PaperlayerView exact revision on arXiv

Source-linked AI summary

On the Error Analysis of Triangular Transport Maps for Conditional Generation via MMD-based Objectives

Michele Martino, Bamdad Hosseini, Amir Sagiv, Amirhossein Taghvaei

arXiv:2610.33610v1stat.MLcs.LGmath.STstat.TH

TL;DR

The paper addresses limited theoretical guarantees for the conditional measures produced by triangular transport generative models. It analyzes conditional and joint MMD objectives, finding generic conditional inconsistency for joint minimizers under misspecification and dimension-free parametric rates for the direct conditional estimator.

  • Problem

    The conditional quality of triangular generative models has limited theoretical guarantees, despite training data typically consisting of joint samples rather than repeated observations at each conditioning value.

    Source: §1 · pp. 1–2

  • Method

    The paper analyzes average conditional error for triangular maps, comparing joint MMD minimization with a conditional-criterion estimator and its sample-based surrogate.

    Source: §3 · p. 11, §4 · p. 16, §6 · p. 31

  • Results

    Joint MMD minimizers are generically inconsistent for conditional error under misspecification, whereas the direct estimator has decomposed error bounds and ambient-dimension-free parametric rates.

    Source: §1.2 · p. 5, §4 · p. 16, §6 · p. 31

  • Takeaways & Limitations

    The results support targeting conditional error rather than relying on joint divergence alone when assessing conditional generative models.

    Source: §6 · p. 31

    Extensions to hierarchical conditional transport and dimension-reduction techniques remain future directions.

    Source: §6 · p. 32

Abstract

from arXiv · show

Conditional generative models parameterized by triangular transport maps are widely used for conditional sampling, yet theoretical guarantees on the quality of the conditional measures they produce remain scarce. We address this question for minimum divergence estimators of triangular maps. We use MMD-based objectives as a tractable setting in which the gap between joint and conditional errors can be analyzed precisely, and we measure performance through the average conditional estimation error, defined as the squared maximum mean discrepancy (MMD) between the estimated and the true conditionals, averaged over the conditioning variable. We first show that the common practice of minimizing a joint MMD may fail to recover the conditionals: writing the joint and the conditional criteria as quadratic energies of the same conditional mean embedding residual, we prove that, for parametric classes, joint minimizers are generically inconsistent in the misspecified setting. We then analyze an estimator that directly minimizes the conditional criterion through a surrogate loss computable from joint target samples and reference samples. We prove a non-asymptotic error bound that decomposes the average conditional estimation error into approximation, estimation, and reference sampling errors, with fast rates independent of the ambient dimension in the parametric setting. Finally, for joint MMDs built from concentrating Gaussian kernels on the conditioning variable, we show that the joint criterion approximates a density-reweighted version of the conditional one, and we derive an error bound for the joint minimizer whose rate suffers from a curse of dimensionality in the conditioning variable. Our results suggest that training criteria for conditional generative models should target the conditional rather than the joint error, a principle that extends to other divergences and to dynamic transport.

1. Introduction.

The paper asks whether triangular transport models trained on joint samples can recover conditional measures, and shows that joint fit may not ensure conditional fit. It develops conditional-error analysis for direct conditional and joint MMD objectives.

1. Introduction.

  • Joint measures can be close while their conditionals remain far apart, exposing a gap in assessing conditional generation through joint fit alone.The introduction illustrates this with two distributions whose conditionals stay different as α grows.Source: §1 · p. 2

1.1. Minimum divergence estimation of triangular maps.

  • Triangular maps connect joint pushforward matching to conditional measures and provide a structured model class for conditional generation.With an independent reference coupling and F = Id, the conditional model reduces to T(y, ·)#ηV.Source: §1.1 · p. 3

1.2. Overview of main results.

  • Under generic conditions in the parametric setting, joint MMD minimizers are inconsistent for average conditional error in the misspecified case.The gap can persist with irreducible error d > 0 even when the estimator is consistent for the joint criterion.Source: §1.2 · pp. 4–5
  • A surrogate estimator that directly minimizes the conditional criterion receives a non-asymptotic error bound decomposing approximation bias and statistical generalization error.In the parametric setting, the reported rates are ambient-dimension-free.Source: §1.2 · p. 5
  • Joint MMD can approach a conditioning-density-reweighted conditional criterion with concentrating kernels, while its smoothing can obscure rapidly varying conditionals.For the joint estimator, the resulting error bound suffers a curse of dimensionality in the conditioning variable.Source: §1.2 · p. 5

1.4. Related literature.

  • Prior work studied triangular transport and conditional estimation, while this paper directly analyzes conditional-error estimation and contrasts it with joint MMD minimization.The paper situates its estimator alongside [38] and contrasts its guarantees with nearest-neighbor approaches in [21].Source: §1.4 · p. 7

2. Preliminaries.

This section establishes the measure and kernel notation used to compare distributions, then reviews how triangular transports connect joint pushforwards to conditional measures. Under a matching-marginal reference assumption, the conditional map component alone recovers the target conditionals; existence of a ground-truth triangular map is assumed rather than guaranteed.

2.1. Basic notation and definitions.

  • The preliminaries fix separable Hilbert spaces, Borel probability measures, marginals, regular conditional measures, and Bochner-space notation, with measurability assumptions ensuring the quantities are well defined.The section also establishes shorthand for maxima, minima, and bounds up to numerical constants.Source: §2.1 · p. 8

2.2. Scalar-valued RKHSs, mean embeddings, and MMD.

  • An MMD is the RKHS norm between kernel mean embeddings, but it is a divergence only when the kernel is characteristic.The review also introduces PSD kernels, feature maps, and the associated RKHS framework.Source: §2.2 · pp. 8–9

2.3. Triangular transport maps.

  • For a triangular map T_F^△ pushing η to ν, F#η_Z = ν_Y, and ν(·|y) is a mixture of T(y,·)#η(·|z) over η(dz|F(z)=y), for ν_Y-almost every y.If F is invertible, the conditional uses η(·|F⁻¹(y)); if η = η_Z ⊗ η_V is independent, even if F is not invertible, it reduces to T(y,·)#η_V = ν(·|y), for ν_Y-almost every y.Source: §2.3 · pp. 9–10
  • Under an independent reference coupling with the target conditioning marginal, the triangular map can leave the conditioning variable unchanged, and its lower component pushes the reference conditional to the target conditional almost everywhere.The section assumes a ground-truth triangular map exists, although such maps are not guaranteed in general.Source: §2.3 · p. 10

3. Problem setup.

The section defines average conditional estimation error as the target for conditional generation and contrasts minimizing it directly with minimizing a joint MMD. It also explains why joint and conditional errors can diverge, and introduces a computable surrogate for direct conditional-MMD minimization.

3. Problem setup.

  • The average conditional divergence measures the conditional discrepancy averaged over conditioning values, and controlling it yields high-probability control at an independently sampled conditioning value.The paper takes the conditional divergence to be squared MMD.Source: §3 · p. 11

Define the pointwise bias and the pointwise-to-averaged bias gap

  • Finite-dimensional parametric classes are typically not decomposable, so improving fit at some conditioning values can worsen it elsewhere and leave a positive pointwise-to-averaged bias gap.Classes dense in a decomposable class can make this gap arbitrarily small.Source: Define the pointwise bias and the… · p. 12

3.1. Estimating triangular maps by minimizing average conditional MMDs.

  • Directly minimizing the average conditional divergence selects maps that minimize the averaged discrepancy between target conditionals and map-generated conditionals.When the class contains an exact conditional map, the population loss can be zero.Source: Define the pointwise bias and the… · p. 12, §3.1 · p. 12
  • The surrogate loss M can be computed using joint target samples and simulated reference samples, avoiding the need to estimate each target conditional embedding accurately from sparse observations.Its gap from the averaged conditional divergence depends only on the target distribution, so it shares the same minimizer and convergence rate.Source: §3.1 · p. 13, Then our goal is to find · p. 14

3.2. Estimating triangular maps by minimizing joint MMDs.

  • A joint MMD can recover the exact target conditionals when its value is zero, but small joint error alone need not control average conditional error.For integral probability metrics such as MMD and Wasserstein distance, unlike KL divergence with a shared conditioning marginal, joint and average conditional errors can be far apart.Source: Then the goal is to find · p. 16

Then the goal is to find

  • The paper formalizes inconsistency of joint-MMD minimization and studies when additional structure on the joint divergence and conditioning marginal can yield conditional-error control.The joint objective does not directly minimize average conditional error; finite-sample or model-class bias can therefore leave conditional performance uncontrolled.Source: Then the goal is to find · pp. 15–16

4. Error analysis of the EMCMD2 minimizer.

The section bounds the average conditional error of the empirical conditional-MMD estimator by approximation and stochastic errors, with an additional reference-sampling error for the practical estimator. Its proof lifts the problem to regression of conditional mean embeddings and uses empirical-process tools to obtain fast rates.

4.1. Assumptions and setup.

  • The stated analysis assumes bounded and Lipschitz kernels and complexity controls on the map classes, while the finite-reference bound additionally requires stronger embedding and entropy conditions.Boundedness and Lipschitz assumptions can be relaxed for light-tailed target and reference measures, at the price of additional logarithmic factors.Source: §4.1 · pp. 17–18

4.2. Average conditional estimation error bound of bTm and bTm,n.

  • Theorem 4.5 bounds average conditional error by approximation and estimation terms, adding an embedding-approximation term when reference samples are finite.The bound holds with high probability under Assumptions 4.1 and, for the finite-reference-sample estimator, 4.3.Source: §4.2 · p. 19
  • The reference-sampling term need not worsen the rate when the reference sample count grows suitably with the target sample count; n proportional to m suffices, and for γ > 0, sublinear n can suffice.The total number mn of map evaluations can be of order m^(4/(2+γ)), up to logarithmic factors.Source: §4.2 · pp. 19–20

4.3. High level strategy of Theorem 4.5.

  • The proof treats the lifted conditional-mean-embedding estimator as an empirical risk minimizer whose excess risk equals average conditional error, then applies localization and empirical-process bounds.Finite reference samples make the estimator an approximate minimizer, with the extra error controlled separately.Source: §4.3 · pp. 20–21

5. Error analysis of the JMMD2

This section analyzes how minimizing the joint MMD criterion relates to average conditional error, including conditional inconsistency under misspecification and the criterion’s density-reweighted target. It studies Gaussian smoothing and dynamic transport, and bounds the Gaussian-smoothed estimator’s conditional error using approximation, smoothing, and sampling terms, while noting that normalization to target the unweighted error is data-dependent.

5.1. Energy representation of DAC and DG.

  • DAC and DG are energies of the same conditional mean-embedding residual, but DG discounts the tail of AG's spectrum and is not proportional to DAC in infinite-dimensional settings.When the model is well-specified and AG is injective, DG minimization remains consistent; in misspecified cases, the spectral bound may provide no quantitative upper bound when the relevant lower eigenvalue is zero.Source: §5.1 · pp. 22–24

5.2. Inconsistency of DG minimizer for the average conditional estimation error.

  • Under generic parametric conditions, DG minimizers have positive excess conditional error over the approximation bias for targets outside a closed, nowhere-dense, Haar-null set.The result assumes regularity, interior DAC minimizers, and no invariance of the model tangent spaces under AG; it also implies no DG minimizer is a DAC minimizer.Source: §5.2 · pp. 24–25

5.3. Average conditional estimation error bound of bTm Gσ under Gaussian smoothing.

  • Concentrating Gaussian kernels make the joint criterion approximate a density-reweighted conditional criterion, with the smoothed discrepancy increasing to that criterion as σ decreases to zero.The Gaussian multiplier exponentially discounts frequencies much higher than 1/σ, making rapidly varying conditional discrepancies nearly invisible; the resulting misspecified-error bound suffers a curse of dimensionality in the conditioning variable.Source: §5 · p. 21, §5.3 · pp. 26–27
  • The Gaussian smoothing analysis extends to translation-invariant kernels, including Matérn and Laplace types, whose finite spectral measures also discount sufficiently high frequencies.The stated extension requires the normalized spectral density to be radially nonincreasing.Source: §5.3 · pp. 26–27
  • For dynamic conditional transport, the analysis motivates controlling averaged local errors with conditioning-dependent weights rather than amplifying error by a worst-case joint Lipschitz constant.Motion in the conditioning variable affects velocity regularity and model approximation, while schedules adapted to that variable can alter local constants and training errors.Source: §5.3 · p. 27
  • As the Gaussian length-scale vanishes, the joint criterion targets DRAC, which reweights conditional error toward regions with higher conditioning-variable density.The analysis obtains a DAC bound by addressing the conversion gap between DAC and DRAC and the smoothing gap as σ decreases.Source: §5.3 · pp. 28–29
  • A symmetric positive semidefinite kernel normalized by the conditioning marginal may target DAC as σ vanishes, but estimating that unknown marginal makes the criterion data-dependent and less implementable.The analysis does not cover the additional estimation error introduced by this normalization.Source: §5.3 · p. 28
  • Theorem 5.17 bounds DAC by approximation bias, CΞσ^{2ζ}, and reference-sampling terms proportional to E_BiasR(m,t)σ^{-dY/2} and R(m,t)^2σ^{-dY}, under its stated assumptions.The theorem gives a probability guarantee of at least 1 − e^{-t}; the smoothing terms involve negative powers of the conditioning dimension dY.Source: §5.3 · p. 30

2. (Bias-agnostic length-scale) Let

The displayed fragment labels a component “Pure sampling.”

2. (Bias-agnostic length-scale) Let

  • The displayed decomposition labels a component “Pure sampling.”Source: §2 · p. 30

3. (Oracle length-scale) Let

The oracle length-scale choices balance smoothing against sampling terms, with different knowledge requirements and rates depending on misspecification.

3. (Oracle length-scale) Let

  • The oracle choice requires knowledge of E_Bias and is optimal over σ up to a factor of 4, while the bias-agnostic choice needs no E_Bias knowledge.The oracle choice is never worse than the bias-agnostic choice up to constants.Source: §3 · p. 31
  • When E_Bias is fixed and dominates the sampling terms, the mixed-term rate improves from m^-ζ/(2ζ+dY) to m^-2ζ/(4ζ+dY), up to logarithmic factors.The bias-agnostic and oracle choices have mixed terms of orders E_Bias eR2(m,t) and eR1(m,t), respectively.Source: §3 · p. 31
  • In the well-specified case, the bias-agnostic and oracle bounds coincide up to constants, but both rates retain exponents depending on dY.The bound’s constants depend on CΞ, ∥pνY∥∞, dY, and ζ.Source: §3 · p. 31
  • The curse of dimensionality is attributed to the joint criterion’s blindness to fast variations in y, and the authors conjecture it is unavoidable.A matching lower bound is left for future research.Source: §3 · p. 31

6. Conclusion.

The paper contrasts direct conditional MMD minimization with joint minimization and identifies extensions of its analysis as open directions.

6. Conclusion.

  • Directly minimizing the conditional criterion through a joint-sample surrogate yields non-asymptotic bounds decomposing error into approximation, estimation, and reference-sampling terms.These bounds have fast rates in terms of hypothesis-class complexity; joint MMD minimization is generically inconsistent under misspecification.Source: §6 · p. 31
  • Concentrating the conditioning-variable kernel can recover conditional guarantees from joint minimization, but the rates deteriorate with the conditioning dimension.Source: §6 · p. 31
  • Open directions include extending the analysis to other divergences, model classes, dynamic transport, hierarchical maps, and dimension-reduction settings.The authors also propose studying how low-dimensional structure in conditioned and conditioning variables affects conditional estimation error.Source: §6 · pp. 31–32

Appendix B. Proofs of preliminary results and problem setup of Section 2 and Section 3.

The appendix establishes how triangular maps represent conditional measures and develops a conditional-mean-embedding regression formulation for analyzing their error. It also outlines a high-probability connection between average conditional error and pointwise conditional error.

Appendix B. Proofs of preliminary results and problem setup of Section 2 and Section 3.

  • The master theorem represents ν(· | y) through transported reference conditionals, using disintegration and essential uniqueness; when F is invertible, the conditional reference measure is concentrated at F⁻¹(y).The appendix identifies this theorem as a generalization of [6, Thm. 2.4].Source: Appendix B · pp. 37–38
  • When the reference measure factors as ηZ ⊗ ηV, the transported conditional T(y, ·)#ηV does not depend on z.This follows because η(· | z) = ηV for every z.Source: Appendix B · p. 38
  • Controlling DAC(ν ∥ bν) yields a high-probability bound on the pointwise conditional error DCond.(ν(· | y), bν(· | y)).The supplied passage states this result without giving the bound's explicit form.Source: Appendix B · p. 38

C.1. Lifted regression problem over conditional mean embeddings.

  • The conditional criterion lifts to regression over conditional mean embeddings, with g† minimizing the population risk and excess risk equal to squared average embedding error.The lifted empirical risk also preserves approximate minimizers of M(νm, ·) over T.Source: §C.1 · pp. 39–40
  • The proof of Theorem 4.5 uses a one-shot localization technique attributed to van de Geer, adapted to bounds for approximate minimizers in A.The variation is based on the version reported in.Source: §C.1 · p. 41

C.2. Preliminary results from empirical process theory.

This section develops empirical-process bounds for localized approximate empirical minimizers, controlling localized loss-class complexity and fluctuations; it completes the local complexity argument by bounding the remaining terms and establishing (C.17) and (C.18).

C.2. Preliminary results from empirical process theory.

  • With probability at least 1−e^−t, Theorem C.3 localizes all approximate empirical minimizers when the comparator’s excess risk, optimization tolerance, and fluctuation term satisfy its radius condition.The guarantee holds simultaneously for every a* with Rm(a*) ≤ Rm(ā) + ϱ; the stated condition includes w(ℓ(ā) + ϱ + Zm) ≤ τ/4.Source: §C.2 · pp. 41–42
  • The section assembles entropy-integral and supremum-concentration tools to control the fluctuations needed in the estimator bounds.The Dudley-type bound addresses uniformly weighted bounded classes, while the rescaled Bousquet inequality applies to centered classes with bounded L2 and supremum norms.Source: §C.2 · pp. 43–44

C.3. Local complexity control of the lifted class.

  • On the segment hull, the loss is bounded by 4B_K^2 and is pointwise Lipschitz in the embedding with constant 4B_K.These properties follow under Item 1 of Assumption 4.1 and support the localized-class analysis.Source: §C.3 · pp. 46–47
  • Under the stated assumptions, the localized loss class has entropy bounded by C_L δ^−γ log(e + δ^−1).The bound is obtained by covering the segment hull around the conditional mean embedding and transferring that cover to the loss class.Source: §C.3 · pp. 47–48
  • The local fluctuation bound controls empirical-versus-population loss differences, and the subsequent τ-selection result chooses a radius for applying one-shot localization.The fluctuation control makes explicit the complexity term used for the estimator bounds.Source: §C.3 · pp. 48, 50
  • Bounding the four terms by (τ*)^2/4 establishes (C.17), and the resulting bound on (τ*)^2 yields (C.18).The passage gives the final combination of the termwise bounds.Source: §C.3 · pp. 50–51

C.4. Error bound for approximate DAC minimizers and proof for bTm .

This section derives an average conditional error bound for approximate minimizers of the empirical criterion. The bound is applied to the idealized estimator, with a separate bias bound supplied by a lemma.

C.4. Error bound for approximate DAC minimizers and proof for bTm .

  • Theorem 4.5's proof combines the localization result with supporting propositions to bound DAC for all approximate minimizers of M(νm, ·).Setting ϱ = 0 gives the first claim for the idealized estimator bTm; the result is later used with ϱ > 0 for bTm,n.Source: §C.4 · p. 51
  • The empirical-process argument accommodates approximate-minimization tolerance ϱ by treating T* with M(νm, T*) ≤ infT M(νm, T) + ϱ as an approximate risk minimizer.The risk correspondence transfers this tolerance to the lifted empirical regression problem.Source: §C.4 · p. 52
  • For m ≥ 3, the idealized estimator bTm satisfies the claimed bound with probability at least 1 − e⁻ᵗ, and Lemma C.2 bounds EBias.The supplied passages do not include the explicit bound formula.Source: §C.4 · p. 52

C.5. Reference sampling error control.

The appendix controls the error from approximating reference-measure integrals with samples, then combines that control with the approximate-minimizer bound for the fully empirical estimator.

C.5. Reference sampling error control.

  • The reference-sampling discrepancy is centered, and its summand is bounded by 12B²K.The proof expresses the centered discrepancy using empirical averaging over sampled reference variables.Source: §C.5 · pp. 53–54
  • Lipschitz control of the kernel and transport maps bounds changes in the sampled embedding terms by Λ∞∥T − S∥∞.This increment control supports uniform sampling-error bounds over the map class.Source: §C.5 · p. 54
  • Uniform reference-sampling error is controlled with probability at least 1 − e⁻ᵗ under the stated assumptions and n ≥ 2.The bound concerns the uniform control of the reference-sampling error across T.Source: §C.5 · p. 55

C.6. Proof for the error bound of bTm,n.

  • Combining the approximate-minimizer and reference-sampling results yields the average conditional estimation error bound for bTm,n.The proof applies the two bounds together with a union bound.Source: §C.6 · p. 57

Appendix D. Proofs of the results in Section 5.

  • Appendix D begins proving the energy representations of the MMD objectives, assuming separability and measurability of the relevant spaces and integrands.Source: Appendix D · pp. 57–58

D.1. Proof of Subsection 5.1, Energy representations of DAC and DG.

The proof represents the conditional and joint criteria through the same conditional mean-embedding residual, then relates their energies through the spectrum of the conditioning kernel. This relation yields conditions for transferring joint-objective guarantees to conditional error and for consistency in the well-specified case.

Energy representations

  • The conditional error is the squared L2 norm of the conditional embedding residual, while the joint criterion has a spectral representation using the conditioning-kernel operator.The operator is bounded, self-adjoint, and positive; it need not be compact when the embedding space is infinite-dimensional.Source: §D.1 · pp. 58–60

Relation between criteria

  • The joint energy discounts the tail of the conditional-energy spectrum and cannot be proportional to it when L2(Y, R; νY) is infinite-dimensional.This spectral difference limits direct equivalence between the two criteria.Source: §D.1 · p. 60

Transfer of guarantees

  • When the conditioning-kernel operator has a positive lower spectral bound, minimizing the joint criterion controls conditional error by the approximation error, up to κ_G.The stated transfer bound applies when λ−_G > 0.Source: §D.1 · p. 60

Well-specified case

  • If the model class contains a map with zero conditional error and the operator is injective, every joint minimizer also has zero conditional error.The proof uses that zero joint energy forces the residual to vanish.Source: §D.1 · pp. 60–61

Parametric generalization bound

  • For compact parametric classes with the stated regularity and Lipschitz conditions, the proof establishes the generalization bound used to control joint-minimizer error.For Gaussian conditioning kernels, the stated rate is independent of σ.Source: §D.1 · pp. 61–62

D.2. Proofs of Subsection 5.2. Inconsistency of DG minimizer for the average conditional estimation error.

The proof shows that, in the misspecified parametric setting under the stated assumptions, the joint- and conditional-error minimizers generically differ. It establishes this by showing that their shared minimizers lie in a negligible exceptional set and that the joint minimizer has a positive conditional-error gap.

Mechanism

  • A common stationary minimizer would require the residual to be orthogonal to both E_θ and A_G E_θ, imposing at least p + 1 independent linear conditions on a p-parameter class.The gradient conditions for both objectives yield this orthogonality requirement.Source: §D.2 · pp. 62–63

Exceptional set

  • The exceptional set of residuals for which both criteria have a common stationary minimizer is closed, nowhere dense, and Haar-null.These properties support the generic inconsistency conclusion under the theorem's assumptions.Source: §D.2 · pp. 63–64

Inconsistency statements

  • The joint-criterion minimizer has conditional error strictly above the best achievable conditional error when the two criteria have no common minimizer.The proof identifies a positive gap d_G between these quantities.Source: §D.2 · p. 64

D.3. Proofs of Subsection 5.3. Average conditional estimation error bound of bTm

The proof controls the conditional error of the Gaussian-kernel joint minimizer through conversion and smoothing bounds. It also analyzes length-scale choices that balance smoothing error against terms involving approximation bias and estimation error.

Gaussian smoothing

  • For every σ, the Gaussian-smoothed joint energy is no greater than the reweighted conditional energy, and it increases to that energy as σ ↓ 0.The proof obtains this comparison through Fourier representations and monotone convergence.Source: §D.3 · p. 65

Conversion gap

  • The conversion-gap argument bounds conditional error using the reweighted criterion under a bounded-density assumption and a margin condition.It also transfers an approximate minimizer of the reweighted criterion to a conditional-error bound.Source: §D.3 · p. 66

Smoothing gap

  • Under the stated density and regularity assumptions, the smoothing gap is bounded, with the Gaussian moment estimate contributing a d_Y^ζ factor.The proof uses E[∥Z∥^(2ζ)] ≤ d_Y^ζ for Z ∼ N(0, I_dY).Source: §D.3 · p. 67

Length-scale selection

  • The oracle length-scale balances the smoothing term C_Ξσ^(2ζ) against terms Aσ^(-d_Y/2) and Bσ^(-d_Y).The proof also treats a bias-agnostic selection separately from this oracle choice.Source: §D.3 · p. 69

Related papers

Loading 2610.33610v1…