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Denote by Bayes the quotient category of ^Bayes by the following equivalence relation: Two morphisms T, T′: (X, µX)→(Y, µY) are equivalent if and only if T∗µX= T′ ∗µX. Then Bayes is a groupoid. 4.Bayesian supervised learning In this section, using the results obtained in the previous section, we propose a mathematical ... | https://arxiv.org/abs/2502.15408v2 |
mm//P(Ym).(4.3) (2) For a training sample Sn∈(X × Y )n, the posterior distribution µΘ|Sn∈ P(Θ) after seeing Snis the value q(n) ΠX(Sn) ΠY(Sn) of a Bayesian inversion q(n) ΠX(Sn):Yn→ P(Θ) of the Markov kernel mn◦ΠΠX(Sn)◦p: Θ→ P(Yn) relative to µΘ. (3) For Tm= (t1, . . . , t m)∈ Xm, the predictive distribution PTm∈ P(... | https://arxiv.org/abs/2502.15408v2 |
is commutative P(Y)Xmn◦ΠXn// RAP(Yn) P(Y)Amn◦ΠA Xn66 where ΠA Xn:P(Y)A→ P(Y)nis the evaluation map defined in (4.1) but forh∈ P(Y)A. As a corollary of Lemma 4.5, we obtain immediately the following. Corollary 4.6. Letq(n):Yn→ P (P(Y)X)be a Bayesian inversion of mn◦ΠXn:P(YX)→ P (Yn). Then (RA)∗◦q(n):Yn→ P (P(Y)A)is a ... | https://arxiv.org/abs/2502.15408v2 |
P(VX)mm◦ΠXm//P(Vm) where δis the Dirac map : δ(x) :=δx, whose measurability is well-known, see Subsection 3.1. The required commutativity can be verified straightforward. □ PROBABILISTIC MORPHISMS 17 Since ⊗m i=1(δf(xi)∗νε) = (⊗m i=1δf(xi))∗νm εwe obtain immediately from Lemma 4.8 the following Corollary 4.9. The assum... | https://arxiv.org/abs/2502.15408v2 |
. . , t m)∈ Xm, see, e.g., [RW2006], [Bishop2006]. This formula is obtained by the same pro- cedure explained in Corollary 4.6, which can be described in the following commutative diagram (4.14) RA,(RA)∗µ (ΠTm×p(n,ε) Xn) (Rm×Rn, µm,nε) ΠRm vvΠRn )) Rm,(ΠTm)∗µ Rn,(p(n,ε) Xn)∗µ .(ΠTm)◦(RV A)◦q(n,ε) XnooRV A◦q(n... | https://arxiv.org/abs/2502.15408v2 |
morphism, Mathematical Structures in Computer Science 31(8), 850-897 (2021) [FGBR20] T. Fritz, T. Gonda, P. Perrone, E. F. Rischel. Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability. Theoretical Computer Science Volume 961, 15 June 2023, 113896 (2023) [Giry82] M. Giry.... | https://arxiv.org/abs/2502.15408v2 |
Dimension-free bounds in high-dimensional linear regression via error-in-operator approach Fedor Noskov* Nikita Puchkin†Vladimir Spokoiny‡ Abstract We consider a problem of high-dimensional linear regression with random design. We suggest a novel approach referred to as error-in-operator which does not estimate the des... | https://arxiv.org/abs/2502.15437v1 |
. . . . . . . . . . . . . . . . . . . . . . . . . 44 C Results from the proof of Theorem 2.3 46 C.1 Proof of Lemma 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 C.2 Proof of Proposition C.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 C.3 Proof of Pro... | https://arxiv.org/abs/2502.15437v1 |
. . . . . . . . . . . . . . . . . . . . 75 D.8.1 Proof of Lemma D.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 D.8.2 Proof of Lemma D.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 D.8.3 Proof of Lemma D.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... | https://arxiv.org/abs/2502.15437v1 |
. . . . . 93 H.4 Proof of Proposition H.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 I Proof of Theorem 3.3 97 2 1 Introduction Recent advances in supervised machine learning devoted to understanding of deep neural networks revealed surprising effects going beyond the classical statistic... | https://arxiv.org/abs/2502.15437v1 |
to the implicit regularization. Another line of research (see, for instance, [Dobriban and Wager, 2018, El Karoui, 2018, Taheri et al., 2021, Richards et al., 2021, Mahdaviyeh and Naulet, 2019, Derezinski et al., 2020, Hastie et al., 2022, Wu and Xu, 2020, Bach, 2024]) examined the least squares and ridge regression es... | https://arxiv.org/abs/2502.15437v1 |
respect to operator helpsbθto be less susceptible to double descent. Following [Spokoiny, 2023], we call bθerror-in-operator estimate . Contribution. The main contributions can be summarized as follows. 4 • We suggest a novel approach for the problem of random design linear regression. We specify leading terms in the b... | https://arxiv.org/abs/2502.15437v1 |
that the covariates X,X1, . . . ,Xnmeet the following requirements. Assumption 2.1. There exists a positive constant CXsuch that, for any matrix Mand any λsatisfying the inequality CX|λ|⩽1/∥M∥F, it holds that logEexpn λ(Σ−1/2X)⊤M(Σ−1/2X)o ⩽λTr(M) +C2 Xλ2∥M∥2 F,where Σ =EXX⊤. In [Puchkin and Rakhuba, 2024], the authors ... | https://arxiv.org/abs/2502.15437v1 |
exploit in our analysis is that its gradient differs from ∇L(υ)by a parameter-free vector z: ∇L(υ) =∇L(υ)−z,where z= 0 Z−EZ vec(bΣ−Σ) ∈R2d+d2. In other words, for any υthe difference L(υ)− L(υ)is an affine function of (Z−EZ)and(bΣ−Σ). We are ready to present bias and variance expansions of the error-in-operator est... | https://arxiv.org/abs/2502.15437v1 |
ζfrom Theorem 2.4 help us to quantify the excess risk of the error-in-operator estimate in terms of rq(k) =X j>kσj σk+1q , k∈ {1, . . . , d }, (11) where σ1⩾σ2⩾. . .⩾σddenote the eigenvalues of Σ. In Appendix F, we prove the following upper bound on the norm of Σ1/2(bθ−θ◦)with explicit constants. Corollary 2.7. Assum... | https://arxiv.org/abs/2502.15437v1 |
. Since rq(k)monotonously decreases in q >0,Var bθ X exhibits a better behaviour than Var bθ(R) X . At the same time, the first term in the right-hand side of (16) is of the same order as the upper bound on the bias of bθ(R)obtained in [Cheng and Montanari, 2024, Proposition 2.2]. Moreover, one can easily show that... | https://arxiv.org/abs/2502.15437v1 |
2.2. Fix arbitrary positive numbers δ <1andρ0⩽1/8. Suppose thatn⩾212(1 +CX)2(r(Σ) + log(2 /δ))and assume that the parameters µandλsatisfy the conditions ∥θ◦∥⩽ρ0µ/7,∥Σ∥∥θ◦∥⩽ρ0µ√ λ/(56·18), (21) λ⩾213(1 +CX)∥Σ∥2∥θ◦∥ ρ0µr 4r(Σ) + log(2 /δ) n, (22) and λ ∥Σ∥∧√ λ⩾211σ∥Σ∥1/2 ρ0µr r(Σ) + log(4 /δ) n. (23) Then, with probabili... | https://arxiv.org/abs/2502.15437v1 |
of finite µon the risk curve when λ⩾λopt(n), but finite µcan significantly mitigate the effect of double descent if λ < λ opt. 5 Proof of Theorem 2.3 We start with studying the difference υ∗−υ◦. For any t∈[0,1], let us define u(t)= (1−t)υ◦+tυ∗. Due to the Newton-Leibniz formula, we have ∇L(υ∗)−∇L(υ◦) =∇L(u(1))−∇L(u(0))... | https://arxiv.org/abs/2502.15437v1 |
satisfied with high probability. This yields that both bυ andυ∗belong to a convex set around υ◦where, according to Lemma A.1, the Hessian ∇2L(υ)is positive definite. Hence, if the penalization parameters µandλare chosen in agreement with (7), the difference (bυ−υ∗)can be approximated by the standardized score H−1 ∗zwhe... | https://arxiv.org/abs/2502.15437v1 |
log(4 /δ) n,∥(bΣ−Σ)θ◦∥⩽4CX∥Σ∥∥θ◦∥r r(Σ) + log(4 /δ) n,(31) and ∥U∥⩽8σ∥Σ∥1/2r r(Σ) + log(4 /δ) n. (32) We denote the corresponding event by E1. Then the inequalities (9) ensure that 395∥bΣ−Σ∥F⩽√ λ 2,395∥U∥⩽µ√ λ, and 395∥(bΣ−Σ)θ◦∥⩽∥θ◦∥√ λ 2. Taking into account the triangle inequality ∥Z−EZ∥⩽∥U∥+∥(bΣ−Σ)θ◦∥, we obtain tha... | https://arxiv.org/abs/2502.15437v1 |
Lemma B.2 (see (ii) and (iv)), we have ∥A∗θ∗−η∗∥⩽√ λ∥θ◦∥,∥θ∗∥⩽∥θ◦∥, and ∥θ∗−θ◦∥⩽∥bλ∥ 1 +3∥θ◦∥ µ ⩽∥θ◦∥ 1 +3 111 =38∥θ◦∥ 37. This means that θ∗belongs to Υ(ρ∗)withρ∗=∥θ◦∥/µ⩽1/112. Hence, there exists an event E∗ 2such that P(E∗ 2)⩾(1−5δ/2)and the inequalities ∥(bΣ−Σ)θ∗∥⩽4CX∥Σ∥1/2∥Σ1/2θ∗∥r r(Σ) + log(4 /δ) n, (41) ∥(b... | https://arxiv.org/abs/2502.15437v1 |
Ψ(n, δ) = 196 σ∥Σ∥1/2 ∥θ◦∥+ 2CX∥Σ∥!2 ·r(Σ)2+ log(4 /δ) n. We postpone the proof of Lemma 6.4 to Section D.5 and finish the proof of Theorem 2.4 first. The next auxiliary lemmata ensure that Hθχ(υ∗)H−1 χχ(υ∗)zχis close to −1 2 Σ(Z−EZ)−Σ(bΣ−Σ)θ∗−(bΣ−Σ)Σbλ . 22 Lemma 6.5. Assume that the parameters µandλfulfil (7)and th... | https://arxiv.org/abs/2502.15437v1 |
of Statistics , 52(6):2879–2912, 2024. G. Chinot and M. Lerasle. On the robustness of the minimum ℓ2interpolator. Bernoulli (to appear), arXiv:2003.05838, 2025+. G. Chinot, M. L ¨offler, and S. van de Geer. On the robustness of minimum norm interpolators and regularized empirical risk minimizers. The Annals of Statisti... | https://arxiv.org/abs/2502.15437v1 |
International Conference on Artificial Intelligence and Statistics , pages 3889–3897. PMLR, 2021. V . Spokoiny. Mixed laplace approximation for marginal posterior and bayesian inference in error-in-operator model. Preprint. ArXiv:2305.09336, 2023. H. Taheri, R. Pedarsani, and C. Thrampoulidis. Fundamental limits of rid... | https://arxiv.org/abs/2502.15437v1 |
−H−1 χχHχθ˘H−1 θθ˘H−1 χχ! . (53) We will be particularly interested in ˘Hθθ=Hθθ−HθχH−1 χχHχθ=H/H χχ, which is nothing but the Schur complement of Hχχ. Before we proceed, we have to introduce additional notation. Let m(·)be an arbitrary probability measure on the space of triplets υ= (θ,η, A). For a function f(υ)(scalar... | https://arxiv.org/abs/2502.15437v1 |
mis clear from the context, we omit it throughout the proof and write ⟨H⟩and⟨θ⟩, instead of ⟨H⟩mand⟨θ⟩mfor brevity. Let us note that, according to (52), the matrix Hχχ(υ)admits the following form: Hχχ=2Id Id⊗θ⊤ Id⊗θId⊗(µ2Id+θθ⊤) . Then, due to the definition of ⟨H⟩it holds that ⟨H⟩χχ= 2Id Id⊗ ⟨θ⟩⊤ Id⊗ ⟨θ⟩Id⊗ µ2Id+ ... | https://arxiv.org/abs/2502.15437v1 |
For this reason, we split the rest of the proof into two steps and show that |T |⩽4 µ0√λ0∥D0υ∥∥D0u∥2and|Q|⩽2 µ2 0λ0∥D0υ∥2∥D0u∥2. Step 1: upper bound on |Q|.We start with a simpler inequality |Q|⩽2 µ2 0λ0∥D0u∥2∥D0υ∥2 and postpone the proof of the upper bound on the absolute value Tto the next step. It is enough to show ... | https://arxiv.org/abs/2502.15437v1 |
0λ0∥D0υ∥2∥D0u∥2. □ A.4 Proof of Lemma A.4 The proofs of the upper bounds ∥HηAH−1/2 AA∥⩽ρand∥H−1/2 θθHθAH−1/2 AA∥⩽ρ√ 2is based on a simple observation that the operator norm of a matrix Bdoes not exceed ρif and only if B⊤B⪯ρ2Id. In our case, it is enough to show that H−1/2 AAHηAHAηH−1/2 AA⪯ρ2Idand H−1/2 θθHθAH−1 AAHAθH−... | https://arxiv.org/abs/2502.15437v1 |
Proof of Lemma B.1 First, let us show that θ∗∈Θ∗. Applying Lemma B.2(iv) with ρ0⩽1/7in place of ρ, we obtain that ∥θ∗−θ◦∥⩽2∥θ◦∥. (70) Next, taking the condition (ii) of the lemma into account and using Lemma B.2(iv), we observe that ∥Σθ∗−Σθ◦∥⩽∥Σ∥(∥θ∗∥+∥θ◦∥)⩽2∥Σ∥∥θ◦∥⩽ρ0µ√ λ/3. (71) The inequalities (70) and (71) yield t... | https://arxiv.org/abs/2502.15437v1 |
λ1 2Σ2+λId−1/2 (Id+B)−1B1 2Σ2+λId−1/2 θ◦ ⩽λ·1√ λ·∥B∥ 1− ∥B∥·∥θ◦∥√ λ=4 3(˜ρ√ 2 + ˜ρ2)∥θ◦∥⩽9˜ρ∥θ◦∥ 4, (79) while the third one does not exceed 1 21 2Σ2+λId−1/2 (Id+B)−11 2Σ2+λId−1/2 Σ(A∗−Σ)θ◦ ⩽1 2√ λ·1 1− ∥B∥· 1 2Σ2+λId−1/2 Σ ∥A∗−Σ∥∥θ◦∥ (80) ⩽1 2√ λ·1 1− ∥B∥·∥θ◦∥2√ λ µ⩽4 3·˜ρ∥θ◦∥ 2=2˜ρ∥θ◦∥ 3. 40 The inequalitie... | https://arxiv.org/abs/2502.15437v1 |
we introduce ρ:=7∥θ◦∥ µ. Due to Proposition C.2, Lemma B.2(iv) and the assumption ∥θ◦∥⩽ρµ/7, we have ∥⟨Aθ−η⟩∥⩽ρµ√ λ/2 + 3∥Σ∥∥bλ∥and∥θ∗∥⩽∥θ◦∥⩽ρµ/7. This implies that ∥⟨(Aθ−η)⟩(θ∗−θ◦)⊤R3∥⩽ ρµ√ λ/2 + 3∥Σ∥∥bλ∥ ·(2ρµ)· ∥R3∥/7 ⩽ρ2√ λ/3 + 2 ρ∥Σ∥∥bλ∥/µ, (83) where we use ∥R3∥⩽2µ−2due to Proposition A.2. Then we bound last tw... | https://arxiv.org/abs/2502.15437v1 |
8√ 2 ⩽ 8√ 2 3+ 1 +1 8√ 2! 14∥θ◦∥ µ2 +35∥Σ∥∥θ◦∥∥bλ∥ µ2√ λ! ∥θ◦∥√ λ ⩽5∥θ◦∥√ λ 14∥θ◦∥ µ2 +35∥Σ∥∥θ◦∥∥bλ∥ µ2√ λ! . This and the inequalities (86) and (87) yield that A∗θ∗−η∗−1 2Σbλ ⩽3∥θ◦∥√ λ 14∥θ◦∥ µ2 +35∥Σ∥∥θ◦∥∥bλ∥ µ2√ λ! . □ C Results from the proof of Theorem 2.3 C.1 Proof of Lemma 5.1 Since the measure mis fixed,... | https://arxiv.org/abs/2502.15437v1 |
(96) approximately equals Id. To do so, we use the following proposition. 48 Proposition C.1. Assume that ρ:=7∥θ◦∥ µ⩽1/7and 7ρ∥Σ∥⩽√ λ. Then the following holds. For any s, t∈[0,1], we have ∥A(s)−Σ∥⩽4ρ2√ λ+ 5ρ∥Σ∥∥bλ∥/µ and∥(A(s))⊤A(t)−Σ2∥⩽3ρλ+ 3∥Σ∥∥bλ∥√ λ/µ. Applying Proposition C.1 and using the bound ∥(Σ2/2 +λId)−1∥⩽λ... | https://arxiv.org/abs/2502.15437v1 |
∥Σ∥∥bλ∥⩽∥Σ∥∥θ◦∥⩽µ√ λ/24due to (25), we derive ∥(Σ2/2 +λId)−1R⊤ (92)∥⩽7ρ/2 +∥Σ∥∥bλ∥ µ√ λ. (102) Step 4. Bounding the remainder II. Next, we analyze ∥(Σ2/2 +λId)−1R(93)∥. The first three terms of (93), can be bounded in the same way: (Σ2/2 +λId)−1n ∥⟨Aθ−η⟩∥2R3+R3D θ⟨(Aθ−η)⟩⊤AE +D A⊤⟨(Aθ−η)⟩θ⊤E R3o ⩽1 λ ∥⟨Aθ−η⟩∥2· ∥R3∥+ ... | https://arxiv.org/abs/2502.15437v1 |
that 3ρ∥Σ∥∥θ◦∥⩽ρµ√ λ/8. Finally, the chain of the inequalities ∥A∗−Σ∥⩽µ−1√ λ∥θ◦∥⩽ρ√ λ/7 following from Lemma B.2, implies that ∥A(t)θ(t)−η(t)∥⩽ρµ√ λ/2 + 3∥Σ∥∥bλ∥. □ D Results from the proof of Theorem 2.4 This section contained proof of auxiliary results related to Theorem 2.4. D.1 Preliminaries We start with the follo... | https://arxiv.org/abs/2502.15437v1 |
∗∇L(υ∗+H−1 ∗z) . 56 Let us recall that D2 ∗= diag (A∗)⊤A∗+λId,2Id, Id⊗ µ2Id+θ∗(θ∗)⊤ and that µ0does not exceed µby the definition. Then it is straightforward to observe that D2 ∗⪰D2 0and D−2 ∗⪯D−2 0. Taking these inequalities into account, we obtain that S bυ−υ∗−H−1 ∗z ⩽ SH−1 ∗D∗ D−1 0∇L(υ∗+H−1 ∗z) 1 + D−1 ∗R(H∗... | https://arxiv.org/abs/2502.15437v1 |
This yields that (1−2ρ0)H−1 ∗⪯4D−2 ∗and then ∥SH−1 ∗D∗∥=∥SD−1 ∗D∗H−1 ∗D∗∥⩽∥SD−1 ∗∥∥D∗H−1 ∗D∗∥⩽4 1−2ρ0∥SD−1 ∗∥. Let us recall that S= diag Σ1/2, Id+1⊗Od and D2 ∗= diag (A∗)⊤A∗+λId,2Id, Id⊗ µ2Id+θ∗(θ∗)⊤ . Hence, it holds that ∥SD−1 ∗∥= Σ1/2 (A∗)⊤A∗+λId−1/2 = (A∗)⊤A∗+λId−1/2 Σ (A∗)⊤A∗+λId−1/2 1/2 ⩽ (A∗)⊤A∗+... | https://arxiv.org/abs/2502.15437v1 |
that (A⊤A+λId)−1/2θ ⩽∥θ∥√ λ. Since 1 2A⊤A+λId−1/2 A⊤ ⩽√ 2and 1 2A⊤A+λId−1/2 ⩽1√ λ, the inequality (113) can be simplified even further: 1 2A⊤A+λId−1/2 HθχH−1 χχzχ ⩽∥U∥√ 2+∥(bΣ−Σ)(θ−θ◦)∥√ 2+∥(bΣ−Σ)(Aθ−η)∥√ λ +√ 2ρ2∥Z−EZ∥+ 2√ 2ρ2 (bΣ−Σ)θ +λ−1/2∥θ∥ µ2+∥θ∥2 (Aθ−η)⊤(Z−EZ) (114) +λ−1/2∥θ∥ 2µ2+∥θ∥2 (Aθ−η)⊤(bΣ−Σ)θ . In o... | https://arxiv.org/abs/2502.15437v1 |
2ρ2∥H1/2 AAw∥2 ⩽((1 +√ 2)ρ+ 2ρ2) ∥v∥2+∥H1/2 AAw∥2 . 63 Hence, we proved that ∥D−1 χQD−1 χ∥⩽ diag( Id, H−1/2 AA)Qdiag( Id, H−1/2 AA) ⩽(1 +√ 2)ρ+ 2ρ2. (115) Step 2: upper bound on ∥eD˘H−1 χχzχ∥.With the inequality (115) the upper bound on the norm of eD˘H−1 χχzχ is straightforward. Indeed, it holds that ˘H−1 χχzχ= D2 ... | https://arxiv.org/abs/2502.15437v1 |
following lemma. Lemma D.4. Let us fix arbitrary A∈Rd×dandλ >0. Then it holds that 1 2 1 2Σ2+λId−1/2 A⊤A−Σ21 2Σ2+λId−1/2 ⩽∥A−Σ∥r 2 λ+∥A−Σ∥2 2λ. We postpone the proof of Lemma D.4 to Appendix D.8.3 and proceed with the proof of Lemma 6.4. Taking into account (7) and applying Lemma B.3, we note that ∥A∗−Σ∥√ λ⩽14∥θ... | https://arxiv.org/abs/2502.15437v1 |
that ∥A∗θ∗−η∗∥⩽√ λ∥θ◦∥and∥θ∗∥⩽∥θ◦∥. This means that θ∗belongs to Υ(ρ∗)withρ∗=∥θ◦∥/µ⩽1/112. Applying Lemma D.3 with υ=υ∗and ρ=ρ∗, we obtain that Hθχ(υ∗)H−1 χχ(υ∗)zχ+1 2(A∗)⊤U−1 2(A∗)⊤(bΣ−Σ)(θ∗−θ◦)−(bΣ−Σ)(A∗θ∗−η∗) =− r1−r2µ2∥θ∗∥2 µ2+∥θ∗∥2 (A∗)⊤(Z−EZ) +r2µ2 µ2+∥θ∗∥2 (A∗θ∗−η∗)⊤(Z−EZ) θ∗ −∥θ∥2 2µ2+∥θ∥2−∥θ∥2 2(µ2+∥θ∥2)+... | https://arxiv.org/abs/2502.15437v1 |
□ D.7 Proof of Lemma 6.6 Note that, due to the definitions of τandζ, we have τ−ζ= (A∗−Σ)U+ (A∗)⊤(bΣ−Σ)(θ∗−θ◦−bλ) −(A∗−Σ)⊤(bΣ−Σ)bλ−(bΣ−Σ)(2A∗θ∗−2η∗−bλ). 72 Then the triangle inequality yields that Σ1/2 Σ2+ 2λId−1(τ−ζ) ⩽ Σ1/2 Σ2+ 2λId−1 ∥A∗−Σ∥ ∥U∥+ (bΣ−Σ)bλ + Σ1/2 Σ2+ 2λId−1(A∗)⊤ (bΣ−Σ)(θ∗−θ◦−bλ) + Σ1/2 Σ2+ 2λI... | https://arxiv.org/abs/2502.15437v1 |
Then the first term in the right-hand side of (129) does not exceed 3ρ2+2ρ2 25 1 2A⊤A+λId−1/2 A⊤A1 2A⊤A+λId−1/2 ⩽6ρ2+ 2ρ4⩽6ρ2+4ρ2 25. Second, it holds that ∥Aθ−η∥2 µ2 1 2A⊤A+λId−1 ⩽ρ2λ·1 λ=ρ2. Finally, let us consider the latter term in the right-hand side of (129). The triangle inequality and the submultiplic... | https://arxiv.org/abs/2502.15437v1 |
A⊤(bΣ−Σ)θ −1 2µ2+∥θ∥2 (Aθ−η)⊤(bΣ−Σ)θ θ. □ D.8.3 Proof of Lemma D.4 Step 1: a bound on the operator norm. Let us start with an upper bound on the operator norm of 1 21 2Σ2+λId−1/2 A⊤A−Σ21 2Σ2+λId−1/2 . 78 Due to the definition of Sand the triangle inequality, it holds that 1 2 1 2Σ2+λId−1/2 A⊤A−Σ21 2Σ2+λId... | https://arxiv.org/abs/2502.15437v1 |
random matrix Ψ =1√βΣ1/2B⊤ZAΣ1/2, where Zis a matrix of size (q×d)with i.i.d. N(0,1)entries and βis defined in (138). Note that vec(Ψ)∼ N 0, β−1(Σ1/2A⊤AΣ1/2)⊗(Σ1/2B⊤BΣ1/2) in this case (see, for instance, [Leng and Pan, 2018] for the details). Denote its density (with respect to the volume measure on the image of (Σ1... | https://arxiv.org/abs/2502.15437v1 |
E.2 with P= Σ1/2B⊤,Q= 1, and β= 2r(Σ1/2B⊤BΣ1/2), we obtain that KL(ρu, µ)⩽log 2 + r(Σ1/2B⊤BΣ1/2) (Σ1/2B⊤)†u 2 ⩽log 2 + r(Σ1/2B⊤BΣ1/2). Besides, since ∥Σ−1/2X1ε1∥ψ1=σand λ=1 4σs r(Σ1/2B⊤BΣ1/2) + log(2 /δ) ∥Σ∥n⩽1 4σp ∥Σ1/2B⊤BΣ1/2∥⩽1 2σ∥ξ∥, Lemma 2 from Zhivotovskiy [2024] yields that logEX,εeλξ⊤Σ−1/2Xε⩽4λ2σ2∥ξ∥2⩽8λ2σ2 ∥... | https://arxiv.org/abs/2502.15437v1 |
with probability at least (1−δ), we simultaneously have Σ1/2(Σ2+ 2λId)−1Σ(bΣ−Σ)bλ ⩽4CX Σ2(Σ2+ 2λId)−1 ∥Σ1/2bλ∥r r(Σ4(Σ2+ 2λId)−2) + log(4 /δ) n =4CX∥Σ∥2 ∥Σ∥2+ 2λ∥Σ1/2bλ∥r r(Σ4(Σ2+ 2λId)−2) + log(4 /δ) n and Σ1/2(Σ2+ 2λId)−1ΣU ⩽8σ Σ2(Σ2+ 2λId)−1 r r(Σ4(Σ2+ 2λId)−2) + log(4 /δ) n =8σ∥Σ∥2 ∥Σ∥2+ 2λr r(Σ4(Σ2+ 2λId)−2) + log... | https://arxiv.org/abs/2502.15437v1 |
Corollary 2.7, we denote k∗=k∗(λ) = max k∈N:σ2 k⩾2λ . The first result relates the squared norms of Σ1/2bλandΣ3/2bλwith the ones of θ◦ ⩽k∗andθ◦ >k∗. Lemma G.1. With the notation introduced above, it holds that Σ1/2bλ 2 ⩽σ2 k∗ 4 Σ−1/2θ◦ ⩽k∗ 2 + Σ1/2θ◦ >k∗ 2 and Σ3/2bλ 2 2λ⩽σ2 k∗ Σ−1/2θ◦ ⩽k∗ 2 +1 4 Σ1/2θ◦ >k∗ 2 . Proof.... | https://arxiv.org/abs/2502.15437v1 |
implies ∥bθ∥2⩽∥Z∥2/λ. Next, we have ∥Z∥⩽∥bΣθ◦∥+∥U∥⩽∥bΣ−Σ∥∥θ∥+∥Σ∥∥θ◦∥+∥U∥ Using ∥bΣ−Σ∥⩽∥Σ∥from condition (ii), ∥Σ∥∥θ◦∥⩽ρ0µ√ λ/(18·8)from condition (i) and ∥U∥⩽ ρ0µ√ λ/18from condition (iii), we infer ∥Z∥⩽2∥Σ∥∥θ◦∥+∥U∥⩽ρ0µ√ λ/6. (150) 91 It yields ∥bθ∥⩽∥Z∥/√ λ⩽ρ0µ/6. (151) Note that bound (150) also implies that (0,Z,bΣ)∈... | https://arxiv.org/abs/2502.15437v1 |
θ∗). Note that it possesses the following identities for any two matrix-valued functions f, gsuch that their product is defined: ∆(fg) = ∆( f)g(bθ,bη,bA) +f(θ∗,η∗, A∗) ∆(g); ∆(fg) =f(bθ,bη,bA) ∆(g) + ∆( f)g(θ∗,η∗, A∗). (153) Using ∇L(bθ,bη,bA)− ∇L (θ∗,η∗, A∗) = 0 , we obtain −∆(A⊤η) + ∆( A⊤Aθ) +λ∆(θ) = 0 −(Z−EZ)... | https://arxiv.org/abs/2502.15437v1 |
∥ ψ2stand for ψ1andψ2Orlicz norms. Lemma H.5 (Lemma B.3 of Puchkin and Rakhuba [2024]) .Suppose that a random vector Xsatisfies Assumption 2.1. Then, for any vector u∈Rd, it holds that ∥u⊤X∥2 ψ2⩽1 +CX log 2·u⊤Σu. Then, we deduce the concentration bound on bΣfrom the following theorem. Lemma H.6 (Theorem 1 of Zhivotovsk... | https://arxiv.org/abs/2502.15437v1 |
prove by induction, that ∥∆t(A)∥⩽ρ2t 0∥bA−bΣ∥. (166) 98 Clearly, the base t= 0holds, by the definition of A0=bΣ. Assume that (166) holds for t, and let us prove it for t+ 1. We have ∥∆t+1(A)∥⩽µ−2 2" ∥bA∥∥bθ∥+∥bAbθ−Z∥+∥bA∥" ∥bθ∥√ λ+∥bAbθ−Z∥ 2λ# · ∥∆t(A)∥# ×" ∥bθ∥√ λ+∥bAbθ−Z∥ 2λ# ∥∆t(A)∥ ⩽µ−2 2" ∥bA∥∥bθ∥+∥bAbθ−Z∥+∥bA∥" ∥... | https://arxiv.org/abs/2502.15437v1 |
Improving variable selection properties by leveraging external data PAUL ROGNON-VAEL1,a, DAVID ROSSELL1,band PIOTR ZWIERNIK2,c 1Department of Economics and Business, Universitat Pompeu Fabra ,apaul.rognon@gmail.com, brosselldavid@gmail.com 2Department of Statistical Sciences, University or Toronto ,cpiotr.zwiernik@utor... | https://arxiv.org/abs/2502.15584v2 |
Dunson (2022) used meta- covariates to determine non-zero loadings in factor models. In causal analysis, the inclusion of control covariates may be driven by their degree of association with the covariates of interest (referred to as treatments) (Antonelli and Dominici, 2021, Belloni, Chernozhukov and Hansen, 2014). Co... | https://arxiv.org/abs/2502.15584v2 |
show that our results are tight with respect to known limits on exact support recovery (Butucea et al., 2018, Wainwright, 2010). Nonlinear ℓ0penalties have however been shown to be optimal for estimation and prediction in the Gaussian sequence model (Wu and Zhou, 2013) and linear regression (Bunea, Tsybakov and Improvi... | https://arxiv.org/abs/2502.15584v2 |
true sparsity and betamin conditions are known for all blocks. Section 4 studies block ℓ0penalties in linear regression, and shows analogous benefits to those in Section 3. These results can be extended to a wide class of Bayesian variable selection methods. Section 5 presents data-based procedures, motivated by empiri... | https://arxiv.org/abs/2502.15584v2 |
the essence of the benefits of ˆ𝑆𝑏over standard selectors before moving to the setting of interest, linear regression. We start by introducing the sequence model, its connection to orthogonal linear regression and dis- cuss that ˆ𝑆𝑏simplifies to performing block-based thresholding. We then provide the variable sele... | https://arxiv.org/abs/2502.15584v2 |
in addition, lim𝑛→∞√𝑛(𝛽∗ min,𝑗−𝜏𝑗)/√︁ (𝜋/2)ln(𝑠𝑗)≤1 then lim𝑛→∞𝑃(ˆ𝑆𝑏⊇𝑆)<1. Proposition 3.2 (i) to (iii) extend to ˆ𝑆𝑏results known for the standard thresholding ˆ𝑆(ˆ𝑆𝑏with𝑏=1) in orthogonal linear regression (Bogdan et al., 2015, Bühlmann and van de Geer, 2011, Wainwright, 2019) and in the sequence ... | https://arxiv.org/abs/2502.15584v2 |
discuss two types of benefits: softening the conditions for model selection consistency and improving the associated convergence rates. Conditions for variable selection consistency. Assumptions A4–A5 give ranges of threshold values that are sufficient and essentially necessary for asymptotic support recovery. For the ... | https://arxiv.org/abs/2502.15584v2 |
et al. (2018) which shows the near-optimality of 𝜏∗ in the worst case. Under the less rigid assumption of unequal 𝛽∗ min,𝑗’s,𝑂𝑅𝑏 𝑜𝑟𝑡ℎ/𝑂𝑅𝑜𝑟𝑡ℎ is however bounded away from 1 in (14) and the oracle rate for the 𝜏∗ 𝑗is strictly better. This highlights the crucial role that varying signal strength plays in t... | https://arxiv.org/abs/2502.15584v2 |
converges to 0. In Example 4, with non-discriminative blocks, the ratio converges to 1, i.e. the benefits in the recoverable signal fade as 𝑛grows. Example 3 illustrates how highly discriminative blocks can also bring significant benefits in terms of signal recoverable in a regime that is only somewhat sparse. 10 Exam... | https://arxiv.org/abs/2502.15584v2 |
only present the most important ideas. In Section 4.2 we state our main theorem on sufficient conditions for variable selection consistency for ˆ𝑆𝑏and oracle convergence rates. To assess the tightness of said sufficient conditions, Section 4.3 gives related necessary conditions. Section 4.4 discusses the gains of blo... | https://arxiv.org/abs/2502.15584v2 |
by leveraging external data 13 The quantity 𝜌(𝑿)is nonnegative and relates to how distinguishable the other models 𝑀∈M are from 𝑆(Wainwright, 2010). More specifically,1 𝑛𝑿⊤ 𝑆\𝑀(𝐼𝑛−𝑃𝑀)𝑿𝑆\𝑀is the sample covariance matrix of the residuals when regressing 𝑿𝑆\𝑀on𝑿𝑀. In an orthonormal case where 𝑿⊤𝑿=𝑛I... | https://arxiv.org/abs/2502.15584v2 |
we can bound the convergence rate of 𝑃(ˆ𝑆𝑏≠𝑆), given in Theo- rem 4.6. We also give oracle block penalties that approximately optimize the bound, and provide the resulting oracle rate of convergence. In the statement, 𝛿<1 and𝑟>1 should be understood as being arbitrarily close to 1. THEOREM 4.6. Assume A1, A6, A7.... | https://arxiv.org/abs/2502.15584v2 |
𝑛¯𝜆|𝛽∗ 𝑖|=𝑜 √𝜅𝑗) 𝑆𝐿 𝑗(𝜅):=( 𝛽∗ 𝑖∈𝑆𝑗 √︂ (1−𝛾)𝑛𝜌(𝑿) 6|𝛽∗ 𝑖|−√𝜅𝑗=√︃ ln(𝑠𝑗)+𝑔𝑗) (24) 𝑆𝐼 𝑗(𝜅):=𝑆𝑗\ 𝑆𝐿 𝑗(𝜅)∪𝑆𝑆 𝑗(𝜅). The subset𝑆𝑆 𝑗(𝜅)gathers signals in 𝑆𝑗that are small with respect to the penalty 𝜅𝑗,𝑆𝐿 𝑗(𝜅)those that are large in that they satisfy Assumption A7, and 𝑆... | https://arxiv.org/abs/2502.15584v2 |
If lim𝑛→∞ √ 𝑛¯𝜆𝛽∗ min/ 𝜆√︁ ln(𝑝−𝑠)=0then𝑃(ˆ𝑆= 𝑆)̸→1and𝑃(ˆ𝑆𝑏=𝑆)→1. Observe that (27) and (28) are analogous to (11) and (12) for the sequence model, up to rescaling by√︁ 2/𝑛, the factor𝜌(𝑿)and sequences(𝑓𝑗,𝑔𝑗)growing arbitrarily slowly. The gains in terms of valid thresholds for consistency discus... | https://arxiv.org/abs/2502.15584v2 |
truly active variables in each block by the average posterior inclusion probabilities in that block. This provides a straightforward way to adapt penalties to sparsity in each block. Relying on the BIC approx- imation also allows the use of fast Bayesian computational methods that overcome the intractability of ℓ0penal... | https://arxiv.org/abs/2502.15584v2 |
have varying strength depending on the unknown 𝛽∗ min,𝑗and number of active signals in each block 𝑠𝑗, and we just saw that the latter can be reliably estimated. We propose a two-step procedure. First, we use a standard (non-block-based) penalty 𝜅◦and estimate the number of active signals 𝑠𝑗in each block 𝑗with ˆ... | https://arxiv.org/abs/2502.15584v2 |
Step 2 a single common penalty 𝜅𝐸𝐵=ln(𝑝/ˆ𝑠−1)+1 2ln(𝑛)is variable selection consistent under the betamin assumption: √︂ (1−𝜓′)𝑛𝜌(𝑿) 6𝛽∗ min−√︄ ln𝑝 |𝑆𝐿(𝜅◦)|−1 +1 2ln(𝑛)=√︁ ln(𝑠)+𝑎′ 𝑗. (34) where𝜓′=1 2 1+ln(𝑝−𝑠)/ ln(𝑝/𝑠−1)+ln(𝑛)/2 and𝑎′ 𝑗→∞ . We have √︄ ln 𝑝 |𝑆𝐿(𝜅◦)|−1 +1 2ln(𝑛)+√︁ ... | https://arxiv.org/abs/2502.15584v2 |
externally-informed selector converges faster and under milder conditions than the standard ℓ0oracle. In particular, it softens the stringent conditions on signal strength. We also provided concrete data analysis methods that incorporate external informa- tion to improve variable selection properties without requiring ... | https://arxiv.org/abs/2502.15584v2 |
and D OMINICI , F. (2021). Bayesian model averaging in causal inference. In Handbook of Bayesian Variable Selection (Chapter 9) 201–226. Chapman and Hall/CRC. BASU, P., C AI, T. T., D AS, K. and S UN, W. (2018). Weighted False Discovery Rate Control in Large-Scale Multiple Testing. Journal of the American Statistical A... | https://arxiv.org/abs/2502.15584v2 |
from https: //imjohnstone.su.domains/GE_09_16_19.pdf. LUO, S. and C HEN, Z. (2013). Extended BIC for linear regression models with diverging number of relevant features and high or ultra-high feature spaces. Journal of Statistical Planning and Inference 143494-504. https: //doi.org/10.1016/j.jspi.2012.08.015 NARISETTY ... | https://arxiv.org/abs/2502.15584v2 |
(2022). Dimension-Free Mixing for High-Dimensional Bayesian Variable Selection. Journal of the Royal Statistical Society Series B: Statistical Methodology 841751-1784. https://doi.org/10.1111/rssb.12546 26 Supplementary material to "Improving variable selection properties by leveraging external data" We provide the fol... | https://arxiv.org/abs/2502.15584v2 |
for any 𝑖∈𝐵𝑗we have ˆ𝛽𝑖=0 if and only if|˜𝛽𝑖|≤𝜆𝑗/(|𝛽◦ 𝑖|𝑛). If𝜷◦=˜𝜷, then for any 𝑖∈𝐵𝑗ˆ𝛽𝑖=0 if and only if|˜𝛽𝑖|≤√︁ 𝜆𝑗/𝑛. If𝜷◦is a LASSO estimate with penalization 𝜆◦, then for any 𝑖∈𝐵𝑗ˆ𝛽𝑖=0 if and only if ˜𝛽2 𝑖+sign(𝜆◦/𝑛−˜𝛽𝑖)𝜆◦/(𝑛˜𝛽𝑖)−𝜆𝑗/𝑛≤0. Equivalently, for any 𝑖∈𝐵𝑗ˆ𝛽�... | https://arxiv.org/abs/2502.15584v2 |
𝑆\𝑀(𝐼𝑛−𝑃𝑀)𝑿𝑆\𝑀 . Finally, by definition of 𝜌(𝑿) in (19) we have that 𝜆min 1 𝑛𝑿⊤ 𝑆\𝑀(𝐼𝑛−𝑃𝑀)𝑿𝑆\𝑀 ≥𝜌(𝑿), and further noting that ∥𝛽∗ 𝑇\𝑀∥2≥ Í𝑏 𝑗=1|𝑇𝑗\𝑀𝑗|min𝑖∈𝑇𝑗\𝑀𝑗𝛽∗ 𝑖2gives the desired result. LEMMA S1.4. Let𝑊∼𝜒2 𝜈(𝜇)with𝜇≥0, then for any 𝑤>𝜇+𝜈 𝑃(𝑊>𝑤)≤𝑒− 𝑤+𝜇 2−√ 2... | https://arxiv.org/abs/2502.15584v2 |
recovery with ˆ𝑆𝑏that applies independently on whether the 𝑠𝑗are fixed or diverging. It is analogous to a necessary condition for recovery with ˆ𝑆shown in Abraham, Castillo and Roquain (2023). LEMMA S1.8. In the sequence model (4), assume A1 and A2. Suppose that 𝜏𝑗< 𝛽∗ min,𝑗satisfies lim𝑛→∞√𝑛𝜏𝑗/√︁ 2 ln(𝑝�... | https://arxiv.org/abs/2502.15584v2 |
that E(𝑁𝐶(𝑀))≤∫1 0𝑃 1 2𝐿𝑄𝑇𝑇≥ln𝑒𝐴𝑇 1 𝑢−1 d𝑢+𝑃 1 2𝐿𝑄𝑇𝑀<𝛾′(𝜇𝑄𝑇𝑀−6Δ𝑀𝑇) . (S52) The third and final step of the proof is to upper bound each of the terms in the right-hand side of (S52). The intuition is that both 𝑇and𝑀are nested within 𝑄𝑇, and therefore 𝐿𝑄𝑇𝑇and𝐿𝑄𝑇𝑀follow chi-squar... | https://arxiv.org/abs/2502.15584v2 |
write 1 2∥√𝑛˜𝜷𝑀∥2−𝑏∑︁ 𝑗=1𝜅𝑗|𝑀𝑗|=𝑏∑︁ 𝑗=1∑︁ 𝑖∈𝑀𝑗(𝑛 2˜𝛽2 𝑖−𝜅𝑗). Then (S58) can be maximized with respect to each 𝑀𝑗by including 𝑖∈ˆ𝑆𝑗whenever𝑛˜𝛽2 𝑖≥2𝜅𝑗. S2.2. Proof of Proposition 3.2 S2.2.1. Part (i) By the union bound and by Lemma S1.2 (i), 𝑃 ˆ𝑆𝑏⊈𝑆 ≤𝑏∑︁ 𝑗=1𝑃 max 𝑖∈𝐵𝑗\𝑆𝑗|𝑦𝑖/√... | https://arxiv.org/abs/2502.15584v2 |
not possible asymptotically. Suppose now that lim 𝑛→∞√𝑛𝜏𝑗/√︁ 2 ln(𝑝𝑗−𝑠𝑗)≥1, then it holds that lim𝑛→∞√𝑛(𝛽∗ min,𝑗−𝜏𝑗)√︁ 𝜋ln(𝑠𝑗)/2<1. By Proposition 3.2 (iv), lim 𝑛→∞𝑃(ˆ𝑆𝑏⊇𝑆)<1 and recovery is not possible asymptotically. S2.4. Proof of Theorem 3.4 We start by showing the bound in (8). By the union ... | https://arxiv.org/abs/2502.15584v2 |
2∥𝒚∥2−1 2∥𝑿𝑀˜𝜷(𝑀)∥2, where in the last equality we used that ˜𝜷(𝑀)=(𝑿𝑇 𝑀𝑿𝑀)−1𝑿𝑇 𝑀𝒚. The maximization in (3) can be then replaced with the maximization of 𝐶(𝑀)=1 2∥𝑿𝑀˜𝜷(𝑀)∥2−Í𝑏 𝑗=1𝜅𝑗|𝑀𝑗|which is equivalent to maximizing 𝑁𝐶(𝑀). S3.2. Proof of Lemma 4.2 Part (i) follows directly from Lemma S... | https://arxiv.org/abs/2502.15584v2 |
min,𝑗2−𝛾𝜅𝑗 . Then by (S71) and (S72) , we get E(𝑁𝐶(𝑀))≤ exp −𝑏∑︁ 𝑗=1|𝑀𝑗\𝑆𝑗|(ln(𝑝𝑗−𝑠𝑗)+𝛿𝑓𝑗 2)−𝑏∑︁ 𝑗=1|𝑆𝑗\𝑀𝑗|(ln(𝑠𝑗)+𝛿𝑔′ 𝑗 2) . (S73) For the final step of the proof, denote S=Í 𝑀∈M\{𝑆}E(𝑁𝐶(𝑀))for convenience. By (S73) we have S≤∑︁ 𝑀∈M\{𝑆}𝑒−Í𝑏 𝑗=1|𝑀𝑗\𝑆𝑗| ln(�... | https://arxiv.org/abs/2502.15584v2 |
𝑛for all𝑗. Then each of the 22𝑏−2𝑏−1 terms inR is bounded above byÍ𝑏 𝑗=1 𝑆(𝑢𝑗)+𝑆(𝑤𝑗) and we get, for every 𝑛large enough, ∑︁ 𝑀∈M\{𝑆}E(𝑁𝐶(𝑀))≤(22𝑏−2𝑏)𝑏∑︁ 𝑗=1 𝑑𝑗1−𝑑𝑝𝑗−𝑠𝑗 𝑗 1−𝑑𝑗+ℎ𝑗1−ℎ𝑠𝑗 𝑗 1−ℎ𝑗 . (S78) We have𝑑𝑗→0 andℎ𝑗→0, then for every 𝑛large enough1−𝑑𝑝𝑗−𝑠𝑗 𝑗 1−𝑑𝑗→1 an... | https://arxiv.org/abs/2502.15584v2 |
(2014) and our Assumption A2, for any𝜀>0, 𝑃 max 𝑀∈𝑂𝑗𝑍𝑀≥𝜆𝑗√︃ 2 ln(𝑝𝑗−𝑠𝑗)(1−𝜀)→1, where𝜆𝑗is as defined prior to the statement of Proposition 4.7. If lim 𝑛→∞𝜅𝑗 𝜆2 𝑗ln(𝑝𝑗−𝑠𝑗)<1, then lim𝑛→∞𝑃 max𝑀∈𝑂𝑗𝑍𝑀≥√︁2𝜅𝑗=1. Hence, by (S84) we have that lim 𝑛→∞𝑃(ˆ𝑆𝑏=𝑆)=0, as we wished to prove. Im... | https://arxiv.org/abs/2502.15584v2 |
4.5, with suitable adjustments. The first step is to use Lemma S1.10 to bound E(𝑁𝐶(𝑀))for every𝑀∉T(𝜅). The main difference is that in the proof of The- orem 4.5 we took 𝑇=𝑆in Lemma S1.10, whereas now we take a model 𝑇=𝑇𝑀∈T(𝜅)that depends on 𝑀. Intuitively,𝑇𝑀contains large truly non-zero parameters that ar... | https://arxiv.org/abs/2502.15584v2 |
𝑗𝛽∗ 𝑖2 𝛾Í𝑏 𝑗=1|𝑀𝑗\𝑇𝑀 𝑗|𝜅𝑗. Observe that for all 𝑗=1,...,𝑏 ,𝑀𝑗\𝑇𝑀 𝑗⊆(𝑆𝑗∩𝑀𝑗)\𝑇𝑀 𝑗and then|𝑀𝑗\𝑇𝑀 𝑗|≥|(𝑆𝑗∩𝑀𝑗)\𝑇𝑀 𝑗|. We then get 𝜇𝑄𝑇𝑀𝑇𝑀 𝐴𝑇𝑀≤𝑛¯𝜆Í𝑏 𝑗=1|(𝑆𝑗∩𝑀𝑗)\𝑇𝑀 𝑗|max𝑖∈(𝑆𝑗∩𝑀𝑗)\𝑇𝑀 𝑗𝛽∗ 𝑖2 𝛾Í𝑏 𝑗=1|(𝑆𝑗∩𝑀𝑗)\𝑇𝑀 𝑗|𝜅𝑗. Moreover, since 𝑀\𝑇𝑀⊆(𝑆𝐼(𝜅... | https://arxiv.org/abs/2502.15584v2 |
2 −Í𝑏 𝑗=1|𝑇𝑗\𝑀𝑗| ln(𝑠𝑗)+𝛿𝑔′ 𝑗 2 . (S99) The right hand-side of (S99) is composed of a double sum over 𝑇∈T(𝜅)and over𝑀∈M(𝑇). Con- sider the sum over 𝑀∈M(𝑇), add𝑇to it, and denote it S(𝑇)=∑︁ 𝑀∈M(𝑇)∪𝑇𝑒−Í𝑏 𝑗=1|𝑀𝑗\𝑇𝑗| ln(𝑝𝑗−𝑠𝑗)+𝛿𝑓𝑗 2 −Í𝑏 𝑗=1|𝑇𝑗\𝑀𝑗| ln(𝑠𝑗)+𝛿𝑔′ 𝑗 2 . (S100... | https://arxiv.org/abs/2502.15584v2 |
𝑗(𝜅)| 𝑝𝑗∑︁ 𝑀∈T(𝜅)E(𝑁𝐶(𝑀)). By Theorem 4.12, lim 𝑛→∞Í 𝑀∈T(𝜅)E(𝑁𝐶(𝑀))=1. It follows that lim 𝑛→∞Eb𝑠𝑗 𝑝𝑗 ≥|𝑆𝐿 𝑗(𝜅)| 𝑝𝑗for every 𝑗=1,...,𝑏 . Improving variable selection properties by leveraging external data 57 We now prove the upper bound. Recall that T(𝜅)by definition includes models that ... | https://arxiv.org/abs/2502.15584v2 |
immediately from the definition of T(𝜅)in (29)) and thereforeÍ 𝑀∈M||𝑀𝑗|>𝑠𝑗𝑁𝐶◦(𝑀)≤Í 𝑀∈M\T(𝜅◦)𝑁𝐶◦(𝑀). Moreover,𝜅◦satisfies Assumption A6 and the as- sumptions of Theorem 4.12 are met for 𝜅◦. Then, by Theorem 4.12, lim 𝑛→∞Í 𝑀∈M\T(𝜅◦)𝑁𝐶◦(𝑀)= lim𝑛→∞Í 𝑀∈M||𝑀𝑗|>𝑠𝑗𝑁𝐶◦(𝑀)=0,𝑝𝑗𝑂◦ 𝑗 𝑝𝑗−𝑠𝑗van... | https://arxiv.org/abs/2502.15584v2 |
+ln 1+𝑠𝑗−|𝑆𝐿 𝑗(𝜅◦)| 𝑝𝑗−𝑠𝑗 1−𝑠𝑗−|𝑆𝐿 𝑗(𝜅◦)| 𝑠𝑗! =ln 𝑝𝑗−𝑠𝑗+ln√𝑛 𝑠𝑗 +ln 𝑝𝑗−|𝑆𝐿 𝑗(𝜅◦)| 𝑝𝑗−𝑠𝑗 |𝑆𝐿 𝑗(𝜅◦)| 𝑠𝑗! =ln 𝑝𝑗/|𝑆𝐿 𝑗(𝜅◦)|−1+1 2ln(𝑛) which shows (S114) holds with probability going to 1 and that Assumption A9 implies Assumption A7 holds for the 𝜅𝐸𝐵 𝑗with probabilit... | https://arxiv.org/abs/2502.15584v2 |
all 𝑗,𝑠𝑗≤𝑘𝑗for some constant 𝑘𝑗. Changing Assumption A3 for Assumption A11 implies redeveloping results relative to the probability of false negatives and consequently sufficient and necessary betamin assumptions. Proposition 3.1 (on the equivalence between block penalties and thresholding in the Gaussian sequen... | https://arxiv.org/abs/2502.15584v2 |
min,𝑜𝑟𝑡ℎand𝛽∗ min,𝑜𝑟𝑡ℎin the diverging 𝑠𝑗’s case, up to loga- rithmic terms in the number of active signals, and up to 𝑔andℎwhich can grow arbitrarily slowly with 𝑛. Note that in Examples 1, 2 and 4 in Section 3.4, ln (𝑠𝑗)=𝑜(ln(𝑝𝑗−𝑠𝑗))for all𝑗. The discussion of the asymptotic behavior of the ratio �... | https://arxiv.org/abs/2502.15584v2 |
𝑛, √︂ (1−𝛾)𝑛𝜌(𝑿) 6𝛽∗ min,𝑗−√︃ max 𝑀∈M˜𝜅𝑗(𝑀)=√︃ ln(𝑠𝑗)+𝑙𝑗 where𝛾:=1 2(1+max𝑗ln(𝑝𝑗−𝑠𝑗) ln(𝑝𝑗−𝑠𝑗)+min𝑀:|𝑀𝑗\𝑆𝑗|>0𝑓𝑗(𝑀))∈(1 2,1). We can now state the main result of this section. THEOREM S6.1. Under Assumptions A1, A13 and, A14, we have ∑︁ 𝑀∈M\{𝑆}E(𝑁𝐶(𝑀))→0 and𝑃(ˆ𝑆𝑏=𝑆)→1. Theorem S... | https://arxiv.org/abs/2502.15584v2 |
That is 𝐴∗ 𝑆:=𝛾𝑏∑︁ 𝑗=1|𝑀𝑗\𝑆𝑗|˜𝜅𝑗(𝑀)+𝑏∑︁ 𝑗=1|𝑆𝑗\𝑀𝑗|1−𝛾 6𝑛𝜌(𝑿)𝛽∗ min,𝑗2−𝛾˜𝜅𝑗(𝑀) . 68 By (S131), we have for all 𝑛large enough, E(𝑁𝐶(𝑀))≤𝑒−𝜓𝐴∗ 𝑆. (S132) Assumption A14 implies that there exists 𝑔′ 𝑗→∞ such that (1−𝛾)𝑛𝜌(𝑿) 6𝛽∗ min,𝑗2−˜𝜅𝑗(𝑀)=ln(𝑠𝑗)+𝑔′ 𝑗. (S133) Let𝛿∈(0,1... | https://arxiv.org/abs/2502.15584v2 |
2−1! ª®® ¬© «1+𝑠𝑗∑︁ 𝑤𝑗=1𝑒−𝑤𝑗 𝛿𝑔′ 𝑗 2−1! ª®® ¬. (S140) Denote 𝑑𝑗=𝑒1−𝛿min𝑀:|𝑀𝑗\𝑆𝑗|>0𝑓𝑗(𝑀) 2, ℎ𝑗=𝑒1−𝛿𝑔′ 𝑗 2. where both expressions go to zero as 𝑛increases since min 𝑀:|𝑀𝑗\𝑆𝑗|>0𝑓𝑗(𝑀)→∞ and𝑔′ 𝑗→∞ . For every𝑗, by the properties of geometric sums, we have 1+𝑝𝑗−𝑠𝑗∑︁ 𝑢𝑗=1𝑒−𝑢𝑗... | https://arxiv.org/abs/2502.15584v2 |
if (S130) holds then Assumption A14 holds for 𝜂, it suffices to show that the following two inequalities −√︃ max 𝑀∈M˜𝜅(𝑀)≥−√︂ 𝜁ln 𝑝−𝑠+1+𝜁−1+1 2ln(𝑛),and (S152) √︂ (1−𝛾)𝑛𝜌(𝑿) 6𝛽∗ min,𝑗≥vuuut ln √𝑛 (1+𝑠)𝜁+𝜁𝑠ln(1−𝑠−1) ln(𝑝−𝑠)+ln √𝑛 (1+𝑠)𝜁+𝜁𝑠ln(1−𝑠−1)𝑛𝜌(𝑿) 12𝛽∗ min,𝑗(S153) hold. We star... | https://arxiv.org/abs/2502.15584v2 |
for example. Table S3 gives, for some regimes of interest, the scalings of (S156), (S157), and (S158) where we assume 𝜆and ¯𝜆are bounded for simplicity. The scalings implied by our sufficient conditions match or improve those Table S3. Scaling of conditions for variable selection consistency RegimeOur sufficient cond... | https://arxiv.org/abs/2502.15584v2 |
Local geometry of high-dimensional mixture models: Effective spectral theory and dynamical transitions Gerard Ben Arous∗, Reza Gheissari†, Jiaoyang Huang‡, Aukosh Jagannath§ Abstract We study the local geometry of empirical risks in high dimensions via the spectral theory of their Hessian and information matrices. We f... | https://arxiv.org/abs/2502.15655v2 |
two decades to the article of LeCun et al [34]. Around a decade ago, a belief emerged in the machine learning literature that the spectra of these random matrices, evaluated along the SGD trajectory, describes a local geometry that has many “flat” directions in which the SGD diffuses, and a hidden low-dimensional struc... | https://arxiv.org/abs/2502.15655v2 |
between the bulk and outliers of matrices formed from the loss landscape, with the low- dimensional subspace of summary statistics of the parameter, through which we also describe the 2 typical trajectories taken by SGD. Mathematically, this requires refinements of arguments from random matrix theory, to understand the... | https://arxiv.org/abs/2502.15655v2 |
of the summary statistic values G(xtn)are typically asymptotically autonomous, meaning they converge to the solution of an autonomous system of ODEs/SDEs in the high-dimensional limit (in particular this is the case in all theexamplesmentionedearlier, pere.g.,[11]). Together, thisgivesadimension-freecharacterization of... | https://arxiv.org/abs/2502.15655v2 |
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