Consistency of semi-supervised learning, stochastic tug-of-war games, and the p-Laplacian

In this paper we give a broad overview of the intersection of partial differential equations (PDEs) and graph-based semi-supervised learning. The overview is focused on a large body of recent work on PDE continuum limits of graph-based learning, which have been used to prove well-posedness of semi-supervised learning algorithms in the large data limit. We highlight some interesting research directions revolving around consistency of graph-based semi-supervised learning, and present some new results on the consistency of p-Laplacian semi-supervised learning using the stochastic tug-of-war game interpretation of the p-Laplacian. We also present the results of some numerical experiments that illustrate our results and suggest directions for future work.

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.

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.

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.

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.

The Batch Artifact Scanning Protocol: A new method using computed tomography (CT) to rapidly create three-dimensional models of objects from large collections en masse

Within anthropology, the use of three-dimensional (3D) imaging has become increasingly standard and widespread since it broadens the available avenues for addressing a wide range of key issues. The ease with which 3D models can be shared has had major impacts for research, cultural heritage, education, science communication, and public engagement, as well as contributing to the preservation of the physical specimens and archiving collections in widely accessible data bases. Current scanning protocols have the ability to create the required research quality 3D models; however, they tend to be time and labor intensive and not practical when working with large collections. Here we describe a streamlined, Batch Artifact Scanning Protocol we have developed to rapidly create 3D models using a medical CT scanner. Though this method can be used on a variety of material types, we use a large collection of experimentally broken ungulate limb bones. Using the Batch Artifact Scanning Protocol, we were able to efficiently create 3D models of 2,474 bone fragments at a rate of less than $3$ minutes per specimen, as opposed to an average of 50 minutes per specimen using structured light scanning.

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.

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.

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.