Coupling probability measures lies at the core of many problems in statistics and machine learning, from domain adaptation to transfer learning and causal inference. Yet, even when restricted to deterministic transports, such couplings are not identifiable: two atomless marginals admit infinitely many transport maps. The common recourse to optimal transport, motivated by cost minimization and cyclical monotonicity, obscures the fact that several distinct notions of multivariate monotone matchings coexist. In this work, we first carry a comparative analysis of three constructions of transport maps: cyclically monotone, quantile-preserving and triangular monotone maps. We establish necessary and sufficient conditions for their equivalence, thereby clarifying their respective structural properties. In parallel, we formulate counterfactual reasoning within the framework of structural causal models as a problem of selecting transport maps between fixed marginals, which makes explicit the role of untestable assumptions in counterfactual reasoning. Then, we are able to connect these two perspectives by identifying conditions on causal graphs and structural equations under which counterfactual maps coincide with classical statistical transports. In this way, we delineate the circumstances in which causal assumptions support the use of a specific structure of transport map. Taken together, our results aim to enrich the theoretical understanding of families of transport maps and to clarify their possible causal interpretations. We hope this work contributes to establishing new bridges between statistical transport and causal inference.

Accurate segmentation of punctate white matter lesions (PWMLs) are fundamental for the timely diagnosis and treatment of related developmental disorders. Automated PWMLs segmentation from infant brain MR images is challenging, considering that the lesions are typically small and low-contrast, and the number of lesions may dramatically change across subjects. Existing learning-based methods directly apply general network architectures to this challenging task, which may fail to capture detailed positional information of PWMLs, potentially leading to severe under-segmentations. In this paper, we propose to leverage the idea of counterfactual reasoning coupled with the auxiliary task of brain tissue segmentation to learn fine-grained positional and morphological representations of PWMLs for accurate localization and segmentation. A simple and easy-to-implement deep-learning framework (i.e., DeepPWML) is accordingly designed. It combines the lesion counterfactual map with the tissue probability map to train a lightweight PWML segmentation network, demonstrating state-of-the-art performance on a real-clinical dataset of infant T1w MR images.

Existing studies on disease diagnostic models focus either on diagnostic model learning for performance improvement or on the visual explanation of a trained diagnostic model. We propose a novel learn-explain-reinforce (LEAR) framework that unifies diagnostic model learning, visual explanation generation (explanation unit), and trained diagnostic model reinforcement (reinforcement unit) guided by the visual explanation. For the visual explanation, we generate a counterfactual map that transforms an input sample to be identified as an intended target label. For example, a counterfactual map can localize hypothetical abnormalities within a normal brain image that may cause it to be diagnosed with Alzheimer's disease (AD). We believe that the generated counterfactual maps represent data-driven and model-induced knowledge about a target task, i.e., AD diagnosis using structural MRI, which can be a vital source of information to reinforce the generalization of the trained diagnostic model. To this end, we devise an attention-based feature refinement module with the guidance of the counterfactual maps. The explanation and reinforcement units are reciprocal and can be operated iteratively. Our proposed approach was validated via qualitative and quantitative analysis on the ADNI dataset. Its comprehensibility and fidelity were demonstrated through ablation studies and comparisons with existing methods.

There exists an apparent negative correlation between performance and interpretability of deep learning models. In an effort to reduce this negative correlation, we propose Born Identity Network (BIN), which is a post-hoc approach for producing multi-way counterfactual maps. A counterfactual map transforms an input sample to be classified as a target label, which is similar to how humans process knowledge through counterfactual thinking. Thus, producing a better counterfactual map may be a step towards explanation at the level of human knowledge. For example, a counterfactual map can localize hypothetical abnormalities from a normal brain image that may cause it to be diagnosed with a disease. Specifically, our proposed BIN consists of two core components: Counterfactual Map Generator and Target Attribution Network. The Counterfactual Map Generator is a variation of conditional GAN which can synthesize a counterfactual map conditioned on an arbitrary target label. The Target Attribution Network works in a complementary manner to enforce target label attribution to the synthesized map. We have validated our proposed BIN in qualitative, quantitative analysis on MNIST, 3D Shapes, and ADNI datasets, and show the comprehensibility and fidelity of our method from various ablation studies.