标题： Lasso误差有界当且仅当在比例情况下其活动集大小远离n 显示英文标题

标题： The Lasso error is bounded iff its active set size is bounded away from n in the proportional regime

摘要： 本注释对线性模型中的Lasso \( \hat b\)进行了分析，在真实回归向量没有任何稀疏性或L1假设的情况下，在维度 \( p \) 和样本 \( n \) 同阶的比例情况下。 在高斯设计和协方差矩阵谱远离0和 $+\infty$的情况下，证明了当且仅当所选变量的数量远离 \( n \) 时，L2风险是随机有界的，即 $$ (1-\|\hat b\|_0/n)^{-1} = O_P(1) \Longleftrightarrow \|\hat b- b^*\|_2 = O_P(1) $$ 当 \( n,p\to+\infty \) 时。 右到左的蕴含规则排除了密集Lasso估计（具有接近$n$个活跃变量的估计）的常数风险，这可以用来排除导致密集估计的调参。 我们然后重新引入稀疏性，重新审视精确的相变，该相变表征了导致Lasso风险无界的真正回归向量的稀疏模式——或者根据上述等价性，导致密集Lasso估计的模式。 这个精确的相变是由\citet{miolane2018distribution,celentano2020lasso}通过等价序列模型中的不动点方程建立的。 这里提供了一种该相变的替代证明，使用简单的论据而不依赖于不动点方程或等价序列模型。 对广为人知的限制特征值论点进行修改，使得分析可以扩展到任何常数阶的小调参，从而在相变的一侧得到有界风险。 在相变的另一侧，当无噪声问题中基追踪无法恢复该符号模式时，证明了Lasso风险可以是无界的。 显示英文摘要

摘要： This note develops an analysis of the Lasso \( \hat b\) in linear models without any sparsity or L1 assumption on the true regression vector, in the proportional regime where dimension \( p \) and sample \( n \) are of the same order. Under Gaussian design and covariance matrix with spectrum bounded away from 0 and $+\infty$, it is shown that the L2 risk is stochastically bounded if and only if the number of selected variables is bounded away from \( n \), in the sense that $$ (1-\|\hat b\|_0/n)^{-1} = O_P(1) \Longleftrightarrow \|\hat b- b^*\|_2 = O_P(1) $$ as \( n,p\to+\infty \). The right-to-left implication rules out constant risk for dense Lasso estimates (estimates with close to $n$ active variables), which can be used to discard tuning parameters leading to dense estimates. We then bring back sparsity in the picture, and revisit the precise phase transition characterizing the sparsity patterns of the true regression vector leading to unbounded Lasso risk -- or by the above equivalence to dense Lasso estimates. This precise phase transition was established by \citet{miolane2018distribution,celentano2020lasso} using fixed-point equations in an equivalent sequence model. An alternative proof of this phase transition is provided here using simple arguments without relying on the fixed-point equations or the equivalent sequence model. A modification of the well-known Restricted Eigenvalue argument allows to extend the analysis to any small tuning parameter of constant order, leading to a bounded risk on one side of the phase transition. On the other side of the phase transition, it is established the Lasso risk can be unbounded for a given sign pattern as soon as Basis Pursuit fails to recover that sign pattern in noiseless problems.