MOTIVATION: The Variant Call Format (VCF) is widely used in genome sequencing but scales poorly. For instance, we estimate a 150 000 genome VCF would occupy 900 TiB, making it costly and complicated to produce, analyze, and store. The issue stems from VCF's requirement to densely represent both reference-genotypes and allele-indexed arrays. These requirements lead to unnecessary data duplication and, ultimately, very large files. RESULTS: To address these challenges, we introduce the Scalable Variant Call Representation (SVCR). This representation reduces file sizes by ensuring they scale linearly with samples. SVCR's linear scaling relies on two techniques, both necessary for linearity: local allele indices and reference blocks, which were first introduced by the Genomic Variant Call Format. SVCR is also lossless and mergeable, allowing for N + 1 and N + K incremental joint-calling. We present two implementations of SVCR: SVCR-VCF, which encodes SVCR in VCF format, and VDS, which uses Hail's native format. Our experiments confirm the linear scalability of SVCR-VCF and VDS, in contrast to the super-linear growth seen with standard VCF files. We also discuss the VDS Combiner, a scalable, open-source tool for producing a VDS from GVCFs and unique features of VDS which enable rapid data analysis. SVCR, and VDS in particular, ensure the scientific community can generate, analyze, and disseminate genetics datasets with millions of samples. AVAILABILITY AND IMPLEMENTATION: https://github.com/hail-is/hail/.

Autoencoders are a deep learning model for representation learning. When trained to minimize the distance between the data and its reconstruction, linear autoencoders (LAEs) learn the subspace spanned by the top principal directions but cannot learn the principal directions themselves. In this paper, we prove that $L_2$-regularized LAEs are symmetric at all critical points and learn the principal directions as the left singular vectors of the decoder. We smoothly parameterize the critical manifold and relate the minima to the MAP estimate of probabilistic PCA. We illustrate these results empirically and consider implications for PCA algorithms, computational neuroscience, and the algebraic topology of learning.