Inverse problems correspond to a certain type of optimization problems formulated over appropriate input distributions. Recently, there has been a growing interest in understanding the computational hardness of these optimization problems, not only in the worst case, but in an average-complexity sense under this same input distribution.In this revised note, we are interested in studying another aspect of hardness, related to the ability to learn how to solve a problem by simply observing a collection of previously solved instances. These 'planted solutions' are used to supervise the training of an appropriate predictive model that parametrizes a broad class of algorithms, with the hope that the resulting model will provide good accuracycomplexity tradeoffs in the average sense.We illustrate this setup on the Quadratic Assignment Problem, a fundamental problem in Network Science. We observe that datadriven models based on Graph Neural Networks offer intriguingly good performance, even in regimes where standard relaxation based techniques appear to suffer.

Recently, Bandeira [5] introduced a new type of algorithm (the so-called probably certifiably correct algorithm) that combines fast solvers with the optimality certificates provided by convex relaxations. In this paper, we devise such an algorithm for the problem of k-means clustering. First, we prove that Peng and Wei's semidefinite relaxation of k-means [20] is tight with high probability under a distribution of planted clusters called the stochastic ball model. Our proof follows from a new dual certificate for integral solutions of this semidefinite program. Next, we show how to test the optimality of a proposed k-means solution using this dual certificate in quasilinear time. Finally, we analyze a version of spectral clustering from Peng and Wei [20] that is designed to solve k-means in the case of two clusters. In particular, we show that this quasilinear-time method typically recovers planted clusters under the stochastic ball model.

Single-cell technologies characterize complex cell populations across multiple data modalities at unprecedented scale and resolution. Multi-omic data for single cell gene expression, in situ hybridization, or single cell chromatin states are increasingly available across diverse tissue types. When isolating specific cell types from a sample of disassociated cells or performing in situ sequencing in collections of heterogeneous cells, one challenging task is to select a small set of informative markers that robustly enable the identification and discrimination of specific cell types or cell states as precisely as possible. Given single cell RNA-seq data and a set of cellular labels to discriminate, scGeneFit selects gene markers that jointly optimize cell label recovery using label-aware compressive classification methods. This results in a substantially more robust and less redundant set of markers than existing methods, most of which identify markers that separate each cell label from the rest. When applied to a data set given a hierarchy of cell types as labels, the markers found by our method improves the recovery of the cell type hierarchy with fewer markers than existing methods using a computationally efficient and principled optimization.