arXiv subjects

David F. Gleich

Publications and source records attributed to David F. Gleich.

At least 19 recordsLinked to original sources

arXiv2026

Can Entry-Wise Clipping Give Spectral Control of Stochastic Gradients?

Training instabilities such as loss spikes are frequently the result of stochastic gradient noise. Because of rare expressions in language training data, and multiple layer composition, the noise impact is heavy-tailed and survives mini-batch averaging. Existing remedies trade off structure against cost: vector-norm clipping ignores the matrix structure of weight updates, while spectral normalization (e.g., Muon (Jordan et al., 2024)) respects it at additional cost. We show that this trade-off can be balanced. Real gradient noise appears to be similar to entry-wise heavy-tailed contamination, and a first-order perturbation analysis reveals a localization property of such noise, under which a simple entry-wise method achieves spectral control. Exploiting this, we derive a tractable surrogate for the Bayes-optimal entry-wise estimator under a Gaussian signal prior. We establish $O(\epsilon^{-4})$ convergence guarantee under Cauchy-contaminated noise. Empirically, we find that smooth shrinkage improves Adam on NanoGPT pretraining, saving ${\sim}7\%$ of training tokens. We further find that applying the entry-wise clipping before spectral normalization yields a ${\sim}2\%$ token saving on top of Muon.

cs.LG↗

arXiv2026

Decoupling Variance and Scale-Invariant Updates in Adaptive Gradient Descent for Unified Vector and Matrix Optimization

Adaptive methods like Adam have become the $\textit{de facto}$ standard for large-scale vector and Euclidean optimization due to their coordinate-wise adaptation with a second-order nature. More recently, matrix-based spectral optimizers like Muon (Jordan et al., 2024b) show the power of treating weight matrices as matrices rather than long vectors. Linking these is hard because many natural generalizations are not feasible to implement, and we also cannot simply move the Adam adaptation to the matrix spectrum. To address this, we reformulate the AdaGrad update and decompose it into a variance adaptation term and a scale-invariant term. This decoupling produces $\textbf{DeVA}$ ($\textbf{De}$coupled $\textbf{V}$ariance $\textbf{A}$daptation), a framework that bridges between vector-based variance adaptation and matrix spectral optimization, enabling a seamless transition from Adam to adaptive spectral descent. Extensive experiments across language modeling and image classification demonstrate that DeVA consistently outperforms state-of-the-art methods such as Muon and SOAP (Vyas et al., 2024), reducing token usage by around 6.6\%. Theoretically, we show that the variance adaptation term effectively improves the blockwise smoothness, facilitating faster convergence. Our implementation is available at https://github.com/Tsedao/Decoupled-Variance-Adaptation

cs.LG↗

arXiv2025

Fault Oblivious Eigenvalue Solver

Eigenvalue problems serve as fundamental substrates for applications in large-scale scientific simulations and machine learning, often requiring computation on massively parallel platforms. As these platforms scale to hundreds of thousands of cores, hardware failures become a significant challenge to reliability and efficiency. In this paper, we propose and analyze a novel fault-tolerant eigenvalue solver based on erasure-coded computations -- a technique that enhances resilience by augmenting the system with redundant data a priori. This transformation reformulates the original eigenvalue problem as a generalized eigenvalue problem, enabling fault-oblivious computation while preserving numerical stability and convergence properties. We formulate the augmentation scheme, establish the necessary conditions for the encoded blocks, and prove the relationship between the original and transformed problems. We implement an erasure-coded TraceMin eigensolver and demonstrate its effectiveness in extracting eigenvalues in the presence of faults. Our experimental results show that the proposed solver incurs minimal computational overhead, maintains robust convergence, and scales efficiently with the number of faults, making it a practical solution for resilient eigenvalue computations in large-scale systems.

math.NA↗

arXiv2025

Dominant H-Eigenvectors of Tensor Kronecker Products Do Not Decouple

We illustrate a counterexample to an open question related to the dominant H-eigenvector of a Kronecker product of tensors. For matrices and Z-eigenvectors of tensors, the dominant eigenvector of a Kronecker product decouples into a product of eigenvectors of the tensors underlying the Kronecker product. This does not occur for H-eigenvectors and indeed, the largest H-eigenvalue can exceed the product of the H-eigenvalues of the component tensors. Beyond this general counterexample, we show this decoupling does hold in the case of diagonal tensors as well as nonnegative tensors.

math.NA↗

arXiv2025

How Fast Can Graph Computations Go on Fine-grained Parallel Architectures

Large-scale graph problems are of critical and growing importance and historically parallel architectures have provided little support. In the spirit of co-design, we explore the question, How fast can graph computing go on a fine-grained architecture? We explore the possibilities of an architecture optimized for fine-grained parallelism, natural programming, and the irregularity and skew found in real-world graphs. Using two graph benchmarks, PageRank (PR) and Breadth-First Search (BFS), we evaluate a Fine-Grained Graph architecture, UpDown, to explore what performance codesign can achieve. To demonstrate programmability, we wrote five variants of these algorithms. Simulations of up to 256 nodes (524,288 lanes) and projections to 16,384 nodes (33M lanes) show the UpDown system can achieve 637K GTEPS PR and 989K GTEPS BFS on RMAT, exceeding the best prior results by 5x and 100x respectively.

cs.DC↗

arXiv2025

Powers of Magnetic Graph Matrix: Fourier Spectrum, Walk Compression, and Applications

Magnetic graphs, originally developed to model quantum systems under magnetic fields, have recently emerged as a powerful framework for analyzing complex directed networks. Existing research has primarily used the spectral properties of the magnetic graph matrix to study global and stationary network features. However, their capacity to model local, non-equilibrium behaviors, often described by matrix powers, remains largely unexplored. We present a novel combinatorial interpretation of the magnetic graph matrix powers through directed walk profiles -- counts of graph walks indexed by the number of edge reversals. Crucially, we establish that walk profiles correspond to a Fourier transform of magnetic matrix powers. The connection allows exact reconstruction of walk profiles from magnetic matrix powers at multiple discrete potentials, and more importantly, an even smaller number of potentials often suffices for accurate approximate reconstruction in real networks. This shows the empirical compressibility of the information captured by the magnetic matrix. This fresh perspective suggests new applications; for example, we illustrate how powers of the magnetic matrix can identify frustrated directed cycles (e.g., feedforward loops) and can be effectively employed for link prediction by encoding local structural details in directed graphs.

physics.soc-ph↗

arXiv2025

An Empirical Study of Conjugate Gradient Preconditioners for Solving Symmetric Positive Definite Systems of Linear Equations

Despite hundreds of papers on preconditioned linear systems of equations, there remains a significant lack of comprehensive performance benchmarks comparing various preconditioners for solving symmetric positive definite (SPD) systems. In this paper, we present a comparative study of 79 matrices using a broad range of preconditioners. Specifically, we evaluate 10 widely used preconditoners across 108 configurations to assess their relative performance against using no preconditioner. Our focus is on preconditioners that are commonly used in practice, are available in major software packages, and can be utilized as black-box tools without requiring significant \textit{a priori} knowledge. In addition, we compare these against a selection of classical methods. We primarily compare them without regards to effort needed to compute the preconditioner. Our results show that symmetric positive definite systems are mostly likely to benefit from incomplete symmetric factorizations, such as incomplete Cholesky (IC). Multigrid methods occasionally do exceptionally well. Simple classical techniques, symmetric Gauss Seidel and symmetric SOR, are not productive. We find that including preconditioner construction costs significantly diminishes the advantages of iterative methods compared to direct solvers; although, tuned IC methods often still outperform direct methods. Additionally, ordering strategies such as approximate minimum degree significantly enhance IC effectiveness. We plan to expand the benchmark with larger matrices, additional solvers, and detailed metrics to provide actionable information on SPD preconditioning.

math.NA↗

arXiv2024

A spatial hypergraph model to smoothly interpolate between pairwise graphs and hypergraphs to study higher-order structures

We introduce a spatial graph and hypergraph model that smoothly interpolates between a graph with purely pairwise edges and a graph where all connections are in large hyperedges. The key component is a spatial clustering resolution parameter that varies between assigning all the vertices in a spatial region to individual clusters, resulting in the pairwise case, to assigning all the vertices in a spatial region to a single cluster, which results in the large hyperedge case. An important outcome of this model is that the spatial structure is invariant to the choice of hyperedges. Consequently, this model enables us to study clustering coefficients, graph diffusion, and epidemic spread and how their behavior changes as a function of the higher-order structure in the network with a fixed spatial substrate. We hope that our model will find future uses to distill or explain other behaviors in higher-order networks.

cs.SI↗

arXiv2024

Suboptimality bounds for trace-bounded SDPs enable a faster and scalable low-rank SDP solver SDPLR+

Semidefinite programs (SDPs) and their solvers are powerful tools with many applications in machine learning and data science. Designing scalable SDP solvers is challenging because by standard the positive semidefinite decision variable is an $n \times n$ dense matrix, even though the input is often an $n \times n$ sparse matrix. However, the solution may not require a full-rank matrix, as shown by Barvinok and Pataki. Two decades ago, Burer and Monteiro developed an SDP solver \texttt{SDPLR} that optimizes over a low-rank factorization instead of the full matrix. This greatly decreases the storage cost and works well for many problems. The original solver \texttt{SDPLR} tracks only the primal infeasibility of the solution, preventing early termination at moderate accuracy. We use a suboptimality bound for trace-bounded SDP problems that enables us to track the progress better and perform early termination. We then develop \texttt{SDPLR+}, which starts the optimization with an extremely low-rank factorization and dynamically updates the rank based on the primal infeasibility and suboptimality. This further speeds up the computation and saves storage. Numerical comparisons on Max Cut, Minimum Bisection, Cut Norm, and Lov\'{a}sz Theta problems with many recent memory-efficient scalable SDP solvers demonstrate the scalability of \texttt{SDPLR+} up to problems with million-by-million decision variables. It is often the fastest solver to a moderate accuracy of $10^{-2}$. Further experiments on $\mu$-conductance, matrix completion, and $k$-means clustering show the potential of \texttt{SDPLR+} on a broader range of data science applications.

math.OC↗

arXiv2024

Better than best low-rank approximation with the singular value decomposition

The Eckhart-Young theorem states that the best low-rank approximation of a matrix can be constructed from the leading singular values and vectors of the matrix. Here, we illustrate that the practical implications of this result crucially depend on the organization of the matrix data. In particular, we will show examples where a rank 2 approximation of the matrix data in a different representation more accurately represents the entire matrix than a rank 5 approximation of the original matrix data -- even though both approximations have the same number of underlying parameters. Beyond images, we show examples of how flexible orientation enables better approximation of time series data, which suggests additional applicability of the findings. Finally, we conclude with a theoretical result that the effect of data organization can result in an unbounded improvement to the matrix approximation factor as the matrix dimension grows.

math.NA↗

arXiv2023

Multi-scale Local Network Structure Critically Impacts Epidemic Spread and Interventions

Network epidemic simulation holds the promise of enabling fine-grained understanding of epidemic behavior, beyond that which is possible with coarse-grained compartmental models. Key inputs to these epidemic simulations are the networks themselves. However, empirical measurements and samples of realistic interaction networks typically display properties that are challenging to capture with popular synthetic models of networks. Our empirical results show that epidemic spread behavior is very sensitive to a subtle but ubiquitous form of multi-scale local structure that is not present in common baseline models, including (but not limited to) uniform random graph models (Erdos-Renyi), random configuration models (Chung-Lu), etc. Such structure is not necessary to reproduce very simple network statistics, such as degree distributions or triangle closing probabilities. However, we show that this multi-scale local structure impacts, critically, the behavior of more complex network properties, in particular the effect of interventions such as quarantining; and it enables epidemic spread to be halted in realistic interaction networks, even when it cannot be halted in simple synthetic models. Key insights from our analysis include how epidemics on networks with widespread multi-scale local structure are easier to mitigate, as well as characterizing which nodes are ultimately not likely to be infected. We demonstrate that this structure results from more than just local triangle structure in the network, and we illustrate processes based on homophily or social influence and random walks that suggest how this multi-scale local structure arises.

cs.SI↗

arXiv2023

Densest Subhypergraph: Negative Supermodular Functions and Strongly Localized Methods

Dense subgraph discovery is a fundamental primitive in graph and hypergraph analysis which among other applications has been used for real-time story detection on social media and improving access to data stores of social networking systems. We present several contributions for localized densest subgraph discovery, which seeks dense subgraphs located nearby given seed sets of nodes. We first introduce a generalization of a recent $\textit{anchored densest subgraph}$ problem, extending this previous objective to hypergraphs and also adding a tunable locality parameter that controls the extent to which the output set overlaps with seed nodes. Our primary technical contribution is to prove when it is possible to obtain a strongly-local algorithm for solving this problem, meaning that the runtime depends only on the size of the input set. We provide a strongly-local algorithm that applies whenever the locality parameter is not too small, and show via counterexample why strongly-local algorithms are impossible below a certain threshold. Along the way to proving our results for localized densest subgraph discovery, we also provide several advances in solving global dense subgraph discovery objectives. This includes the first strongly polynomial time algorithm for the densest supermodular set problem and a flow-based exact algorithm for a heavy and dense subgraph discovery problem in graphs with arbitrary node weights. We demonstrate our algorithms on several web-based data analysis tasks.

cs.DS↗

arXiv2023

Theoretical bounds on the network community profile from low-rank semi-definite programming

We study a new connection between a technical measure called $\mu$-conductance that arises in the study of Markov chains for sampling convex bodies and the network community profile that characterizes size-resolved properties of clusters and communities in social and information networks. The idea of $\mu$-conductance is similar to the traditional graph conductance, but disregards sets with small volume. We derive a sequence of optimization problems including a low-rank semi-definite program from which we can derive a lower bound on the optimal $\mu$-conductance value. These ideas give the first theoretically sound bound on the behavior of the network community profile for a wide range of cluster sizes. The algorithm scales up to graphs with hundreds of thousands of nodes and we demonstrate how our framework validates the predicted structures of real-world graphs.

cs.SI↗

arXiv2023

A Cheeger Inequality for Size-Specific Conductance

The $\mu$-conductance measure proposed by Lov\'asz and Simonovits is a size-specific conductance score that identifies the set with smallest conductance while disregarding those sets with volume smaller than a $\mu$ fraction of the whole graph. Using $\mu$-conductance enables us to study the network structures in new ways. In this manuscript we study a modified spectral cut for $\mu$-conductance that is a natural relaxation of the integer program of $\mu$-conductance and show that the optimum of this program has a two-sided Cheeger inequality with $\mu$-conductance.

cs.DM↗

arXiv2022

Topological structure of complex predictions

Complex prediction models such as deep learning are the output from fitting machine learning, neural networks, or AI models to a set of training data. These are now standard tools in science. A key challenge with the current generation of models is that they are highly parameterized, which makes describing and interpreting the prediction strategies difficult. We use topological data analysis to transform these complex prediction models into pictures representing a topological view. The result is a map of the predictions that enables inspection. The methods scale up to large datasets across different domains and enable us to detect labeling errors in training data, understand generalization in image classification, and inspect predictions of likely pathogenic mutations in the BRCA1 gene.

cs.LG↗

arXiv2022

A flexible PageRank-based graph embedding framework closely related to spectral eigenvector embeddings

We study a simple embedding technique based on a matrix of personalized PageRank vectors seeded on a random set of nodes. We show that the embedding produced by the element-wise logarithm of this matrix (1) are related to the spectral embedding for a class of graphs where spectral embeddings are significant, and hence useful representation of the data, (2) can be done for the entire network or a smaller part of it, which enables precise local representation, and (3) uses a relatively small number of PageRank vectors compared to the size of the networks. Most importantly, the general nature of this embedding strategy opens up many emerging applications, where eigenvector and spectral techniques may not be well established, to the PageRank-based relatives. For instance, similar techniques can be used on PageRank vectors from hypergraphs to get "spectral-like" embeddings.

cs.SI↗

arXiv2021

fauci-email: a json digest of Anthony Fauci's released emails

A collection of over 3000 pages of emails sent by Anthony Fauci and his staff were released in an effort to understand the United States government response to the COVID-19 pandemic. We describe how this email data was translated into a resource consisting of json files that make many future studies easy. Findings from our processed data include (i) successful organizational partitions using the simple mincut techniques in Zachary's karate club methodology, (ii) a natural example where the normalized cut and minimum conductance set are extremely different, and (iii) organizational groups identified by optimum modularity clusters that illustrate a working hierarchy. These example uses suggest the data will be useful for future research and pedagogical uses in terms of human and system behavioral interactions. We explain a number of ways to turn email information into a network, a hypergraph, a temporal sequence, and a tensor for subsequent analysis as well as a few examples of such analysis.

cs.SI↗

arXiv2021

Strongly Local Hypergraph Diffusions for Clustering and Semi-supervised Learning

Hypergraph-based machine learning methods are now widely recognized as important for modeling and using higher-order and multiway relationships between data objects. Local hypergraph clustering and semi-supervised learning specifically involve finding a well-connected set of nodes near a given set of labeled vertices. Although many methods for local clustering exist for graphs, there are relatively few for localized clustering in hypergraphs. Moreover, those that exist often lack flexibility to model a general class of hypergraph cut functions or cannot scale to large problems. To tackle these issues, this paper proposes a new diffusion-based hypergraph clustering algorithm that solves a quadratic hypergraph cut based objective akin to a hypergraph analog of Andersen-Chung-Lang personalized PageRank clustering for graphs. We prove that, for graphs with fixed maximum hyperedge size, this method is strongly local, meaning that its runtime only depends on the size of the output instead of the size of the hypergraph and is highly scalable. Moreover, our method enables us to compute with a wide variety of cardinality-based hypergraph cut functions. We also prove that the clusters found by solving the new objective function satisfy a Cheeger-like quality guarantee. We demonstrate that on large real-world hypergraphs our new method finds better clusters and runs much faster than existing approaches. Specifically, it runs in few seconds for hypergraphs with a few million hyperedges compared with minutes for flow-based technique. We furthermore show that our framework is general enough that can also be used to solve other p-norm based cut objectives on hypergraphs. Our code is available \url{github.com/MengLiuPurdue/LHQD}.

cs.SI↗