Wang and Leng (2016), High-Dimensional Ordinary Least-Squares Projection for Screening Variables, Journal of The Royal Statistical Society Series B, 78, 589–611

原文摘要

The conclusion in Lemma 5 in the supplementary material (SM) of Wang and Leng (2016), on which Theorems 1–3 rely, is incorrect. Here we provide an additional condition under which the conclusion of this lemma holds. We assume that the rows of the matrix X follow a Gaussian distribution with precision matrix Ω = Σ−1 that satisfies c 4 - 1 n - τ / 2 ≤ λ min ( Ω ) ≤ λ max ( Ω ) ≤ c 4 n τ / 2 . This implies the assumption on the condition number of Σ in Assumption A3. We drop the assumption that Σii = 1 (SM, Proposition 5). We append the assumptions in Wang and Leng (2016) with condition (C1) max j , k ≠ i | Ω ij - Ω ik | ≤ c 0 n 5 τ / 4 p for some constant c0 > 0 and max j ≠ k | Ω jj - Ω kk | ≤ c 5 n τ / 4 p for some constant c5 > 0. We discuss this assumption in more detail in the final paragraph. Suppose 1 − 15τ/2 − 2κ − ν = ϵ for some ϵ > 0, p ≥ c′·n3/2 for some c′ > 0 and log p = o ( n 1 - 11 τ / 2 - 2 κ - ν - ϵ ¯ ) , where 0 < ϵ ¯ < ϵ . Then, with probability 1 - O ( exp { - n 1 - 11 τ / 2 - 2 κ - ν - ϵ ¯ } ) , for all i ∈ S, | e i H H T β | ≥ c · n 1 - τ - κ p and for all j ∉ S, | e j H H T β | ≤ c ~ · n 1 - τ - κ - ϵ ¯ / 2 p for some ( c , c ~ ) > 0 . This implies that Lemma 5 holds. Proof: Write H ( i , j ) = e i T X T ( X X T ) - 1 X e j . As HHT is a projection matrix, ∑ j = 1 p H ( i , j ) 2 = H ( i , i ) and hence, min j ≠ i | H ( i , j ) | ≤ H ( i , i ) / p . By Lemma 4 in Wang and Leng (2016), H ( i , i ) ≤ c 2 ′ n 1 + τ / p with probability at least 1 − 4exp(−Cn), and using the reverse triangle inequality max j ≠ i | H ( i , j ) | = min j ≠ i | H ( i , j ) | + max j , k ≠ i ( | H ( i , j ) | - | H ( i , k ) | ) ≤ min j ≠ i | H ( i , j ) | + max j , k ≠ i | H ( i , j ) - H ( i , k ) | ≤ ( c 2 ′ ) 1 / 2 n 1 / 2 + τ / 2 / p + max j , k ≠ i | H ( i , j ) - H ( i , k ) | with probability at least 1 − 4exp(−Cn). We will first show that for any C > 0 and t > 0, max j , k ≠ i | H ( i , j ) - H ( i , k ) | = O n 1 / 2 + 9 τ / 4 + n 1 / 2 + 5 τ / 4 t p with probability 1 − 9p2 exp(−Cn) − 2p2 exp(−t2/2). Let ϵ denote ϵ2 or ϵ3. For (II) and (III), using the Woodbury identity, x 3 T ( X X T ) - 1 ϵ = x 3 T ( X 2 X 2 T ) - 1 ϵ - x 3 T ( X 2 X 2 T ) - 1 X 1 ( I + X 1 T ( X 2 X 2 T ) - 1 X 1 ) - 1 X 1 T ( X 2 X 2 T ) - 1 ϵ , where the first item can be bounded by using the classical tail bound as | x 3 T ( X 2 X 2 T ) - 1 ϵ | ≤ x 3 T ( X 2 X 2 T ) - 2 x 3 · | Θ 11 | - 1 / 2 t = O n 1 / 2 + 5 τ / 4 p · t with probability 1 − 2 exp(−t2/2) − 5 exp(−Cn). The second item is bounded as x 3 T ( X 2 X 2 T ) - 1 X 1 ( I + X 1 T ( X 2 X 2 T ) - 1 X 1 ) - 1 X 1 T ( X 2 X 2 T ) - 1 ϵ ≤ x 3 T ( X 2 X 2 T ) - 1 x 3 · ϵ T ( X 2 X 2 T ) - 1 ϵ · λ max X 1 T ( X 2 X 2 T ) - 1 X 1 ( I + X 1 T ( X 2 X 2 T ) - 1 X 1 ) - 1 = O n 1 / 2 + 9 τ / 4 p with probability at least 1 − 4 exp(−Cn) and the last equality uses that p ≥ c′·n3/2 for some c′ > 0. Applying a union bound, we conclude that max j , k ≠ i | H ( i , j ) - H ( i , k ) | ≤ O n 1 / 2 + 9 τ / 4 + n 1 / 2 + 5 τ / 4 t p with probability 1 − 9p2 exp(−Cn) − 2p2 exp(−t2/2). For i ∈ S, | e i T H H T β | = | H ( i , i ) β i + ∑ j ≠ i H ( i , j ) β j | ≥ c 1 - 1 c 2 n 1 - τ - κ p - max i ≠ j | H ( i , j ) | s ‖ β ‖ with probability at least 1 − 4 exp(−Cn). Under Assumption A3 and using the previously established result on max i ≠ j | H ( i , j ) | , the second term is negligible compared to the first when 1 − 15τ/2 − 2κ − ν = ϵ for some ϵ > 0, and t = nα with α = 1 / 2 - 11 τ / 4 - κ - ν / 2 - ϵ ¯ / 2 > 0 where 0 < ϵ ¯ < ϵ . If log p = o(n2α), then for all i ∈ S, | e i H H T β | ≥ c n 1 - τ - κ p for some c > 0 with probability 1 − O( exp {−n2α}). For i ∉ S, we then have | e i T H H T β | = ∑ i ≠ j H ( i , j ) β j ≤ c ~ · n 1 - τ - κ - ϵ ¯ / 2 p for some c ~ > 0 with the same probability. This concludes the proof. Condition (C1) rules out the general covariance structure stated in the paper, but it helps to explain why HOLP works well for the numerical examples in the paper. In particular, (C1) holds for the examples in 4.1.1 (independent predictors) and 4.1.2 (equicorrelated predictors). Example 4.1.4 is a factor model X = FΛT + V, with F a random n × r matrix, E[FTF/n] = Ir, E[VTV/n] = cIp for some constant c > 0, E[VTF] = 0. If we consider Λ fixed and for simplicity assume that ΛTΛ/p = Ir, we have Ω = c - 1 I p - 1 c + p Λ Λ T so that (C1) holds if the elements of Λ are bounded and r is fixed. Though (C1) does not hold for the examples in 4.1.5 and 4.1.6, the variables in these examples can be partitioned into groups such that variables in the same group satisfy (C1). Helpful discussion with Rajen Shah is gratefully acknowledged.

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