arXiv subjects

Avanti Athreya

Publications and source records attributed to Avanti Athreya.

At least 19 recordsLinked to original sources

arXiv2026

Recovering manifold structure in LLM responses through a joint Euclidean mirror

Understanding the behavior of black-box large language models and determining effective means of comparing their performance is a key task in modern machine learning. We consider how large language models respond to a specific query by analyzing how the distributions of responses vary over different values of tuning parameters. We frame this problem in a general mathematical setting, treating the mapping from model parameters to response distributions as a structured family of probability measures, endowed with a geometry via a dissimilarity measure. We show how dissimilarities between response distributions can be represented in low-dimensional Euclidean space through a joint Euclidean mirror surface encoding the underlying geometry, which permits both qualitative and quantitative analysis of large language models and provides insight into predicting response distributions for different values of tuning parameters. We propose an estimation procedure for the underlying joint Euclidean mirror based on observed samples from the response distributions, and we prove its asymptotic properties. Additionally, we propose a statistically consistent procedure to infer the value of an unknown model parameter based on samples from the corresponding response distribution and the estimated joint Euclidean mirror. In an experimental setting with large language models, we find that changes in different tuning parameter values correspond to distinct directions in the embedding space, making it possible to estimate the tuning parameters that were used to generate a given response.

stat.ME↗

arXiv2026

Vertex misalignment and changepoint localization in network time series

Inference for time series of networks often relies on accurate vertex correspondence between network realizations at different times. In practice, however, such vertex alignments can be misspecified or unknown. We study the impact of vertex alignment on changepoint localization for dynamic networks through two illustrative models, each with a similar changepoint, with the key distinction being whether changepoint information is contained in marginal or joint distributions of the time-varying latent positions. We compare localization techniques ranging from the simple network statistic of average degree to the modern procedure of Euclidean mirrors. In one model, vertex misalignment causes little error, and in the other, it impairs localization in ways that cannot be corrected through graph matching or optimal transport, which we show are closely related in this setting. Our results demonstrate that robust network inference necessitates reckoning with the subtle interplay of marginal and joint information in the observed network time series.

math.ST↗

arXiv2026

Gaussian mixture models as a proxy for interacting language models

Large language models (LLMs) are powerful tools that, in a number of settings, overlap with the results of human pattern recognition and reasoning. Retrieval-augmented generation (RAG) further allows LLMs to produce tailored output depending on the contents of their RAG databases. However, LLMs depend on complex, computationally expensive algorithms. In this paper, we introduce interacting Gaussian mixture models (GMMs) as a proxy for interacting LLMs. We construct a model of interacting GMMs, complete with an analogue to RAG updating, under which GMMs can generate, exchange, and update data and parameters. We show that this interacting system of Gaussian mixture models, which can be implemented at minimal computational cost, mimics certain aspects of experimental simulations of interacting LLMs whose iterative responses depend on feedback from other LLMs. We build a Markov chain from this system of interacting GMMs; formalize and interpret the notion of polarization for such a chain; and prove lower bounds on the probability of polarization. This provides theoretical insight into the use of interacting Gaussian mixture models as a computationally efficient proxy for interacting large language models.

cs.CL↗

arXiv2026

Euclidean mirrors and first-order changepoints in network time series

We describe a model for a network time series whose evolution is governed by an underlying stochastic process, known as the latent position process, in which network evolution can be represented in Euclidean space by a curve, called the Euclidean mirror. We define the notion of a first-order changepoint for a time series of networks, and construct a family of latent position process networks with underlying first-order changepoints. We prove that a spectral estimate of the associated Euclidean mirror localizes these changepoints, even when the graph distribution evolves continuously, but at a rate that changes. Simulated and real data examples on organoid networks show that this localization captures empirically significant shifts in network evolution.

stat.ME↗

arXiv2026

Dynamic networks clustering via mirror distance

The classification of different patterns of network evolution, for example in brain connectomes or social networks, is a key problem in network inference and modern data science. Building on the notion of a network's Euclidean mirror, which captures its evolution as a curve in Euclidean space, we develop the Dynamic Network Clustering through Mirror Distance (DNCMD), an algorithm for clustering dynamic networks based on a distance measure between their associated mirrors. We provide theoretical guarantees for DNCMD to achieve exact recovery of distinct evolutionary patterns for latent position random networks both when underlying vertex features change deterministically and when they follow a stochastic process. We validate our theoretical results through numerical simulations and demonstrate the application of DNCMD to understand edge functions in Drosophila larval connectome data, as well as to analyze temporal patterns in dynamic trade networks.

stat.ME↗

arXiv2025

Procrustes Problems on Random Matrices

Meaningful comparison between sets of observations often necessitates alignment or registration between them, and the resulting optimization problems range in complexity from those admitting simple closed-form solutions to those requiring advanced and novel techniques. We compare different Procrustes problems in which we align two sets of points after various perturbations by minimizing the norm of the difference between one matrix and an orthogonal transformation of the other. The minimization problem depends significantly on the choice of matrix norm; we highlight recent developments in nonsmooth Riemannian optimization and characterize which choices of norm work best for each perturbation. We show that in several applications, from low-dimensional alignments to hypothesis testing for random networks, when Procrustes alignment with the spectral or robust norm is the appropriate choice, it is often feasible to replace the computationally more expensive spectral and robust minimizers with their closed-form Frobenius-norm counterpart. Our work reinforces the synergy between optimization, geometry, and statistics.

stat.ME↗

arXiv2024

Multiple Network Embedding for Anomaly Detection in Time Series of Graphs

This paper considers the graph signal processing problem of anomaly detection in time series of graphs. We examine two related, complementary inference tasks: the detection of anomalous graphs within a time series, and the detection of temporally anomalous vertices. We approach these tasks via the adaptation of statistically principled methods for joint graph inference, specifically \emph{multiple adjacency spectral embedding} (MASE). We demonstrate that our method is effective for our inference tasks. Moreover, we assess the performance of our method in terms of the underlying nature of detectable anomalies. We further provide the theoretical justification for our method and insight into its use. Applied to the Enron communication graph, a large-scale commercial search engine time series of graphs, and a larval Drosophila connectome data, our approaches demonstrate their applicability and identify the anomalous vertices beyond just large degree change.

stat.ME↗

arXiv2024

Euclidean mirrors and dynamics in network time series

Analyzing changes in network evolution is central to statistical network inference, as underscored by recent challenges of predicting and distinguishing pandemic-induced transformations in organizational and communication networks. We consider a joint network model in which each node has an associated time-varying low-dimensional latent vector of feature data, and connection probabilities are functions of these vectors. Under mild assumptions, the time-varying evolution of the latent vectors exhibits low-dimensional manifold structure under a suitable notion of distance. This distance can be approximated by a measure of separation between the observed networks themselves, and there exist Euclidean representations for underlying network structure, as characterized by this distance, at any given time. These Euclidean representations, called Euclidean mirrors, permit the visualization of network evolution and transform network inference questions such as change-point and anomaly detection into a classical setting. We illustrate our methodology with real and synthetic data, and identify change points corresponding to massive shifts in pandemic policies in a communication network of a large organization.

stat.ME↗

arXiv2024

Random line graphs and edge-attributed network inference

We extend the latent position random graph model to the line graph of a random graph, which is formed by creating a vertex for each edge in the original random graph, and connecting each pair of edges incident to a common vertex in the original graph. We prove concentration inequalities for the spectrum of a line graph, as well as limiting distribution results for the largest eigenvalue and the empirical spectral distribution in certain settings. For the stochastic blockmodel, we establish that although naive spectral decompositions can fail to extract necessary signal for edge clustering, there exist signal-preserving singular subspaces of the line graph that can be recovered through a carefully-chosen projection. Moreover, we can consistently estimate edge latent positions in a random line graph, even though such graphs are of a random size, typically have high rank, and possess no spectral gap. Our results demonstrate that the line graph of a stochastic block model exhibits underlying block structure, and in simulations, we synthesize and test our methods against several commonly-used techniques, including tensor decompositions, for cluster recovery and edge covariate inference. By naturally incorporating information encoded in both vertices and edges, the random line graph improves network inference.

cs.SI↗

arXiv2023

Discovering a change point and piecewise linear structure in a time series of organoid networks via the iso-mirror

Recent advancements have been made in the development of cell-based in-vitro neuronal networks, or organoids. In order to better understand the network structure of these organoids, a super-selective algorithm has been proposed for inferring the effective connectivity networks from multi-electrode array data. In this paper, we apply a novel statistical method called spectral mirror estimation to the time series of inferred effective connectivity organoid networks. This method produces a one-dimensional iso-mirror representation of the dynamics of the time series of the networks which exhibits a piecewise linear structure. A classical change point algorithm is then applied to this representation, which successfully detects a change point coinciding with the neuroscientifically significant time inhibitory neurons start appearing and the percentage of astrocytes increases dramatically. This finding demonstrates the potential utility of applying the iso-mirror dynamic structure discovery method to inferred effective connectivity time series of organoid networks.

stat.AP↗

arXiv2022

Distance-based Positive and Unlabeled Learning for Ranking

Learning to rank -- producing a ranked list of items specific to a query and with respect to a set of supervisory items -- is a problem of general interest. The setting we consider is one in which no analytic description of what constitutes a good ranking is available. Instead, we have a collection of representations and supervisory information consisting of a (target item, interesting items set) pair. We demonstrate analytically, in simulation, and in real data examples that learning to rank via combining representations using an integer linear program is effective when the supervision is as light as "these few items are similar to your item of interest." While this nomination task is quite general, for specificity we present our methodology from the perspective of vertex nomination in graphs. The methodology described herein is model agnostic.

cs.LG↗

arXiv2021

The Phantom Alignment Strength Conjecture: Practical use of graph matching alignment strength to indicate a meaningful graph match

The alignment strength of a graph matching is a quantity that gives the practitioner a measure of the correlation of the two graphs, and it can also give the practitioner a sense for whether the graph matching algorithm found the true matching. Unfortunately, when a graph matching algorithm fails to find the truth because of weak signal, there may be "phantom alignment strength" from meaningless matchings that, by random noise, have fewer disagreements than average (sometimes substantially fewer); this alignment strength may give the misleading appearance of significance. A practitioner needs to know what level of alignment strength may be phantom alignment strength and what level indicates that the graph matching algorithm obtained the true matching and is a meaningful measure of the graph correlation. The {\it Phantom Alignment Strength Conjecture} introduced here provides a principled and practical means to approach this issue. We provide empirical evidence for the conjecture, and explore its consequences.

math.OC↗

arXiv2021

On spectral algorithms for community detection in stochastic blockmodel graphs with vertex covariates

In network inference applications, it is often desirable to detect community structure, namely to cluster vertices into groups, or blocks, according to some measure of similarity. Beyond mere adjacency matrices, many real networks also involve vertex covariates that carry key information about underlying block structure in graphs. To assess the effects of such covariates on block recovery, we present a comparative analysis of two model-based spectral algorithms for clustering vertices in stochastic blockmodel graphs with vertex covariates. The first algorithm uses only the adjacency matrix, and directly estimates the block assignments. The second algorithm incorporates both the adjacency matrix and the vertex covariates into the estimation of block assignments, and moreover quantifies the explicit impact of the vertex covariates on the resulting estimate of the block assignments. We employ Chernoff information to analytically compare the algorithms' performance and derive the information-theoretic Chernoff ratio for certain models of interest. Analytic results and simulations suggest that the second algorithm is often preferred: we can often better estimate the induced block assignments by first estimating the effect of vertex covariates. In addition, real data examples also indicate that the second algorithm has the advantages of revealing underlying block structure and taking observed vertex heterogeneity into account in real applications. Our findings emphasize the importance of distinguishing between observed and unobserved factors that can affect block structure in graphs.

cs.SI↗

arXiv2021

The Importance of Being Correlated: Implications of Dependence in Joint Spectral Inference across Multiple Networks

Spectral inference on multiple networks is a rapidly-developing subfield of graph statistics. Recent work has demonstrated that joint, or simultaneous, spectral embedding of multiple independent networks can deliver more accurate estimation than individual spectral decompositions of those same networks. Such inference procedures typically rely heavily on independence assumptions across the multiple network realizations, and even in this case, little attention has been paid to the induced network correlation in such joint embeddings. Here, we present a generalized omnibus embedding methodology and provide a detailed analysis of this embedding across both independent and correlated networks, the latter of which significantly extends the reach of such procedures. We describe how this omnibus embedding can itself induce correlation, leading us to distinguish between inherent correlation -- the correlation that arises naturally in multisample network data -- and induced correlation, which is an artifice of the joint embedding methodology. We show that the generalized omnibus embedding procedure is flexible and robust, and prove both consistency and a central limit theorem for the embedded points. We examine how induced and inherent correlation can impact inference for network time series data, and we provide network analogues of classical questions such as the effective sample size for more generally correlated data. Further, we show how an appropriately calibrated generalized omnibus embedding can detect changes in real biological networks that previous embedding procedures could not discern, confirming that the effect of inherent and induced correlation can be subtle and transformative, with import in theory and practice.

stat.ME↗

arXiv2021

On a complete and sufficient statistic for the correlated Bernoulli random graph model

Inference on vertex-aligned graphs is of wide theoretical and practical importance.There are, however, few flexible and tractable statistical models for correlated graphs, and even fewer comprehensive approaches to parametric inference on data arising from such graphs. In this paper, we consider the correlated Bernoulli random graph model (allowing different Bernoulli coefficients and edge correlations for different pairs of vertices), and we introduce a new variance-reducing technique -- called \emph{balancing} -- that can refine estimators for model parameters. Specifically, we construct a disagreement statistic and show that it is complete and sufficient; balancing can be interpreted as Rao-Blackwellization with this disagreement statistic. We show that for unbiased estimators of functions of model parameters, balancing generates uniformly minimum variance unbiased estimators (UMVUEs). However, even when unbiased estimators for model parameters do {\em not} exist -- which, as we prove, is the case with both the heterogeneity correlation and the total correlation parameters -- balancing is still useful, and lowers mean squared error. In particular, we demonstrate how balancing can improve the efficiency of the alignment strength estimator for the total correlation, a parameter that plays a critical role in graph matchability and graph matching runtime complexity.

math.ST↗

arXiv2020

Inference for multiple heterogeneous networks with a common invariant subspace

The development of models for multiple heterogeneous network data is of critical importance both in statistical network theory and across multiple application domains. Although single-graph inference is well-studied, multiple graph inference is largely unexplored, in part because of the challenges inherent in appropriately modeling graph differences and yet retaining sufficient model simplicity to render estimation feasible. This paper addresses exactly this gap, by introducing a new model, the common subspace independent-edge (COSIE) multiple random graph model, which describes a heterogeneous collection of networks with a shared latent structure on the vertices but potentially different connectivity patterns for each graph. The COSIE model encompasses many popular network representations, including the stochastic blockmodel. The model is both flexible enough to meaningfully account for important graph differences and tractable enough to allow for accurate inference in multiple networks. In particular, a joint spectral embedding of adjacency matrices - the multiple adjacency spectral embedding (MASE) - leads, in a COSIE model, to simultaneous consistent estimation of underlying parameters for each graph. Under mild additional assumptions, MASE estimates satisfy asymptotic normality and yield improvements for graph eigenvalue estimation and hypothesis testing. In both simulated and real data, the COSIE model and the MASE embedding can be deployed for a number of subsequent network inference tasks, including dimensionality reduction, classification, hypothesis testing and community detection. Specifically, when MASE is applied to a dataset of connectomes constructed through diffusion magnetic resonance imaging, the result is an accurate classification of brain scans by patient and a meaningful determination of heterogeneity across scans of different subjects.

stat.ME↗

arXiv2020

On estimation and inference in latent structure random graphs

We define a latent structure model (LSM) random graph as a random dot product graph (RDPG) in which the latent position distribution incorporates both probabilistic and geometric constraints, delineated by a family of underlying distributions on some fixed Euclidean space, and a structural support submanifold from which the latent positions for the graph are drawn. For a one-dimensional latent structure model with known structural support, we show how spectral estimates of the latent positions of an RDPG can be used for efficient estimation of the paramaters of the LSM. We describe how to estimate or learn the structural support in cases where it is unknown, with an illustrative focus on graphs with latent positions along the Hardy-Weinberg curve. Finally, we use the latent structure model formulation to test bilateral homology in the Drosophila connectome.

stat.ME↗

arXiv2020

Numerical tolerance for spectral decompositions of random matrices

We precisely quantify the impact of statistical error in the quality of a numerical approximation to a random matrix eigendecomposition, and under mild conditions, we use this to introduce an optimal numerical tolerance for residual error in spectral decompositions of random matrices. We demonstrate that terminating an eigendecomposition algorithm when the numerical error and statistical error are of the same order results in computational savings with no loss of accuracy. We also repair a flaw in a ubiquitous termination condition, one in wide employ in several computational linear algebra implementations. We illustrate the practical consequences of our stopping criterion with an analysis of simulated and real networks. Our theoretical results and real-data examples establish that the tradeoff between statistical and numerical error is of significant import for data science.

stat.CO↗