. Let F be a probability distribution on Rd which admits a bounded density. We investigate the Euler characteristic of the Cech complex on n points sampled from F i.i.d. as n -> infinity in the thermodynamic limit regime. As a main result, we identify a condition for two probability distributions to yield the same expected Euler characteristic under this construction. Namely, this happens if and only if their densities admit the same excess mass transform. Building on work of Bobrowski, we establish a connection between the limiting expected Euler characteristic of any such probability distribution F and the one of the uniform distribution on [0, 1]d through an integral transform. Our approach relies on constructive proofs, offering explicit calculations of expected Euler characteristics in lower dimensions as well as reconstruction of a distribution from its limiting Euler characteristic. In the context of topological data analysis, where the Euler characteristic serves as a summary of the shape of data, we address the inverse problem and determine what can be discriminated using this invariant. This research sheds light on the relationship between a probability distribution and topological properties of the Cech complex on its samples in the thermodynamic limit.
Algebraic topology has proved to be a very useful tool in addressing the computability of rendezvous problems in distributed computing. To date, the tools used include the fundamental group and ordinary homology. This paper introduces more sophisticated techniques from homotopy theory involving the Hilton-Milnor theorem in order to analyze whether rendezvous problems are solvable. A criterion is proved that ensures a solution given certain hypotheses. Families of examples are given when the hypotheses are satisfied and when they are not satisfied; these include new cases when homology alone suffices to ensure a solution, and cases when homology needs to be supplemented by additional properties to obtain a solution.
. Nonlinear dynamical systems are ubiquitous in various scientific areas. The Koopman operator theory provides a way to analyze nonlinear dynamical systems with a linear operator, whose spectra contain rich information in understanding the behaviors of dynamical systems. In this paper, we aim to use a data-driven method to construct the principal eigenfunctions of the Koopman operator. Our approach relies on the decomposition of the Ko opman eigenfunction where the linear part associates with the left eigenvectors of the linearization of the system at the equilibrium, and the nonlinear part satisfies a linear PDE. Then, we solve the linear PDE using a random features model, which is a data-driven machine learning technique for large-scale kernel approximation. In addition, the random features model can be viewed as a randomized shallow neural network where only the linear readout needs to be trained, which significantly reduces the computational complexity. We provide a computational framework for the approximation of Koopman eigenfunctions and an error analysis for our proposed method. Finally, we present simulation results for learning the principal eigenfunctions and constructing the Lyapunov function using learned principal eigenfunctions. Numerical experiments indicate that our method is fast and accurate. As such, it can serve as a benchmark for approximating Koopman eigenfunctions.
. Tensors, or n-way arrays, store data that can be indexed along an arbitrary number of dimensions. Tensor analysis using decomposition techniques can reveal latent factors and useful information about the underlying data. Tensors can grow extremely large, often containing millions of entries, and may be sparse, with the majority of the values equal to zero, making explicit storage impractical. Interest in tensor analysis using decomposition has expanded across diverse fields, including data mining, signal processing, computer vision, and machine learning. However, few sparse tensor formats provide an efficient method for locating arbitrary non-zero values. A motivating example comes from Lowe's textual influence model (2018), which applied non-negative sparse tensor decomposition to measure authors' contributions. Building the tensor required counting n-grams in a sliding window, necessitating frequent updates and insertions. Lowe discovered the need for a sparse tensor format that allowed rapid updates and insertions without regenerating the entire structure. We recognized the potential for a new sparse tensor storage format for applications that require frequent and rapid tensor lookup and insertion. This observation motivated the proposal of Hashed Coordinate Storage (HaCOO) in 2021, a mode-agnostic sparse tensor format that stores indexes and values in a separate chaining hash table, allowing constanttime insertion and access to arbitrary entries. HaCOO also balances storage efficiency by adopting ALTO's adaptive encoding scheme, which encodes index metadata using the minimum number of bits while supporting rapid structural updates.
We propose a multi-dimensional persistent sheaf Laplacian (MPSL) framework on simplicial complexes for image analysis. The proposed method is motivated by the strong sensitivity of commonly used dimensionality reduction techniques, such as principal component analysis (PCA), to the choice of reduced dimension. Rather than selecting a single reduced dimension or averaging results across dimensions, we exploit complementary advantages of multiple reduced dimensions. At a given dimension, image samples are regarded as simplicial complexes, and persistent sheaf Laplacians are utilized to extract a multiscale localized topological spectral representation for individual image samples. Statistical summaries of the resulting spectra are then aggregated across scales and dimensions to form multiscale multi-dimensional image representations. We evaluate the proposed framework on the COIL20 and ETH80 image datasets using standard classification protocols. Experimental results show that the proposed method provides more stable performance across a wide range of reduced dimensions and achieves consistent improvements to PCA-based baselines in moderate dimensional regimes.
Modern business and economic datasets often exhibit nonlinear, multi-scale structures that traditional linear tools under-represent. Topological Data Analysis (TDA) offers a geometric lens for uncovering robust patterns, such as connected components, loops and voids, across scales. This paper provides an intuitive, figure-driven introduction to persistent homology and a practical, reproducible TDA pipeline for applied analysts. Through comparative case studies in consumer behavior, equity markets (SAX/eSAX vs. TDA) and foreign exchange dynamics, we demonstrate how topological features can reveal segmentation patterns and structural relationships beyond classical statistical methods. We discuss methodological choices regarding distance metrics, complex construction and interpretation, and we introduce the Topological Stability Index (TSI), a simple yet interpretable indicator of structural variability derived from persistence lifetimes. We conclude with practical guidelines for TDA implementation, visualization and communication in business and economic analytics.
While persistent homology (PH) is widely used for data shape analysis, persistent commutative algebra (PCA) has seen limited adoption in machine learning (ML) and data science. Unlike PH, which delivers topological invariants in the form of Betti numbers, PCA provides both algebraic invariants and graded Betti numbers. In this work, we introduce a first-of-its-kind application of commutative algebra graded Betti numbers in ML and data science. Specifically, we present Graded Betti Number Learning (GBNL) for protein-nucleic acid binding prediction. Protein-DNA/RNA interactions are fundamental to cellular processes such as replication, transcription, translation, and gene regulation, however their understanding and prediction remain challenging. GBNL represents each nucleic acid sequence as a family of k-merspecific sets and derives persistent graded Betti invariants from PCA, generating multiscale topological representations of local nucleotide organization. To incorporate cross-molecule context, these graded Betti representations are paired with transformer-based protein embeddings, linking nucleotide-level signals with global protein patterns. The proposed framework effectively detects single-site mutations and distinguish complete mutation patterns. Operating on primary sequences with minimal preprocessing, GBNL bridges commutative algebra, reduced algebraic topology, combinatorics, and ML, establishing a new paradigm for comparative sequence analysis. Numerical studies using three datasets highlight the success of GBNL in protein-nucleic acid binding prediction.
. The computationally cheap machine learning architecture of random feature maps can be viewed as a single-layer feedforward network in which the weights of the hidden layer are random but fixed and only the outer weights are learned via linear regression. The internal weights are typically chosen from a prescribed distribution. The choice of the internal weights significantly impacts the accuracy of random feature maps. We address here the task of how to best select the internal weights. In particular, we consider the forecasting problem where random feature maps are used to learn a one-step propagator map for a dynamical system. We provide a computationally cheap hit-andrun algorithm to select good internal weights which lead to good forecasting skill. We show that the number of good features is the main factor controlling the forecasting skill of random feature maps and acts as an effective feature dimension. Lastly, we compare random feature maps with single-layer feedforward neural networks in which the internal weights are now learned using gradient descent. We find that random feature maps have superior forecasting capabilities whilst having several orders of magnitude lower computational cost.
Across scientific domains, a fundamental challenge is to characterize and compute the mappings from underlying physical processes to observed signals and measurements. While nonlinear neural networks have achieved considerable success, they remain theoretically opaque, which hinders adoption in contexts where interpretability is paramount. In contrast, linear neural networks serve as a simple yet effective foundation for gaining insight into these complex relationships. In this work, we develop a unified theoretical framework for analyzing linear encoder-decoder architectures through the lens of Bayes risk minimization for solving data-driven scientific machine learning problems. We derive closed-form, rank-constrained linear and affine linear optimal mappings for forward modeling and inverse recovery tasks. Our results generalize existing formulations by accommodating rank-deficiencies in data, forward operators, and measurement processes. We validate our theoretical results by conducting numerical experiments on datasets from simple biomedical imaging, financial factor analysis, and simulations involving nonlinear fluid dynamics via the shallow water equations. This work provides a robust baseline for understanding and benchmarking learned neural network models for scientific machine learning problems.
. We develop a novel stability theory for Sinkhorn semigroups based on Lyapunov techniques and quantitative contraction coefficients, and establish exponential convergence of Sinkhorn iterations on weighted Banach spaces. This operator-theoretic framework yields explicit exponential decay rates of Sinkhorn iterates toward Schrodinger bridges with respect to a broad class of phi-divergences and Kantorovich-type distances, including relative entropy, squared Hellinger integrals, alpha-divergences, weighted total variation norms, and Wasserstein distances. To the best of our knowledge, these results provide the first systematic contraction inequalities of this kind for entropic transport and the Sinkhorn algorithm. We further introduce Lyapunov contraction principles under minimal regularity assumptions, leading to quantitative exponential stability estimates for a large family of Sinkhorn semigroups. The framework applies to models with polynomially growing potentials and heavy-tailed marginals on general normed spaces, as well as to more structured boundary state-space models, including semicircle transitions and Beta, Weibull, and exponential marginals, together with semi-compact settings. Finally, our approach extends naturally to statistical finite mixtures of such models, including kernel-based density estimators arising in modern generative modeling.
Directed graphs arise in many applications where computing persistent homology helps to encode the shape and structure of the input information. However, there are only a few ways to turn the directed graph information into an undirected simplicial complex filtration required by the standard persistent homology framework. In this paper, we present a new filtration constructed from a directed graph, called the walk-length filtration. This filtration mirrors the behavior of small walks visiting certain collections of vertices in the directed graph. We show that, while the persistence is not stable under the usual L_∞-style network distance, a generalized L_1-style distance is, indeed, stable. We further provide an algorithm for its computation, and investigate the behavior of this filtration in examples, including cycle networks and synthetic hippocampal networks with a focus on comparison to the often used Dowker filtration.
Gaussian processes (GPs) are crucial in machine learning for quantifying uncertainty in predictions. However, their associated covariance matrices, defined by kernel functions, are typically dense and large-scale, posing significant computational challenges. This paper introduces a matrix-free method that utilizes the Non-equispaced Fast Fourier Transform (NFFT) to achieve nearly linear complexity in the multiplication of kernel matrices and their derivatives with vectors for a predetermined accuracy level. To address high-dimensional problems, we propose an additive kernel approach. Each sub-kernel in this approach captures lower-order feature interactions, allowing for the efficient application of the NFFT method and potentially increasing accuracy across various real-world datasets. Additionally, we implement a preconditioning strategy that accelerates hyperparameter tuning, further improving the efficiency and effectiveness of GPs.
Topological data analysis (TDA) has emerged as an effective approach in data science, with its key technique, persistent homology, rooted in algebraic topology. Although alternative approaches based on differential topology, geometric topology, and combinatorial Laplacians have been proposed, combinatorial commutative algebra has hardly been developed for machine learning and data science. In this work, we introduce persistent Stanley-Reisner theory to bridge commutative algebra, combinatorial algebraic topology, machine learning, and data science. We propose persistent h-vectors, persistent f-vectors, persistent graded Betti numbers, persistent facet ideals, and facet persistence modules. Stability analysis indicates that these algebraic invariants are stable against geometric perturbations. We employ a machine learning prediction on a molecular dataset to demonstrate the utility of the proposed persistent Stanley-Reisner theory for practical applications.
In this article we propose and develop a new methodology which is inspired from Kalman filtering and multilevel Monte Carlo (MLMC), entitle the multilevel localized ensemble Kalman–Bucy Filter (MLLEnKBF). Based on the work of Chada et al. , we provide an important extension on this which is to include the technique of covariance localization. Localization is important as it can induce stability and remove long spurious correlations, particularly with a small ensemble size. Our resulting algorithm is used for both state and parameter estimation, for the later we exploit our method for normalizing constant estimation. As of yet, MLMC has only been applied to localized data assimilation methods in a discrete-time setting, therefore this work acts as a first in the continuous-time setting. Numerical results indicate its performance, and benefit through a range of model problems, which include a linear Ornstein–Uhlenbeck process, of moderately high dimension, and the Lorenz 96 model, for parameter estimation. Our results demonstrate improved stability, and that with MLMC, one can reduce the computational complexity to attain an order is MSE 𝒪(ϵ^2), for ϵ>0.
. Entropic optimal transport problems are regularized versions of optimal transport problems. These models play an increasingly important role in machine learning and generative modelling. For finite spaces, these problems are commonly solved using Sinkhorn algorithm (a.k.a. iterative proportional fitting procedure). However, in more general settings the Sinkhorn iterations are based on nonlinear conditional/conjugate transformations and exact finitedimensional solutions cannot be computed. This article presents a finite-dimensional recursive formulation of the iterative proportional fitting procedure for general Gaussian multivariate models. As expected, this recursive formulation is closely related to the celebrated Kalman filter and related Riccati matrix difference equations, and it yields algorithms that can be implemented in practical settings without further approximations. We extend this filtering methodology to develop a refined and self-contained convergence analysis of Gaussian Sinkhorn algorithms, including closed form expressions of entropic transport maps and Schrodinger bridges.
Inverse problems, which involve estimating parameters from incomplete or noisy observations, arise in various fields such as medical imaging, geophysics, and signal processing. These problems are often ill-posed, requiring regularization techniques to stabilize the solution. In this work, we employ Flow Matching (FM), a generative framework that integrates a deterministic processes to map a simple reference distribution, such as a Gaussian, to the target distribution. Our method DAWN-FM: Data-AWare and Noise-Informed Flow Matching incorporates data and noise embedding, allowing the model to access representations about the measured data explicitly and also account for noise in the observations, making it particularly robust in scenarios where data is noisy or incomplete. By learning a time-dependent velocity field, FM not only provides accurate solutions but also enables uncertainty quantification by generating multiple plausible outcomes. Unlike pretrained diffusion models, which may struggle in highly ill-posed settings, our approach is trained specifically for each inverse problem and adapts to varying noise levels. We validate the effectiveness and robustness of our method through extensive numerical experiments on tasks such as image deblurring and tomography. The code is available at: https://github.com/ahxmeds/DAWN-FM.git.
Graphs are fundamental tools for modeling pairwise interactions in complex systems. However, many real-world systems involve multi-way interactions that cannot be fully captured by standard graphs. Hypergraphs, which generalize graphs by allowing edges to connect any number of vertices, offer a more expressive framework. In this paper, we introduce a new metric on the space of hypergraphs, inspired by the Gromov-Hausdorff distance for metric spaces. We establish Lipschitz properties of common hypergraph transformations, which send hypergraphs to graphs, including a novel graphification method with ties to single linkage hierarchical clustering. Additionally, we derive lower bounds for the hypergraph distance via invariants coming from basic summary statistics and from topological data analysis techniques. Finally, we explore stability properties of cost functions in the context of optimal transport. Our results in this direction consider Lipschitzness of the Hausdorff map and conservation of the non-negative cross curvature property under limits of cost functions.
We present a new sampling-based approach for enabling efficient computation of low-rank Bayesian matrix completion and quantifying the associated uncertainty. Firstly, we design a new prior model based on the singular-value-decomposition (SVD) parametrization of low-rank matrices. Our prior is analogous to the seminal nuclear-norm regularization used in non-Bayesian setting and enforces orthogonality in the factor matrices by constraining them to Stiefel manifolds. Then, we design a geodesic Hamiltonian Monte Carlo (-within-Gibbs) algorithm for generating posterior samples of the SVD factor matrices. We demonstrate that our approach resolves the sampling difficulties encountered by standard Gibbs samplers for the common two-matrix factorization used in matrix completion. More importantly, the geodesic Hamiltonian sampler allows for sampling in cases with more general likelihoods than the typical Gaussian likelihood and Gaussian prior assumptions adopted in most of the existing Bayesian matrix completion literature. We demonstrate an applications of our approach to fit the categorical data of a mice protein dataset and the MovieLens recommendation problem. Numerical examples demonstrate superior sampling performance, including better mixing and faster convergence to a stationary distribution. Moreover, they demonstrate improved accuracy on the two real-world benchmark problems we considered.
This paper presents an enhanced adaptive random Fourier features (ARFF) training algorithm for shallow neural networks, building upon the work introduced in "Adaptive Random Fourier Features with Metropolis Sampling", Kammonen et al., Foundations of Data Science, 2(3):309-332, 2020. This improved method uses a particle filter-type resampling technique to stabilize the training process and reduce the sensitivity to parameter choices. The Metropolis test can also be omitted when resampling is used, reducing the number of hyperparameters by one and reducing the computational cost per iteration compared to the ARFF method. We present comprehensive numerical experiments demonstrating the efficacy of the proposed algorithm in function regression tasks as a stand-alone method and as a pretraining step before gradient-based optimization, using the Adam optimizer. Furthermore, we apply the proposed algorithm to a simple image regression problem, illustrating its utility in sampling frequencies for the random Fourier features (RFF) layer of coordinate-based multilayer perceptrons. In this context, we use the proposed algorithm to sample the parameters of the RFF layer in an automated manner.
Unlike traditional cameras which synchronously register pixel intensity, neuromorphic sensors only register 'changes' at pixels where a change is occurring asynchronously. This enables neuromorphic sensors to sample at a microsecond level and efficiently capture the dynamics. Since only sequences of asynchronous event changes are recorded rather than brightness intensities over time, many traditional image processing techniques cannot be directly applied. Furthermore, existing approaches, including the ones recently introduced by the authors, use traditional images combined with neuromorphic event data to carry out reconstructions. The aim of this work is to introduce an optimization-based approach to reconstruct images and dynamics only from the neuromorphic event data without any additional knowledge of the events. Each pixel is modeled temporally. The experimental results on real data demonstrate the viability of the presented approach for processing neuromorphic sensor data across diverse real-world scenarios.
