arXiv subjects

Florentina Bunea

Publications and source records attributed to Florentina Bunea.

At least 19 recordsLinked to original sources

arXiv2026

The EM-algorithm and the Method of Moments in Softmax Mixture Models

Softmax Mixture Models (SMMs) are discrete $K$-component mixture models for the probabilities of selecting one of $p$ candidate feature vectors $X_1,\ldots,X_p\in\mathbb{R}^L$ in heterogeneous populations and are widely used in econometrics and scientific applications. Related softmax mixture mechanisms also appear in modern LLM architectures. We provide a theoretical and methodological study of SMMs, focusing on the Expectation-Maximization (EM) algorithm and the Method of Moments (MoM). We show that EM recovers the mixture atoms at the parametric rate, up to logarithmic factors, after $\mathcal{O}(\log N)$ iterations, provided atom separation is at least of order $\log K$. This improves on separation conditions in existing analyses of EM for high-dimensional Gaussian mixtures. We also develop MoM procedures for parameter and subspace estimation. Although MoM parameter estimates converge more slowly than EM and can deteriorate with $K$, they provide provable warm starts for EM and are useful for small $K$. For general $K$, we estimate the atom subspace via MoM and recommend running EM from multiple random initializations within this subspace. Finally, as $p\to\infty$, we show that SMMs approximate mixtures of exponential tilts of the feature distribution, yielding asymptotic identifiability.

stat.ML↗

arXiv2026

Fidelity-Diversity Metrics for Text

As language modeling technology matures, there is an increasing research focus on the composition and curation of datasets used to train these models. For instance, practitioners commonly seek to augment high-quality datasets with additional text to enhance the performance of models trained on that data. However, informed decisions about data augmentation require more nuanced assessments about data quality. We build on work measuring the precision and recall of generative models to develop a pair of metrics that quantify (1) fidelity, capturing how closely candidate text resembles reference data, and (2) diversity, capturing how well it covers the modes of the reference dataset. Our metrics are based on optimal transport divergence functionals between discrete text summaries. In experiments on M2D2 text datasets, we show that these metrics are able to disentangle a lack of fidelity from a lack of diversity in deficient candidate text. In further experiments, our metrics detect diversity deficits in synthetic GSM8K-style math datasets, which correlate with degradations in downstream accuracy of language models finetuned on this synthetic data.

cs.CL↗

arXiv2025

Beyond entropic regularization: Debiased Gaussian estimators for discrete optimal transport and general linear programs

This work proposes new estimators for discrete optimal transport plans that enjoy Gaussian limits centered at the true solution. This behavior stands in stark contrast with the performance of existing estimators, including those based on entropic regularization, which are asymptotically biased and only satisfy a CLT centered at a regularized version of the population-level plan. We develop a new regularization approach based on a different class of penalty functions, which can be viewed as the duals of those previously considered in the literature. The key feature of these penalty schemes it that they give rise to preliminary estimates that are asymptotically linear in the penalization strength. Our final estimator is obtained by constructing an appropriate linear combination of two penalized solutions corresponding to two different tuning parameters so that the bias introduced by the penalization cancels out. Unlike classical debiasing procedures, therefore, our proposal entirely avoids the delicate problem of estimating and then subtracting the estimated bias term. Our proofs, which apply beyond the case of optimal transport, are based on a novel asymptotic analysis of penalization schemes for linear programs. As a corollary of our results, we obtain the consistency of the naive bootstrap for fully data-driven inference on the true optimal solution. Simulation results and two data analyses support strongly the benefits of our approach relative to existing techniques.

math.ST↗

arXiv2024

Estimation and inference for the Wasserstein distance between mixing measures in topic models

The Wasserstein distance between mixing measures has come to occupy a central place in the statistical analysis of mixture models. This work proposes a new canonical interpretation of this distance and provides tools to perform inference on the Wasserstein distance between mixing measures in topic models. We consider the general setting of an identifiable mixture model consisting of mixtures of distributions from a set $\mathcal{A}$ equipped with an arbitrary metric $d$, and show that the Wasserstein distance between mixing measures is uniquely characterized as the most discriminative convex extension of the metric $d$ to the set of mixtures of elements of $\mathcal{A}$. The Wasserstein distance between mixing measures has been widely used in the study of such models, but without axiomatic justification. Our results establish this metric to be a canonical choice. Specializing our results to topic models, we consider estimation and inference of this distance. Though upper bounds for its estimation have been recently established elsewhere, we prove the first minimax lower bounds for the estimation of the Wasserstein distance in topic models. We also establish fully data-driven inferential tools for the Wasserstein distance in the topic model context. Our results apply to potentially sparse mixtures of high-dimensional discrete probability distributions. These results allow us to obtain the first asymptotically valid confidence intervals for the Wasserstein distance in topic models.

math.ST↗

arXiv2023

Asymptotic confidence sets for random linear programs

Motivated by the statistical analysis of the discrete optimal transport problem, we prove distributional limits for the solutions of linear programs with random constraints. Such limits were first obtained by Klatt, Munk, & Zemel (2022), but their expressions for the limits involve a computationally intractable decomposition of $\mathbb{R}^m$ into a possibly exponential number of convex cones. We give a new expression for the limit in terms of auxiliary linear programs, which can be solved in polynomial time. We also leverage tools from random convex geometry to give distributional limits for the entire set of random optimal solutions, when the optimum is not unique. Finally, we describe a simple, data-driven method to construct asymptotically valid confidence sets in polynomial time.

math.ST↗

arXiv2022

Likelihood estimation of sparse topic distributions in topic models and its applications to Wasserstein document distance calculations

This paper studies the estimation of high-dimensional, discrete, possibly sparse, mixture models in topic models. The data consists of observed multinomial counts of $p$ words across $n$ independent documents. In topic models, the $p\times n$ expected word frequency matrix is assumed to be factorized as a $p\times K$ word-topic matrix $A$ and a $K\times n$ topic-document matrix $T$. Since columns of both matrices represent conditional probabilities belonging to probability simplices, columns of $A$ are viewed as $p$-dimensional mixture components that are common to all documents while columns of $T$ are viewed as the $K$-dimensional mixture weights that are document specific and are allowed to be sparse. The main interest is to provide sharp, finite sample, $\ell_1$-norm convergence rates for estimators of the mixture weights $T$ when $A$ is either known or unknown. For known $A$, we suggest MLE estimation of $T$. Our non-standard analysis of the MLE not only establishes its $\ell_1$ convergence rate, but reveals a remarkable property: the MLE, with no extra regularization, can be exactly sparse and contain the true zero pattern of $T$. We further show that the MLE is both minimax optimal and adaptive to the unknown sparsity in a large class of sparse topic distributions. When $A$ is unknown, we estimate $T$ by optimizing the likelihood function corresponding to a plug in, generic, estimator $\hat{A}$ of $A$. For any estimator $\hat{A}$ that satisfies carefully detailed conditions for proximity to $A$, the resulting estimator of $T$ is shown to retain the properties established for the MLE. The ambient dimensions $K$ and $p$ are allowed to grow with the sample sizes. Our application is to the estimation of 1-Wasserstein distances between document generating distributions. We propose, estimate and analyze new 1-Wasserstein distances between two probabilistic document representations.

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arXiv2021

Prediction in latent factor regression: Adaptive PCR and beyond

This work is devoted to the finite sample prediction risk analysis of a class of linear predictors of a response $Y\in \mathbb{R}$ from a high-dimensional random vector $X\in \mathbb{R}^p$ when $(X,Y)$ follows a latent factor regression model generated by a unobservable latent vector $Z$ of dimension less than $p$. Our primary contribution is in establishing finite sample risk bounds for prediction with the ubiquitous Principal Component Regression (PCR) method, under the factor regression model, with the number of principal components adaptively selected from the data -- a form of theoretical guarantee that is surprisingly lacking from the PCR literature. To accomplish this, we prove a master theorem that establishes a risk bound for a large class of predictors, including the PCR predictor as a special case. This approach has the benefit of providing a unified framework for the analysis of a wide range of linear prediction methods, under the factor regression setting. In particular, we use our main theorem to recover known risk bounds for the minimum-norm interpolating predictor, which has received renewed attention in the past two years, and a prediction method tailored to a subclass of factor regression models with identifiable parameters. This model-tailored method can be interpreted as prediction via clusters with latent centers. To address the problem of selecting among a set of candidate predictors, we analyze a simple model selection procedure based on data-splitting, providing an oracle inequality under the factor model to prove that the performance of the selected predictor is close to the optimal candidate. We conclude with a detailed simulation study to support and complement our theoretical results.

stat.ML↗

arXiv2021

Interpolating Predictors in High-Dimensional Factor Regression

This work studies finite-sample properties of the risk of the minimum-norm interpolating predictor in high-dimensional regression models. If the effective rank of the covariance matrix $Σ$ of the $p$ regression features is much larger than the sample size $n$, we show that the min-norm interpolating predictor is not desirable, as its risk approaches the risk of trivially predicting the response by 0. However, our detailed finite-sample analysis reveals, surprisingly, that this behavior is not present when the regression response and the features are {\it jointly} low-dimensional, following a widely used factor regression model. Within this popular model class, and when the effective rank of $Σ$ is smaller than $n$, while still allowing for $p \gg n$, both the bias and the variance terms of the excess risk can be controlled, and the risk of the minimum-norm interpolating predictor approaches optimal benchmarks. Moreover, through a detailed analysis of the bias term, we exhibit model classes under which our upper bound on the excess risk approaches zero, while the corresponding upper bound in the recent work arXiv:1906.11300 diverges. Furthermore, we show that the minimum-norm interpolating predictor analyzed under the factor regression model, despite being model-agnostic and devoid of tuning parameters, can have similar risk to predictors based on principal components regression and ridge regression, and can improve over LASSO based predictors, in the high-dimensional regime.

stat.ML↗

arXiv2021

Inference in latent factor regression with clusterable features

Regression models, in which the observed features $X \in \R^p$ and the response $Y \in \R$ depend, jointly, on a lower dimensional, unobserved, latent vector $Z \in \R^K$, with $K< p$, are popular in a large array of applications, and mainly used for predicting a response from correlated features. In contrast, methodology and theory for inference on the regression coefficient $β$ relating $Y$ to $Z$ are scarce, since typically the un-observable factor $Z$ is hard to interpret. Furthermore, the determination of the asymptotic variance of an estimator of $β$ is a long-standing problem, with solutions known only in a few particular cases. To address some of these outstanding questions, we develop inferential tools for $β$ in a class of factor regression models in which the observed features are signed mixtures of the latent factors. The model specifications are practically desirable, in a large array of applications, render interpretability to the components of $Z$, and are sufficient for parameter identifiability. Without assuming that the number of latent factors $K$ or the structure of the mixture is known in advance, we construct computationally efficient estimators of $β$, along with estimators of other important model parameters. We benchmark the rate of convergence of $β$ by first establishing its $\ell_2$-norm minimax lower bound, and show that our proposed estimator is minimax-rate adaptive. Our main contribution is the provision of a unified analysis of the component-wise Gaussian asymptotic distribution of $\wh β$ and, especially, the derivation of a closed form expression of its asymptotic variance, together with consistent variance estimators. The resulting inferential tools can be used when both $K$ and $p$ are independent of the sample size $n$, and when both, or either, $p$ and $K$ vary with $n$, while allowing for $p > n$.

stat.ME↗

arXiv2020

Detecting approximate replicate components of a high-dimensional random vector with latent structure

High-dimensional feature vectors are likely to contain sets of measurements that are approximate replicates of one another. In complex applications, or automated data collection, these feature sets are not known a priori, and need to be determined. This work proposes a class of latent factor models on the observed high-dimensional random vector $X \in \mathbb{R}^p$, for defining, identifying and estimating the index set of its approximately replicate components. The model class is parametrized by a $p \times K$ loading matrix $A$ that contains a hidden sub-matrix whose rows can be partitioned into groups of parallel vectors. Under this model class, a set of approximate replicate components of $X$ corresponds to a set of parallel rows in $A$: these entries of $X$ are, up to scale and additive error, the same linear combination of the $K$ latent factors; the value of $K$ is itself unknown. The problem of finding approximate replicates in $X$ reduces to identifying, and estimating, the location of the hidden sub-matrix within $A$, and of the partition of its row index set $H$. Both $H$ and its partiton can be fully characterized in terms of a new family of criteria based on the correlation matrix of $X$, and their identifiability, as well as that of the unknown latent dimension $K$, are obtained as consequences. The constructive nature of the identifiability arguments enables computationally efficient procedures, with consistency guarantees. When $A$ has the errors-in-variable parametrization, the difficulty of the problem is elevated. The task becomes that of separating out groups of parallel rows that are proportional to canonical basis vectors from other dense parallel rows in $A$. This is met under a scale assumption, via a principled way of selecting the target row indices, guided by the succesive maximization of Schur complements of appropriate covariance matrices.

stat.ME↗

arXiv2020

High-Dimensional Inference for Cluster-Based Graphical Models

Motivated by modern applications in which one constructs graphical models based on a very large number of features, this paper introduces a new class of cluster-based graphical models, in which variable clustering is applied as an initial step for reducing the dimension of the feature space. We employ model assisted clustering, in which the clusters contain features that are similar to the same unobserved latent variable. Two different cluster-based Gaussian graphical models are considered: the latent variable graph, corresponding to the graphical model associated with the unobserved latent variables, and the cluster-average graph, corresponding to the vector of features averaged over clusters. Our study reveals that likelihood based inference for the latent graph, not analyzed previously, is analytically intractable. Our main contribution is the development and analysis of alternative estimation and inference strategies, for the precision matrix of an unobservable latent vector $Z$. We replace the likelihood of the data by an appropriate class of empirical risk functions, that can be specialized to the latent graphical model and to the simpler, but under-analyzed, cluster-average graphical model. The estimators thus derived can be used for inference on the graph structure, for instance on edge strength or pattern recovery. Inference is based on the asymptotic limits of the entry-wise estimates of the precision matrices associated with the conditional independence graphs under consideration. While taking the uncertainty induced by the clustering step into account, we establish Berry-Esseen central limit theorems for the proposed estimators. It is noteworthy that, although the clusters are estimated adaptively from the data, the central limit theorems regarding the entries of the estimated graphs are proved under the same conditions one would use if the clusters were known....

stat.ML↗

arXiv2020

Optimal estimation of sparse topic models

Topic models have become popular tools for dimension reduction and exploratory analysis of text data which consists in observed frequencies of a vocabulary of $p$ words in $n$ documents, stored in a $p\times n$ matrix. The main premise is that the mean of this data matrix can be factorized into a product of two non-negative matrices: a $p\times K$ word-topic matrix $A$ and a $K\times n$ topic-document matrix $W$. This paper studies the estimation of $A$ that is possibly element-wise sparse, and the number of topics $K$ is unknown. In this under-explored context, we derive a new minimax lower bound for the estimation of such $A$ and propose a new computationally efficient algorithm for its recovery. We derive a finite sample upper bound for our estimator, and show that it matches the minimax lower bound in many scenarios. Our estimate adapts to the unknown sparsity of $A$ and our analysis is valid for any finite $n$, $p$, $K$ and document lengths. Empirical results on both synthetic data and semi-synthetic data show that our proposed estimator is a strong competitor of the existing state-of-the-art algorithms for both non-sparse $A$ and sparse $A$, and has superior performance is many scenarios of interest.

stat.ML↗

arXiv2019

A fast algorithm with minimax optimal guarantees for topic models with an unknown number of topics

We propose a new method of estimation in topic models, that is not a variation on the existing simplex finding algorithms, and that estimates the number of topics K from the observed data. We derive new finite sample minimax lower bounds for the estimation of A, as well as new upper bounds for our proposed estimator. We describe the scenarios where our estimator is minimax adaptive. Our finite sample analysis is valid for any number of documents (n), individual document length (N_i), dictionary size (p) and number of topics (K), and both p and K are allowed to increase with n, a situation not handled well by previous analyses. We complement our theoretical results with a detailed simulation study. We illustrate that the new algorithm is faster and more accurate than the current ones, although we start out with a computational and theoretical disadvantage of not knowing the correct number of topics K, while we provide the competing methods with the correct value in our simulations.

stat.ML↗

arXiv2019

Adaptive Estimation in Structured Factor Models with Applications to Overlapping Clustering

This work introduces a novel estimation method, called LOVE, of the entries and structure of a loading matrix A in a sparse latent factor model X = AZ + E, for an observable random vector X in Rp, with correlated unobservable factors Z \in RK, with K unknown, and independent noise E. Each row of A is scaled and sparse. In order to identify the loading matrix A, we require the existence of pure variables, which are components of X that are associated, via A, with one and only one latent factor. Despite the fact that the number of factors K, the number of the pure variables, and their location are all unknown, we only require a mild condition on the covariance matrix of Z, and a minimum of only two pure variables per latent factor to show that A is uniquely defined, up to signed permutations. Our proofs for model identifiability are constructive, and lead to our novel estimation method of the number of factors and of the set of pure variables, from a sample of size n of observations on X. This is the first step of our LOVE algorithm, which is optimization-free, and has low computational complexity of order p2. The second step of LOVE is an easily implementable linear program that estimates A. We prove that the resulting estimator is minimax rate optimal up to logarithmic factors in p. The model structure is motivated by the problem of overlapping variable clustering, ubiquitous in data science. We define the population level clusters as groups of those components of X that are associated, via the sparse matrix A, with the same unobservable latent factor, and multi-factor association is allowed. Clusters are respectively anchored by the pure variables, and form overlapping sub-groups of the p-dimensional random vector X. The Latent model approach to OVErlapping clustering is reflected in the name of our algorithm, LOVE.

stat.ME↗

arXiv2018

Model Assisted Variable Clustering: Minimax-optimal Recovery and Algorithms

Model-based clustering defines population level clusters relative to a model that embeds notions of similarity. Algorithms tailored to such models yield estimated clusters with a clear statistical interpretation. We take this view here and introduce the class of G-block covariance models as a background model for variable clustering. In such models, two variables in a cluster are deemed similar if they have similar associations will all other variables. This can arise, for instance, when groups of variables are noise corrupted versions of the same latent factor. We quantify the difficulty of clustering data generated from a G-block covariance model in terms of cluster proximity, measured with respect to two related, but different, cluster separation metrics. We derive minimax cluster separation thresholds, which are the metric values below which no algorithm can recover the model-defined clusters exactly, and show that they are different for the two metrics. We therefore develop two algorithms, COD and PECOK, tailored to G-block covariance models, and study their minimax-optimality with respect to each metric. Of independent interest is the fact that the analysis of the PECOK algorithm, which is based on a corrected convex relaxation of the popular K-means algorithm, provides the first statistical analysis of such algorithms for variable clustering. Additionally, we contrast our methods with another popular clustering method, spectral clustering, specialized to variable clustering, and show that ensuring exact cluster recovery via this method requires clusters to have a higher separation, relative to the minimax threshold. Extensive simulation studies, as well as our data analyses, confirm the applicability of our approach.

stat.ME↗

arXiv2016

PECOK: a convex optimization approach to variable clustering

The problem of variable clustering is that of grouping similar components of a $p$-dimensional vector $X=(X_{1},\ldots,X_{p})$, and estimating these groups from $n$ independent copies of $X$. When cluster similarity is defined via $G$-latent models, in which groups of $X$-variables have a common latent generator, and groups are relative to a partition $G$ of the index set $\{1, \ldots, p\}$, the most natural clustering strategy is $K$-means. We explain why this strategy cannot lead to perfect cluster recovery and offer a correction, based on semi-definite programing, that can be viewed as a penalized convex relaxation of $K$-means (PECOK). We introduce a cluster separation measure tailored to $G$-latent models, and derive its minimax lower bound for perfect cluster recovery. The clusters estimated by PECOK are shown to recover $G$ at a near minimax optimal cluster separation rate, a result that holds true even if $K$, the number of clusters, is estimated adaptively from the data. We compare PECOK with appropriate corrections of spectral clustering-type procedures, and show that the former outperforms the latter for perfect cluster recovery of minimally separated clusters.

math.ST↗

arXiv2015

On the sample covariance matrix estimator of reduced effective rank population matrices, with applications to fPCA

This work provides a unified analysis of the properties of the sample covariance matrix $Σ_n$ over the class of $p\times p$ population covariance matrices $Σ$ of reduced effective rank $r_e(Σ)$. This class includes scaled factor models and covariance matrices with decaying spectrum. We consider $r_e(Σ)$ as a measure of matrix complexity, and obtain sharp minimax rates on the operator and Frobenius norm of $Σ_n-Σ$, as a function of $r_e(Σ)$ and $\|Σ\|_2$, the operator norm of $Σ$. With guidelines offered by the optimal rates, we define classes of matrices of reduced effective rank over which $Σ_n$ is an accurate estimator. Within the framework of these classes, we perform a detailed finite sample theoretical analysis of the merits and limitations of the empirical scree plot procedure routinely used in PCA. We show that identifying jumps in the empirical spectrum that consistently estimate jumps in the spectrum of $Σ$ is not necessarily informative for other goals, for instance for the selection of those sample eigenvalues and eigenvectors that are consistent estimates of their population counterparts. The scree plot method can still be used for selecting consistent eigenvalues, for appropriate threshold levels. We provide a threshold construction and also give a rule for checking the consistency of the corresponding sample eigenvectors. We specialize these results and analysis to population covariance matrices with polynomially decaying spectra, and extend it to covariance operators with polynomially decaying spectra. An application to fPCA illustrates how our results can be used in functional data analysis.

math.ST↗

arXiv2014

Convex Banding of the Covariance Matrix

We introduce a new sparse estimator of the covariance matrix for high-dimensional models in which the variables have a known ordering. Our estimator, which is the solution to a convex optimization problem, is equivalently expressed as an estimator which tapers the sample covariance matrix by a Toeplitz, sparsely-banded, data-adaptive matrix. As a result of this adaptivity, the convex banding estimator enjoys theoretical optimality properties not attained by previous banding or tapered estimators. In particular, our convex banding estimator is minimax rate adaptive in Frobenius and operator norms, up to log factors, over commonly-studied classes of covariance matrices, and over more general classes. Furthermore, it correctly recovers the bandwidth when the true covariance is exactly banded. Our convex formulation admits a simple and efficient algorithm. Empirical studies demonstrate its practical effectiveness and illustrate that our exactly-banded estimator works well even when the true covariance matrix is only close to a banded matrix, confirming our theoretical results. Our method compares favorably with all existing methods, in terms of accuracy and speed. We illustrate the practical merits of the convex banding estimator by showing that it can be used to improve the performance of discriminant analysis for classifying sound recordings.

math.ST↗