The computational resource growth in natural science motivates the use of machine learning for automated scientific discovery. However, unstructured empirical datasets are often high dimensional, unlabeled, and imbalanced. Therefore, discarding irrelevant (i.e., noisy and information-poor) features is essential for the automated discovery of governing parameters in scientific environments. We present Gaussian Stochastic Gates (STG), which rely on a probabilistic relaxation of the L0 norm of the number of selected features to address this challenge. By applying the Stochastic Gates to a neural network’s input layer, we derive a flexible, fully differentiable model that simultaneously identifies the most relevant features and learns complex nonlinear models. The STG neural network outperforms the state-of-the-art feature selection methods, both in terms of predictive power and its ability to correctly identify the correct subset of informative features. The model was successfully applied for critical biological tasks such as COX proportional hazards model and differential expression analysis on HIV and Melanoma patients. Next, using a linear model, we provide a theoretical basis for optimizing the STG objective using small batches (i.e., SGD). In particular, we present an approximation bound for estimating an unknown signal based on noisy observations. Finally, we develop an extension of the STG model for unsupervised feature selection. The new model is trained to select highly correlated features with the leading eigenvectors of a gated graph Laplacian. The gating mechanism allows us to re-evaluate the Laplacian for different subsets of features and unmask informative structures buried by nuisance features. I will demonstrate that the proposed approach outperforms several unsupervised feature selection baselines.

Abstract:

A discrete Schrödinger operator $H_V = \varepsilon\Delta + V$ on $\ell^2(\mathbb{Z})$ is called Anderson localized if it exhibits a basis of exponentially decaying eigenvectors. If $V_n$ is sampled from a potential function by Diophantine rotations on the one-dimensional torus, $H_V$ is known to be almost-surely Anderson localized for sufficiently small $\varepsilon$ if the potential is either analytic, or cosine-like and symmetric. In this talk, we discuss a new perturbative proof of almost-sure localization for Schrödinger operators with potential sampled from any $C^2$-smooth Morse function with two monotonicity intervals along a Diophantine rotation orbit on the circle. This is a joint work with Tom VandenBoom.

Abstract:

In this talk, we will discuss symmetry inheritance in the Ricci flow in higher dimensions under suitable curvature positivity assumptions. After some background, we will cover the three main ingredients of the proof: a linear analysis on the cylinder, the important neck improvement theorem, and the contradiction argument for cap symmetry (closely following the argument in 3D). Time permitting, we will compare the argument to its counterpart for mean curvature flow and mention some potential applications to the mean curvature flow in higher codimension. This is joint work with Simon Brendle.