21 problems

Consider sparse regression routines applied to highly correlated linear systems arising in equation learning. Correlated-system thresholding conjecture. Thresholding-based sparse r…

Consider filtering noisy discrete data before constructing the weak-form WSINDy linear system, with the simple moving-average filter used in the article as the baseline. Local-poly…

Let $\pmb{\nu}^{(m)}$ be a discrete filter at resolution level $m$, and let $\lVert\pmb{\nu}^{(m)}\rVert_{\overrightarrow{2}}^2$ denote the squared $\ell_2$ norm of its coefficient…

Consider sparse regression for the weak-form linear systems arising in the continuum limit of equation-learning problems, where spurious terms may appear. Thresholding advantage co…

Let MSTLS denote the modified sequential thresholding least-squares algorithm, and distinguish its original formulation from the simplified hard-thresholding formulation analyzed i…

The WSINDy setting combines weak derivative evaluation with classical derivative evaluation in sparse equation learning. Hybrid derivative consistency conjecture. Results on weak-f…

Let thresholded SCAD denote the smoothly clipped absolute deviation variable-selection method followed by a thresholding post-processing step, and let $\rho$ denote the relevant po…

Overparameterized randomly trained networks use a large feature space, and sparse regression seeks a subset of features that still approximates a target function accurately. Sparse…

The atmospheric dynamics are assumed to be explainable by low-order polynomials, with unpolluted-environment kinetics described by the equations for nitrogen dioxide and ozone. Sec…

Consider instances of best subset selection for which a mixed-integer optimization solver does not prove optimality within the allotted computation time, and compare their optimali…

Consider sparse linear regression with $n$ observations and sparsity level $k$, and let the algorithmic sample-size scale be $n_{\mathrm{alg}}=k\log p$. The Gamarnik–Zadik absence-…

Let $\beta^*$ be an exactly $k$-sparse vector with $|\beta^*|_{\mathrm{min}}\geq 1$, and consider the near-optimal sparse regression solutions whose objective is the residual norm…

Let $\beta^*$ be a general $k$-sparse vector with $|\beta^*|_{\mathrm{min}}\geq 1$, and let $n^*$ and $n_{\mathrm{alg}}$ denote the information-theoretic and algorithmic sample-siz…

Let $X\in\mathbb{R}^{n\times p}$ have independent standard Gaussian entries, let $\beta^*$ be a $k$-sparse vector, and observe $Y=X\beta^*+W$, where $W$ has independent…

Strong support-recovery impossibility conjecture. Below $n^*$, recovery of $\beta^*$ is impossible in the very strong information-theoretic sense that even obtaining a fraction of…

Greedy-recovery conjecture. In this regime, straightforward greedy algorithms based on one-step improvements might be able to recover $\beta^*$.

Let $\hat{\boldsymbol{\beta}}_{l_0}(\gamma_0)$ and $\hat{\boldsymbol{\beta}}_{l_1}(\gamma_1)$ be the $l_0$ and $l_1$ penalized regression estimators, respectively, with tuning para…

Consider the $\ell_0$-based sparse regression problem and an exact reformulation of such $\ell_0$-regularized problems as optimization problems involving convex functions in Boolea…

The paper considers sparse linear prediction and compares polynomial-time procedures with methods whose computational cost may be exponential in the dimension. Polynomial-time gap…

Large-sparsity minimax-rate conjecture. The minimax rate of estimation is larger than

For integers $k\leq p$, let $\mathcal{M}(k,p)$ be the class of all subsets of $\{1,\ldots,p\}$ of cardinality $k$, and let $\Theta[k,p]$ be the set of vectors in $\mathbb{R}^p$ wit…