Neural networks have been used prominently in several machine learning and statistics applications. In general, the underlying optimization of neural networks is non-convex which makes analyzing their performance challenging. In this paper, we take another approach to this problem by constraining the network such that the corresponding optimization landscape has good theoretical properties without significantly compromising performance. In particular, for two-layer neural networks we introduce Porcupine Neural Networks (PNNs) whose weight vectors are constrained to lie over a finite set of lines. We show that most local optima of PNN optimizations are global while we have a characterization of regions where bad local optimizers may exist. Moreover, our theoretical and empirical results suggest that an unconstrained neural network can be approximated using a polynomially-large PNN.

BACKGROUND: With the recent proliferation of single-cell RNA-Seq experiments, several methods have been developed for unsupervised analysis of the resulting datasets. These methods often rely on unintuitive hyperparameters and do not explicitly address the subjectivity associated with clustering. RESULTS: In this work, we present DendroSplit, an interpretable framework for analyzing single-cell RNA-Seq datasets that addresses both the clustering interpretability and clustering subjectivity issues. DendroSplit offers a novel perspective on the single-cell RNA-Seq clustering problem motivated by the definition of "cell type", allowing us to cluster using feature selection to uncover multiple levels of biologically meaningful populations in the data. We analyze several landmark single-cell datasets, demonstrating both the method's efficacy and computational efficiency. CONCLUSION: DendroSplit offers a clustering framework that is comparable to existing methods in terms of accuracy and speed but is novel in its emphasis on interpretabilty. We provide the full DendroSplit software package at https://github.com/jessemzhang/dendrosplit .