arXiv subjects

Nathan Mankovich

Publications and source records attributed to Nathan Mankovich.

11 recordsLinked to original sources

arXiv2026

Disentangling Forced and Internal Climate Variability in Single Realizations using Dynamic Mode Decomposition with Control

We show that a single climate realization can be decomposed into forced and internal components by treating external forcing as a dynamical driver within a linear stochastic system, an idea grounded in pullback attractor theory. In doing so, we address a central methodological challenge in climate science with direct implications for climate projection and the detection and attribution of the forced response, disentangling the forced climate response from internal variability in a single observed record. Statistical methods range from approaches trained on large ensembles to techniques operating on single realizations. The latter often rely on linear frameworks such as linear inverse models (LIMs) and linear regression. LIMs ignore forcing predictors, whereas linear regression omits climate system dynamics. Here we introduce PullbackDMDc, a method grounded in non-autonomous dynamical systems theory and dynamic mode decomposition with control (DMDc), incorporating pullback attractor estimation to decompose a single climate realization into spatial modes and their associated forced and internal components, yielding a physically interpretable picture of the underlying dynamics. We illustrate the utility of PullbackDMDc for Earth System Model (ESM) evaluation by applying it to near-surface air temperature and sea-level pressure from reanalysis and four ESM large ensembles. PullbackDMDc estimates the forced response with skill matching or exceeding established baselines and identifies optimal forcing predictors against model-based ground truth. Its internal variability components reveal that ESMs qualitatively capture interannual and decadal modes while exhibiting systematic differences relative to each other and to observations. Skillful forced response estimation and a novel decomposition position PullbackDMDc as a practical tool for single-realization climate analysis and ESM evaluation.

stat.ML↗

arXiv2025

Feature Extraction in the Remote Sensing Data Value Chain: A Systematic Review of Methods and Applications

Earth observation involves collecting, analyzing, and processing an ever-growing mass of data. This planetary data is crucial for addressing relevant societal, economic, and environmental challenges, ranging from environmental monitoring to urban planning and disaster management. However, its high dimensionality entails significant feature redundancy and computational overhead, limiting the effectiveness of machine learning models. Feature extraction (FE) techniques address these challenges by preserving essential data properties while reducing redundancy and enhancing tasks in Remote Sensing (RS). The landscape of FE for RS is diverse, disorganized, and rapidly evolving. We offer a practical guide for this landscape by introducing a framework of FE. Using this framework, we trace the evolution of FE across the data value chain in RS. Finally, we synthesize these trends and offer perspectives for the future of FE in RS by first characterizing this shift from single-task models to unified representations, then identifying two perspectives in the foundation model era: the need for robust and interpretable FE and the potential of bridging classical FE with modern representation learning.

cs.CV↗

arXiv2025

A Flag Decomposition for Hierarchical Datasets

Flag manifolds encode nested sequences of subspaces and serve as powerful structures for various computer vision and machine learning applications. Despite their utility in tasks such as dimensionality reduction, motion averaging, and subspace clustering, current applications are often restricted to extracting flags using common matrix decomposition methods like the singular value decomposition. Here, we address the need for a general algorithm to factorize and work with hierarchical datasets. In particular, we propose a novel, flag-based method that decomposes arbitrary hierarchical real-valued data into a hierarchy-preserving flag representation in Stiefel coordinates. Our work harnesses the potential of flag manifolds in applications including denoising, clustering, and few-shot learning.

cs.CV↗

arXiv2024

Featurizing Koopman Mode Decomposition For Robust Forecasting

This article introduces an advanced Koopman mode decomposition (KMD) technique -- coined Featurized Koopman Mode Decomposition (FKMD) -- that uses delay embedding and a learned Mahalanobis distance to enhance analysis and prediction of high dimensional dynamical systems. The delay embedding expands the observation space to better capture underlying manifold structure, while the Mahalanobis distance adjusts observations based on the system's dynamics. This aids in featurizing KMD in cases where good features are not a priori known. We show that FKMD improves predictions for a high-dimensional linear oscillator, a high-dimensional Lorenz attractor that is partially observed, and a cell signaling problem from cancer research.

math.DS↗

arXiv2024

Fun with Flags: Robust Principal Directions via Flag Manifolds

Principal component analysis (PCA), along with its extensions to manifolds and outlier contaminated data, have been indispensable in computer vision and machine learning. In this work, we present a unifying formalism for PCA and its variants, and introduce a framework based on the flags of linear subspaces, ie a hierarchy of nested linear subspaces of increasing dimension, which not only allows for a common implementation but also yields novel variants, not explored previously. We begin by generalizing traditional PCA methods that either maximize variance or minimize reconstruction error. We expand these interpretations to develop a wide array of new dimensionality reduction algorithms by accounting for outliers and the data manifold. To devise a common computational approach, we recast robust and dual forms of PCA as optimization problems on flag manifolds. We then integrate tangent space approximations of principal geodesic analysis (tangent-PCA) into this flag-based framework, creating novel robust and dual geodesic PCA variations. The remarkable flexibility offered by the 'flagification' introduced here enables even more algorithmic variants identified by specific flag types. Last but not least, we propose an effective convergent solver for these flag-formulations employing the Stiefel manifold. Our empirical results on both real-world and synthetic scenarios, demonstrate the superiority of our novel algorithms, especially in terms of robustness to outliers on manifolds.

cs.CV↗

arXiv2024

Optimal codes in the Stiefel manifold

We consider the coding problem in the Stiefel manifold with chordal distance. After considering various low-dimensional instances of this problem, we use Rankin's bounds on spherical codes to prove upper bounds on the minimum distance of a Stiefel code, and then we construct several examples of codes that achieve equality in these bounds.

math.MG↗

arXiv2024

Recovering Latent Confounders from High-dimensional Proxy Variables

Detecting latent confounders from proxy variables is an essential problem in causal effect estimation. Previous approaches are limited to low-dimensional proxies, sorted proxies, and binary treatments. We remove these assumptions and present a novel Proxy Confounder Factorization (PCF) framework for continuous treatment effect estimation when latent confounders manifest through high-dimensional, mixed proxy variables. For specific sample sizes, our two-step PCF implementation, using Independent Component Analysis (ICA-PCF), and the end-to-end implementation, using Gradient Descent (GD-PCF), achieve high correlation with the latent confounder and low absolute error in causal effect estimation with synthetic datasets in the high sample size regime. Even when faced with climate data, ICA-PCF recovers four components that explain $75.9\%$ of the variance in the North Atlantic Oscillation, a known confounder of precipitation patterns in Europe. Code for our PCF implementations and experiments can be found here: https://github.com/IPL-UV/confound_it. The proposed methodology constitutes a stepping stone towards discovering latent confounders and can be applied to many problems in disciplines dealing with high-dimensional observed proxies, e.g., spatiotemporal fields.

stat.ML↗

arXiv2024

Out-of-distribution robustness for multivariate analysis via causal regularisation

We propose a regularisation strategy of classical machine learning algorithms rooted in causality that ensures robustness against distribution shifts. Building upon the anchor regression framework, we demonstrate how incorporating a straightforward regularisation term into the loss function of classical multivariate analysis algorithms, such as (orthonormalized) partial least squares, reduced-rank regression, and multiple linear regression, enables out-of-distribution generalisation. Our framework allows users to efficiently verify the compatibility of a loss function with the regularisation strategy. Estimators for selected algorithms are provided, showcasing consistency and efficacy in synthetic and real-world climate science problems. The empirical validation highlights the versatility of anchor regularisation, emphasizing its compatibility with multivariate analysis approaches and its role in enhancing replicability while guarding against distribution shifts. The extended anchor framework advances causal inference methodologies, addressing the need for reliable out-of-distribution generalisation.

stat.ML↗

arXiv2023

Chordal Averaging on Flag Manifolds and Its Applications

This paper presents a new, provably-convergent algorithm for computing the flag-mean and flag-median of a set of points on a flag manifold under the chordal metric. The flag manifold is a mathematical space consisting of flags, which are sequences of nested subspaces of a vector space that increase in dimension. The flag manifold is a superset of a wide range of known matrix spaces, including Stiefel and Grassmanians, making it a general object that is useful in a wide variety computer vision problems. To tackle the challenge of computing first order flag statistics, we first transform the problem into one that involves auxiliary variables constrained to the Stiefel manifold. The Stiefel manifold is a space of orthogonal frames, and leveraging the numerical stability and efficiency of Stiefel-manifold optimization enables us to compute the flag-mean effectively. Through a series of experiments, we show the competence of our method in Grassmann and rotation averaging, as well as principal component analysis. We release our source code under https://github.com/nmank/FlagAveraging.

cs.CV↗

arXiv2022

Randomized Extended Kaczmarz is a Limit Point of Sketch-and-Project

The sketch-and-project (SAP) framework for solving systems of linear equations has unified the theory behind popular projective iterative methods such as randomized Kaczmarz, randomized coordinate descent, and variants thereof. The randomized extended Kaczmarz (REK) method is a popular extension of randomized Kaczmarz for solving inconsistent systems, which has not yet been shown to lie within the SAP framework. In this work we show that, in a certain sense, REK may be expressed as the limit point of a family of SAP methods, but we argue that it is unlikely that REK can be translated into a SAP method itself. We provide an extensive theoretical analysis of the family of methods comprising said limit, including convergence guarantees and further connections to REK. We follow this with an array of experiments demonstrating these methods and their connections in practice.

math.NA↗

arXiv2022

The Flag Median and FlagIRLS

Finding prototypes (e.g., mean and median) for a dataset is central to a number of common machine learning algorithms. Subspaces have been shown to provide useful, robust representations for datasets of images, videos and more. Since subspaces correspond to points on a Grassmann manifold, one is led to consider the idea of a subspace prototype for a Grassmann-valued dataset. While a number of different subspace prototypes have been described, the calculation of some of these prototypes has proven to be computationally expensive while other prototypes are affected by outliers and produce highly imperfect clustering on noisy data. This work proposes a new subspace prototype, the flag median, and introduces the FlagIRLS algorithm for its calculation. We provide evidence that the flag median is robust to outliers and can be used effectively in algorithms like Linde-Buzo-Grey (LBG) to produce improved clusterings on Grassmannians. Numerical experiments include a synthetic dataset, the MNIST handwritten digits dataset, the Mind's Eye video dataset and the UCF YouTube action dataset. The flag median is compared the other leading algorithms for computing prototypes on the Grassmannian, namely, the $\ell_2$-median and to the flag mean. We find that using FlagIRLS to compute the flag median converges in $4$ iterations on a synthetic dataset. We also see that Grassmannian LBG with a codebook size of $20$ and using the flag median produces at least a $10\%$ improvement in cluster purity over Grassmannian LBG using the flag mean or $\ell_2$-median on the Mind's Eye dataset.

stat.ML↗