Lasso and Partially-Rotated Designs

We consider the sparse linear regression model $\mathbf{y} = X \beta +\mathbf{w}$, where $X \in \mathbb{R}^{n \times d}$ is the design, $\beta \in \mathbb{R}^{d}$ 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 $\beta$. In this setting, the Lasso estimate achieves prediction error $O(k \log d / \gamma n)$, where $\gamma$ is the restricted eigenvalue (RE) constant of $X$ with respect to $\mathrm{support}(\beta)$. 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 $\mathrm{support}(\beta)$. Let $\lambda_{\min}$ be the smallest eigenvalue of $\frac{1}{n} X_{\mathrm{support}(\beta)}^\top X_{\mathrm{support}(\beta)}$, where $X_{\mathrm{support}(\beta)}$ is the restriction of $X$ to the columns indexed by $\mathrm{support}(\beta)$. In this setting, our results imply that Lasso achieves prediction error $O(k \log d / \lambda_{\min} n)$ with high probability. This prediction error bound is independent of the arbitrary columns of $X$ not indexed by $\mathrm{support}(\beta)$, 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].