically inspired features of the protein $\text{C}_\alpha$ backbone. Across ESM-2 8M layers, an FDR-controlled discovery analysis shows that local geometry is significantly associated with many SAE features, with varying levels of predictive strength, expanding coverage beyond database and sequence-based methods. In particular, geometry can distinguish SAE features sharing the same database annotation, revealing substructure within known biological labels. A significant portion of SAE features activate on unannotated metagenomic protein sequences enabling us to use our SAE annotations to better understand these sequences. In addition, ablation experiments at the level of contact prediction show that removing found geometric features shifts ESM-2's predicted contact maps in the direction of the descriptor. This provides a robust method of annotating proteins activated within SAE neurons at a residue level, providing a bridge between mechanistic interpretability and structural biology.

q-bio.QM↗

arXiv2026

The Geometry of Cochains on Sampled Vietoris-Rips Complexes

We study the geometry of simplicial cochains on Vietoris-Rips complexes built from i.i.d. samples of a compact embedded manifold. By integrating differential forms over affine simplices, we define a cochain map from smooth forms to simplicial cochains and equip the latter with kernel-weighted inner products determined by ambient pairwise distances. At scales where the sampled complex recovers the homotopy type of the manifold, we prove quantitative high-probability convergence of these inner products and, under uniform sampling, of the associated codifferential energies to their continuum counterparts. As consequences, we obtain spectral upper bounds for the discrete Hodge Laplacian and, for orientable manifolds, convergence of harmonic representatives of discretized harmonic one-forms and consistency of harmonic smoothing for circular coordinates.

math.DG↗

arXiv2026

Asymptotically Optimal Ergodic Coverage on Generalized Motion Fields

Autonomous robotic exploration in remote and extreme environments allows scientists to model complex transport phenomena and collective behaviors described by continuously deforming flow fields. Although these environments are naturally modeled as time-varying domains, most adaptive exploration methods assume static environments and fail to provide adequate coverage or satisfy any formal guarantees. This is especially the case in oceanography where autonomous underwater systems (UxS) have highly restrictive compute and payload requirements that necessitate path planning methods that yield robust data collection strategies in open-loop and underactuated settings. In this work, to address the aforementioned issues, we propose to formulate adaptive search as an ergodic coverage problem and investigate certifying coverage in the ergodic sense over evolving domains with flow-induced dynamics. We expand upon recent work demonstrating maximum mean discrepancy (MMD) as a functional ergodic metric, and derive a flow-adaptive formulation that explicitly accounts for domain evolution within the coverage objective. We show that this approach preserves ergodic coverage guarantees in ambient flows and enables effective exploration in under-actuated, and even open-loop planning settings by integrating environment dynamics. Experiments validate that our method generalizes to diverse spatiotemporal processes including ocean exploration, and tracking human and cattle movement. Physical experiments on aerial and legged robotic platforms validate our ability to obtain ergodic coverage in non-convex, flow-restricted environments while respecting robot dynamics.

cs.RO↗

arXiv2026

Orthogonal polynomials on path-space

We consider the orthogonalisation of the signature of a stochastic process as the analogue of orthogonal polynomials on path-space. Under an infinite radius of convergence assumption, we prove density of linear functions on the signature in $L^p$ functions on grouplike elements, making it possible to represent a square-integrable function on (rough) paths as an $L^2$-convergent series. By viewing the shuffle algebra as commutative polynomials on the free Lie algebra, we revisit much of the theory of classical orthogonal polynomials in several variables, such as the recurrence relation and Favard's theorem. Finally, we restrict our attention to the case of Brownian motion with and without drift, and prove that dimension-independent orthogonal signature exists with drift but not without. We end with numerical examples of how orthogonal signature polynomials of Brownian motion can be applied for the approximation of functions on paths sampled from the Wiener measure.

math.PR↗

arXiv2025

CTransformer: Deep-transformer-based 3D cell membrane tracking with subcellular-resolved molecular quantification

Deep learning segmentation and fluorescence imaging techniques allow the cellular morphology of living embryos to be constructed spatiotemporally. These development processes involve numerous molecules distributed at the subcellular scale, such as cell adhesion (E-cadherin), which accumulate at cell-cell interfaces to regulate intercellular connection. However, quantifying molecular distributions within specific subcellular regions across the entire embryo, where cell movement and molecular redistribution occur rapidly, is challenging due to the need for simultaneous cell morphology reconstruction and lineage tracing due to photobleaching and phototoxicity. We report a transformer-based pipeline, CTransformer, that establishes a 4D cellular morphology map before the 550-cell (late) stage. CTransformer constructed 4D cellular morphology atlases, reaching 80% accuracy at the 550-cell stage. Through this advanced architecture, we use only one channel to reconstruct cell morphology and achieve cell tracing. With each cell's morphology as a reference, the distribution of specific molecules throughout the cell body and at cell interfaces can be quantitatively measured in another fluorescence channel. We apply this methodology to track E-cadherin during embryonic development of the worm Caenorhabditis elegans, from fertilization to gastrulation. Our results reveal that E-cadherin is tightly regulated across individual embryos, both within single cells and at cell-cell interfaces, displaying an anterior-posterior gradient and cell- and lineage-specific patterns. Furthermore, its spatiotemporal heterogeneity influences cell mechanics and embryonic morphogenesis, helping explain how C. elegans achieves stereotypical developmental patterns at cellular resolution.

physics.bio-ph↗

arXiv2025

Thin homotopy and the signature of piecewise linear surfaces

We introduce a crossed module of piecewise linear surfaces and study the signature homomorphism, defined as the surface holonomy of a universal translation invariant $2$-connection. This provides a transform whereby surfaces are represented by formal series of tensors. Our main result is that the signature uniquely characterizes a surface up to translation and thin homotopy, also known as tree-like equivalence in the case of paths. This generalizes a result of Chen and positively answers a question of Kapranov in the setting of piecewise linear surfaces. As part of this work, we provide several equivalent definitions of thin homotopy, generalizing the plethora of definitions which exist in the case of paths. Furthermore, we develop methods for explicitly and efficiently computing the surface signature.

math.AT↗

arXiv2025

Path-Dependent SDEs: Solutions and Parameter Estimation

We develop a consistent method for estimating the parameters of a rich class of path-dependent SDEs, called signature SDEs, which can model general path-dependent phenomena. Path signatures are iterated integrals of a given path with the property that any sufficiently nice function of the path can be approximated by a linear functional of its signatures. This is why we model the drift and diffusion of our signature SDE as linear functions of path signatures. We provide conditions that ensure the existence and uniqueness of solutions to a general signature SDE. We then introduce the Expected Signature Matching Method (ESMM) for linear signature SDEs, which enables inference of the signature-dependent drift and diffusion coefficients from observed trajectories. Furthermore, we prove that ESMM is consistent: given sufficiently many samples and Picard iterations used by the method, the parameters estimated by the ESMM approach the true parameter with arbitrary precision. Finally, we demonstrate on a variety of empirical simulations that our ESMM accurately infers the drift and diffusion parameters from observed trajectories. While parameter estimation is often restricted by the need for a suitable parametric model, this work makes progress toward a completely general framework for SDE parameter estimation, using signature terms to model arbitrary path-independent and path-dependent processes.

math.ST↗

arXiv2025

Communities in the Kuramoto Model: Dynamics and Detection via Path Signatures

The behavior of multivariate dynamical processes is often governed by underlying structural connections that relate the components of the system. For example, brain activity, which is often measured via time series is determined by an underlying structural graph, where nodes represent neurons or brain regions and edges cortical connectivity. Existing methods for inferring structural connections from observed dynamics, such as correlation-based or spectral techniques, may fail to fully capture complex relationships in high-dimensional time series in an interpretable way. Here, we propose the use of path signatures, a mathematical framework that encodes geometric and temporal properties of continuous paths, to address this problem. Path signatures provide a reparametrization-invariant characterization of dynamical data and can be used to compute the lead matrix, which reveals lead-lag phenomena. We showcase our approach on time series from coupled oscillators in the Kuramoto model defined on a stochastic block model graph, termed the Kuramoto Stochastic Block Model (KSBM). Using mean-field theory and Gaussian approximations, we analytically derive reduced models of KSBM dynamics in different temporal regimes and theoretically characterize the lead matrix in these settings. Leveraging these insights, we propose a novel signature-based community detection algorithm, achieving exact recovery of structural communities from observed time series in multiple KSBM instances. We also explored the performance of our community detection on a stochastic variant of the KSBM as well as on real neuropixels of cortical recordings to demonstrate applicability on real-world data. Our results demonstrate that path signatures provide a novel perspective on analyzing complex neural data and other high-dimensional systems, explicitly exploiting temporal functional relationships to infer underlying structure.

stat.ML↗

arXiv2025

Towards Scalable Topological Regularizers

Latent space matching, which consists of matching distributions of features in latent space, is a crucial component for tasks such as adversarial attacks and defenses, domain adaptation, and generative modelling. Metrics for probability measures, such as Wasserstein and maximum mean discrepancy, are commonly used to quantify the differences between such distributions. However, these are often costly to compute, or do not appropriately take the geometric and topological features of the distributions into consideration. Persistent homology is a tool from topological data analysis which quantifies the multi-scale topological structure of point clouds, and has recently been used as a topological regularizer in learning tasks. However, computation costs preclude larger scale computations, and discontinuities in the gradient lead to unstable training behavior such as in adversarial tasks. We propose the use of principal persistence measures, based on computing the persistent homology of a large number of small subsamples, as a topological regularizer. We provide a parallelized GPU implementation of this regularizer, and prove that gradients are continuous for smooth densities. Furthermore, we demonstrate the efficacy of this regularizer on shape matching, image generation, and semi-supervised learning tasks, opening the door towards a scalable regularizer for topological features.

cs.LG↗

arXiv2024

Ergodic Trajectory Optimization on Generalized Domains Using Maximum Mean Discrepancy

We present a novel formulation of ergodic trajectory optimization that can be specified over general domains using kernel maximum mean discrepancy. Ergodic trajectory optimization is an effective approach that generates coverage paths for problems related to robotic inspection, information gathering problems, and search and rescue. These optimization schemes compel the robot to spend time in a region proportional to the expected utility of visiting that region. Current methods for ergodic trajectory optimization rely on domain-specific knowledge, e.g., a defined utility map, and well-defined spatial basis functions to produce ergodic trajectories. Here, we present a generalization of ergodic trajectory optimization based on maximum mean discrepancy that requires only samples from the search domain. We demonstrate the ability of our approach to produce coverage trajectories on a variety of problem domains including robotic inspection of objects with differential kinematics constraints and on Lie groups without having access to domain specific knowledge. Furthermore, we show favorable computational scaling compared to existing state-of-the-art methods for ergodic trajectory optimization with a trade-off between domain specific knowledge and computational scaling, thus extending the versatility of ergodic coverage on a wider application domain.

cs.RO↗

arXiv2024

The Surface Signature and Rough Surfaces

Parallel transport, or path development, provides a rich characterization of paths which preserves the underlying algebraic structure of concatenation. The path signature is universal among such maps: any (translation-invariant) parallel transport factors uniquely through the path signature. Furthermore, the path signature is a central object in the theory of rough paths, which provides an integration theory for highly irregular paths. A fundamental result is Lyons' extension theorem, which allows us to compute the signature of rough paths, and in turn provides a way to compute parallel transport of arbitrarily irregular paths. In this article, we consider the notion of surface holonomy, a generalization of parallel transport to the higher dimensional setting of surfaces parametrized by rectangular domains, which preserves the higher algebraic structures of horizontal and vertical concatenation. Building on work of Kapranov, we introduce the surface signature, which is universal among surface holonomy maps with respect to continuous 2-connections. Furthermore, we introduce the notion of a rough surface and prove a surface extension theorem, which allows us to compute the signature of rough surfaces. By exploiting the universal property of the surface signature, this provides a method to compute the surface holonomy of arbitrarily irregular surfaces.

math.FA↗

arXiv2024

Stein Variational Ergodic Search

Exploration requires that robots reason about numerous ways to cover a space in response to dynamically changing conditions. However, in continuous domains there are potentially infinitely many options for robots to explore which can prove computationally challenging. How then should a robot efficiently optimize and choose exploration strategies to adopt? In this work, we explore this question through the use of variational inference to efficiently solve for distributions of coverage trajectories. Our approach leverages ergodic search methods to optimize coverage trajectories in continuous time and space. In order to reason about distributions of trajectories, we formulate ergodic search as a probabilistic inference problem. We propose to leverage Stein variational methods to approximate a posterior distribution over ergodic trajectories through parallel computation. As a result, it becomes possible to efficiently optimize distributions of feasible coverage trajectories for which robots can adapt exploration. We demonstrate that the proposed Stein variational ergodic search approach facilitates efficient identification of multiple coverage strategies and show online adaptation in a model-predictive control formulation. Simulated and physical experiments demonstrate adaptability and diversity in exploration strategies online.

cs.RO↗

arXiv2023

Random Surfaces and Higher Algebra

We introduce a characteristic function for laws of random surfaces $\mathbf{X}: [0,s] \times [0,t] \to \mathbb{R}^d$, in the spirit of expected path developments for one-dimensional stochastic processes into matrix groups. A key property is that path development is structure preserving: path concatenation becomes matrix multiplication. The main challenge is to account for two distinct concatenation operations for surfaces: horizontal and vertical. To address this, we use the notion of surface holonomy from higher geometry to define surface developments, and study this in a stochastic context. We generalize surface developments to the Young setting of $\rho$-H\"older surfaces, where $\rho > \frac12$, show that such developments characterize parametrized surfaces. Our main result shows that the resulting expected surface development provides a computable and structured description of laws of random surfaces and leads to a natural metric on the space of probability measures on surfaces.

math.PR↗

arXiv2023

The Signature Kernel

The signature kernel is a positive definite kernel for sequential data. It inherits theoretical guarantees from stochastic analysis, has efficient algorithms for computation, and shows strong empirical performance. In this short survey paper for a forthcoming Springer handbook, we give an elementary introduction to the signature kernel and highlight these theoretical and computational properties.

math.PR↗

arXiv2023

Convex Hulls of Curves: Volumes and Signatures

Taking the convex hull of a curve is a natural construction in computational geometry. On the other hand, path signatures, central in stochastic analysis, capture geometric properties of curves, although their exact interpretation for levels larger than two is not well understood. In this paper, we study the use of path signatures to compute the volume of the convex hull of a curve. We present sufficient conditions for a curve so that the volume of its convex hull can be computed by such formulae. The canonical example is the classical moment curve, and our class of curves, which we call cyclic, includes other known classes such as $d$-order curves and curves with totally positive torsion. We also conjecture a necessary and sufficient condition on curves for the signature volume formula to hold. Finally, we give a concrete geometric interpretation of the volume formula in terms of lengths and signed areas.

math.MG↗

arXiv2022

Capturing Graphs with Hypo-Elliptic Diffusions

Convolutional layers within graph neural networks operate by aggregating information about local neighbourhood structures; one common way to encode such substructures is through random walks. The distribution of these random walks evolves according to a diffusion equation defined using the graph Laplacian. We extend this approach by leveraging classic mathematical results about hypo-elliptic diffusions. This results in a novel tensor-valued graph operator, which we call the hypo-elliptic graph Laplacian. We provide theoretical guarantees and efficient low-rank approximation algorithms. In particular, this gives a structured approach to capture long-range dependencies on graphs that is robust to pooling. Besides the attractive theoretical properties, our experiments show that this method competes with graph transformers on datasets requiring long-range reasoning but scales only linearly in the number of edges as opposed to quadratically in nodes.

cs.LG↗

arXiv2022

A Topological Approach to Mapping Space Signatures

A common approach for describing classes of functions and probability measures on a topological space $\mathcal{X}$ is to construct a suitable map $\Phi$ from $\mathcal{X}$ into a vector space, where linear methods can be applied to address both problems. The case where $\mathcal{X}$ is a space of paths $[0,1] \to \mathbb{R}^n$ and $\Phi$ is the path signature map has received much attention in stochastic analysis and related fields. In this article we develop a generalized $\Phi$ for the case where $\mathcal{X}$ is a space of maps $[0,1]^d \to \mathbb{R}^n$ for any $d \in \mathbb{N}$, and show that the map $\Phi$ generalizes many of the desirable algebraic and analytic properties of the path signature to $d \ge 2$. The key ingredient to our approach is topological; in particular, our starting point is a generalisation of K-T Chen's path space cochain construction to the setting of cubical mapping spaces.

math.FA↗

arXiv2021

Signatures, Lipschitz-free spaces, and paths of persistence diagrams

Paths of persistence diagrams provide a summary of the dynamic topological structure of a one-parameter family of metric spaces. These summaries can be used to study and characterize the dynamic shape of data such as swarming behavior in multi-agent systems, time-varying fMRI scans from neuroscience, and time-dependent scalar fields in hydrodynamics. While persistence diagrams can provide a powerful topological summary of data, the standard space of persistence diagrams lacks the sufficient algebraic and analytic structure required for many theoretical and computational analyses. We enrich the space of persistence diagrams by isometrically embedding it into a Lipschitz-free space, a Banach space built from a universal construction. We utilize the Banach space structure to define bounded variation paths of persistence diagrams, which can be studied using the path signature, a reparametrization-invariant characterization of paths valued in a Banach space. The signature is universal and characteristic which allows us to theoretically characterize measures on the space of paths and motivates its use in the context of kernel methods. However, kernel methods often require a feature map into a Hilbert space, so we introduce the moment map, a stable and injective feature map for static persistence diagrams, and compose it with the discrete path signature, producing a computable feature map into a Hilbert space. Finally, we demonstrate the efficacy of our methods by applying this to a parameter estimation problem for a 3D model of swarming behavior.

math.AT↗