21 problems

Let Forward Stagewise regression and the Lasso be the two fitting procedures discussed in the source, and let least-squares boosting be the procedure that repeatedly fits regressio…

Lasso active-set constraints. The Lasso choice must satisfy ${\mathcal A}_1\subseteq {\mathcal A}$ and ${\mathcal A}\subseteq {\mathcal A}_{10}$; moreover, $w_{\mathcal A}$ cannot…

Let $(X_i,Y_i)_{i=1}^{n+1}$ follow the data-generating process described in Section 2, let $\hat{\beta}$ be the LASSO estimator fit on all $n+1$ observations, and let…

Let $c$ parameterize the design correlation, and let the known-sign and unknown-sign summary criteria be denoted by $\psi_\Lambda(c)$, $\psi_{\max}(c)$, $\psi_\Lambda^{\pm}(c)$, an…

Let ${\cal F}^{(0)}_M((0,1]^d)$ denote the class of order-zero functions with variation bounded by $M$, and let $({\cal R}_{\phi}(d,J):J)$ and $({\cal R}_{\tilde{\phi}}(d,J):J)$ be…

Let $M_1$ and $M_2$ be the Lipschitz constants controlling the first two and second two arguments, respectively, of the parameter functional $\phi_\theta$. Let $n_1,n_2$ be the sam…

Let $\boldsymbol\theta_k$, $\boldsymbol\theta_k^{\mathrm d}$, $\boldsymbol\theta_k^f$, and $\boldsymbol\theta_k^{f,\mathrm d}$ denote the original, debiased, fixed-design, and fixe…

Let $(\mathrm{BP}_{\sigma,\mathcal{K}})$ denote the generalized basis-pursuit program with gauge or structural set $\mathcal{K}$, and let parameter sensitivity refer to the depende…

The model concerns confidence intervals for a regression coefficient $\theta^*_j$, including exact intervals obtained by inverting tests, leave-one-out confidence intervals, and de…

Consider a high-dimensional change point regression model with $K$ true change points, and suppose the Lasso estimator produces no more than $2K$ estimated change points. A variant…

Incoherence-relaxation conjecture. The incoherence condition could be relaxed for the trimmed $\ell_1$ problem, similarly to the case of non-convex $\mu$-amenable regularizers such…

Let $\hat{\beta}^{\lambda}(p)$ denote the LASSO estimator at problem size $p$. Let $\eta$ be the scalar soft-thresholding function, let $B\sim p_\beta$ and $W\sim N(0,1)$ be indepe…

Let $\operatorname{NSE}(\sigma)$ denote the normalized squared reconstruction error of the constrained Generalized LASSO, and define … For the isotropically random orthogonal measu…

Let $\mathbf{x}_0\in\mathbb{R}^n$, let $\mathbf{y}=\mathbf{A}\mathbf{x}_0+\sigma\mathbf{v}$, and let $\hat{\mathbf{x}}$ be the reconstruction produced by the constrained Generalize…

Let an unknown signal be $\mathbf{x}_0\in\mathbb{R}^n$, let measurements be $\mathbf{y}=\mathbf{A}\mathbf{x}_0+\sigma\mathbf{v}\in\mathbb{R}^m$, and let $\hat{\mathbf{x}}$ be the s…

Universality conjecture. When $A$ has independent sub-Gaussian entries, the results of this paper should still hold, although with an extra error term that becomes large when $x$ i…

Let $\boldsymbol y=\mathbf X\boldsymbol{\beta}^*+\boldsymbol{\xi}$ be a Gaussian linear regression model with deterministic design matrix $\mathbf X\in\mathbb{R}^{n\times p}$, and…

Phase-transition universality conjecture. The choice of measurement matrix ensemble does not affect the phase transition of c-LASSO and CAMP.

Universality conjecture. The results are universal and hold for a “larger” class of random matrices.

Let $\{x(n),w(n),A(n)\}_{n\in\mathbb{N}}$ be a converging sequence of instances whose matrices $A(n)$ have i.i.d. entries with mean $\mathbb{E}\{A_{ij}\}=0$, variance…

Let $A$ be the measurement matrix, let AMP denote the approximate message-passing algorithm studied in the paper, and let LASSO denote the corresponding $\ell_1$-regularized least-…