arXiv subjects

Michael W. Trosset

Publications and source records attributed to Michael W. Trosset.

14 recordsLinked to original sources

arXiv2026

Out-of-Sample Embedding with Proximity Data: Projection versus Restricted Reconstruction

The problem of using proximity (similarity or dissimilarity) data for the purpose of "adding a point to a vector diagram" was first studied by J.C. Gower in 1968. Since then, a number of methods -- mostly kernel methods -- have been proposed for solving what has come to be called the problem of *out-of-sample embedding*. We survey the various kernel methods that we have encountered and show that each can be derived from one or the other of two competing strategies: *projection* or *restricted reconstruction*. Projection can be analogized to a well-known formula for adding a point to a principal component analysis. Restricted reconstruction poses a different challenge: how to best approximate redoing the entire multivariate analysis while holding fixed the vector diagram that was previously obtained. This strategy results in a nonlinear optimization problem that can be simplified to a unidimensional search. Various circumstances may warrant either projection or restricted reconstruction.

stat.ML↗

arXiv2026

Learning Submanifolds for Subsequent Inference on Random Dot Product Graphs, Part 1: Theory

We propose a framework for restricted inference on random dot product graphs whose latent positions lie on an unknown low-dimensional support manifold. For general decision problems, we propose semisupervised decision rules that use auxiliary data to learn the support manifold. Specifically, our rules use the Isomap manifold learning procedure to construct a low-dimensional Euclidean representation of the observed graph, in which space an isometrically invariant function maps configurations of points to actions. We study the behavior of the proposed rules as the quantity of auxiliary data sampled from the unknown support manifold increases. We show that, as the auxiliary sample size increases, the risk of the semisupervised rule converges to the risk of an oracle rule that relies on the maximal amount of low-dimensional Euclidean structure that can be extracted from the support manifold. Examples, applications, and simulation studies are deferred to a sequel.

stat.ML↗

arXiv2025

Consistent estimation of generative model representations in the data kernel perspective space

Generative models, such as large language models and text-to-image diffusion models, produce relevant information when presented a query. Different models may produce different information when presented the same query. As the landscape of generative models evolves, it is important to develop techniques to study and analyze differences in model behaviour. In this paper we present novel theoretical results for embedding-based representations of generative models in the context of a set of queries. In particular, we establish sufficient conditions for the consistent estimation of the model embeddings in situations where the query set and the number of models grow.

cs.LG↗

arXiv2025

Convergence guarantees for response prediction for latent structure network time series

In this article, we propose a technique to predict the response associated with an unlabeled time series of networks in a semisupervised setting. Our model involves a collection of time series of random networks of growing size, where some of the time series are associated with responses. Assuming that the collection of time series admits an unknown lower dimensional structure, our method exploits the underlying structure to consistently predict responses at the unlabeled time series of networks. Each time series represents a multilayer network on a common set of nodes, and raw stress embedding, a popular dimensionality reduction tool, is used for capturing the unknown latent low dimensional structure. Apart from establishing theoretical convergence guarantees and supporting them with numerical results, we demonstrate the use of our method in the analysis of real-world biological learning circuits of larval Drosophila.

stat.ME↗

arXiv2024

Continuous Multidimensional Scaling

Multidimensional scaling (MDS) is the act of embedding proximity information about a set of $n$ objects in $d$-dimensional Euclidean space. As originally conceived by the psychometric community, MDS was concerned with embedding a fixed set of proximities associated with a fixed set of objects. Modern concerns, e.g., that arise in developing asymptotic theories for statistical inference on random graphs, more typically involve studying the limiting behavior of a sequence of proximities associated with an increasing set of objects. Here we are concerned with embedding dissimilarities by minimizing Kruskal's (1964) raw stress criterion. Standard results from the theory of point-to-set maps can be used to establish that, if $n$ is fixed and a sequence of dissimilarity matrices converges, then the limit of their embedded structures is the embedded structure of the limiting dissimilarity matrix. But what if $n$ increases? It then becomes necessary to reformulate MDS so that the entire sequence of embedding problems can be viewed as a sequence of optimization problems in a fixed space. We present such a reformulation, {\em continuous MDS}. Within the continuous MDS framework, we derive two $L^p$ consistency results, one for embedding without constraints on the configuration, the other for embedding subject to {\em approximate Lipschitz constraints}\/ that encourage smoothness of the embedding function. The latter approach, {\em Approximate Lipschitz Embedding}\/ (ALE) is new. Finally, we demonstrate that embedded structures produced by ALE can be interpolated in a way that results in uniform convergence.

stat.ML↗

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

Popularity Adjusted Block Models are Generalized Random Dot Product Graphs

We connect two random graph models, the Popularity Adjusted Block Model (PABM) and the Generalized Random Dot Product Graph (GRDPG), by demonstrating that the PABM is a special case of the GRDPG in which communities correspond to mutually orthogonal subspaces of latent vectors. This insight allows us to construct new algorithms for community detection and parameter estimation for the PABM, as well as improve an existing algorithm that relies on Sparse Subspace Clustering. Using established asymptotic properties of Adjacency Spectral Embedding for the GRDPG, we derive asymptotic properties of these algorithms. In particular, we demonstrate that the absolute number of community detection errors tends to zero as the number of graph vertices tends to infinity. Simulation experiments illustrate these properties.

stat.ML↗

arXiv2021

Learning 1-Dimensional Submanifolds for Subsequent Inference on Random Dot Product Graphs

A random dot product graph (RDPG) is a generative model for networks in which vertices correspond to positions in a latent Euclidean space and edge probabilities are determined by the dot products of the latent positions. We consider RDPGs for which the latent positions are randomly sampled from an unknown $1$-dimensional submanifold of the latent space. In principle, restricted inference, i.e., procedures that exploit the structure of the submanifold, should be more effective than unrestricted inference; however, it is not clear how to conduct restricted inference when the submanifold is unknown. We submit that techniques for manifold learning can be used to learn the unknown submanifold well enough to realize benefit from restricted inference. To illustrate, we test $1$- and $2$-sample hypotheses about the Fréchet means of small communities of vertices, using the complete set of vertices to infer latent structure. We propose test statistics that deploy the Isomap procedure for manifold learning, using shortest path distances on neighborhood graphs constructed from estimated latent positions to estimate arc lengths on the unknown $1$-dimensional submanifold. Unlike conventional applications of Isomap, the estimated latent positions do not lie on the submanifold of interest. We extend existing convergence results for Isomap to this setting and use them to demonstrate that, as the number of auxiliary vertices increases, the power of our test converges to the power of the corresponding test when the submanifold is known. Finally, we apply our methods to an inference problem that arises in studying the connectome of the Drosophila larval mushroom body. The univariate learnt manifold test rejects ($p<0.05$), while the multivariate ambient space test does not ($p\gg0.05$), illustrating the value of identifying and exploiting low-dimensional structure for subsequent inference.

stat.ML↗

arXiv2021

Rehabilitating Isomap: Euclidean Representation of Geodesic Structure

Manifold learning techniques for nonlinear dimension reduction assume that high-dimensional feature vectors lie on a low-dimensional manifold, then attempt to exploit manifold structure to obtain useful low-dimensional Euclidean representations of the data. Isomap, a seminal manifold learning technique, is an elegant synthesis of two simple ideas: the approximation of Riemannian distances with shortest path distances on a graph that localizes manifold structure, and the approximation of shortest path distances with Euclidean distances by multidimensional scaling. We revisit the rationale for Isomap, clarifying what Isomap does and what it does not. In particular, we explore the widespread perception that Isomap should only be used when the manifold is parametrized by a convex region of Euclidean space. We argue that this perception is based on an extremely narrow interpretation of manifold learning as parametrization recovery, and we submit that Isomap is better understood as constructing Euclidean representations of geodesic structure. We reconsider a well-known example that was previously interpreted as evidence of Isomap's limitations, and we re-examine the original analysis of Isomap's convergence properties, concluding that convexity is not required for shortest path distances to converge to Riemannian distances.

stat.ML↗

arXiv2019

Approximate Information Tests on Statistical Submanifolds

Parametric inference posits a statistical model that is a specified family of probability distributions. Restricted inference, e.g., restricted likelihood ratio testing, attempts to exploit the structure of a statistical submodel that is a subset of the specified family. We consider the problem of testing a simple hypothesis against alternatives from such a submodel. In the case of an unknown submodel, it is not clear how to realize the benefits of restricted inference. To do so, we first construct information tests that are locally asymptotically equivalent to likelihood ratio tests. Information tests are conceptually appealing but (in general) computationally intractable. However, unlike restricted likelihood ratio tests, restricted information tests can be approximated even when the statistical submodel is unknown. We construct approximate information tests using manifold learning procedures to extract information from samples of an unknown (or intractable) submodel, thereby providing a roadmap for computational solutions to a class of previously impenetrable problems in statistical inference. Examples illustrate the efficacy of the proposed methodology.

math.ST↗

arXiv2017

Identifiability of two-component skew normal mixtures with one known component

We give sufficient identifiability conditions for estimating mixing proportions in two-component mixtures of skew normal distributions with one known component. We consider the univariate case as well as two multivariate extensions: a multivariate skew normal distribution (MSN) by Azzalini and Dalla Valle (1996) and the canonical fundamental skew normal distribution (CFUSN) by Arellano-Valle and Genton (2005). The characteristic function of the CFUSN distribution is additionally derived.

math.ST↗

arXiv2016

Fast Embedding for JOFC Using the Raw Stress Criterion

The Joint Optimization of Fidelity and Commensurability (JOFC) manifold matching methodology embeds an omnibus dissimilarity matrix consisting of multiple dissimilarities on the same set of objects. One approach to this embedding optimizes the preservation of fidelity to each individual dissimilarity matrix together with commensurability of each given observation across modalities via iterative majorization of a raw stress error criterion by successive Guttman transforms. In this paper, we exploit the special structure inherent to JOFC to exactly and efficiently compute the successive Guttman transforms, and as a result we are able to greatly speed up the JOFC procedure for both in-sample and out-of-sample embedding. We demonstrate the scalability of our implementation on both real and simulated data examples.

stat.ML↗

arXiv2016

On the Power of Likelihood Ratio Tests in Dimension-Restricted Submodels

Likelihood ratio tests are widely used to test statistical hypotheses about parametric families of probability distributions. If interest is restricted to a subfamily of distributions, then it is natural to inquire if the restricted LRT is superior to the unrestricted LRT. Marden's general LRT conjecture posits that any restriction placed on the alternative hypothesis will increase power. The only published counterexample to this conjecture is rather technical and involves a restriction that maintains the dimension of the alternative. We formulate the dimension-restricted LRT conjecture, which posits that any restriction that replaces a parametric family with a subfamily of lower dimension will increase power. Under standard regularity conditions, we then demonstrate that the restricted LRT is asymptotically more powerful than the unrestricted LRT for local alternatives. Remarkably, however, even the dimension-restricted LRT conjecture fails in the case of finite samples. Our counterexamples involve subfamilies of multinomial distributions. In particular, our study of the Hardy-Weinberg subfamily of trinomial distributions provides a simple and elegant demonstration that restrictions may not increase power.

math.ST↗

arXiv2016

Nonparametric semi-supervised learning of class proportions

The problem of developing binary classifiers from positive and unlabeled data is often encountered in machine learning. A common requirement in this setting is to approximate posterior probabilities of positive and negative classes for a previously unseen data point. This problem can be decomposed into two steps: (i) the development of accurate predictors that discriminate between positive and unlabeled data, and (ii) the accurate estimation of the prior probabilities of positive and negative examples. In this work we primarily focus on the latter subproblem. We study nonparametric class prior estimation and formulate this problem as an estimation of mixing proportions in two-component mixture models, given a sample from one of the components and another sample from the mixture itself. We show that estimation of mixing proportions is generally ill-defined and propose a canonical form to obtain identifiability while maintaining the flexibility to model any distribution. We use insights from this theory to elucidate the optimization surface of the class priors and propose an algorithm for estimating them. To address the problems of high-dimensional density estimation, we provide practical transformations to low-dimensional spaces that preserve class priors. Finally, we demonstrate the efficacy of our method on univariate and multivariate data.

stat.ML↗