An official website of the United States government
We present a novel framework for learning cost-efficient latent representations in problems with highdimensional state spaces through nonlinear dimension reduction. By enriching linear state approximations with low-order polynomial terms we account for key nonlinear interactions existing in the data thereby reducing the problem’s intrinsic dimensionality. Two methods are introduced for learning the representation of such low-dimensional, polynomial manifolds for embedding the data. The manifold parametrization coefficients can be obtained by regression via either a proper orthogonal decomposition or an alternating minimization based approach. Our numerical results focus on the one-dimensional Korteweg-de Vries equation where accounting for nonlinear correlations in the data was found to lower the representation error by up to two orders of magnitude compared to linear dimension reduction techniques.
more »
« less
This content will become publicly available on September 21, 2026
LIT-LVM: Structured Regularization for Interaction Terms in Linear Predictors using Latent Variable Models
Some of the simplest, yet most frequently used predictors in statistics and machine learning use weighted linear combinations of features. Such linear predictors can model non-linear relationships between features by adding interaction terms corresponding to the products of all pairs of features. We consider the problem of accurately estimating coefficients for interaction terms in linear predictors. We hypothesize that the coefficients for different interaction terms have an approximate low-dimensional structure and represent each feature by a latent vector in a low-dimensional space. This low-dimensional representation can be viewed as a structured regularization approach that further mitigates overfitting in high-dimensional settings beyond standard regularizers such as the lasso and elastic net. We demonstrate that our approach, called LIT-LVM, achieves superior prediction accuracy compared to the elastic net, hierarchical lasso, and factorization machines on a wide variety of simulated and real data, particularly when the number of interaction terms is high compared to the number of samples. LIT-LVM also provides low-dimensional latent representations for features that are useful for visualizing and analyzing their relationships.
more »
« less
- Award ID(s):
- 2318751
- PAR ID:
- 10663941
- Publisher / Repository:
- Transactions on machine learning research
- Date Published:
- Journal Name:
- Transactions on machine learning research
- ISSN:
- 2835-8856
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Summary We develop a Bayesian methodology aimed at simultaneously estimating low-rank and row-sparse matrices in a high-dimensional multiple-response linear regression model. We consider a carefully devised shrinkage prior on the matrix of regression coefficients which obviates the need to specify a prior on the rank, and shrinks the regression matrix towards low-rank and row-sparse structures. We provide theoretical support to the proposed methodology by proving minimax optimality of the posterior mean under the prediction risk in ultra-high-dimensional settings where the number of predictors can grow subexponentially relative to the sample size. A one-step post-processing scheme induced by group lasso penalties on the rows of the estimated coefficient matrix is proposed for variable selection, with default choices of tuning parameters. We additionally provide an estimate of the rank using a novel optimization function achieving dimension reduction in the covariate space. We exhibit the performance of the proposed methodology in an extensive simulation study and a real data example.more » « less
-
Summary The lasso has been studied extensively as a tool for estimating the coefficient vector in the high-dimensional linear model; however, considerably less is known about estimating the error variance in this context. In this paper, we propose the natural lasso estimator for the error variance, which maximizes a penalized likelihood objective. A key aspect of the natural lasso is that the likelihood is expressed in terms of the natural parameterization of the multi-parameter exponential family of a Gaussian with unknown mean and variance. The result is a remarkably simple estimator of the error variance with provably good performance in terms of mean squared error. These theoretical results do not require placing any assumptions on the design matrix or the true regression coefficients. We also propose a companion estimator, called the organic lasso, which theoretically does not require tuning of the regularization parameter. Both estimators do well empirically compared to pre-existing methods, especially in settings where successful recovery of the true support of the coefficient vector is hard. Finally, we show that existing methods can do well under fewer assumptions than previously known, thus providing a fuller story about the problem of estimating the error variance in high-dimensional linear models.more » « less
-
Elucidating the complex relationships between genotypes, phenotypes, and fitness remains one of the fundamental challenges in evolutionary biology. Part of the difficulty arises from the enormous number of possible genotypes and the lack of understanding of the underlying phenotypic differences driving adaptation. Here we present a computational method that takes advantage of modern high-throughput fitness measurements to learn a map from high-dimensional fitness profiles to a low-dimensional latent space in a geometry-informed manner. We demonstrate that our approach using a Riemannian Hamiltonian variational autoencoder outperforms traditional linear dimensionality reduction techniques by capturing the nonlinear structure of the phenotype-fitness map. When applied to simulated adaptive dynamics, we show that the learned latent space retains information about the underlying adaptive phenotypic space and accurately reconstructs complex fitness landscapes. We then apply this method to a dataset of high-throughput fitness measurements of under different antibiotic pressures and demonstrate superior predictive power for out-of-sample data compared to linear approaches. Our work provides a data-driven implementation of Fisher's geometric model of adaptation, transforming it from a theoretical framework into an empirically grounded approach for understanding evolutionary dynamics using modern deep learning methods.more » « less
-
The problem of classifying multiple categorical responses is fundamental in modern machine learning and statistics, with diverse applications in fields such as bioinformatics and imaging. This manuscript investigates linear discriminant analysis (LDA) with high-dimensional predictors and multiple multi-class responses. Specifically, we first examine two different classification scenarios under the bivariate LDA model: joint classification of the two responses and conditional classification of one response while observing the other. To achieve optimal classification rules for both scenarios, we introduce two novel tensor formulations of the discriminant coefficients and corresponding regularization strategies. For joint classification, we propose an overlapping group lasso penalty and a blockwise coordinate descent algorithm to efficiently compute the joint discriminant coefficient tensors. For conditional classification, we utilize an alternating direction method of multipliers (ADMM) algorithm to compute the discriminant coefficient tensors under new constraints. We then extend our method and algorithms to general multivariate responses. Finally, we validate the effectiveness of our approach through simulation studies and applications to benchmark datasets.more » « less
