Support union recovery in high-dimensional multivariate regression
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高维多元回归中的支持集并集恢复
- 研究问题
- 在多个响应共享协变量的高维回归中，多元组 Lasso 何时能够准确恢复非零系数行，其样本需求如何受稀疏重叠影响？
- 方法与思路
- 作者研究按系数行进行分块正则化的多元组 Lasso，用稀疏重叠函数描述样本复杂度，并推导恢复成功与失败的阈值。
- 材料陈述的贡献
- 摘要称，标准高斯设计下阈值刻画精确；合适的非相关设计与系数结构可降低样本需求，但一般设计下普通 Lasso 可能更好。
- 阅读关注点
- 应核对设计与噪声假设、稀疏重叠函数的定义、成功和失败阈值，以及相对普通 Lasso 的改善所需的正交条件。
- 阅读边界
- 依据版本未注明的 OpenAlex 摘要；比较优势有明确设计和系数结构限制，不能推广到所有高维回归或其他推断目标。
原文摘要
In multivariate regression, a K-dimensional response vector is regressed upon a common set of p covariates, with a matrix B∗ ∈ ℝp × K of regression coefficients. We study the behavior of the multivariate group Lasso, in which block regularization based on the ℓ1 ∕ ℓ2 norm is used for support union recovery, or recovery of the set of s rows for which B∗ is nonzero. Under high-dimensional scaling, we show that the multivariate group Lasso exhibits a threshold for the recovery of the exact row pattern with high probability over the random design and noise that is specified by the sample complexity parameter θ(n, p, s) := n ∕ [2ψ(B∗) log(p − s)]. Here n is the sample size, and ψ(B∗) is a sparsity-overlap function measuring a combination of the sparsities and overlaps of the K-regression coefficient vectors that constitute the model. We prove that the multivariate group Lasso succeeds for problem sequences (n, p, s) such that θ(n, p, s) exceeds a critical level θu, and fails for sequences such that θ(n, p, s) lies below a critical level θℓ. For the special case of the standard Gaussian ensemble, we show that θℓ = θu so that the characterization is sharp. The sparsity-overlap function ψ(B∗) reveals that, if the design is uncorrelated on the active rows, ℓ1 ∕ ℓ2 regularization for multivariate regression never harms performance relative to an ordinary Lasso approach and can yield substantial improvements in sample complexity (up to a factor of K) when the coefficient vectors are suitably orthogonal. For more general designs, it is possible for the ordinary Lasso to outperform the multivariate group Lasso. We complement our analysis with simulations that demonstrate the sharpness of our theoretical results, even for relatively small problems.