arXiv subjects

Thomas Strohmer

Publications and source records attributed to Thomas Strohmer.

At least 19 recordsLinked to original sources

arXiv2026

Observation geometry for uncertainty-aware Hamiltonian inference and experimental design in quantum magnets

Determining microscopic interactions from spectroscopic and scattering measurements is central to understanding quantum materials, yet it often remains unclear which interactions can be reliably revealed by the available experimental data and how additional experimental modalities should be designed to resolve the remaining ambiguities. Here we present an artificial intelligence-enabled framework for uncertainty-aware Hamiltonian inference and adaptive experimental design. By combining Hamiltonian-conditioned neural surrogates with Bayesian inference and observation geometry, the framework characterizes how measurements constrain Hamiltonian parameter space, quantifies the identifiability of microscopic interactions, and propagates posterior uncertainty directly in the physical Hamiltonian parameter space rather than an abstract learned representation. Using multimodal powder and single-crystal inelastic neutron scattering measurements of the quantum magnet NiPS$_{3}$, we demonstrate physically interpretable Hamiltonian inference, modality-aware uncertainty quantification, and adaptive experimental design. The framework provides a general strategy for uncertainty-aware microscopic characterization and multimodal experimental design across quantum materials.

cond-mat.str-el↗

arXiv2026

Mathematics of Data Science

This book is about the mathematical foundations of data science. 1. Introduction 2. Curses, Blessings, and Surprises in High Dimensions 3. Singular Value Decomposition and Principal Component Analysis 4. Linear Regression and Regularization 5. Graphs, Networks, and Clustering 6. Nonlinear Dimension Reduction and Diffusion Maps 7. Linear Dimension Reduction via Random Projections 8. Optimization for Data Science 9. Classification 10. A Mathematical Introduction to Deep Learning 11. Large Sample Limit of Graph Laplacians 12. Community 13. Concentration of Measure and Gaussian Analysis 14. Matrix Concentration Inequalities 15. Compressive Sensing and Sparsity 16. Low-Rank Matrix Recovery

cs.LG↗

arXiv2026

The "I Don't Know" Filter: Enhancing Agentic Reliability in Function Calling

The language models that underpin agents have seen a rapid rise in performance on function calling benchmarks. However, the metrics used in the training and evaluation of these models often encourage models to make positive claims even when the answer is uncertain, leading to hallucinations. Such hallucinations can be disastrous when language models are trusted to use function calls to make decisions in high stakes applications. To that end, we propose an agent evaluation metric that takes into account the negative outcomes associated with incorrect function calls. Further, to catch hallucinations before they can cause harm, we propose a lightweight trainable filter that can quantify a language model's uncertainty and remove potentially harmful function calls. By training that filter to detect and suppress uncertain function calls without modifying the underlying model, we demonstrate a practical path toward agents that know when to say "I don't know," a property we argue is essential to production reliability.

cs.SE↗

arXiv2026

Towards Multiscale Graph-based Protein Learning with Geometric Secondary Structural Motifs

Graph neural networks (GNNs) have emerged as powerful tools for learning protein structures by capturing spatial relationships at the residue level. However, existing GNN-based methods often face challenges in learning multiscale representations and modeling long-range dependencies efficiently. In this work, we propose an efficient multiscale graph-based learning framework tailored to proteins. Our proposed framework contains two crucial components: (1) It constructs a hierarchical graph representation comprising a collection of fine-grained subgraphs, each corresponding to a secondary structure motif (e.g., $\alpha$-helices, $\beta$-strands, loops), and a single coarse-grained graph that connects these motifs based on their spatial arrangement and relative orientation. (2) It employs two GNNs for feature learning: the first operates within individual secondary motifs to capture local interactions, and the second models higher-level structural relationships across motifs. Our modular framework allows a flexible choice of GNN in each stage. Theoretically, we show that our hierarchical framework preserves the desired maximal expressiveness, ensuring no loss of critical structural information. Empirically, we demonstrate that integrating baseline GNNs into our multiscale framework remarkably improves prediction accuracy and reduces computational cost across various benchmarks.

cs.LG↗

arXiv2025

Forgetting-MarI: LLM Unlearning via Marginal Information Regularization

As AI models are trained on ever-expanding datasets, the ability to remove the influence of specific data from trained models has become essential for privacy protection and regulatory compliance. Unlearning addresses this challenge by selectively removing parametric knowledge from the trained models without retraining from scratch, which is critical for resource-intensive models such as Large Language Models (LLMs). Existing unlearning methods often degrade model performance by removing more information than necessary when attempting to ''forget'' specific data. We introduce Forgetting-MarI, an LLM unlearning framework that provably removes only the additional (marginal) information contributed by the data to be unlearned, while preserving the information supported by the data to be retained. By penalizing marginal information, our method yields an explicit upper bound on the unlearn dataset's residual influence in the trained models, providing provable undetectability. Extensive experiments confirm that our approach outperforms current state-of-the-art unlearning methods, delivering reliable forgetting and better preserved general model performance across diverse benchmarks. This advancement represents an important step toward making AI systems more controllable and compliant with privacy and copyright regulations without compromising their effectiveness.

cs.AI↗

arXiv2025

The Future of Artificial Intelligence and the Mathematical and Physical Sciences (AI+MPS)

This community paper developed out of the NSF Workshop on the Future of Artificial Intelligence (AI) and the Mathematical and Physics Sciences (MPS), which was held in March 2025 with the goal of understanding how the MPS domains (Astronomy, Chemistry, Materials Research, Mathematical Sciences, and Physics) can best capitalize on, and contribute to, the future of AI. We present here a summary and snapshot of the MPS community's perspective, as of Spring/Summer 2025, in a rapidly developing field. The link between AI and MPS is becoming increasingly inextricable; now is a crucial moment to strengthen the link between AI and Science by pursuing a strategy that proactively and thoughtfully leverages the potential of AI for scientific discovery and optimizes opportunities to impact the development of AI by applying concepts from fundamental science. To achieve this, we propose activities and strategic priorities that: (1) enable AI+MPS research in both directions; (2) build up an interdisciplinary community of AI+MPS researchers; and (3) foster education and workforce development in AI for MPS researchers and students. We conclude with a summary of suggested priorities for funding agencies, educational institutions, and individual researchers to help position the MPS community to be a leader in, and take full advantage of, the transformative potential of AI+MPS.

cs.AI↗

arXiv2025

FedOSAA: Improving Federated Learning with One-Step Anderson Acceleration

Federated learning (FL) is a distributed machine learning approach that enables multiple local clients and a central server to collaboratively train a model while keeping the data on their own devices. First-order methods, particularly those incorporating variance reduction techniques, are the most widely used FL algorithms due to their simple implementation and stable performance. However, these methods tend to be slow and require a large number of communication rounds to reach the global minimizer. We propose FedOSAA, a novel approach that preserves the simplicity of first-order methods while achieving the rapid convergence typically associated with second-order methods. Our approach applies one Anderson acceleration (AA) step following classical local updates based on first-order methods with variance reduction, such as FedSVRG and SCAFFOLD, during local training. This AA step is able to leverage curvature information from the history points and gives a new update that approximates the Newton-GMRES direction, thereby significantly improving the convergence. We establish a local linear convergence rate to the global minimizer of FedOSAA for smooth and strongly convex loss functions. Numerical comparisons show that FedOSAA substantially improves the communication and computation efficiency of the original first-order methods, achieving performance comparable to second-order methods like GIANT.

cs.LG↗

arXiv2025

Machine Unlearning via Information Theoretic Regularization

How can we effectively remove or ``unlearn'' undesirable information, such as specific features or the influence of individual data points, from a learning outcome while minimizing utility loss and ensuring rigorous guarantees? We introduce a unified mathematical framework based on information-theoretic regularization to address both data-point unlearning and feature unlearning. For data-point unlearning, we introduce the \emph{Marginal Unlearning Principle}, an auditable and provable framework. Moreover, we provide an information-theoretic unlearning definition based on the proposed principle and provable guarantees on sufficiency and necessity of marginal unlearning. We then show that the proposed framework provides a natural solution to the marginal unlearning problem and yields auditable high-probability marginal-unlearning guarantees. For feature unlearning, the framework applies to deep learning with flexible training objectives. By combining flexibility in learning objectives with simplicity in regularization design, our approach is highly adaptable and practical for a wide range of machine learning and AI applications. From a mathematical perspective, we provide a unified analytic solution to the optimal feature unlearning problem with a variety of information-theoretic training objectives. Our theoretical analysis reveals intriguing connections between machine unlearning, information theory, optimal transport, and extremal sigma algebras. Numerical simulations support our theoretical findings.

cs.LG↗

arXiv2024

WHOMP: Optimizing Randomized Controlled Trials via Wasserstein Homogeneity

We investigate methods for partitioning datasets into subgroups that maximize diversity within each subgroup while minimizing dissimilarity across subgroups. We introduce a novel partitioning method called the $\textit{Wasserstein Homogeneity Partition}$ (WHOMP), which optimally minimizes type I and type II errors that often result from imbalanced group splitting or partitioning, commonly referred to as accidental bias, in comparative and controlled trials. We conduct an analytical comparison of WHOMP against existing partitioning methods, such as random subsampling, covariate-adaptive randomization, rerandomization, and anti-clustering, demonstrating its advantages. Moreover, we characterize the optimal solutions to the WHOMP problem and reveal an inherent trade-off between the stability of subgroup means and variances among these solutions. Based on our theoretical insights, we design algorithms that not only obtain these optimal solutions but also equip practitioners with tools to select the desired trade-off. Finally, we validate the effectiveness of WHOMP through numerical experiments, highlighting its superiority over traditional methods.

stat.ML↗

arXiv2024

Differentially Private Synthetic High-dimensional Tabular Stream

While differentially private synthetic data generation has been explored extensively in the literature, how to update this data in the future if the underlying private data changes is much less understood. We propose an algorithmic framework for streaming data that generates multiple synthetic datasets over time, tracking changes in the underlying private data. Our algorithm satisfies differential privacy for the entire input stream (continual differential privacy) and can be used for high-dimensional tabular data. Furthermore, we show the utility of our method via experiments on real-world datasets. The proposed algorithm builds upon a popular select, measure, fit, and iterate paradigm (used by offline synthetic data generation algorithms) and private counters for streams.

cs.CR↗

arXiv2024

Quantum paving: When sphere packings meet Gabor frames

We introduce the new problems of quantum packing, quantum covering, and quantum paving. These problems arise naturally when considering an algebra of non-commutative operators that is deeply rooted in quantum physics as well as in Gabor analysis. Quantum packing and quantum covering show similarities with energy minimization and the dual problem of polarization. Quantum paving, in turn, aims to simultaneously optimize both quantum packing and quantum covering. Classical sphere packing and covering hint the optimal configurations for our new problems. We present solutions in certain cases, state several conjectures related to quantum paving and discuss some applications.

quant-ph↗

arXiv2024

Convergence Analysis of the Alternating Anderson-Picard Method for Nonlinear Fixed-point Problems

Anderson Acceleration (AA) has been widely used to solve nonlinear fixed-point problems due to its rapid convergence. This work focuses on a variant of AA in which multiple Picard iterations are performed between each AA step, referred to as the Alternating Anderson-Picard (AAP) method. Despite introducing more ``slow'' Picard iterations, this method has been shown to be efficient and even more robust in both linear and nonlinear cases. However, there is a lack of theoretical analysis for AAP in the nonlinear case. In this paper, we address this gap by establishing the equivalence between AAP and a multisecant-GMRES method that uses GMRES to solve a multisecant linear system at each iteration. From this perspective, we show that AAP ``converges'' to the Newton-GMRES method. Specifically, as the residual approaches zero, the multisecant matrix, the approximate Jacobian inverse, the search direction, and the optimization gain of AAP converge to their counterparts in the Newton-GMRES method. These connections provide insights for analyzing the asymptotic convergence properties of AAP. Consequently, we show that AAP is locally $q$-linear convergent and provide an upper bound for the convergence factor of AAP. To validate the theoretical results, numerical examples are provided.

math.NA↗

arXiv2024

Metric geometry of the privacy-utility tradeoff

Synthetic data are an attractive concept to enable privacy in data sharing. A fundamental question is how similar the privacy-preserving synthetic data are compared to the true data. Using metric privacy, an effective generalization of differential privacy beyond the discrete setting, we raise the problem of characterizing the optimal privacy-accuracy tradeoff by the metric geometry of the underlying space. We provide a partial solution to this problem in terms of the "entropic scale", a quantity that captures the multiscale geometry of a metric space via the behavior of its packing numbers. We illustrate the applicability of our privacy-accuracy tradeoff framework via a diverse set of examples of metric spaces.

cs.CR↗

arXiv2024

Improving Autoencoder Image Interpolation via Dynamic Optimal Transport

Autoencoders are important generative models that, among others, have the ability to interpolate image sequences. However, interpolated images are usually not semantically meaningful.In this paper, motivated by dynamic optimal transport, we consider image interpolation as a mass transfer problem and propose a novel regularization term to penalize non-smooth and unrealistic changes in the interpolation result. Specifically, we define the path energy function for each path connecting the source and target images. The autoencoder is trained to generate the $L^2$ optimal transport geodesic path when decoding a linear interpolation of their latent codes. With a simple extension, this model can handle complicated environments, such as allowing mass transfer between obstacles and unbalanced optimal transport. A key feature of the proposed method is that it is physics-driven and can generate robust and realistic interpretation results even when only very limited training data are available.

math.OC↗

arXiv2024

Private measures, random walks, and synthetic data

Differential privacy is a mathematical concept that provides an information-theoretic security guarantee. While differential privacy has emerged as a de facto standard for guaranteeing privacy in data sharing, the known mechanisms to achieve it come with some serious limitations. Utility guarantees are usually provided only for a fixed, a priori specified set of queries. Moreover, there are no utility guarantees for more complex - but very common - machine learning tasks such as clustering or classification. In this paper we overcome some of these limitations. Working with metric privacy, a powerful generalization of differential privacy, we develop a polynomial-time algorithm that creates a private measure from a data set. This private measure allows us to efficiently construct private synthetic data that are accurate for a wide range of statistical analysis tools. Moreover, we prove an asymptotically sharp min-max result for private measures and synthetic data for general compact metric spaces. A key ingredient in our construction is a new superregular random walk, whose joint distribution of steps is as regular as that of independent random variables, yet which deviates from the origin logarithmicaly slowly.

cs.CR↗

arXiv2024

Semi-Supervised Clustering of Sparse Graphs: Crossing the Information-Theoretic Threshold

The stochastic block model is a canonical random graph model for clustering and community detection on network-structured data. Decades of extensive study on the problem have established many profound results, among which the phase transition at the Kesten-Stigum threshold is particularly interesting both from a mathematical and an applied standpoint. It states that no estimator based on the network topology can perform substantially better than chance on sparse graphs if the model parameter is below a certain threshold. Nevertheless, if we slightly extend the horizon to the ubiquitous semi-supervised setting, such a fundamental limitation will disappear completely. We prove that with an arbitrary fraction of the labels revealed, the detection problem is feasible throughout the parameter domain. Moreover, we introduce two efficient algorithms, one combinatorial and one based on optimization, to integrate label information with graph structures. Our work brings a new perspective to the stochastic model of networks and semidefinite program research.

stat.ML↗

arXiv2024

An Algorithm for Streaming Differentially Private Data

Much of the research in differential privacy has focused on offline applications with the assumption that all data is available at once. When these algorithms are applied in practice to streams where data is collected over time, this either violates the privacy guarantees or results in poor utility. We derive an algorithm for differentially private synthetic streaming data generation, especially curated towards spatial datasets. Furthermore, we provide a general framework for online selective counting among a collection of queries which forms a basis for many tasks such as query answering and synthetic data generation. The utility of our algorithm is verified on both real-world and simulated datasets.

cs.DB↗

arXiv2024

Auto-Calibration and Biconvex Compressive Sensing with Applications to Parallel MRI

We study an auto-calibration problem in which a transform-sparse signal is acquired via compressive sensing by multiple sensors in parallel, but with unknown calibration parameters of the sensors. This inverse problem has an important application in pMRI reconstruction, where the calibration parameters of the receiver coils are often difficult and costly to obtain explicitly, but nonetheless are a fundamental requirement for high-precision reconstructions. Most auto-calibration strategies for this problem involve solving a challenging biconvex optimization problem, which lacks reconstruction guarantees. In this work, we transform the auto-calibrated parallel compressive sensing problem to a convex optimization problem using the idea of `lifting'. By exploiting sparsity structures in the signal and the redundancy introduced by multiple sensors, we solve a mixed-norm minimization problem to recover the underlying signal and the sensing parameters simultaneously. Our method provides robust and stable recovery guarantees that take into account the presence of noise and sparsity deficiencies in the signals. As such, it offers a theoretically guaranteed approach to auto-calibrated parallel imaging in MRI under appropriate assumptions. Applications in compressive sensing pMRI are discussed, and numerical experiments using real and simulated MRI data are presented to support our theoretical results.

math.OC↗