证明
Proposition 17 ([The prediction curve]). For the strict and gap critical…Lemma 16 ([Entropy loss under a Gaussian observation]). Let p and q be smooth…观点
We first prove the strict-case spectral estimates. If a random vector S has covariance C and mean a , the affine estimator a+C(+C)^ -1 (S+G-a) of S from S+G is its least-squares affine predictor. Its mean squared norm is |a|^2+(C^2(+C)^ -1 ) , by direct expansion of the covariances. Conditional expectation has at least this squared norm, because it is the orthogonal projection onto all functions of the observation. The same calculation holds conditionally on any side information, using fresh independent Gaussian noise. By the Gaussian signal derivative formula and its monotonicity, |[w L_1]|^2 ≤2(d(2)-d(1))≤2d(2)=o(I). Apply the affine prediction bound to S=w and C=_ηw . For every fixed ε>0 , (C\1_ C>ε )=o(I), since t^2/(1+t)≥εt/(1+ε) for t>ε . For every projection of rank at most cI , |Πw|^2 ≤εcI+(C\1_ C>ε ) +| w|^2. First take the sequence limit, then ε0 . Equation proves Equation uniformly in Π . In the gap case a projection of rank at most cI is eventually zero, which proves the same assertion. Conditionally on U_r , the additional observation L_1 has signal w and covariance C_r . Orthogonality of successive conditional expectations and the conditional affine prediction bound yield I(p_1(r)-m(r)) ≥(C_r^2(+C_r)^ -1 ). Proposition 17 ([The prediction curve]) makes the left side o(I) . The same scalar eigenvalue inequality proves Equation. We finally bound the dimensions of the deterministic outlier spaces. Let D=_η[XX^* X+√sG=Y_0], where the inner posterior kernel always refers to the original prior μ . Conditional Jensen and the definition of M_s imply _ηb_sb_s^* D, _ηM_s D/s. For each deterministic projection Q of rank l≥1 , the original subgaussian assumption gives _μ((|QX|^2)/(Ca^2))≤Cl. Indeed choose a 1/2 -net of the unit sphere in ranQ with at most 5^l points. The norm |QX| is at most twice the largest absolute linear form over this net, and bounding the exponential of a maximum by the sum proves the claim. In the strict case, form the joint law by first drawing Y_0 from η and then drawing X from the original posterior kernel. Its entropy relative to the original joint law of (X,X+√sG) is exactly H_0=I/(2λ) . Data processing therefore bounds the entropy of its X -marginal relative to μ by H_0 . Applying the entropy variational inequality to Equation gives (QD)≤Ca^2(H_0+l). If the spectral subspace of D above τ>0 has dimension l , then τl≤Ca^2(H_0+l) . For sufficiently small a this implies l≤2Ca^2H_0/τ=O_τ(I) ; the assertion is trivial when l=0 . The spectral subspace of ww^* above ε has dimension at most I/ε , since its trace is I . Take V_ε to be the span of this subspace and the spectral subspace of D above sε . Its dimension is at most C_εI , and Equation gives all three bounds in Equation. In the gap case, averaging the original posterior under η=μ_s gives D=_μXX^* Ca^2 . Thus Equation gives the two claimed operator bounds directly. Also \| ww^*\|_ ≤I→0 , so V_ε= 0 suffices eventually. This completes all assertions.