arXiv subjects

Jeff Calder

Publications and source records attributed to Jeff Calder.

At least 37 records · Page 2Linked to original sources

arXiv2023

Novel Batch Active Learning Approach and Its Application to Synthetic Aperture Radar Datasets

Active learning improves the performance of machine learning methods by judiciously selecting a limited number of unlabeled data points to query for labels, with the aim of maximally improving the underlying classifier's performance. Recent gains have been made using sequential active learning for synthetic aperture radar (SAR) data arXiv:2204.00005. In each iteration, sequential active learning selects a query set of size one while batch active learning selects a query set of multiple datapoints. While batch active learning methods exhibit greater efficiency, the challenge lies in maintaining model accuracy relative to sequential active learning methods. We developed a novel, two-part approach for batch active learning: Dijkstra's Annulus Core-Set (DAC) for core-set generation and LocalMax for batch sampling. The batch active learning process that combines DAC and LocalMax achieves nearly identical accuracy as sequential active learning but is more efficient, proportional to the batch size. As an application, a pipeline is built based on transfer learning feature embedding, graph learning, DAC, and LocalMax to classify the FUSAR-Ship and OpenSARShip datasets. Our pipeline outperforms the state-of-the-art CNN-based methods.

cs.LG↗

arXiv2022

Poisson Reweighted Laplacian Uncertainty Sampling for Graph-based Active Learning

We show that uncertainty sampling is sufficient to achieve exploration versus exploitation in graph-based active learning, as long as the measure of uncertainty properly aligns with the underlying model and the model properly reflects uncertainty in unexplored regions. In particular, we use a recently developed algorithm, Poisson ReWeighted Laplace Learning (PWLL) for the classifier and we introduce an acquisition function designed to measure uncertainty in this graph-based classifier that identifies unexplored regions of the data. We introduce a diagonal perturbation in PWLL which produces exponential localization of solutions, and controls the exploration versus exploitation tradeoff in active learning. We use the well-posed continuum limit of PWLL to rigorously analyze our method, and present experimental results on a number of graph-based image classification problems.

stat.ML↗

arXiv2022

Use and Misuse of Machine Learning in Anthropology

Machine learning (ML), being now widely accessible to the research community at large, has fostered a proliferation of new and striking applications of these emergent mathematical techniques across a wide range of disciplines. In this paper, we will focus on a particular case study: the field of paleoanthropology, which seeks to understand the evolution of the human species based on biological and cultural evidence. As we will show, the easy availability of ML algorithms and lack of expertise on their proper use among the anthropological research community has led to foundational misapplications that have appeared throughout the literature. The resulting unreliable results not only undermine efforts to legitimately incorporate ML into anthropological research, but produce potentially faulty understandings about our human evolutionary and behavioral past. The aim of this paper is to provide a brief introduction to some of the ways in which ML has been applied within paleoanthropology; we also include a survey of some basic ML algorithms for those who are not fully conversant with the field, which remains under active development. We discuss a series of missteps, errors, and violations of correct protocols of ML methods that appear disconcertingly often within the accumulating body of anthropological literature. These mistakes include use of outdated algorithms and practices; inappropriate train/test splits, sample composition, and textual explanations; as well as an absence of transparency due to the lack of data/code sharing, and the subsequent limitations imposed on independent replication. We assert that expanding samples, sharing data and code, re-evaluating approaches to peer review, and, most importantly, developing interdisciplinary teams that include experts in ML are all necessary for progress in future research incorporating ML within anthropology.

cs.LG↗

arXiv2022

Uniform Convergence Rates for Lipschitz Learning on Graphs

Lipschitz learning is a graph-based semi-supervised learning method where one extends labels from a labeled to an unlabeled data set by solving the infinity Laplace equation on a weighted graph. In this work we prove uniform convergence rates for solutions of the graph infinity Laplace equation as the number of vertices grows to infinity. Their continuum limits are absolutely minimizing Lipschitz extensions with respect to the geodesic metric of the domain where the graph vertices are sampled from. We work under very general assumptions on the graph weights, the set of labeled vertices, and the continuum domain. Our main contribution is that we obtain quantitative convergence rates even for very sparsely connected graphs, as they typically appear in applications like semi-supervised learning. In particular, our framework allows for graph bandwidths down to the connectivity radius. For proving this we first show a quantitative convergence statement for graph distance functions to geodesic distance functions in the continuum. Using the "comparison with distance functions" principle, we can pass these convergence statements to infinity harmonic functions and absolutely minimizing Lipschitz extensions.

math.NA↗

arXiv2022

Using machine learning on new feature sets extracted from 3D models of broken animal bones to classify fragments according to break agent

Distinguishing agents of bone modification at paleoanthropological sites is at the root of much of the research directed at understanding early hominin exploitation of large animal resources and the effects those subsistence behaviors had on early hominin evolution. However, current methods, particularly in the area of fracture pattern analysis as a signal of marrow exploitation, have failed to overcome equifinality. Furthermore, researchers debate the replicability and validity of current and emerging methods for analyzing bone modifications. Here we present a new approach to fracture pattern analysis aimed at distinguishing bone fragments resulting from hominin bone breakage and those produced by carnivores. This new method uses 3D models of fragmentary bone to extract a much richer dataset that is more transparent and replicable than feature sets previously used in fracture pattern analysis. Supervised machine learning algorithms are properly used to classify bone fragments according to agent of breakage with average mean accuracy of 77% across tests.

cs.CV↗

arXiv2022

Rates of convergence for the continuum limit of nondominated sorting

Nondominated sorting is a discrete process that sorts points in Euclidean space according to the coordinatewise partial order, and is used to rank feasible solutions to multiobjective optimization problems. It was previously shown that nondominated sorting of random points has a Hamilton-Jacobi equation continuum limit. We prove quantitative error estimates for the convergence of nondominated sorting to its continuum limit Hamilton-Jacobi equation. Our proof uses the maximum principle and viscosity solution machinery, along with new semiconvexity estimates for domains with corner singularities.

math.AP↗

arXiv2022

Boundary Estimation from Point Clouds: Algorithms, Guarantees and Applications

We investigate identifying the boundary of a domain from sample points in the domain. We introduce new estimators for the normal vector to the boundary, distance of a point to the boundary, and a test for whether a point lies within a boundary strip. The estimators can be efficiently computed and are more accurate than the ones present in the literature. We provide rigorous error estimates for the estimators. Furthermore we use the detected boundary points to solve boundary-value problems for PDE on point clouds. We prove error estimates for the Laplace and eikonal equations on point clouds. Finally we provide a range of numerical experiments illustrating the performance of our boundary estimators, applications to PDE on point clouds, and tests on image data sets.

math.NA↗

arXiv2022

Graph-based Active Learning for Semi-supervised Classification of SAR Data

We present a novel method for classification of Synthetic Aperture Radar (SAR) data by combining ideas from graph-based learning and neural network methods within an active learning framework. Graph-based methods in machine learning are based on a similarity graph constructed from the data. When the data consists of raw images composed of scenes, extraneous information can make the classification task more difficult. In recent years, neural network methods have been shown to provide a promising framework for extracting patterns from SAR images. These methods, however, require ample training data to avoid overfitting. At the same time, such training data are often unavailable for applications of interest, such as automatic target recognition (ATR) and SAR data. We use a Convolutional Neural Network Variational Autoencoder (CNNVAE) to embed SAR data into a feature space, and then construct a similarity graph from the embedded data and apply graph-based semi-supervised learning techniques. The CNNVAE feature embedding and graph construction requires no labeled data, which reduces overfitting and improves the generalization performance of graph learning at low label rates. Furthermore, the method easily incorporates a human-in-the-loop for active learning in the data-labeling process. We present promising results and compare them to other standard machine learning methods on the Moving and Stationary Target Acquisition and Recognition (MSTAR) dataset for ATR with small amounts of labeled data.

cs.LG↗

arXiv2022

Hamilton-Jacobi equations on graphs with applications to semi-supervised learning and data depth

Shortest path graph distances are widely used in data science and machine learning, since they can approximate the underlying geodesic distance on the data manifold. However, the shortest path distance is highly sensitive to the addition of corrupted edges in the graph, either through noise or an adversarial perturbation. In this paper we study a family of Hamilton-Jacobi equations on graphs that we call the $p$-eikonal equation. We show that the $p$-eikonal equation with $p=1$ is a provably robust distance-type function on a graph, and the $p\to \infty$ limit recovers shortest path distances. While the $p$-eikonal equation does not correspond to a shortest-path graph distance, we nonetheless show that the continuum limit of the $p$-eikonal equation on a random geometric graph recovers a geodesic density weighted distance in the continuum. We consider applications of the $p$-eikonal equation to data depth and semi-supervised learning, and use the continuum limit to prove asymptotic consistency results for both applications. Finally, we show the results of experiments with data depth and semi-supervised learning on real image datasets, including MNIST, FashionMNIST and CIFAR-10, which show that the $p$-eikonal equation offers significantly better results compared to shortest path distances.

math.AP↗

arXiv2022

Analysis and algorithms for $\ell_p$-based semi-supervised learning on graphs

This paper addresses theory and applications of $\ell_p$-based Laplacian regularization in semi-supervised learning. The graph $p$-Laplacian for $p>2$ has been proposed recently as a replacement for the standard ($p=2$) graph Laplacian in semi-supervised learning problems with very few labels, where Laplacian learning is degenerate. In the first part of the paper we prove new discrete to continuum convergence results for $p$-Laplace problems on $k$-nearest neighbor ($k$-NN) graphs, which are more commonly used in practice than random geometric graphs. Our analysis shows that, on $k$-NN graphs, the $p$-Laplacian retains information about the data distribution as $p\to \infty$ and Lipschitz learning ($p=\infty$) is sensitive to the data distribution. This situation can be contrasted with random geometric graphs, where the $p$-Laplacian forgets the data distribution as $p\to \infty$. We also present a general framework for proving discrete to continuum convergence results in graph-based learning that only requires pointwise consistency and monotonicity. In the second part of the paper, we develop fast algorithms for solving the variational and game-theoretic $p$-Laplace equations on weighted graphs for $p>2$. We present several efficient and scalable algorithms for both formulations, and present numerical results on synthetic data indicating their convergence properties. Finally, we conduct extensive numerical experiments on the MNIST, FashionMNIST and EMNIST datasets that illustrate the effectiveness of the $p$-Laplacian formulation for semi-supervised learning with few labels. In particular, we find that Lipschitz learning ($p=\infty$) performs well with very few labels on $k$-NN graphs, which experimentally validates our theoretical findings that Lipschitz learning retains information about the data distribution (the unlabeled data) on $k$-NN graphs.

math.NA↗

arXiv2021

Lipschitz regularity of graph Laplacians on random data clouds

In this paper we study Lipschitz regularity of elliptic PDEs on geometric graphs, constructed from random data points. The data points are sampled from a distribution supported on a smooth manifold. The family of equations that we study arises in data analysis in the context of graph-based learning and contains, as important examples, the equations satisfied by graph Laplacian eigenvectors. In particular, we prove high probability interior and global Lipschitz estimates for solutions of graph Poisson equations. Our results can be used to show that graph Laplacian eigenvectors are, with high probability, essentially Lipschitz regular with constants depending explicitly on their corresponding eigenvalues. Our analysis relies on a probabilistic coupling argument of suitable random walks at the continuum level, and an interpolation method for extending functions on random point clouds to the continuum manifold. As a byproduct of our general regularity results, we obtain high probability $L^\infty$ and approximate $\mathcal{C}^{0,1}$ convergence rates for the convergence of graph Laplacian eigenvectors towards eigenfunctions of the corresponding weighted Laplace-Beltrami operators. The convergence rates we obtain scale like the $L^2$-convergence rates established by two of the authors in previous work.

math.AP↗

arXiv2021

A continuum limit for the PageRank algorithm

Semi-supervised and unsupervised machine learning methods often rely on graphs to model data, prompting research on how theoretical properties of operators on graphs are leveraged in learning problems. While most of the existing literature focuses on undirected graphs, directed graphs are very important in practice, giving models for physical, biological, or transportation networks, among many other applications. In this paper, we propose a new framework for rigorously studying continuum limits of learning algorithms on directed graphs. We use the new framework to study the PageRank algorithm, and show how it can be interpreted as a numerical scheme on a directed graph involving a type of normalized graph Laplacian. We show that the corresponding continuum limit problem, which is taken as the number of webpages grows to infinity, is a second-order, possibly degenerate, elliptic equation that contains reaction, diffusion, and advection terms. We prove that the numerical scheme is consistent and stable and compute explicit rates of convergence of the discrete solution to the solution of the continuum limit PDE. We give applications to proving stability and asymptotic regularity of the PageRank vector. Finally, we illustrate our results with numerical experiments and explore an application to data depth.

math.AP↗

arXiv2020

Online Prediction With History-Dependent Experts: The General Case

We study the problem of prediction of binary sequences with expert advice in the online setting, which is a classic example of online machine learning. We interpret the binary sequence as the price history of a stock, and view the predictor as an investor, which converts the problem into a stock prediction problem. In this framework, an investor, who predicts the daily movements of a stock, and an adversarial market, who controls the stock, play against each other over $N$ turns. The investor combines the predictions of $n\geq 2$ experts in order to make a decision about how much to invest at each turn, and aims to minimize their regret with respect to the best-performing expert at the end of the game. We consider the problem with history-dependent experts, in which each expert uses the previous $d$ days of history of the market in making their predictions. We prove that the value function for this game, rescaled appropriately, converges as $N\to \infty$ at a rate of $O(N^{-1/6})$ to the viscosity solution of a nonlinear degenerate elliptic PDE, which can be understood as the Hamilton-Jacobi-Issacs equation for the two-person game. As a result, we are able to deduce asymptotically optimal strategies for the investor. Our results extend those established by the first author and R.V.Kohn [13] for $n=2$ experts and $d\leq 4$ days of history. To appear in Communications on Pure and Applied Mathematics.

math.AP↗

arXiv2020

The Virtual Goniometer: A new method for measuring angles on 3D models of fragmentary bone and lithics

The contact goniometer is a commonly used tool in lithic and zooarchaeological analysis, despite suffering from a number of shortcomings due to the physical interaction between the measuring implement, the object being measured, and the individual taking the measurements. However, lacking a simple and efficient alternative, researchers in a variety of fields continue to use the contact goniometer to this day. In this paper, we present a new goniometric method that we call the virtual goniometer, which takes angle measurements virtually on a 3D model of an object. The virtual goniometer allows for rapid data collection, and for the measurement of many angles that cannot be physically accessed by a manual goniometer. We compare the intra-observer variability of the manual and virtual goniometers, and find that the virtual goniometer is far more consistent and reliable. Furthermore, the virtual goniometer allows for precise replication of angle measurements, even among multiple users, which is important for reproducibility of goniometric-based research. The virtual goniometer is available as a plug-in in the open source mesh processing packages Meshlab and Blender, making it easily accessible to researchers exploring the potential for goniometry to improve archaeological methods and address anthropological questions.

math.NA↗

arXiv2020

Asymptotically optimal strategies for online prediction with history-dependent experts

We establish sharp asymptotically optimal strategies for the problem of online prediction with history dependent experts. The prediction problem is played (in part) over a discrete graph called the $d$ dimensional de Bruijn graph, where $d$ is the number of days of history used by the experts. Previous work [11] established $O(\varepsilon)$ optimal strategies for $n=2$ experts and $d\leq 4$ days of history, while [10] established $O(\varepsilon^{1/3})$ optimal strategies for all $n\geq 2$ and all $d\geq 1$, where the game is played for $N$ steps and $\varepsilon=N^{-1/2}$. In this paper, we show that the optimality conditions over the de Bruijn graph correspond to a graph Poisson equation, and we establish $O(\varepsilon)$ optimal strategies for all values of $n$ and $d$.

math.OC↗

arXiv2020

Poisson Learning: Graph Based Semi-Supervised Learning At Very Low Label Rates

We propose a new framework, called Poisson learning, for graph based semi-supervised learning at very low label rates. Poisson learning is motivated by the need to address the degeneracy of Laplacian semi-supervised learning in this regime. The method replaces the assignment of label values at training points with the placement of sources and sinks, and solves the resulting Poisson equation on the graph. The outcomes are provably more stable and informative than those of Laplacian learning. Poisson learning is efficient and simple to implement, and we present numerical experiments showing the method is superior to other recent approaches to semi-supervised learning at low label rates on MNIST, FashionMNIST, and Cifar-10. We also propose a graph-cut enhancement of Poisson learning, called Poisson MBO, that gives higher accuracy and can incorporate prior knowledge of relative class sizes.

cs.LG↗

arXiv2020

Improved spectral convergence rates for graph Laplacians on epsilon-graphs and k-NN graphs

In this paper we improve the spectral convergence rates for graph-based approximations of Laplace-Beltrami operators constructed from random data. We utilize regularity of the continuum eigenfunctions and strong pointwise consistency results to prove that spectral convergence rates are the same as the pointwise consistency rates for graph Laplacians. In particular, for an optimal choice of the graph connectivity $\varepsilon$, our results show that the eigenvalues and eigenvectors of the graph Laplacian converge to those of the Laplace-Beltrami operator at a rate of $O(n^{-1/(m+4)})$, up to log factors, where $m$ is the manifold dimension and $n$ is the number of vertices in the graph. Our approach is general and allows us to analyze a large variety of graph constructions that include $\varepsilon$-graphs and $k$-NN graphs.

math.PR↗

arXiv2020

Rates of Convergence for Laplacian Semi-Supervised Learning with Low Labeling Rates

We study graph-based Laplacian semi-supervised learning at low labeling rates. Laplacian learning uses harmonic extension on a graph to propagate labels. At very low label rates, Laplacian learning becomes degenerate and the solution is roughly constant with spikes at each labeled data point. Previous work has shown that this degeneracy occurs when the number of labeled data points is finite while the number of unlabeled data points tends to infinity. In this work we allow the number of labeled data points to grow to infinity with the number of labels. Our results show that for a random geometric graph with length scale $\varepsilon>0$ and labeling rate $β>0$, if $β\ll\varepsilon^2$ then the solution becomes degenerate and spikes form, and if $β\gg \varepsilon^2$ then Laplacian learning is well-posed and consistent with a continuum Laplace equation. Furthermore, in the well-posed setting we prove quantitative error estimates of $O(\varepsilonβ^{-1/2})$ for the difference between the solutions of the discrete problem and continuum PDE, up to logarithmic factors. We also study $p$-Laplacian regularization and show the same degeneracy result when $β\ll \varepsilon^p$. The proofs of our well-posedness results use the random walk interpretation of Laplacian learning and PDE arguments, while the proofs of the ill-posedness results use $Γ$-convergence tools from the calculus of variations. We also present numerical results on synthetic and real data to illustrate our results.

math.ST↗