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Minimizers are k-mer sampling schemes designed to generate sketches for large sequences that preserve sufficiently long matches between sequences. Despite their widespread application, learning an effective minimizer scheme with optimal sketch size is still an open question. Most work in this direction focuses on designing schemes that work well on expectation over random sequences, which have limited applicability to many practical tools. On the other hand, several methods have been proposed to construct minimizer schemes for a specific target sequence. These methods, however, require greedy approximations to solve an intractable discrete optimization problem on the permutation space of k-mer orderings. To address this challenge, we propose: (a) a reformulation of the combinatorial solution space using a deep neural network re-parameterization; and (b) a fully differentiable approximation of the discrete objective. We demonstrate that our framework, DEEPMINIMIZER, discovers minimizer schemes that significantly outperform state-of-the-art constructions on genomic sequences. 

Combinatorial group testing and compressed sensing both focus on recovering a sparse vector of dimensionality n from a much smaller number 𝑚<𝑛 of measurements. In the first approach, the problem is defined over the Boolean field – the goal is to recover a Boolean vector and measurements are Boolean; in the second approach, the unknown vector and the measurements are over the reals. Here, we focus on real-valued group testing setting that more closely fits modern testing protocols relying on quantitative measurements, such as qPCR, where the goal is recovery of a sparse, Boolean vector and the pooling matrix needs to be Boolean and sparse, but the unknown input signal vector and the measurement outcomes are nonnegative reals, and the matrix algebra implied in the test protocol is over the reals. With the recent renewed interest in group testing, focus has been on quantitative measurements resulting from qPCR, but the method proposed for sample pooling were based on matrices designed with Boolean measurements in mind. Here, we investigate constructing pooling matrices dedicated for the real-valued group testing. We provide conditions for pooling matrices to guarantee unambiguous decoding of positives in this setting. We also show a deterministic algorithm for constructing matrices meeting the proposed condition, for small matrix sizes that can be implemented using a laboratory robot. Using simulated data, we show that the proposed approach leads to matrices that can be applied for higher positivity rates than combinatorial group testing matrices considered for viral testing previously. We also validate the approach through wet lab experiments involving SARS-CoV-2 nasopharyngeal swab samples. 

We consider species tree estimation from multiple loci subject to intralocus recombination. We focus on 𝑅∗, a summary coalescent-based method using rooted triplets. We demonstrate analytically that intralocus recombination gives rise to an inconsistency zone, in which correct inference is not assured even in the limit of infinite amount of data. In addition, we validate and characterize this inconsistency zone through a simulation study that suggests that differential rates of recombination between closely related taxa can amplify the effect of incomplete lineage sorting and contribute to inconsistency.