We consider the sparse linear regression model $\mathbf{y} = X β+\mathbf{w}$, where $X \in \mathbb{R}^{n \times d}$ is the design, $β\in \mathbb{R}^{d}$ is a $k$-sparse secret, and $\mathbf{w} \sim N(0, I_n)$ is the noise. Given input $X$ and $\mathbf{y}$, the goal is to estimate $β$. In this setting, the Lasso estimate achieves prediction error $O(k \log d / γn)$, where $γ$ is the restricted eigenvalue (RE) constant of $X$ with respect to $\mathrm{support}(β)$. In this paper, we introduce a new $\textit{semirandom}$ family of designs -- which we call $\textit{partially-rotated}$ designs -- for which the RE constant with respect to the secret is bounded away from zero even when a subset of the design columns are arbitrarily correlated among themselves. As an example of such a design, suppose we start with some arbitrary $X$, and then apply a random rotation to the columns of $X$ indexed by $\mathrm{support}(β)$. Let $λ_{\min}$ be the smallest eigenvalue of $\frac{1}{n} X_{\mathrm{support}(β)}^\top X_{\mathrm{support}(β)}$, where $X_{\mathrm{support}(β)}$ is the restriction of $X$ to the columns indexed by $\mathrm{support}(β)$. In this setting, our results imply that Lasso achieves prediction error $O(k \log d / λ_{\min} n)$ with high probability. This prediction error bound is independent of the arbitrary columns of $X$ not indexed by $\mathrm{support}(β)$, and is as good as if all of these columns were perfectly well-conditioned. Technically, our proof reduces to showing that matrices with a certain deterministic property -- which we call $\textit{restricted normalized orthogonality}$ (RNO) -- lead to RE constants that are independent of a subset of the matrix columns. This property is similar but incomparable with the restricted orthogonality condition of [CT05].

For partial differential equations on domains of arbitrary shapes, existing works of neural operators attempt to learn a mapping from geometries to solutions. It often requires a large dataset of geometry-solution pairs in order to obtain a sufficiently accurate neural operator. However, for many industrial applications, e.g., engineering design optimization, it can be prohibitive to satisfy the requirement since even a single simulation may take hours or days of computation. To address this issue, we propose reference neural operators (RNO), a novel way of implementing neural operators, i.e., to learn the smooth dependence of solutions on geometric deformations. Specifically, given a reference solution, RNO can predict solutions corresponding to arbitrary deformations of the referred geometry. This approach turns out to be much more data efficient. Through extensive experiments, we show that RNO can learn the dependence across various types and different numbers of geometry objects with relatively small datasets. RNO outperforms baseline models in accuracy by a large lead and achieves up to 80% error reduction.

Tipping points are abrupt, drastic, and often irreversible changes in the evolution of non-stationary and chaotic dynamical systems. For instance, increased greenhouse gas concentrations are predicted to lead to drastic decreases in low cloud cover, referred to as a climatological tipping point. In this paper, we learn the evolution of such non-stationary dynamical systems using a novel recurrent neural operator (RNO), which learns mappings between function spaces. After training RNO on only the pre-tipping dynamics, we employ it to detect future tipping points using an uncertainty-based approach. In particular, we propose a conformal prediction framework to forecast tipping points by monitoring deviations from physics constraints (such as conserved quantities and partial differential equations), enabling forecasting of these abrupt changes along with a rigorous measure of uncertainty. We illustrate our proposed methodology on non-stationary ordinary and partial differential equations, such as the Lorenz-63 and Kuramoto-Sivashinsky equations. We also apply our methods to forecast a climate tipping point in stratocumulus cloud cover. In our experiments, we demonstrate that even partial or approximate physics constraints can be used to accurately forecast future tipping points.