To quantify phenotypic distances between Heliconius butterflies, a deep convolutional neural network was trained to classify photographs of Heliconius butterflies by subspecies, with 1500 of the 2468 total images used for network training and the remainder for testing. The training method (21, 22) used triplets of images, each replicate showing the network two images sampled from the same subspecies and one sampled from a different subspecies. Image classification and spatial embedding were performed using a 15-layer deep learning network (Supplementary Computer Code), which we name ButterflyNet (fig. S1). This makes use of a triplet embedding loss function (21, 22) to train a network to organize its inputs (images) in a space such that proximity in that space (Euclidean distance) is highly correlated with identity (in this case subspecies). The learned embedding was then passed through an additional small network to perform direct categorical subspecies classification. Overall, the total network optimizes the sum of the triplet loss and the categorical cross entropy (eqs. S1 and S2). The computer code used for machine learning is provided as a Python script (Supplementary Computer Code), which makes use of the PyTorch, Scikit-l arn, and Adam packages (for further details, see Supplementary Methods).

After the network was trained on 1500 images randomly sampled from the 2468 images in the dataset, network testing was performed on the remainder (968 images). Testing presents the trained network with new images, which it has not encountered before. The network then classifies the new images by subspecies, image classifications are compared to the known subspecies identities, and the overall accuracy of test classifications is reported. Additional testing was performed using an SVC trained on the embeddings from the main network (ButterflyNet) to determine the accuracy of classification of specimens to subspecies based on their locations in the phenotypic spatial embedding.

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