Neural Information Processing Systems http://nips.cc/

Non-negative matrix factorization is a popular tool for decomposing data into feature and weight matrices under non-negativity constraints. It enjoys practical success but is poorly understood theoretically. This paper proposes an algorithm that alternates between decoding the weights and updating the features, and shows that assuming a generative model of the data, it provably recovers the groundtruth under fairly mild conditions. In particular, its only essential requirement on features is linear independence. Furthermore, the algorithm uses ReLU to exploit the non-negativity for decoding the weights, and thus can tolerate adversarial noise that can potentially be as large as the signal, and can tolerate unbiased noise much larger than the signal. The analysis relies on a carefully designed coupling between two potential functions, which we believe is of independent interest.

Neural Information Processing Systems http://nips.cc/

We study linear regression and classification in a setting where the learning algorithm is allowed to access only a limited number of attributes per example, known as the limited attribute observation model. In this well-studied model, we provide the first lower bounds giving a limit on the precision attainable by any algorithm for several variants of regression, notably linear regression with the absolute loss and the squared loss, as well as for classification with the hinge loss. We complement these lower bounds with a general purpose algorithm that gives an upper bound on the achievable precision limit in the setting of learning with missing data.

Neural Information Processing Systems http://nips.cc/

This paper investigates the asymptotic distribution of the maximum-likelihood estimate (MLE) in multinomial logistic models in the high-dimensional regime where dimension and sample size are of the same order. While classical largesample theory provides asymptotic normality of the MLE under certain conditions, such classical results are expected to fail in high-dimensions as documented for the binary logistic case in the seminal work of Sur and Candès [2019]. We address this issue in classification problems with 3 or more classes, by developing asymptotic normality and asymptotic chi-square results for the multinomial logistic MLE (also known as cross-entropy minimizer) on null covariates. Our theory leads to a new methodology to test the significance of a given feature. Extensive simulation studies on synthetic data corroborate these asymptotic results and confirm the validity of proposed p-values for testing the significance of a given feature.