In many medical image classification problems, labeled data is scarce while unlabeled data is more available. Semi-supervised learning and self-supervised learning are two different research directions that can improve accuracy by learning from extra unlabeled data. Recent methods from both directions have reported significant gains on traditional benchmarks. Yet past benchmarks do not focus on medical tasks and rarely compare self- and semi- methods together on equal footing. Furthermore, past benchmarks often handle hyperparameter tuning suboptimally. First, they may not tune hyperparameters at all, leading to underfitting. Second, when tuning does occur, it often unrealistically uses a labeled validation set much larger than the train set. Both cases make previously published rankings of methods difficult to translate to practical settings. This study contributes a systematic evaluation of self- and semi- methods with a unified experimental protocol intended to guide a practitioner with scarce overall labeled data and a limited compute budget. We answer two key questions: Can hyperparameter tuning be effective with realistic-sized validation sets? If so, when all methods are tuned well, which self- or semi-supervised methods reach the best accuracy? Our study compares 13 representative semi- and self-supervised methods to strong labeled-set-only baselines on 4 medical datasets. From 20000+ total GPU hours of computation, we provide valuable best practices to resource-constrained, results-focused practitioners.

We study the problem of designing hard negative sampling distributions for unsupervised contrastive representation learning. We propose and analyze a novel min-max framework that seeks a representation which minimizes the maximum (worst-case) generalized contrastive learning loss over all couplings (joint distributions between positive and negative samples subject to marginal constraints) and prove that the resulting min-max optimum representation will be degenerate. This provides the first theoretical justification for incorporating additional regularization constraints on the couplings. We re-interpret the min-max problem through the lens of Optimal Transport (OT) theory and utilize regularized transport couplings to control the degree of hardness of negative examples. Through experiments we demonstrate that the negative samples generated from our designed negative distribution are more similar to the anchor than those generated from the baseline negative distribution. We also demonstrate that entropic regularization yields negative sampling distributions with parametric form similar to that in a recent state-of-the-art negative sampling design and has similar performance in multiple datasets. Utilizing the uncovered connection with OT, we propose a new ground cost for designing the negative distribution and show improved performance of the learned representation on downstream tasks compared to the representation learned when using squared Euclidean cost.

For a widely-studied data model and general loss and sample-hardening functions we prove that the Supervised Contrastive Learning (SCL), Hard-SCL (HSCL), and Unsupervised Contrastive Learning (UCL) risks are minimized by representations that exhibit Neural Collapse (NC), i.e., the class means form an Equianglular Tight Frame (ETF) and data from the same class are mapped to the same representation. We also prove that for any representation mapping, the HSCL and Hard-UCL (HUCL) risks are lower bounded by the corresponding SCL and UCL risks. Although the optimality of ETF is known for SCL, albeit only for InfoNCE loss, its optimality for HSCL and UCL under general loss and hardening functions is novel. Moreover, our proofs are much simpler, compact, and transparent. We empirically demonstrate, for the first time, that ADAM optimization of HSCL and HUCL risks with random initialization and suitable hardness levels can indeed converge to the NC geometry if we incorporate unit-ball or unit-sphere feature normalization. Without incorporating hard negatives or feature normalization, however, the representations learned via ADAM suffer from dimensional collapse (DC) and fail to attain the NC geometry.

This paper considers the problem of measure estimation under the barycentric coding model (BCM), in which an unknown measure is assumed to belong to the set of Wasserstein-2 barycenters of a finite set of known measures. Estimating a measure under this model is equivalent to estimating the unknown barycenteric coordinates. We provide novel geometrical, statistical, and computational insights for measure estimation under the BCM, consisting of three main results. Our first main result leverages the Riemannian geometry of Wasserstein-2 space to provide a procedure for recovering the barycentric coordinates as the solution to a quadratic optimization problem assuming access to the true reference measures. The essential geometric insight is that the parameters of this quadratic problem are determined by inner products between the optimal displacement maps from the given measure to the reference measures defining the BCM. Our second main result then establishes an algorithm for solving for the coordinates in the BCM when all the measures are observed empirically via i.i.d. samples. We prove precise rates of convergence for this algorithm -- determined by the smoothness of the underlying measures and their dimensionality -- thereby guaranteeing its statistical consistency. Finally, we demonstrate the utility of the BCM and associated estimation procedures in three application areas: (i) covariance estimation for Gaussian measures; (ii) image processing; and (iii) natural language processing.