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2(t)+1 2edF− 2(t)for large d. We shall argue that F+ 2(t)> F− 2(t), and therefore Z2(t)≈1 2edF+ 2(t)for large d(this asymmetry arises because xis close to x1generated from q+). The collapse time can then be found from e−d/2≈1 2edF+ 2(tC) for large d, which gives the condition F+ 2(tC) =−1 2. We expect to have the conce... | https://arxiv.org/abs/2502.04339v1 |
different non-linearities. The curve tCrefers to the collapse time obtained using (28). where Σ = ht(ηtFFT+Id). For the determinant, we have 1 dlog det Σ = log( ht) +1 dlog det ηtFFT p+Id . Thus, we find collapse time by the condition α−lim d→+∞1 2dlog det ηtCFFT/p+Id = 0. (27) Now, we specialize to the cases of ra... | https://arxiv.org/abs/2502.04339v1 |
following argument, we find thatg′ t(1) =−1/2: −g′ t(1) =−lim d→∞1 dEx1 P+ t(x)∂Pt,λ(x) ∂λ λ=1 = lim d→∞1 dExE" ∥x−atx1∥2 2ht x# = lim d→∞1 2dE∥z∥2=1 2. This implies that t∗andtccalculated using (24) are the same. Thus, the approximation made in (24) is valid for t≥tc. ACKNOWLEDGMENT The work of A. J. G and R. V has ... | https://arxiv.org/abs/2502.04339v1 |
Empirical Risk,” IEEE Transactions on Information Theory , vol. 66, no. 1, pp. 401–418, Jan. 2020, conference Name: IEEE Transactions on Information Theory. [15] C. Luneau and N. Macris, “Tensor Estimation With Structured Priors,” IEEE Journal on Selected Areas in Information Theory , vol. 1, no. 3, pp. 705–722, Nov. 2... | https://arxiv.org/abs/2502.04339v1 |
arXiv:2502.04422v1 [math.ST] 6 Feb 2025THE MAXIMUM LIKELIHOOD DEGREE OF FARLIE-GUMBEL-MORGENSTERN BIVARIATE EXPONENTIAL DISTRIBUTION POOJA YADAV1. TANUJA SRIVASTAVA2 Abstract. The maximum likelihood degree of a statistical model refers to the number of solutions, where the derivative of the log-l ikelihood function is ... | https://arxiv.org/abs/2502.04422v1 |
θ≤(n−1). For the ML-degree of θ, the points of concern are V(h)\(V(h)∩V(k)) =V(h)\V(h,k). Theorem 2.1. V(h,k)/\e}atio\slash=∅if and only if there exists l/\e}atio\slash=m∈ {1,2,...,n} such that cl=cm. Proof. . Suppose h(θ)andk(θ)have a common zero, say α, that is,h(α) = 0 andk(α) = 0. Consider k(α) =n/productdisplay i=... | https://arxiv.org/abs/2502.04422v1 |
MLE will be either 1or−1since −1≤θ≤1. Whenmis equal to n, then the value of lis at least 2. Hence, the ML-degree of the association parameter θis greater than or equal to 1. 6 POOJA YADAV. TANUJA SRIVASTAVA Acknowledgements The first author would like to thank the University Grants Com mission, India, for providing finan... | https://arxiv.org/abs/2502.04422v1 |
arXiv:2502.04491v1 [cs.LG] 6 Feb 2025Provable Sample-Efficient Transfer Learning Conditional Di ffusion Models via Representation Learning Ziheng Cheng∗Tianyu Xie†Shiyue Zhang‡Cheng Zhang§ Abstract While conditional diffusion models have achieved remarkable success in various applications, they require abundant data to tra... | https://arxiv.org/abs/2502.04491v1 |
model (see Table 1). Tasks Backbone Score Network Condition Encoder Text-to-Image [ Esser et al. ,2024] 2-8B 4.7B Text-to-Audio [ Liu et al. ,2024] 350-750M 750M Reinforcement Learning [ Chi et al. ,2023] 9M 20-45M Table 1: Comparing the number of parameters of different parts in CDMs. While this paradigm has demonstrat... | https://arxiv.org/abs/2502.04491v1 |
5.1) and behavior cloning (Theorem 5.2), and present guarantees in terms of posterior estimation and optimality gap, laying the theoretical foundations of t ransfer learning CDMs in practice. We also conduct numerical experiments in Section 6to verify our results. 1.1 Related Works Score Approximation and Distribution ... | https://arxiv.org/abs/2502.04491v1 |
denote the data and conditions, respectively. The blackb oard bold letterPrepresents the joint distribution of ( x,y), while the lowercase pdenotes its density function. The superscript kindicates the task index, and the subscript imeans the sample index. The norm ∝⌊a∇⌈⌊l·∝⌊a∇⌈⌊lrefers to the ℓ2-norm for vectors and th... | https://arxiv.org/abs/2502.04491v1 |
(ERM), /hatwidef,/hatwideh= argmin f∈F⊗K,h∈H1 nKK/summationdisplay k=1n/summationdisplay i=1ℓ(xk i,yk i,sfk,h). (2.6) Then for the fine-tuning task, we solve /hatwidef0:= argmin f∈F01 mm/summationdisplay i=1ℓ(x0 i,y0 i,sf,/hatwideh). (2.7) Heresf,h(x,y,t) :=f(x,h(y),t) forf:Rdx×[0,1]dy×[T0,T]→Rdxandh: [0,1]Dy→[0,1]dyand... | https://arxiv.org/abs/2502.04491v1 |
generally impossible to characterize the error from an imperfect representation map h. Hence it is inevitable to show the smoothness of pt(·|y) to some extent. Fortunately, even with assumptions merely on the initial da ta distribution, we are still able to prove smoothness of the forward process in any bounded regio n... | https://arxiv.org/abs/2502.04491v1 |
εapprox+KlogNF+logNH nK+logNF m/parenrightbigg , by additionally assuming smooth- ness of loss function and applying local Rademacher complex ity techniques. The difference in our analysis lies in the intricacy of time-dependent score matc hing loss, where the Lipschitzness and (or) smoothness need to be re-verified. Des... | https://arxiv.org/abs/2502.04491v1 |
Assumption 3.1,3.2,3.3, to achieve Rf/greaterorsimilarlog1 2(nKMf/δ)and inf h∈H1 KK/summationdisplay k=1inf f∈FE(x,y)∼Pk[ℓPk(x,y,sf,h)] =O/parenleftbig log2(nK/(εδ))ε2/parenrightbig ,(transfer learning) (4.1) inf h∈HEP∼Pmetainf f∈FE(x,y)∼P[ℓP(x,y,sf,h)] =O/parenleftbig log2(nK/(εδ))ε2/parenrightbig ,(meta-learning) (4.... | https://arxiv.org/abs/2502.04491v1 |
case, Yang et al. [2024] is able to bound the TV distance by /tildewideO(m−1 4+n−1−α(n) dx+5), escaping the curse of dimensionality for target task. However, the assumption on shared latent varia ble distribution is stringent and we believe our analysis methods can be extended to this setting as well. 5 Applications We... | https://arxiv.org/abs/2502.04491v1 |
In such s ettings, the state often corresponds to visual observations of the robot’s surroundings, such as hi gh resolution images, and thus typically share a low-dimensional underlying representation. LetMbe the space of decision-making environments, where each M∈ Mis an infinite horizon Markov Decision Process (MDP) s... | https://arxiv.org/abs/2502.04491v1 |
error of different models. In the pre- training phase, the /hatwidefk;1≤k≤Kandˆhare trained on the K= 10 source distributions with 400K iterations and a batch size of 512. In the fine-tuning phase, t he pre-trained representation map /hatwideh is fixed, and the /hatwidef0is trained on the target distribution with 200K ite... | https://arxiv.org/abs/2502.04491v1 |
Chen, Holden Lee, and Jianfeng Lu. Improved analysi s of score-based generative modeling: User-friendly bounds under minimal smoothness assumption s. InInternational Conference on Machine Learning , pages 4735–4763. PMLR, 2023a. 14 Minshuo Chen, Wenjing Liao, Hongyuan Zha, and Tuo Zhao. Dist ribution approximation and ... | https://arxiv.org/abs/2502.04491v1 |
Conference on Computer Vision , pages 7323–7334, 2023. Haoran He, ChenjiaBai, Kang Xu, Zhuoran Yang, Weinan Zhang, Dong Wang, Bin Zhao, and Xue- long Li. Diffusion model is an effective planner and data synthes izer for multi-task reinforcement learning. Advances in neural information processing systems , 36:64896–64917,... | https://arxiv.org/abs/2502.04491v1 |
the IEEE/CVF con- ference on computer vision and pattern recognition , pages 10684–10695, 2022. Nataniel Ruiz, Yuanzhen Li, Varun Jampani, Yael Pritch, Mic hael Rubinstein, and Kfir Aber- man. Dreambooth: Fine tuning text-to-image diffusion models for subject-driven generation. InProceedings of the IEEE/CVF conference on... | https://arxiv.org/abs/2502.04491v1 |
Error bounds for approximations with deep relu networks. Neural networks , 94: 103–114, 2017. Longlin Yu, Tianyu Xie, Yu Zhu, Tong Yang, Xiangyu Zhang, and Cheng Zhang. Hierarchical semi-implicit variational inference with application to d iffusion model acceleration. Advances in Neural Information Processing Systems , ... | https://arxiv.org/abs/2502.04491v1 |
Cxy such that the following holds. For any h∈ Hand(x1,y1),···,(xm,ym)i.i.d.∼P, define the empirical minimizer /hatwidef:= argmin f∈F1 mm/summationdisplay i=1ℓ(xi,yi,sf,h). (A.18) The population loss of /hatwidefcan be bounded by E{(xi,yi)}m i=1∼PE(x,y)∼P[ℓP(x,y,s/hatwidef,h)]≤4 inf f∈FE(x,y)∼P[ℓP(x,y,sf,h)]+Cxylog3(m)rx... | https://arxiv.org/abs/2502.04491v1 |
inf f∈FE(x,y)∼P[ℓP(x,y,sf,h)]+6C7M2 flog3(m/δ)/parenleftbigg r† m+log(1/δ) m/parenrightbigg , (A.39) We conclude the proof by noticing that E[X] =/integraldisplay∞ 0P(X≥x)dxand plugging in the bound above. Proposition A.5 (Prop.3.3).There exists some constant CZ,CRsuch that the following holds. For any P1,···,PK, letxk... | https://arxiv.org/abs/2502.04491v1 |
any f∈ F⊗K,h∈ H, Ez∼/hatwideP(K)[/tildewideℓ(z,f,h)]≤2 nKK/summationdisplay k=1n/summationdisplay i=1/tildewideℓ(zk i,f,h)+C5M/parenleftbigg r∗ K,n+log(log(nK)/δ) nK/parenrightbigg +C5/radicalbigg M2log(log(nK)/δ) nKexp(−C′ 1R2),(A.54) 27 1 nKK/summationdisplay k=1n/summationdisplay i=1/tildewideℓ(zk i,f,h)≤2Ez∼/hatwid... | https://arxiv.org/abs/2502.04491v1 |
inFasf1,f2, respectively. Without loss of generality, suppose L(P,h1)≥ L(P,h2). Then L(P,h1)−L(P,h2)≤Et,xt,y/bracketleftig/vextendsingle/vextendsingle/vextendsingle∝⌊a∇⌈⌊lf2(xt,h1(y),t)−∇xlogpt(xt|y)∝⌊a∇⌈⌊l2−∝⌊a∇⌈⌊lf2(xt,h2(y),t)−∇xlogpt(xt|y)∝⌊a∇⌈⌊l2/vextendsingle/vextendsingle/vextendsingle/bracketrightig ≤Et,xt,y[... | https://arxiv.org/abs/2502.04491v1 |
X∝⌊a∇⌈⌊lxt∝⌊a∇⌈⌊l2)2∝⌊a∇⌈⌊lxt∝⌊a∇⌈⌊l2/bracketrightig +2(B2+(L2+1)C0) ≤C′′ X(∝⌊a∇⌈⌊lx∝⌊a∇⌈⌊l6+1).(A.100) 34 Lemma A.11. LetΦbe a class of functions on domain ΩandPbe a probability distribution over Ω. Suppose that for any ϕ∈Φ,∝⌊a∇⌈⌊lϕ∝⌊a∇⌈⌊lL∞(Ω)≤b,EP[ϕ]≥0, andEP[ϕ2]≤BEP[ϕ] +B0for some b,B,B 0≥0. Letx1,···,xni.i.d.∼Pan... | https://arxiv.org/abs/2502.04491v1 |
(lognK)/δ/parenrightbig nK. (A.122) From now on we reason on the conjunction of ( A.120), (A.121) and (A.122). Define Uj=Bǫj+B0+bRK,n(Φ(j))+/radicaligg b2(Bǫj+B0)log/parenleftbig log(b/ǫj)/δ/parenrightbig nK+b2log/parenleftbig log(b/ǫj)/δ/parenrightbig nK.(A.123) and thus for any ϕ∈Φ(j), we have1 nKK/summationdisplay k... | https://arxiv.org/abs/2502.04491v1 |
[−Rf,Rf]dx×[0,1]dy×[T0,T]. By Proposition B.2,B.3, inf h∈HEP∼Pmetainf f∈FE(x,y)∼P[ℓP(x,y,sf,h)] ≤inf h∈HEP∼PmetaEy∼P[2γ2 f∝⌊a∇⌈⌊lh(y)−h∗(y)∝⌊a∇⌈⌊l2]+16M2 fexp(−C′ 1R2 f) +EP∼Pmetainf f∈F2∝⌊a∇⌈⌊lf(xt,h∗(y),t)−fP ∗(xt,h∗(y),t)∝⌊a∇⌈⌊l2 ≤2 inf h∈Hγ2 f∝⌊a∇⌈⌊lh−h∗∝⌊a∇⌈⌊l2 L∞([0,1]Dy)+16M2 fexp(−C′ 1R2 f) +2EP∼Pmetainf f∈F∝⌊a... | https://arxiv.org/abs/2502.04491v1 |
Now wecharacterize theconfigurationof neuralnetwork /hatwidef(x,w,t). For boundedness,by Lemma A.10, ∝⌊a∇⌈⌊l/hatwidef(x,w,t)∝⌊a∇⌈⌊l ≤ ∝⌊a∇⌈⌊lf∗∝⌊a∇⌈⌊lL∞(ΩRf)+ε≤2C′′ XR6 f=:Mf. (B.24) Hence we can let Rf=O/parenleftbigg log1 2/parenleftbiggnK εδ/parenrightbigg/parenrightbigg to ensure the lower bound of Rfmentioned above... | https://arxiv.org/abs/2502.04491v1 |
∆.(B.35) Theorem B.5 (Thm.4.3).Suppose Assumption 3.1,3.2,3.3hold. For sufficiently large integers n,K,mandδ>0, with proper configuration of neural network family and T,T0, it holds that with probability no less than 1−δ, EP∼PmetaE{(xi,yi)}m i=1∼PEy∼Py[TV(/hatwidePx|y,Px|y)]/lessorsimilarlog5 2(nK/δ)log3(m∧n) (m∧n)1 dx+dy... | https://arxiv.org/abs/2502.04491v1 |
AndAψ(x)≥Aψ(0)−∝⌊a∇⌈⌊lx∝⌊a∇⌈⌊l1≥ −logC− ∝⌊a∇⌈⌊lx∝⌊a∇⌈⌊l1. Further note that the posterior density pθ(x|y) = pφ(x)exp(∝an}⌊∇a⌋k⌉tl⌉{tx,h∗(y)∝an}⌊∇a⌋k⌉t∇i}ht−Aψ(x)) Zθ, where the normalizing constant Zθ(y) is lower bounded by Zθ(y) =/integraldisplay pφ(x)exp(∝an}⌊∇a⌋k⌉tl⌉{tx,h∗(y)∝an}⌊∇a⌋k⌉t∇i}ht−Aψ(x))dx ≥/integraldispl... | https://arxiv.org/abs/2502.04491v1 |
A sliced Wasserstein and diffusion approach to random coefficient models Keunwoo Lim∗, Ting Ye†, and Fang Han‡ April 25, 2025 Abstract We propose a new minimum-distance estimator for linear random coefficient models. This estimator integrates the recently advanced sliced Wasserstein distance with the nearest neighbor m... | https://arxiv.org/abs/2502.04654v2 |
metric have also been introduced and validated in Bonneel et al. (2015) and Tanguy et al. (2024), among many others. Building on these insights, this paper proposes a novel SW-based minimum-distance estimator for the distribution of β, where weights are adaptively chosen over an NN graph to reduce bias. To the best of ... | https://arxiv.org/abs/2502.04654v2 |
the indicator function and SX(V, k) :=n j∈[n] :nX i=11 ∥eXi−V∥2<∥eXj−V∥2 < ko denote the set of indices corresponding to the k-NNs of Vin{eXi;i∈[n]}. Since Ximay take discrete values, ties can occur, causing the cardinality of SX(V, k)to exceed k. In such cases, consistent with Lin and Han (2024b,a), we select an arb... | https://arxiv.org/abs/2502.04654v2 |
of [k]such that aν(1)≤ ··· ≤ aν(k). Additionally, the projection of a vector ψ∈Rkdonto the set sBR(0)kin Euclidean space is defined as Proj sBR(0)k(ψ) = argmin w∈sBR(0)k∥ψ−w∥2. Algorithm 1 is a block coordinate descent method for computing bw. This algorithm is a revised version of Tanguy et al. (2024, Algorithm 1), in... | https://arxiv.org/abs/2502.04654v2 |
equivalent to Assumption 4 in Hoderlein et al. (2010), which naturally holds when Xhas a full-dimensional support. In particular, this assumption is valid when (1.1) does not include an intercept term, i.e., Xi,1= 1for all i∈[n]. Such scenarios are plausible in many cases, including those discussed in Hoderlein et al. ... | https://arxiv.org/abs/2502.04654v2 |
case, nonparametric maximum likelihood estimators) may exist but all converge to the estimand. At this stage, it remains unclear whether the derived rate is minimax optimal or not. Moreover, as different metrics are employed, the obtained results are not directly comparable to those in the literature. Nevertheless, it ... | https://arxiv.org/abs/2502.04654v2 |
P(sBR(0))with density ϱ0∈L∞(sBR(0)). Then, for the minimization problem: eFλ ζ(µ) =1 2SW2 2(µ, ζ) +λH(µ), where ζis a probability measure with positive smooth density, there exists a generalized minimizing movement scheme such that the density (ϱt)tof(µt)tsatisfies the continuity equation: ∂ϱt ∂t=−div(vtϱt) +λ∆ϱt, (5.1... | https://arxiv.org/abs/2502.04654v2 |
be correlated with the unobserved noise U. In practice, however, not all potential outcomes {Y(w);w∈ W}are observable. In the context of arandomized controlled trial , we introduce a random treatment variable W∈ Wand only observe the outcome Y:=Y(W), which corresponds to the potential outcome under the assigned treatme... | https://arxiv.org/abs/2502.04654v2 |
computation time (in seconds) and the empirical distance (in SW2distance) between the original samples and the output, averaged over 100experiments. The empirical SW2distance using the Monte Carlo method is computed with the POT package (Flamary et al., 2021) with 100unit vector samples. For implementing Algorithms 1 a... | https://arxiv.org/abs/2502.04654v2 |
the control group, we examine the effects of combination therapies—zidovudine and didanosine (Therapy 2) and zidovudine and zalcitabine (Therapy 3)—as well as monotherapy with didanosine (Therapy 4), as considered in Hammer et al. (1996, Table 2). We analyze each investigational treatment (Therapies 2-4) separately aga... | https://arxiv.org/abs/2502.04654v2 |
fsXis directly obtained by the Radon-Nikodym theorem. Consider the Borel seteK⊂eD(τ)with area S(eK)and corresponding set K⊂Rdsuch that K=n (0,eT2:d)∈Rd|eT= (eT1,eT2:d)∈eKo with area S(K), and let the set eA(τ)be eA(τ) ={eT2:d∈Rd−1|∥eT2:d∥2<(1−C2 sXτ2α)1/2}. Then, denoting the function l:eA(τ)→Ras l(eT2:d) = (1 − ∥eT2:d... | https://arxiv.org/abs/2502.04654v2 |
(2015, Theorem 1) can still be used to show that EW2 2(µV β,eµV k)≤Ck−1/3,ford≤5, EW2 2(µV β,eµV k)≤Ck−2/d,ford≥6 holds for some constant C=C(R, d). It remains to upper bound the second term in (A.2). To this end, let’s denote the set S(V, γ) and its range bound γ(V, k)forV∈Sd−1and0≤γ≤√ 2as S(V, γ) ={eX∈Sd−1|⟨V,eX⟩... | https://arxiv.org/abs/2502.04654v2 |
uniformly equicontinuous. Then, by Hirsch and Lacombe (2012, Propo- sition 3.2), since the equicontinuous sequence (sFk(η(w)))m∈Nconverges to Fk(η(w))on the dense setH, the sequence converges uniformly to the continuous function Fk(η(w)). It leads to the final convergence result Pσ lim m→∞∥sFk(η(w))− F k(η(w))∥∞= 0 =... | https://arxiv.org/abs/2502.04654v2 |
argmin wq∈sBR(0)Γq(wq)with Γq(U) =m−1(U−L−1 msVq)⊤Lm(U−L−1 msVq)(A.6) and for each iteration of Algorithm 2, the convex optimization formulation is given as eψℓ+1,q= argmin wq∈sBR(0)eΓq(wq)witheΓq(U) =d−1(U−m−1dsVq)⊤(U−m−1dsVq)(A.7) for same values of sVqon1≤q≤ksince ψℓ=eψℓholds. Step 3. Next step is on the uniform con... | https://arxiv.org/abs/2502.04654v2 |
≥ϵ)≤2 exp −mϵ2/8R4 . Then, introducing Am:=|sFk(η(w))− F k(η(w))|, we have E[Am]≤Z∞ 0P(Am≥ϵ) dϵ≤2Z∞ 0exp −mϵ2 8R4 dϵ=r 8πR4 m. The proof is then complete by setting m≥8πR4ϵ−2. The next proposition analyzes the approximation accuracy of Algorithm 2. Proposition A.2. Assume the case of N=k, Assumption 4.2, and ψℓ=eψℓ,... | https://arxiv.org/abs/2502.04654v2 |
inequality f(Z,W)(T, t)≥CZCW(1 +∥T∥2)−κ(1 +|t|)−κ≥CZCW(1 +∥(T, t)∥2)−2κ. Proof of Proposition 6.2: We have the lower bound of fWϵ(t)as fWϵ(t)≥π−1ϵ−1(1 +ϵ−2(MW+|t|)2)−1≥π−1ϵ−1(1 +ϵ−1(MW+|t|))−2 =π−1ϵ(ϵ+MW+|t|)−2≥π−1ϵmax{1, ϵ+MW}−2(1 +|t|)−2, which leads to the bound of f(Z,Wϵ)(T, t)as f(Z,Wϵ)(T, t)≥CZπ−1ϵmax{1, ϵ+MW}−2(... | https://arxiv.org/abs/2502.04654v2 |
V2∈Sd−1with the condition ⟨V1, V2⟩ ≥1−γ2for some 0≤γ≤√ 2. Then, there exists a constant C=C(R)such that |⟨U, V 2⟩ − ⟨U, V 1⟩⟨V1, V2⟩| ≤Cγ. Proof.By direct computation of the objective term, ⟨U, V 2⟩ − ⟨U, V 1⟩⟨V1, V2⟩=⟨U, V 2⟩ − ⟨U, V 2⟩⟨V1, V2⟩2+⟨U, V 1⟩⟨V1, V2⟩2− ⟨U, V 1⟩⟨V1, V2⟩ =⟨U, V 2⟩(1− ⟨V1, V2⟩2) +⟨V1, V2⟩(⟨U,... | https://arxiv.org/abs/2502.04654v2 |
Corenflos, A., Fatras, K., Fournier, N., Gautheron, L., Gayraud, N. T., Janati, H., Rakotoma- monjy, A., Redko, I., Rolet, A., Schutz, A., Seguy, V., Sutherland, D. J., Tavenard, R., Tong, A., and Vayer, T. (2021). Pot: Python optimal transport. Journal of Machine Learning Research , 22(78):1–8. Fournier, N. and Guilli... | https://arxiv.org/abs/2502.04654v2 |
and Andrew, M. E. (2011). K-nearest neighbor based consistent entropy estimation for hyperspherical distributions. Entropy, 13(3):650–667. Lim, K. and Han, F. (2024). Smoothed NPMLEs in nonparametric Poisson mixtures and beyond. arXiv preprint arXiv:2406.08808 . Lin, Z., Ding, P., and Han, F. (2023). Estimation based o... | https://arxiv.org/abs/2502.04654v2 |
EARLY STOPPING FOR REGRESSION TREES BYRATMIR MIFTACHOV1,a, MARKUS REISS2,b 1Institute of Mathematics and School of Business and Economics, Humboldt-Universität zu Berlin , aratmir.miftachov@hu-berlin.de 2Institute of Mathematics, Humboldt-Universität zu Berlin,bmreiss@math.hu-berlin.de We develop early stopping rules f... | https://arxiv.org/abs/2502.04709v2 |
inequalities. 1arXiv:2502.04709v2 [math.ST] 7 Mar 2025 2 far before the tree is fully grown. Unlike previous pre-pruning (or early stopping) methods, our stopping rule monitors the global residual norm, as for the discrepancy principle for in- verse problems (Engl, Hanke and Neubauer, 1996) and its statistical applicat... | https://arxiv.org/abs/2502.04709v2 |
the original Breiman algorithm under further assumptions in a high-dimensional setting with a polynomial convergence rate in n. Subsequently, under the assumption of an additive model, Klusowski and Tian (2024) achieve a logarithmic rate for the regression tree. They control the tree depth using a depth parameter, but ... | https://arxiv.org/abs/2502.04709v2 |
obtained by identifying g1,g2:Rd→Rwith∥g1−g2∥n= 0, that is coinciding on the design. The vectors (Yi),(εi)can be lifted from RntoL2 nviaY(Xi) =Yi,ε(Xi) =εiand, similarly, g∈L2 nis encoded by (g(Xi))i=1,...,n∈Rn. In this sense, L2 nis identified with Rn. 2.2. Regression tree algorithm. We follow the classical algorithm ... | https://arxiv.org/abs/2502.04709v2 |
are at three common sequential ordering mechanisms to build a decision tree. This defines an order to which we apply the residual-based early stopping rule. Initially, the origi- nal CART algorithm is based on a depth-first search principle to grow the tree. However, this approach is statistically unsuitable for implem... | https://arxiv.org/abs/2502.04709v2 |
ˆgglob←g 4: Stop 5: end if 6: Initialize next partition Pkg+1← Pkg 7: forj= 1,...,k g, where |Pkg|=kgdo 8: ifCardinality |A(g) j|= 1then 9: Do not split A(g) j 10: else 11: Split node A(g) jusing CART criteria into A(g+1) jLandA(g+1) jRwhere jL̸=jR 12: Refine partition Pkg+1←(Pkg+1∪ {A(g+1) jL,A(g+1) jR})\ {A(g) j} 13:... | https://arxiv.org/abs/2502.04709v2 |
known about the generalized projection flow (Πt)t∈[0,n]obtained in this case. From a computational perspec- tive, the interpolation comes at no additional cost since the required quantities are calculated in either case. Although we focus on regression tree methods, the early stopping theory developed here applies in m... | https://arxiv.org/abs/2502.04709v2 |
andΠn= Id , respectively, while their continuity in tis a consequence of the continuity of t7→Πt. The monotonicity of ∥Πtε∥2 nfollows from Πs⪯Πtand their commutativity via ∥Πsε∥2 n=⟨Πs(Π1/2 sε),Π1/2 sε⟩n⩽⟨Πt(Π1/2 sε),Π1/2 sε⟩n=⟨Πs(Π1/2 tε),Π1/2 tε⟩n⩽∥Πtε∥2 n. The monotonicity argument for ∥(Id−Πt)f∥2 nis completely ana... | https://arxiv.org/abs/2502.04709v2 |
n−1/2, which represents an intrin- sic information loss due to early stopping, compare the lower bound in Blanchard, Hoffmann and Reiß (2018a). The interpolation error is due to the fact that Πτbis usually not a projection, in which case it vanishes. It must be analyzed for the concrete flow (Πt)under consideration, wh... | https://arxiv.org/abs/2502.04709v2 |
we bound for any δ >0, using 2AB⩽δA2+δ−1B2, 2E |⟨(Id−Πτb)2f,ε⟩n| ⩽δE ∥(Id−Πτb)f∥2 n +δ−1E ⟨(Id−Πτb)2f ∥(Id−Πτb)f∥n,ε⟩2 n , 12 which holds in particular also under convention (4.3). Therefore we find by triangle inequality and Lemma 4.2 E[∥ˆFτ−f∥2 n]⩽2E[∥ˆFτb−f∥2 n] + 2E[∥ˆFτ−ˆFτb∥2 n] ⩽(4 + 2 δ)E[∥(Id−Πτb)f∥2 n] ... | https://arxiv.org/abs/2502.04709v2 |
Now, we allow the projections (Πt)to be data-driven and to depend arbitrarily on the data (Xi,Yi)i=1,...,n. This aligns with the standard CART splitting criterion, analyzed theoretically in Scornet, Biau and Vert (2015), Chi et al. (2022) and Klusowski and Tian (2024). PROPOSITION 4.11. Suppose that εis¯σ-subgaussian a... | https://arxiv.org/abs/2502.04709v2 |
the semi-global early stopping error under independent splitting will attain the same risk as the oracle, up to a small numerical factor. Usual convergence rate results for regression trees are even much slower than the nonpara- metric minimax rates and utilize modified splitting techniques; see, e.g., Genuer (2012), B... | https://arxiv.org/abs/2502.04709v2 |
empirical to the population prediction error depends on the regression tree algorithms and often requires some additional assumptions. General results are obtained by Wager and Walther (2015) in the case where the splitting rule ensures a minimal percentage of the parent node’s sample size also for the child nodes. Oth... | https://arxiv.org/abs/2502.04709v2 |
large hyperparameter results in the root node. Each hyperparameter can be substituted into Equation 5.2, giving a unique pruned regression tree. In our simulation and empirical applications, the optimal hyperparameter λopt∈Λprunis selected using 5-fold cross-validation. The pruned regression tree estimator is then deno... | https://arxiv.org/abs/2502.04709v2 |
regression tree as in Algorithm 3.1, resulting in estimator ˆFˆgglob 2: Apply cost-complexity pruning to get the sequence of cost-complexity hyperparameters for ˆFˆgglob+1, denoted asΛ2step 3: Determine the optimal hyperparameter λ′ opt∈Λ2step by 5-fold cross-validation 4: Output: Two-step regression tree estimator ˆF2... | https://arxiv.org/abs/2502.04709v2 |
2)2+ (x2−1 2)2−0.9(x1−1 2)(x2−1 2) . 6.2. Simulation B: high-dimensional setting. Following the simulation setup of Haris, Simon and Shojaie (2022), the covariates are drawn as Xiiid∼U(−2.5,2.5)30and the noise variables as εi∼N(0,1). We generate n= 1000 observations each for the training and the test set with a regre... | https://arxiv.org/abs/2502.04709v2 |
the smaller semi-global oracle error for these examples, compare Figure 6.3(a) below. The overall semi-global prediction error for these examples, as well as the other low dimensional examples, is in fact better than the global early stopping error, see Table 6.2 below. Third, the relative efficiencies in high dimensio... | https://arxiv.org/abs/2502.04709v2 |
two-step oracle achieves a performance comparable to the pruning oracle while being faster to compute (see Table 6.3). Hence, the two-step procedure effectively combines the 22 Pruning Global Global Int Two-Step Semi Deep Rectangular 0.21 (0.20) 0.33 (0.30) 0.31 (0.30) 0.20 (0.19) 0.30 (0.22) 1.05 Circular 0.25 (0.24) ... | https://arxiv.org/abs/2502.04709v2 |
early stopping for high dimensions. 7. Conclusion. This paper introduces a novel, data-driven stopping rule for regression trees based on the discrepancy principle, offering a computationally efficient alternative to existing post-pruning and pre-pruning methods. Clear theoretical guarantees in terms of or- acle inequa... | https://arxiv.org/abs/2502.04709v2 |
of possible projections until time t. This is inspired by Hucker and Reiß (2024), but the argument differs. By the linear interpolation property Πτb=αΠkg(τb)+ (1−α)Πkg(τb)+1holds, where τb= (1−α)kg(τb)+αkg(τb)+1andα∈[0,1), see Example 3.3. Simple algebra shows (Id−Πτb)2= (1−α)2(Id−Πkg(τb)) + (1−(1−α)2)(Id−Πkg(τb)+1). T... | https://arxiv.org/abs/2502.04709v2 |
and M ATHÉ , P. (2012). Discrepancy principle for statistical inverse problems with application to conjugate gradient iteration. Inverse Problems 28115011. BOUCHERON , S., L UGOSI , G. and M ASSART , P. (2013). Concentration inequalities: A nonasymptotic theory of independence. Oxford University Press . BREIMAN , L. (1... | https://arxiv.org/abs/2502.04709v2 |
S HOJAIE , A. (2022). Generalized sparse additive models. Journal of Machine Learn- ing Research 231–56. HASTIE , T., T IBSHIRANI , R. and F RIEDMAN , J. (2017). The elements of statistical learning: data mining, infer- ence, and prediction . Springer. HUCKER , L. and R EISS, M. (2024). Early stopping for conjugate gra... | https://arxiv.org/abs/2502.04709v2 |
arXiv:2502.05134v1 [math.ST] 7 Feb 2025Information-Theoretic Guarantees for Recovering Low-Ran k Tensors from Symmetric Rank-One Measurements Eren C. Kızılda˘ g∗ February 10, 2025 Abstract In this paper, we investigate the sample complexity of recov ering tensors with low sym- metric rank from symmetric rank-one measur... | https://arxiv.org/abs/2502.05134v1 |
. . . . . . . . 22 8.2 Part (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2 1 Introduction Tensors play a significant role in modern data science, as tensor-va lued datasets frequently emerge in various applications such as neuroscience [ BS05], imaging [ ZLZ13,ZAZD19], and sig... | https://arxiv.org/abs/2502.05134v1 |
for detailed discussions. A comprehensive analysis of cubic sketchin g, specifically focusing on pairwise and triple-wise interactions (the case ℓ= 3), was conducted in [ HZC20]. 1.1 Connections to Learning Two-Layer Polynomial Networks Our setup is closely related to the problem of learning two-layer neur al networks w... | https://arxiv.org/abs/2502.05134v1 |
the data ( Yi,Xi),i∈[N], as defined in ( 6). Even though solving ( 7) is NP-hard [ HL13], its analysis gives a benchmark for evaluating computationally efficient methods, see Section 2.5for details. Our main result establishes a sample size upper bound for ( 7). Theorem 2.1. LetC >2ℓ2be an arbitrary constant and N≥Crd. Th... | https://arxiv.org/abs/2502.05134v1 |
Ti1,...,iℓis determined solely by the d-tuple (α1,...,α d) whereαicounts the occurrences of i∈[d] among ( i1,...,iℓ). This can be incorporated into the optimization by reducing the number of variables to/parenleftbigd+ℓ−1 ℓ/parenrightbig , which is the dimension of Sℓ[CGLM08 ]. 6 2.2 Sample Complexity Lower Bounds In t... | https://arxiv.org/abs/2502.05134v1 |
of these prior models cover. Moreover, their frameworks do not extend to two-layer polynomial networks as discussed earlier. Other related works [ RSS17,CMWX20 ,ARB20,GLM+22,LZ23] study measurement models where Xiconsists of i.i.d.sub-Gaussian entries, or, more generally, satisfies a certain tensor restricted isometry p... | https://arxiv.org/abs/2502.05134v1 |
neural networks. 2.5 Information-Theoretic Bounds and Noiseless Models We now highlight the significance of information-theoretic bounds an d noiseless models, as well as outline future directions. Information-Theoretic Results Information-theoretic guarantees such as ours serve as a foundational step towards computatio... | https://arxiv.org/abs/2502.05134v1 |
in Theorem 4.4). We then rely on a monotonicity property of covering numbers [ Ver18, Exercise 4.2.10]: covering numbers of ζS(2r) is bounded above by that of ζCP(2r). Our approach requires controlling certain probabilistic terms. Since the data distributions in [ENP12] and [MHWG14 ] differ significantly from our setting... | https://arxiv.org/abs/2502.05134v1 |
obtain: Lemma 4.5. LetC >0be an absolute constant. For any d≥2,r >0andǫ∈(0,2], logN/parenleftBig ζS(2r),/bardbl·/bardblF,ǫ/parenrightBig ≤2rℓdlog2 ǫ+Crℓ2dlogd. 4.2 Probability Estimates Our probability estimates crucially rely on Carbery-Wright inequality, a powerful tool for estab- lishing anti-concentration bounds fo... | https://arxiv.org/abs/2502.05134v1 |
bounded by ǫC′ℓrd/parenleftbigg log1 ǫ/parenrightbiggN ℓ ·Oǫ(1), (21) where the terms Oǫ(1) depend only on r,ℓ,dand remain constant as ǫ→0. Since C′>0, (21) tends to zero as ǫ→0. Thus, taking a countable sequence ( ǫn)n∈Nwithǫn→0 and using the continuity of probability measures, we obtain P/bracketleftbigg inf T∈ζS(2r)... | https://arxiv.org/abs/2502.05134v1 |
1≤i≤d /BD{αi=α′ i} = /BD{α=α′}. Thus, the family {Pα}is indeed orthonormal. The following result is folklore and can be proven, e.g., by modifying the argument of [ Arg12] together with an induction over dwhich we skip for simplicity. Proposition 5.2. The family Pαform an orthonormal basis for L2(D⊗d). Proposition 5.2a... | https://arxiv.org/abs/2502.05134v1 |
N} ⊂ {±1}dsuch that •N=ecdǫ2. •For every 1≤i < j≤N,1 d/vextendsingle/vextendsingle/angbracketleftbig v′ i,v′ j/angbracketrightbig/vextendsingle/vextendsingle≤ǫ. Proof of Lemma 6.1.Our proof is based on the probabilistic method [ AS16]. Suppose v′ i(j), i∈[N] andj∈[d] are i.i.d.Rademacher variables: P/bracketleftbig v′ ... | https://arxiv.org/abs/2502.05134v1 |
inequality [ CT06, Theorem 2.10.1], we obtain P/bracketleftbig/hatwideT/\e}atio\slash=T∗/bracketrightbig ≥H/parenleftbig T∗|Y(N),X(N)/parenrightbig −1 log|Ψ|. (43) Next, the chain rule for conditional entropy yields H/parenleftbig T∗,Y(N)|X(N)/parenrightbig =H/parenleftbig T∗|Y(N),X(N)/parenrightbig +H/parenleftbig Y(N... | https://arxiv.org/abs/2502.05134v1 |
regression using low-rank and sparse tucker decompositions , SIAM Journal on Mathematics of Data Science 2(2020), no. 4, 944–966. [Arg12] Martin Argerami, Orthonormal basis for product l2space, Mathematics Stack Ex- change, 2012, https://math.stackexchange.com/q/105486 (version: 2024-07-30). [AS16] Noga Alon and Joel H... | https://arxiv.org/abs/2502.05134v1 |
[CT05b] Richard Caron and Tim Traynor, The zero set of a polynomial , WSMR Report (2005), 05–02. [CT06] Thomas M Cover and Joy A Thomas, Elements of information theory , Wiley- Interscience, 2006. [CW01] Anthony Carbery and James Wright, Distributional and Lqnorm inequalities for polynomials over convex bodies in Rn, M... | https://arxiv.org/abs/2502.05134v1 |
Sparse and low-rank tensor esti- mation via cubic sketchings , International conference on artificial intelligence and statistics, PMLR, 2020, pp. 1319–1330. [Jal19] Shirin Jalali, Toward theoretically founded learning-based compressed s ensing, IEEE Transactions on Information Theory 66(2019), no. 1, 387–400. [JP17] Sh... | https://arxiv.org/abs/2502.05134v1 |
509. [SBRJ19] Arun Sai Suggala, Kush Bhatia, Pradeep Ravikumar, and P rateek Jain, Adap- tive hard thresholding for near-optimal consistent robust regression , Conference on Learning Theory, PMLR, 2019, pp. 2892–2897. [SJL18] Mahdi Soltanolkotabi, Adel Javanmard, and Jason D Lee, Theoretical insights into the optimizat... | https://arxiv.org/abs/2502.05134v1 |
Distribution of singular values in large sample cross-covariance matrices Arabind Swain Department of Physics, Emory University, Atlanta, GA 30322, USA Sean Alexander Ridout Department of Physics, Emory University, Atlanta, GA 30322, USA and Initiative in Theory and Modeling of Living Systems, Atlanta, GA 30322, USA Il... | https://arxiv.org/abs/2502.05254v2 |
To utilize RMT methods, most of which only work forsquaresymmetricmatrices, we focus insteadon eigen- values of CTC=1 T2eXTeYeYTeX. (5) Nonzero eigenvalues of CTC, which we denote as λ, are the same as nonzero eigenvalues of CCT, and their dis- tributionisrelatedtothedistributionofnonzerosingular values of C, denoted a... | https://arxiv.org/abs/2502.05254v2 |
same. To relate densities to each other, we need to sub- tract the δfunctions at zero, and then rescale the densi- ties of nonzero eigenvalues to one in all three cases. With this, we write the finite size Stieltjes transform ofCTC: gCTC,NX(z) =1 NX T1 TTX µ=11 z−λµ+NX−T z! =1 NX T hT(z) +NX−T z (10) =pXhT(z) + (1 −p... | https://arxiv.org/abs/2502.05254v2 |
but unequal in size), the bounds are γ±≈1±√pY+pX√pYpX. (20) We see that, in all of these limits, the center of the singular value distribution is approximately the geomet- ric mean of the inverse aspect ratios,q 1 pXpY=√qXqY. This sets the typical scale of sampling noise singular val- ues at a given sample size T. The ... | https://arxiv.org/abs/2502.05254v2 |
for the whitened cross-correlation matrices, because the self-covariances 5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 pXpY 0.000.250.500.751.001.251.501.752.00()/pXpY pX=pY=1.25 Semi-Analytic Simulation Analytic bounds Figure 3. Distribution of nonzero eigenvalues for T > NX, NY, specifically pX=pY= 1.25. Plotting conventions are th... | https://arxiv.org/abs/2502.05254v2 |
that in many cases the cross- covariance may be the most effective tool for detecting the shared signal in a pair of high-dimensional observa- tions. ACKNOWLEDGMENTS We thank Philipp Fleig, Eslam Abdelaleem, and K. Michael Martini for helpful discussions. This work was funded, in part, by a Simons Foundation Investiga-... | https://arxiv.org/abs/2502.05254v2 |
the nonzero part of the SVD of C, we use: ρA(z) = 2 zρA2(z2), (A20) and the edges obey γ±=p λ±. 1. Spectrum of the empirical cross covariance matrix for T < N X, NY a. Simplified solutions for pX=pY ForpX=pY, the cubic equation for the Stieltjes trans- form, Eq. (A13), reduces to: h3z2pX2+h2z(pX(1−pX) +pX(1−pX)) +h (1... | https://arxiv.org/abs/2502.05254v2 |
to: αh3z2p2 X+h2zpX(α+ 1) + h 1−zαp2 X +αp2 X= 0. (A32) The discriminant of Eq. (A32) is calculated using Eq. (A15). Written as a polynomial in z, it is D= 4z5α4p8 X+z4(α2p6 X−10α3p6 X+α4p6 X−18α3p7 X −18α4p7 X−27α4p8 X) +z3(−2αp4 X+ 8α2p4 X−2α3p4 X −4αp5 X+ 6α2p5 X+ 6α3p5 X−4α4p5 X) +z2(p2 X−2αp2 X+α2p2 X).(A33) Asp... | https://arxiv.org/abs/2502.05254v2 |
TROPICAL FR ´ECHET MEANS BO LIN, KAMILLO FERRY, CARLOS AM ´ENDOLA, ANTHEA MONOD, AND RURIKO YOSHIDA Abstract. The Fr´ echet mean is a key measure of central tendency as a barycenter for a given set of points in a general metric space. It is computed by solving an optimization problem and is a fundamental quantity in st... | https://arxiv.org/abs/2502.05322v1 |
BO LIN, KAMILLO FERRY, CARLOS AM ´ENDOLA, ANTHEA MONOD, AND RURIKO YOSHIDA 1.1.Related Work. A similar statistical quantity of centrality that is defined by a similar optimization problem is the Fermat–Weber point , which is a generalization of the median to general metric spaces. These have been introduced in tropical... | https://arxiv.org/abs/2502.05322v1 |
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