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Sparse Dictionary Learning for Measure-Valued Data via Linearized Optimal Transport: Efficient Algorithms and Sample Complexity

Brendan Mallery, Shuchin Aeron, James Murphy

arXiv:2610.12924v1stat.MLcs.LG

TL;DR

Wasserstein dictionary learning faces computational and statistical challenges. This paper learns sparse dictionaries from linearized optimal-transport embeddings in a Hilbert space, and reports improved computational efficiency and generalization bounds tied to map-representation complexity.

  • Problem

    Wasserstein dictionary learning faces computational and statistical challenges, and it remains unclear which data and modeling tasks benefit from it.

    Source: §1 · p. 1

  • Method

    The paper embeds measures into a Hilbert space through linearized optimal transport and learns dictionaries with sparse autoencoder parameterizations and sparse nonnegative codes.

    Source: §1 · pp. 1–2, §2.2 · p. 3

  • Results

    The authors report improved computational efficiency over existing Wasserstein dictionary learning methods, effective and interpretable atoms and reconstructions, and sample complexity bounds depending on map-representation complexity.

    Source: §7 · p. 13

  • Takeaways & Limitations

    The framework is applied to point clouds, text documents, and hyperspectral image processing, with learned atoms and reconstructions reported as effective and interpretable.

    Source: §1 · p. 2, §7 · p. 13

    The stated generalization results assume either a bounded rectangular domain with density bounded above and below, or a subgaussian base measure, depending on the result.

    Source: §6 · p. 13

Abstract

from arXiv · show

We study dictionary learning for probability measure-valued data by embedding measures into Hilbert spaces using features derived from optimal transport geometry. We learn data-adapted dictionaries through sparse autoencoder parameterizations, producing fast, scalable representations of probability measures as sparse finite-dimensional codes. The resulting framework combines the rich, geometric structure of optimal transport with the interpretability and efficiency of sparse representation learning. We demonstrate its utility across diverse data modalities, including hyperspectral images, geometric point clouds, and document embeddings in large language model latent spaces. We also prove generalization bounds for sparse dictionary learning objectives on broad classes of $L^2$-valued measure embeddings, expressed in terms of the metric complexity of the atom class. For optimal transport and entropic transport map dictionaries, we derive new covering number bounds that yield explicit rates. Together, our computational and theoretical results support a new, scalable model for representation learning in the space of probability measures via embedding into Hilbert spaces.

1 Introduction

Wasserstein dictionary learning offers data-adapted representations of probability measures, but its nonlinear geometry brings computational and statistical challenges. The paper develops sparse linearized models, efficient sparse-autoencoder parameterizations, applications, and generalization bounds.

1 Introduction

  • Wasserstein dictionary learning learns data-representative measure dictionaries for synthesis and representation, but faces computational, statistical, and foundational challenges.The paper identifies questions about which data and modeling tasks benefit from WDL.Source: §1 · p. 1
  • The proposed LWDL formulations promote sparse latent codes by embedding probability measures into a Hilbert space through linearized optimal transport.Sparse nonnegative combinations of learned dictionary elements approximate the embedded measures and can be synthesized into reconstructed measures.Source: §1 · pp. 1–2
  • A multi-layer sparse autoencoder parameterizes dictionary learning in the embedded Hilbert space, enabling faster atom learning than methods based on nonlinear Wasserstein geometry.The paper also describes these parameterizations as enabling faster representation learning than baselines.Source: §1 · p. 2
  • The framework is applied to point clouds, text documents, and hyperspectral image processing to reveal and leverage intrinsic structure.The paper also presents generalization bounds whose dependence reflects the linearization model.Source: §1 · p. 2

2 Models, Methods, and Related Work

The section introduces Wasserstein and entropic transport concepts, then formulates dictionary learning in an embedded Hilbert space. Its LWDL variants use different embeddings and atom classes for different modeling contexts.

2.1 Notation and Background

  • The ρ-linearized optimal transport metric compares measures through the L2(ρ) distance between their transport maps from a fixed base measure.The section also introduces entropic transport maps as smooth surrogates of optimal transport maps that converge to them as the regularization parameter tends to zero under mild assumptions.Source: §2.1 · p. 3

2.2 Linearized Models for WDL

  • Linearized measure-valued dictionary learning learns m atoms whose sparse nonnegative combinations approximate embedded measures, using a sparsity-inducing penalty.The paper calls the Wasserstein-motivated choices of embedding and atom class LWDL.Source: §2.2 · p. 3
  • LWDL-OT preserves exact transport-map structure but is restricted to one dimension, while LWDL-EOT provides finite-dimensional map parameterizations for higher dimensions.LWDL-KP uses scalar potentials instead of vector-valued maps and is suited to very high-dimensional coding.Source: §2.2 · p. 3

3 Algorithms for LWDL

The algorithms jointly train dictionary atoms and a shared sparse encoder in the linearized space, with atom parameterizations matched to each embedding. The related-work discussion contrasts this approach with nonlinear Wasserstein methods and fixed-reference alternatives.

3 Algorithms for LWDL

  • A shared LISTA-type encoder computes sparse codes while method-specific atom parameterizations are trained jointly using the empirical objective.The implementation represents embeddings and atoms by evaluations at finitely supported base-measure points, making inner products, combinations, and reconstruction losses finite-dimensional.Source: §3 · pp. 3–5
  • LWDL-OT uses monotone-map atoms in one dimension, LWDL-EOT uses smooth finite-dimensional entropic-map parameterizations, and LWDL-KP uses centered scalar potentials.For LWDL-EOT, the authors report that displacement fields perform better than the maps themselves and use displacement embeddings.Source: §3 · pp. 4–5

3.1 Related Work

  • Prior nonlinear Wasserstein dictionary methods require iterative synthesis and expensive Wasserstein-distance computations, and use fixed supports for data, atoms, and reconstructions.Other methods learn coefficients for fixed reference measures, including approaches with explicit sparsity penalties.Source: §3.1 · p. 5

4 Numerical Results

The experiments test reconstruction and atom recovery on a synthetic LBCM, then compare the linearized methods with WDL. LWDL achieves lower reconstruction losses across the tested data sets, while the synthetic study also assesses recovery of atoms and coefficients.

4.1 Reconstruction and Identifiability on a Synthetic LBCM

  • The synthetic test-set mean L2(ρ) reconstruction error is 9.22 × 10^-4.The planted data comprise mixtures of five MNIST transport-map atoms.Source: §4.1 · p. 6
  • Although learned atoms weakly resemble the planted digits visually, 24 of 25 displayed nearest neighbors match their corresponding digit, and all five top-ranked neighbors match.The final atom-matching value is 1.74 × 10^-2.Source: §4.1 · p. 6
  • After aligning coordinates, the learned coefficients have an average MSE of 8.50 × 10^-3 against the generating coefficients.The authors report that the coefficients closely approximate the ground truth despite residual error in recovered atoms.Source: §4.1 · p. 6

4.2 Comparison with WDL

  • Across all data sets, LWDL methods achieve mean W2 reconstruction losses roughly 4 to 100 times lower than WDL within the prescribed time budget.The comparison uses a 1000s budget; Table 2 reports the final squared W2 errors.Source: Table 2 · p. 7, §4.2 · p. 7
  • Linearized methods compute the expensive OT or EOT map once, then train with standard autoencoder optimization, enabling parallelization and GPU acceleration.The authors attribute much of WDL's time use to applying dense kernel matrices.Source: §4.2 · p. 7

5 Applications

The applications use modality-specific linearizations of measure-valued data across hyperspectral images, point clouds, and transformer activations. They report reconstruction and classification results, robustness to spectral corruption, and semantic specialization of document atoms.

5.1 Hyperspectral Image Analysis with One-Dimensional LWDL-OT

  • Hyperspectral pixels are normalized as probability measures and embedded using one-dimensional transport maps; three datasets are evaluated, with dictionary size set by non-background class count.The datasets are Pavia Centre, Botswana, and Salinas A.Source: §5.1 · p. 7
  • On Pavia spectra, LWDL-OT outperforms NMF on W2 reconstruction error but is outperformed by NMF on L2; its transport atoms often concentrate mass near wavelength endpoints.The authors relate endpoint concentration to the convex geometry of one-dimensional monotone transport maps.Source: Figure 3 · p. 9, §5.1 · p. 9, Table 3 · p. 10
  • 13 of 15 nonzero DropUniform levels had the highest accuracy with LWDL-OT, which also had the least-negative fitted stability slope on all three datasets.Table 4 reports fixed configurations across corruption levels, with accuracies averaged over five corruption seeds and, for LWDL-OT and N-SAE, five training seeds.Source: Table 4 · p. 10, §5.1 · p. 10

5.2 Point-Cloud Data with LWDL-EOT

  • 96.1% held-out test accuracy was achieved by a linear classifier trained on point-cloud codes, whose learned atoms also capture object parts.Examples include chair seats and backrests, chair legs, and toilet bases.Source: §5.2 · p. 10, Figure 4 · p. 11

5.3.1 Synthetic Textual Corruptions.

  • Across 10 random splits, logistic regression on the first two code principal components exceeds 90% accuracy in separating corruption types, and the first six give about 95%.Atom usage is strongly class-dependent and appears to persist across activation functions and numbers of atoms.Source: Figure 4 · p. 11, §5.3.1 · p. 11

5.3 LWDL-KP on Transformer Activation Distributions

  • Transformer documents are represented by token-level hidden-state measures embedded as centered Kantorovich potentials and modeled with LWDL-KP.The hidden states come from an intermediate layer of Pythia-410M.Source: §5.3 · p. 11

5.3.2 Document-Level Atoms on The Pile.

  • Learned atoms on The Pile often associate with documents from a single source, while contrastive same-source atoms show distinct semantic themes.This provides evidence that the atoms specialize beyond source identity.Source: §5.3.2 · p. 12

6 Generalization Error Bounds for LWDL

The section establishes generalization bounds for LWDL in terms of the complexity of its candidate atom class. Instantiating the bounds for transport-map classes yields an explicit rate under stated regularity conditions.

6 Generalization Error Bounds for LWDL

  • LWDL’s generalization error is bounded through the metric covering numbers of its candidate atom class in L2(ρ).The paper first establishes a general dictionary-learning complexity result, then instantiates it for transport-map classes.Source: §6 · pp. 12–13
  • The generic result assumes compact unit-norm atoms, a moment-bounded data distribution, and a lower-semicontinuous penalty that grows at least linearly in the code’s L1 norm.The bound is stated for sampled measures and applies for ϵ ≥ 0.Source: §6 · p. 12
  • For d > 1, the transport-map bound is O(m^(1/d)n^(-2/(3d))) under the stated support and density assumptions.The constant factors depend on D, d, β, ϑ, and c.Source: §6 · p. 13

7 Conclusion

The paper presents sparse dictionary learning in Wasserstein space through linearized optimal transport. It reports improved computational efficiency and interpretable atoms and reconstructions, alongside sample-complexity results tied to map-representation complexity.

7 Conclusion

  • The approach combines sparse dictionary learning in Wasserstein space with linearized optimal transport.Source: §7 · p. 13
  • Empirical results show improved computational efficiency over existing Wasserstein dictionary-learning methods and efficacy and interpretability of learned atoms and reconstructions.Source: §7 · p. 13
  • The sample-complexity results depend on the complexity of the map representations considered for the data.Source: §7 · p. 13
  • Future work includes guaranteeing atom identifiability under an LWDL generative model and studying the impact of the base measure, including learning it.Source: §7 · p. 13

A.1 Additional Results for the Wasserstein Space

This appendix develops the Wasserstein and linearized optimal-transport foundations of LWDL and describes its variants, parameterizations, shared encoder, and measure encoding and synthesis. It also presents hyperspectral-image encoding and density-estimation details and reports clustering performance under spectral corruption.

A.1 Additional Results for the Wasserstein Space; A.2 Covering Numbers; A.3 Subgaussian Probability Measures

  • Brenier’s theorem gives a unique optimal transport map up to μ-almost-everywhere equivalence when μ is absolutely continuous, and covering numbers contract under Lipschitz maps; subgaussian measures are defined through a bounded subgaussian norm.The appendix defines Gσ(R^d) as the measures with subgaussian norm at most σ.Source: §A.1 · p. 18, §A.2 · p. 19, §A.3 · p. 19

A.4 Additional Background on LOT; A.5 Wasserstein Barycenters

  • LOT embeds a measure as its optimal displacement map in L2(ρ), making ρ-LOT the distance between embeddings in a reference-dependent tangent space; Wasserstein-2 barycenters synthesize measures while preserving geometry.The choice of reference affects practical LOT applications, and barycenter uniqueness follows under mild conditions such as one input measure being absolutely continuous.Source: §A.4 · p. 19, §A.5 · p. 19

A.6 Synthesizing in the LWDL Framework

  • LWDL-OT and LWDL-EOT recover probability-measure atoms by pushforwards and enable out-of-sample synthesis, whereas LWDL-KP is efficient for encoding but does not readily support synthesis.For LWDL-KP, differentiating scalar potential atoms to obtain measures can introduce numerical issues.Source: §A.6 · p. 20

B LWDL as a Linearization of WDL

  • LWDL-OT, LWDL-EOT, and LWDL-KP arise by replacing Wasserstein barycenters with reference-linearized counterparts or linearizing W2 through Kantorovich potentials.For LWDL-KP, the linearization uses L2(ρ) geometry and identifies the first variation with a Kantorovich potential.Source: B LWDL as a Linearization of WDL · p. 20

C.1 LWDL-OT

  • Exact LWDL-OT parameterizes monotone transport atoms using positive increments, but is restricted to one dimension because higher-dimensional convex-gradient constraints are harder to enforce.The implementation assumes measures supported on a compact interval; normalized hyperspectral pixels are given as an example.Source: §C.1 · p. 21

C.2 LWDL-EOT

  • LWDL-EOT applies in any dimension to finitely supported reference measures and uses entropic displacement fields as atoms; experiments observed better performance than using raw maps.The displacement parameterization treats the identity map as a baseline and the learned atoms as residual displacements.Source: §C.2 · pp. 21, 23

C.3 LWDL-KP

  • LWDL-KP centers input Kantorovich potentials and learned atoms by their ρ-means, then reconstructs each centered potential with a sparse nonnegative combination.Centering addresses the additive-constant ambiguity of Kantorovich potentials.Source: §C.3 · p. 23

C.4 LISTA-Type Encoder

  • The shared LISTA-type encoder iterates learned affine transformations and activations to produce sparse nonnegative codes, without requiring its parameters to correspond to a fixed ISTA update.The architecture is trained end-to-end with the stated objective using AdamW and a cosine learning-rate schedule.Source: §C.4 · pp. 23, 26

E Additional Details: HSI

  • A hyperspectral cube indexes image locations by spatial coordinates and wavelength bands, and Algorithm 6 encodes a hyperspectral pixel with an OT map.A selected pixel yields a one-dimensional spectrum and intensities across individual bands.Source: Figure 5 · p. 27, E Additional Details: HSI · p. 27
  • Algorithm 7 estimates a density by drawing uniform source samples, mapping them through Q, and forming a Gaussian kernel density estimate.Its inputs include the map samples, Monte Carlo sample count, bandwidth, and output grid.Source: E Additional Details: HSI · p. 27
  • Table 6 compares DropUniform performance across data sets and k values.Source: Table 6 · p. 28
  • LWDL-OT is particularly robust to uniform band dropout, while clustering accuracy under contiguous dropout and logarithmic spectral warping varies across data sets and methods.The comparison covers Pavia Centre, Botswana, and Salinas A.Source: Figure 6 · p. 29

F Additional Details: 3D Point-Clouds

The point-cloud experiment applies Algorithm 3 to 3D data using an entropic parameter of 0.025 and a 17 × 17 × 17 support grid. Training yields a final mean squared L2(ρ) reconstruction error of 0.00173.

F Additional Details: 3D Point-Clouds

G Details: Synthetic LBCM

The experiment uses an entropic transport-map dictionary and a LISTA encoder; its reconstructions capture large-scale features but not fine detail, which the authors hypothesize reflects the diffuse base measure.

G Details: Synthetic LBCM

  • Algorithm 3 uses five atoms, entropic parameter ϵ = 0.025, a 20-layer LISTA encoder, centered displacements, and a ReLU encoder with ℓ1 penalty c = 10⁻⁴.Training runs for 1000 epochs; the reported checkpoint is epoch 862 from the selected run with seed 0.Source: G Details: Synthetic LBCM · p. 30
  • The LWDL-EOT atoms are free-support empirical measures formed by pushing the base measure ρ through learned entropic maps on a uniform grid in [0, 1]².Each atom uses centered grid potentials as its learned parameters.Source: G Details: Synthetic LBCM · p. 30
  • Reconstructions capture large-scale features but miss fine details obscured by local jitter, which the authors hypothesize results from the more diffuse base measure ρ.They point to additional evidence for this hypothesis in Section 5.2.Source: G Details: Synthetic LBCM · pp. 30–31

H Details: WDL Comparison

The WDL comparison evaluates reconstruction under a shared time budget, while the text-corruption experiments compare code behavior across corruption strengths. LWDL outperforms WDL on the reported Pavia and MNIST reconstruction comparisons, and highly corrupted data show less class overlap.

H Details: WDL Comparison

  • The comparison uses n = 100 or 1000 data points, ten atoms, and a 1000-second budget; LWDL batch size increases from 100 to 256 at n = 1000.WDL is run with γ = 0.5 and 2, using its released implementation without additional implementation-level optimization.Source: H Details: WDL Comparison · p. 31
  • LWDL permits variable support and reconstructs measures by combining learned map atoms and pushing ρ through the combined map; WDL assumes fixed support.LWDL-KP is excluded from direct comparison because it cannot synthesize a probability measure this way.Source: H Details: WDL Comparison · p. 31
  • On Pavia with n = 100, LWDL-OT reaches mean error 5 × 10⁻⁵ versus WDL’s 4.6 × 10⁻³ (γ = 0.5) and 4.9 × 10⁻³ (γ = 2).At n = 1000, WDL terminates from numerical instability at epoch 7 with errors around 6 × 10⁻³, while LWDL-OT reaches 9.3 × 10⁻⁵.Source: H Details: WDL Comparison · p. 31
  • On MNIST, LWDL-EOT reaches W2 reconstruction errors of 2.6 × 10⁻³ at n = 100 and 2.7 × 10⁻³ at n = 1000, versus WDL errors of 1.2 × 10⁻² and 1.0 × 10⁻², then 1.3 × 10⁻².The two WDL values at each sample size correspond to γ = 0.5 and γ = 2 at n = 100, and both runs at n = 1000.Source: H Details: WDL Comparison · p. 31

I Additional Details: LLM Data

  • Pythia-410M has approximately 410 million parameters, a model dimension of 1024, 24 layers, and was trained on a deduplicated version of The Pile.The passage identifies The Pile with reference [19].Source: I Additional Details: LLM Data · p. 31

I.1 Corrupted Text

  • The experiment runs on a MacBook Pro with an Apple M2 Pro processor and 16 GB of unified memory using PyTorch MPS.The same hardware and backend are specified for the synthetic text-corruption experiments.Source: §I.1 · p. 31
  • The corruption experiment generates 19,200 documents from an 850-word review embedded as a measure with support size 1,130.Corruptions include word swaps, within-word letter permutations, word deletion, and word duplication.Source: §I.1.1 · p. 32

I.1.1 Corruptions

  • Highly corrupted data show significantly less class overlap than data with r ≤ 75, aligning with the reported accuracy jump at higher corruption strength.The comparison projects codes onto the first principal component of the TopK40 atoms; TopK trials use zero ℓ1 regularization because of hard sparsity.Source: §I.1.1 · p. 32, Figure 7 · p. 33

I.2 The Pile

In The Pile, learned atoms often specialize beyond source identity, though some remain redundant; n-gram enrichment also identifies atoms associated with specific document themes.

I.2 The Pile

  • Four features associated with Wikipedia at greater than 99% purity activate on distinct themes: athletes, notable figures, population centers, and disambiguation pages.The disambiguation-page category is cited as.Source: §I.2 · p. 33
  • Some highly source-concentrated atoms remain redundant: f98 and f469 have nearly identical top documents.The authors suggest increasing sparsity or filtering correlated features as possible mitigations.Source: §I.2 · p. 33
  • For StackExchange-associated atoms, x² has enrichment E(x², 407) = 133.82 for f407 and E(x², j) = 0 for f134 and f263.The top documents for f407 are dominated by Math StackExchange posts.Source: §I.2 · p. 33

J Impact of Base Measure

The learned atoms inherit qualitative properties of their base measure. The accompanying theory relates generalization and atom-class covering numbers, with a Lipschitz control result for the technical analysis.

J Impact of Base Measure

  • Atoms learned with a uniform-cube base measure versus a point-cloud base measure inherit qualitative properties of their respective bases.The authors interpret atoms as elements of the L2(ρ) tangent space, which locally approximates the Wasserstein neighborhood of ρ.Source: J Impact of Base Measure · p. 33, Figure 8 · p. 35

K.1 Preliminary Discussion of Theorems 6.1 and 6.2

  • Theorem 6.1 controls LWDL-OT and LWDL-EOT generalization error using data moments, regularization strength, and covering numbers of candidate atoms.The comparison result cited from Vainsencher et al. does not apply to these applications because P2(M2(ϑ)) violates its covering-number assumption.Source: §K.1 · pp. 34–35
  • Theorem 6.2 derives OT-map covering numbers from convex-function results and Brenier’s theorem, but the resulting rates are cursed by dimensionality.For EOT maps, rates are parametric because EOT potentials are strongly regular under subgaussian assumptions on base and target measures.Source: §K.1 · p. 35

K.2 Technical Lemmas

  • The technical lemmas establish that hρ(μ, ϵ, T) is Lipschitz in T, with a constant depending on the second moments of μ and the coercivity of g.The stated result is followed by a corollary giving bounds for hρ(μ, ϵ, T) and its expectation over Θ.Source: §K.2 · p. 39

K.3 Proof of Theorem 6.1

This section establishes generalization bounds for sparse dictionary-learning models by controlling dictionary atom complexity and deriving covering-number bounds for optimal-transport and entropic transport maps. It also provides supporting compactness, approximation, equivalence, and convergence arguments.

K.3 Proof of Theorem 6.1

  • The bound degrades as c approaches zero; when c = 0, the argument cannot control the optimal code’s ℓ1 norm, so another stability mechanism is needed.The authors suggest restricting the code to a compact subset as one possible alternative for non-coercive costs.Source: §K.3 · p. 45
  • Theorem 6.1 bounds generalization error using data moments, regularization strength, and the covering number of candidate dictionary atoms.The proof reduces the error to a uniform deviation bound, applies symmetrization and Dudley’s integral bound, and transfers covering numbers through a Lipschitz map from dictionaries to the loss class.Source: §K.3 · pp. 40, 43, 45

K.4 Covering Number Bounds

  • Uniform regularity yields covering-number bounds for smooth function classes, while the subgaussian-data corollary gives bounds for classes with controlled derivatives.The results rely on concentration of the measure to ensure convergence of the covering-number sum.Source: §K.4 · pp. 46–47

K.4.1 Covering Numbers of OT Maps

  • Convex-function covering results yield covering-number bounds for OT maps, including a corollary for absolutely continuous source measures under stated conditions.The authors note that they could not find these exact OT-map results in the literature.Source: §K.4.1 · pp. 47–48
  • For the potential ϕ_U→ν, the cited argument establishes Lipschitz continuity on Ω with constant |Ω| and, since Ω=Qd, the bound |ϕ_U→ν(x)|≤|Ω|² for every x∈Ω.Source: §K.4.1 · p. 49

K.4.2 Covering Numbers for EOT Maps

  • Entropic transport maps have stronger quantitative regularity than general transport maps, yielding covering-number bounds from derivative estimates.The stated bound depends on dimension, regularization, and Gaussian scale parameters.Source: §K.4.2 · pp. 49–51

K.5.1 Sample Complexity of LWDL-OT

  • For d > 1, the LWDL-OT sample-complexity bound is O(m^(1/d)n^(-2/(3d))) under the stated domain and density conditions.The constant factors depend on D, d, β, and min{c, c2}^(-1).Source: §K.5.1 · p. 52

K.5.2 Sample Complexity of LWDL-EOT

  • The LWDL-EOT generalization bound follows by applying Theorem 6.1 to entropic-map covering numbers and the closure of the atom class.The argument establishes compactness from finite covering numbers; the bound’s constants depend on σ, d, ϵ, ϑ, and min{c, c2}^(-1).Source: §K.5.2 · pp. 52–53

K.6 Proof of Theorem 6.2

  • The proof of Theorem 6.2 obtains its result by combining Corollaries K.25 and K.27.Source: §K.6 · p. 53

K.7 Proof of Lemma K.6

  • The proof of Lemma K.6 handles finite function classes first, then extends the result using compactness and δ-cover approximations.The final approximation step sends δ to zero.Source: §K.7 · pp. 53–55

K.8 Proof of Corollary K.12

  • The Corollary K.12 proof partitions the domain into convex pieces and uses superexponential shell decay to obtain its bound, with constants depending on d and q.Source: §K.8 · pp. 55, 57

K.9 Proof of Corollary K.15

  • Under the stated measure conditions, the optimal transport map classes based at U and ρ are in bijection through a Lipschitz embedding.The embedding is injective, and reversing the argument gives surjectivity.Source: §K.9 · pp. 57–58

K.10 Proof of Lemma K.24

  • For Lemma K.24, convergence in probability together with uniform integrability yields convergence in L2(ρ).The proof establishes uniform integrability for the relevant sequence and applies a convergence result to obtain the L2 conclusion.Source: §K.10 · p. 58

K.11 Proof of Lemma K.23

  • For p > 2, optimizing over δ gives a bound asymptotically dominated by C2 log(n)/(2√n).The proof first sets C4 = 1, noting that the full result can be recovered by scaling.Source: §K.11 · p. 59
  • For p < 2, the proof also derives and optimizes a bound over δ.The supplied passages do not state the resulting bound for this regime.Source: §K.11 · pp. 59–60
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