Spacing Test for Fused Lasso

Choosing between classical and Bayesian sparse regression methods involves a real trade-off: penalized estimators like Lasso run in milliseconds but give no uncertainty estimates,while Horseshoe and Spike-and-Slab priors produce full posteriors but need MCMC chains that take minutes per fit.Surprisingly few studies compare these two families head-to-head under the conditions that actually make sparse regression hard -- correlated features, weak signals, and growing dimensionality. We benchmark six methods (OLS, Ridge,Lasso, Elastic Net, Horseshoe, Spike-and-Slab) on synthetic data with three covariance structures (rho up to 0.9), four SNR levels, and p in {20, 50, 100}, plus the Diabetes dataset,totalling over 2,600 experiments. The results are clear on some points and nuanced on others. Bayesian methods win on prediction error (MSE 72 vs. 108-267), and the Horseshoe delivers near-nominal 95% coverage (94.8%). But Spike-and-Slab,despite narrower intervals, under-covers at 91.9% -- its continuous relaxation likely plays a role. For variable selection, Lasso and Spike-and-Slab tie at F1 ~ 0.47, making Lasso the practical default when posteriors are not needed. Code and data are available at https://github.com/xiao98/sparse-bayesian-regression-bench.

We determine the sharp restricted isometry threshold for recovery at global minima of the rank-restricted matrix LASSO. For target rank $r_{\star}$, if the rank-$k$ RIP constant satisfies $δ<δ_{\mathrm{sharp}}(k/r_{\star})$, where $δ_{\mathrm{sharp}}(t)=t/(4-t)$ for $0<t<4/3$ and $δ_{\mathrm{sharp}}(t)=\sqrt{(t-1)/t}$ for $t\ge4/3$, then every global minimizer has Frobenius error $\lesssim\sqrt{r_{\star}}λ$ for all $λ\gtrsim\|\mathcal{A}^{*}(ξ)\|_{\mathrm{op}}$ and at every search rank $r\ge r_{\star}$. The constants depend only on the RIP constant and $t=k/r_{\star}$, and in particular are independent of the search rank. When the rank restriction is inactive, the result specializes to the ordinary convex matrix LASSO. We also obtain the analogous results for sparsity-restricted vector LASSO. Conversely, we show that the threshold $δ<δ_{\mathrm{sharp}}(k/r_{\star})$ cannot be improved, due to the existence of counterexamples whose global minimizers fail to recover the ground truth.

Selective inference (SI) provides statistically valid $p$-values for hypotheses selected by applying an algorithm to the data, correcting for the bias that arises when the same data are used both to select and to test a hypothesis. Developing an SI procedure for a new algorithm, however, has required an expert to derive, and then implement, the selection event, i.e., the conditions under which the hypothesis is selected. Repeating this specialized effort for every new algorithm is why exact SI has so far been available for only a narrow class. We propose AutoSI, a framework that removes this barrier in two ways. First, AutoSI constructs the selection event automatically from the algorithm's individual operations, so the user only writes the algorithm as ordinary NumPy-like code and derives nothing by hand. Second, AutoSI broadens the class of selection events SI can handle: existing exact methods are limited to selection events characterized by linear or quadratic inequalities in the data, whereas AutoSI covers any algorithm expressible through rational functions of the data (ratios of polynomials). We prove that the $p$-values computed by AutoSI are exactly valid in finite samples. We demonstrate AutoSI on three feature-selection methods, each written in a few dozen lines of code. One of these methods, the lasso with its tuning parameter selected by cross-validated $R^2$, cannot be handled within existing exact SI frameworks and is made possible by AutoSI. Experiments on synthetic and real datasets show that the resulting $p$-values control the type I error rate (i.e., the false positive rate) at the nominal level while retaining high power.