arXiv subjects

Kenneth Lange

Publications and source records attributed to Kenneth Lange.

At least 19 recordsLinked to original sources

arXiv2025

Tactics for Improving Least Squares Estimation

This paper deals with tactics for fast computation in least squares regression in high dimensions. These tactics include: (a) the majorization-minimization (MM) principle, (b) smoothing by Moreau envelopes, and (c) the proximal distance principle for constrained estimation. In iteratively reweighted least squares, the MM principle can create a surrogate function that trades case weights for adjusted responses. Reduction to ordinary least squares then permits the reuse of the Gram matrix and its Cholesky decomposition across iterations. This tactic is pertinent to estimation in L2E regression and generalized linear models. For problems such as quantile regression, non-smooth terms of an objective function can be replaced by their Moreau envelope approximations and majorized by spherical quadratics. Finally, penalized regression with distance-to-set penalties also benefits from this perspective. Our numerical experiments validate the speed and utility of deweighting and Moreau envelope approximations. Julia software implementing these experiments is available on our web page.

stat.CO↗

arXiv2024

A Stability Framework for Parameter Selection in the Minimum Covariance Determinant Problem

The Minimum Covariance Determinant (MCD) method is a widely adopted tool for robust estimation and outlier detection. In this paper, we introduce MCD model selection based on the notion of stability. Our best subset method leverages prior best practices such as statistical depths for initialization and concentration steps for subset refinement. Our contribution lies in constructing a bootstrap procedure to estimate the instability of the best subset algorithm. The instability path offers insights into a dataset's inlier/outlier structure and facilitates suitable choice of the subset size. We rigorously benchmark the proposed framework against existing MCD variants and illustrate its practical utility on several real-world datasets.

stat.ME↗

arXiv2022

The upper-crossing/solution (US) algorithm for root-finding with strongly stable convergence

In this paper, we propose a new and broadly applicable root-finding method, called as the upper-crossing/solution (US) algorithm, which belongs to the category of non-bracketing (or open domain) methods. The US algorithm is a general principle for iteratively seeking the unique root $\theta^{*}$ of a non-linear equation $g(\theta)=0$ and its each iteration consists of two steps: an upper-crossing step (U-step) and a solution step (S-step), where the U-step finds an upper-crossing function or a $U$-function $U(\theta|\theta^{(t)})$ [whose form depends on $\theta^{(t)}$ being the $t$-th iteration of $\theta^{*}$] based on a new notion of so-called changing direction inequality, and the S-step solves the simple $U$-equation $U(\theta|\theta^{(t)}) =0$ to obtain its explicit solution $\theta^{(t+1)}$. The US algorithm holds two major advantages: (i) It strongly stably converges to the root $\theta^{*}$; and (ii) it does not depend on any initial values, in contrast to Newton's method. The key step for applying the US algorithm is to construct one simple $U$-function $U(\theta|\theta^{(t)})$ such that an explicit solution to the $U$-equation $U(\theta|\theta^{(t)}) =0$ is available. Based on the first-, second- and third-derivative of $g(\theta)$, three methods are given for constructing such $U$-functions. We show various applications of the US algorithm in such as calculating quantile in continuous distributions, calculating exact $p$-values for skew null distributions, and finding maximum likelihood estimates of parameters in a class of continuous/discrete distributions. The analysis of the convergence rate of the US algorithm and some numerical experiments are also provided. Especially, because of the property of strongly stable convergence, the US algorithm could be one of the powerful tools for solving an equation with multiple roots.

math.NA↗

arXiv2022

A Flexible Quasi-Copula Distribution for Statistical Modeling

Copulas, generalized estimating equations, and generalized linear mixed models promote the analysis of grouped data where non-normal responses are correlated. Unfortunately, parameter estimation remains challenging in these three frameworks. Based on prior work of Tonda, we derive a new class of probability density functions that allow explicit calculation of moments, marginal and conditional distributions, and the score and observed information needed in maximum likelihood estimation. We also illustrate how the new distribution flexibly models longitudinal data following a non-Gaussian distribution. Finally, we conduct a tri-variate genome-wide association analysis on dichotomized systolic and diastolic blood pressure and body mass index data from the UK-Biobank, showcasing the modeling prowess and computational scalability of the new distribution.

stat.ME↗

arXiv2022

Feature Selection for Vertex Discriminant Analysis

We revisit vertex discriminant analysis (VDA) from the perspective of proximal distance algorithms. By specifying sparsity sets as constraints that directly control the number of active features, VDA is able to fit multiclass classifiers with no more than $k$ active features. We combine our sparse VDA approach with repeated cross validation to fit classifiers across the full range of model sizes on a given dataset. Our numerical examples demonstrate that grappling with sparsity directly is an attractive approach to model building in high-dimensional settings. Applications to kernel-based VDA are also considered.

stat.CO↗

arXiv2022

A Sharper Computational Tool for $\text{L}_2\text{E}$ Regression

Building on previous research of Chi and Chi (2022), the current paper revisits estimation in robust structured regression under the $\text{L}_2\text{E}$ criterion. We adopt the majorization-minimization (MM) principle to design a new algorithm for updating the vector of regression coefficients. Our sharp majorization achieves faster convergence than the previous alternating proximal gradient descent algorithm (Chi and Chi, 2022). In addition, we reparameterize the model by substituting precision for scale and estimate precision via a modified Newton's method. This simplifies and accelerates overall estimation. We also introduce distance-to-set penalties to allow constrained estimation under nonconvex constraint sets. This tactic also improves performance in coefficient estimation and structure recovery. Finally, we demonstrate the merits of our improved tactics through a rich set of simulation examples and a real data application.

stat.ME↗

arXiv2022

A unified analysis of convex and non-convex lp-ball projection problems

The task of projecting onto $\ell_p$ norm balls is ubiquitous in statistics and machine learning, yet the availability of actionable algorithms for doing so is largely limited to the special cases of $p = \left\{ 0, 1,2, \infty \right\}$. In this paper, we introduce novel, scalable methods for projecting onto the $\ell_p$ ball for general $p>0$. For $p \geq1 $, we solve the univariate Lagrangian dual via a dual Newton method. We then carefully design a bisection approach for $p<1$, presenting theoretical and empirical evidence of zero or a small duality gap in the non-convex case. The success of our contributions is thoroughly assessed empirically, and applied to large-scale regularized multi-task learning and compressed sensing.

math.OC↗

arXiv2021

Algorithms for Sparse Support Vector Machines

Many problems in classification involve huge numbers of irrelevant features. Model selection reveals the crucial features, reduces the dimensionality of feature space, and improves model interpretation. In the support vector machine literature, model selection is achieved by $\ell_1$ penalties. These convex relaxations seriously bias parameter estimates toward 0 and tend to admit too many irrelevant features. The current paper presents an alternative that replaces penalties by sparse-set constraints. Penalties still appear, but serve a different purpose. The proximal distance principle takes a loss function $L(\boldsymbol{\beta})$ and adds the penalty $\frac{\rho}{2}\mathrm{dist}(\boldsymbol{\beta}, S_k)^2$ capturing the squared Euclidean distance of the parameter vector $\boldsymbol{\beta}$ to the sparsity set $S_k$ where at most $k$ components of $\boldsymbol{\beta}$ are nonzero. If $\boldsymbol{\beta}_\rho$ represents the minimum of the objective $f_\rho(\boldsymbol{\beta})=L(\boldsymbol{\beta})+\frac{\rho}{2}\mathrm{dist}(\boldsymbol{\beta}, S_k)^2$, then $\boldsymbol{\beta}_\rho$ tends to the constrained minimum of $L(\boldsymbol{\beta})$ over $S_k$ as $\rho$ tends to $\infty$. We derive two closely related algorithms to carry out this strategy. Our simulated and real examples vividly demonstrate how the algorithms achieve much better sparsity without loss of classification power.

stat.ME↗

arXiv2021

A Proximal Distance Algorithm for Likelihood-Based Sparse Covariance Estimation

This paper addresses the task of estimating a covariance matrix under a patternless sparsity assumption. In contrast to existing approaches based on thresholding or shrinkage penalties, we propose a likelihood-based method that regularizes the distance from the covariance estimate to a symmetric sparsity set. This formulation avoids unwanted shrinkage induced by more common norm penalties and enables optimization of the resulting non-convex objective by solving a sequence of smooth, unconstrained subproblems. These subproblems are generated and solved via the proximal distance version of the majorization-minimization principle. The resulting algorithm executes rapidly, gracefully handles settings where the number of parameters exceeds the number of cases, yields a positive definite solution, and enjoys desirable convergence properties. Empirically, we demonstrate that our approach outperforms competing methods by several metrics across a suite of simulated experiments. Its merits are illustrated on an international migration dataset and a classic case study on flow cytometry. Our findings suggest that the marginal and conditional dependency networks for the cell signalling data are more similar than previously concluded.

stat.ME↗

arXiv2021

Nonconvex Optimization via MM Algorithms: Convergence Theory

The majorization-minimization (MM) principle is an extremely general framework for deriving optimization algorithms. It includes the expectation-maximization (EM) algorithm, proximal gradient algorithm, concave-convex procedure, quadratic lower bound algorithm, and proximal distance algorithm as special cases. Besides numerous applications in statistics, optimization, and imaging, the MM principle finds wide applications in large scale machine learning problems such as matrix completion, discriminant analysis, and nonnegative matrix factorizations. When applied to nonconvex optimization problems, MM algorithms enjoy the advantages of convexifying the objective function, separating variables, numerical stability, and ease of implementation. However, compared to the large body of literature on other optimization algorithms, the convergence analysis of MM algorithms is scattered and problem specific. This survey presents a unified treatment of the convergence of MM algorithms. With modern applications in mind, the results encompass non-smooth objective functions and non-asymptotic analysis.

math.OC↗

arXiv2021

Poisson Phase Retrieval in Very Low-count Regimes

This paper discusses phase retrieval algorithms for maximum likelihood (ML) estimation from measurements following independent Poisson distributions in very low-count regimes, e.g., 0.25 photon per pixel. To maximize the log-likelihood of the Poisson ML model, we propose a modified Wirtinger flow (WF) algorithm using a step size based on the observed Fisher information. This approach eliminates all parameter tuning except the number of iterations. We also propose a novel curvature for majorize-minimize (MM) algorithms with a quadratic majorizer. We show theoretically that our proposed curvature is sharper than the curvature derived from the supremum of the second derivative of the Poisson ML cost function. We compare the proposed algorithms (WF, MM) with existing optimization methods, including WF using other step-size schemes, quasi-Newton methods such as LBFGS and alternating direction method of multipliers (ADMM) algorithms, under a variety of experimental settings. Simulation experiments with a random Gaussian matrix, a canonical DFT matrix, a masked DFT matrix and an empirical transmission matrix demonstrate the following. 1) As expected, algorithms based on the Poisson ML model consistently produce higher quality reconstructions than algorithms derived from Gaussian noise ML models when applied to low-count data. 2) For unregularized cases, our proposed WF algorithm with Fisher information for step size converges faster than other WF methods, e.g., WF with empirical step size, backtracking line search, and optimal step size for the Gaussian noise model; it also converges faster than the LBFGS quasi-Newton method. 3) In regularized cases, our proposed WF algorithm converges faster than WF with backtracking line search, LBFGS, MM and ADMM.

cs.IT↗

arXiv2020

Extensions to the Proximal Distance Method of Constrained Optimization

The current paper studies the problem of minimizing a loss $f(\boldsymbol{x})$ subject to constraints of the form $\boldsymbol{D}\boldsymbol{x} \in S$, where $S$ is a closed set, convex or not, and $\boldsymbol{D}$ is a matrix that fuses parameters. Fusion constraints can capture smoothness, sparsity, or more general constraint patterns. To tackle this generic class of problems, we combine the Beltrami-Courant penalty method with the proximal distance principle. The latter is driven by minimization of penalized objectives $f(\boldsymbol{x})+\frac{\rho}{2}\text{dist}(\boldsymbol{D}\boldsymbol{x},S)^2$ involving large tuning constants $\rho$ and the squared Euclidean distance of $\boldsymbol{D}\boldsymbol{x}$ from $S$. The next iterate $\boldsymbol{x}_{n+1}$ of the corresponding proximal distance algorithm is constructed from the current iterate $\boldsymbol{x}_n$ by minimizing the majorizing surrogate function $f(\boldsymbol{x})+\frac{\rho}{2}\|\boldsymbol{D}\boldsymbol{x}-\mathcal{P}_{S}(\boldsymbol{D}\boldsymbol{x}_n)\|^2$. For fixed $\rho$ and a subanalytic loss $f(\boldsymbol{x})$ and a subanalytic constraint set $S$, we prove convergence to a stationary point. Under stronger assumptions, we provide convergence rates and demonstrate linear local convergence. We also construct a steepest descent (SD) variant to avoid costly linear system solves. To benchmark our algorithms, we compare against the alternating direction method of multipliers (ADMM). Our extensive numerical tests include problems on metric projection, convex regression, convex clustering, total variation image denoising, and projection of a matrix to a good condition number. These experiments demonstrate the superior speed and acceptable accuracy of our steepest variant on high-dimensional problems.

math.OC↗

arXiv2020

Simple and Scalable Sparse k-means Clustering via Feature Ranking

Clustering, a fundamental activity in unsupervised learning, is notoriously difficult when the feature space is high-dimensional. Fortunately, in many realistic scenarios, only a handful of features are relevant in distinguishing clusters. This has motivated the development of sparse clustering techniques that typically rely on k-means within outer algorithms of high computational complexity. Current techniques also require careful tuning of shrinkage parameters, further limiting their scalability. In this paper, we propose a novel framework for sparse k-means clustering that is intuitive, simple to implement, and competitive with state-of-the-art algorithms. We show that our algorithm enjoys consistency and convergence guarantees. Our core method readily generalizes to several task-specific algorithms such as clustering on subsets of attributes and in partially observed data settings. We showcase these contributions thoroughly via simulated experiments and real data benchmarks, including a case study on protein expression in trisomic mice.

stat.ML↗

arXiv2019

OPENMENDEL: A Cooperative Programming Project for Statistical Genetics

Statistical methods for genomewide association studies (GWAS) continue to improve. However, the increasing volume and variety of genetic and genomic data make computational speed and ease of data manipulation mandatory in future software. In our view, a collaborative effort of statistical geneticists is required to develop open source software targeted to genetic epidemiology. Our attempt to meet this need is called the OPENMENDELproject (https://openmendel.github.io). It aims to (1) enable interactive and reproducible analyses with informative intermediate results, (2) scale to big data analytics, (3) embrace parallel and distributed computing, (4) adapt to rapid hardware evolution, (5) allow cloud computing, (6) allow integration of varied genetic data types, and (7) foster easy communication between clinicians, geneticists, statisticians, and computer scientists. This article reviews and makes recommendations to the genetic epidemiology community in the context of the OPENMENDEL project.

stat.AP↗

arXiv2018

BioSimulator.jl: Stochastic simulation in Julia

Biological systems with intertwined feedback loops pose a challenge to mathematical modeling efforts. Moreover, rare events, such as mutation and extinction, complicate system dynamics. Stochastic simulation algorithms are useful in generating time-evolution trajectories for these systems because they can adequately capture the influence of random fluctuations and quantify rare events. We present a simple and flexible package, BioSimulator.jl, for implementing the Gillespie algorithm, $\tau$-leaping, and related stochastic simulation algorithms. The objective of this work is to provide scientists across domains with fast, user-friendly simulation tools. We used the high-performance programming language Julia because of its emphasis on scientific computing. Our software package implements a suite of stochastic simulation algorithms based on Markov chain theory. We provide the ability to (a) diagram Petri Nets describing interactions, (b) plot average trajectories and attached standard deviations of each participating species over time, and (c) generate frequency distributions of each species at a specified time. BioSimulator.jl's interface allows users to build models programmatically within Julia. A model is then passed to the simulate routine to generate simulation data. The built-in tools allow one to visualize results and compute summary statistics. Our examples highlight the broad applicability of our software to systems of varying complexity from ecology, systems biology, chemistry, and genetics. The user-friendly nature of BioSimulator.jl encourages the use of stochastic simulation, minimizes tedious programming efforts, and reduces errors during model specification.

q-bio.QM↗

arXiv2018

Orthogonal Trace-Sum Maximization: Applications, Local Algorithms, and Global Optimality

This paper studies the problem of maximizing the sum of traces of matrix quadratic forms on a product of Stiefel manifolds. This orthogonal trace-sum maximization (OTSM) problem generalizes many interesting problems such as generalized canonical correlation analysis (CCA), Procrustes analysis, and cryo-electron microscopy of the Nobel prize fame. For these applications finding global solutions is highly desirable but it has been unclear how to find even a stationary point, let alone testing its global optimality. Through a close inspection of Ky Fan's classical result (1949) on the variational formulation of the sum of largest eigenvalues of a symmetric matrix, and a semidefinite programming (SDP) relaxation of the latter, we first provide a simple method to certify global optimality of a given stationary point of OTSM. This method only requires testing whether a symmetric matrix is positive semidefinite. A by-product of this analysis is an unexpected strong duality between Shapiro-Botha (1988) and Zhang-Singer (2017). After showing that a popular algorithm for generalized CCA and Procrustes analysis may generate oscillating iterates, we propose a simple fix that provably guarantees convergence to a stationary point. The combination of our algorithm and certificate reveals novel global optima of various instances of OTSM.

math.OC↗

arXiv2017

Generalized Linear Model Regression under Distance-to-set Penalties

Estimation in generalized linear models (GLM) is complicated by the presence of constraints. One can handle constraints by maximizing a penalized log-likelihood. Penalties such as the lasso are effective in high dimensions, but often lead to unwanted shrinkage. This paper explores instead penalizing the squared distance to constraint sets. Distance penalties are more flexible than algebraic and regularization penalties, and avoid the drawback of shrinkage. To optimize distance penalized objectives, we make use of the majorization-minimization principle. Resulting algorithms constructed within this framework are amenable to acceleration and come with global convergence guarantees. Applications to shape constraints, sparse regression, and rank-restricted matrix regression on synthetic and real data showcase strong empirical performance, even under non-convex constraints.

stat.ML↗

arXiv2016

An MM Algorithm for Split Feasibility Problems

The classical multi-set split feasibility problem seeks a point in the intersection of finitely many closed convex domain constraints, whose image under a linear mapping also lies in the intersection of finitely many closed convex range constraints. Split feasibility generalizes important inverse problems including convex feasibility, linear complementarity, and regression with constraint sets. When a feasible point does not exist, solution methods that proceed by minimizing a proximity function can be used to obtain optimal approximate solutions to the problem. We present an extension of the proximity function approach that generalizes the linear split feasibility problem to allow for non-linear mappings. Our algorithm is based on the principle of majorization-minimization, is amenable to quasi-Newton acceleration, and comes complete with convergence guarantees under mild assumptions. Furthermore, we show that the Euclidean norm appearing in the proximity function of the non-linear split feasibility problem can be replaced by arbitrary Bregman divergences. We explore several examples illustrating the merits of non-linear formulations over the linear case, with a focus on optimization for intensity-modulated radiation therapy.

math.OC↗