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LeJEPA: Provable and Scalable Self-Supervised Learning Without the Heuristics

Randall Balestriero, Yann LeCun

arXiv:2511.08544v3cs.LGcs.AIcs.CVstat.ML

TL;DR

JEPA training lacks principled guidance and often relies on brittle anti-collapse heuristics. LeJEPA combines a JEPA predictive loss with SIGReg, and matches state-of-the-art performance across domains and over 60 architectures while requiring fewer than 50 lines of core implementation.

  • Problem

    JEPA training lacks principled guidance and relies on brittle heuristics to prevent representation collapse.

    Source: §1 · p. 2, §2.2 · p. 4

  • Method

    LeJEPA combines JEPA prediction with SIGReg, which targets isotropic Gaussian embeddings to eliminate collapse by construction.

    Source: §1 · p. 2

  • Results

    LeJEPA matches state-of-the-art performance across domains and over 60 architectures while requiring fewer than 50 lines of core implementation.

    Source: §7 · p. 18

  • Takeaways & Limitations

    LeJEPA provides a mathematically rigorous foundation that directly informs practical JEPA algorithm design.

    Source: §7 · p. 18

    CDF-based distribution tests are unsuitable for gradient optimization and disrupt distributed scalability because they require sorting and synchronization.

    Source: §4.2.2 · p. 9

Abstract

from arXiv · show

Learning manipulable representations of the world and its dynamics is central to AI. Joint-Embedding Predictive Architectures (JEPAs) offer a promising blueprint, but lack of practical guidance and theory has led to ad-hoc R&D. We present a comprehensive theory of JEPAs and instantiate it in {\bf LeJEPA}, a lean, scalable, and theoretically grounded training objective. First, we identify the isotropic Gaussian as the optimal distribution that JEPAs' embeddings should follow to minimize downstream prediction risk. Second, we introduce a novel objective--{\bf Sketched Isotropic Gaussian Regularization} (SIGReg)--to constrain embeddings to reach that ideal distribution. Combining the JEPA predictive loss with SIGReg yields LeJEPA with numerous theoretical and practical benefits: (i) single trade-off hyperparameter, (ii) linear time and memory complexity, (iii) stability across hyper-parameters, architectures (ResNets, ViTs, ConvNets) and domains, (iv) heuristics-free, e.g., no stop-gradient, no teacher-student, no hyper-parameter schedulers, and (v) distributed training-friendly implementation requiring only $\approx$50 lines of code. Our empirical validation covers 10+ datasets, 60+ architectures, all with varying scales and domains. As an example, using imagenet-1k for pretraining and linear evaluation with frozen backbone, LeJEPA reaches 79\% with a ViT-H/14. We hope that the simplicity and theory-friendly ecosystem offered by LeJEPA will reestablish self-supervised pre-training as a core pillar of AI research (\href{https://github.com/rbalestr-lab/lejepa}{GitHub repo}).

1 Introduction

The paper grounds JEPAs in two axioms—predictive agreement and isotropic Gaussian embeddings—to replace brittle anti-collapse heuristics with a principled design. It introduces SIGReg and combines it with the JEPA objective in LeJEPA, a scalable method validated across architectures, datasets, and model scales.

Motivation

  • JEPAs can collapse or become dimensionality-deficient, making current training brittle because it relies on stop-gradients, asymmetric views, teacher-student networks, normalization, whitening, and delicate hyperparameter tuning.These failure modes have shifted attention toward scaling data and models while leaving JEPA theory comparatively unexplored.Source: §1 · p. 2

Theory

  • The paper identifies predictive-task solvability and isotropic Gaussian embeddings as the minimal axioms for JEPA design.It establishes that the isotropic Gaussian uniquely minimizes downstream prediction risk across broad task families, for both linear and nonlinear probes.Source: §1 · p. 2

LeJEPA

  • LeJEPA combines the JEPA predictive objective with SIGReg to eliminate collapse without stop-gradients or teacher-student architectures, using one trade-off hyperparameter across datasets, architectures, and scales.The resulting method also provides provable statistical guarantees and linear memory and computational complexity.Source: §1 · pp. 2–3

SIGReg

  • SIGReg matches embedding distributions to an isotropic Gaussian through random projections and characteristic-function tests, combining statistical guarantees with linear complexity and bounded gradients.Its projection-based construction is designed to avoid the curse of dimensionality while enforcing the target distribution.Source: §1 · p. 2, Figure 2 · p. 3

Validation

  • LeJEPA is validated across ViTs, ConvNeXts, ResNets, MaxViTs, and Swin Transformers at scales approaching 1 billion parameters, matching or exceeding state-of-the-art methods.On Galaxy10 and Food101, direct in-domain pretraining outperforms DINOv2-based transfer learning.Source: §1 · p. 3

2 Background and Notations

This section defines the data and network notation, characterizes JEPAs by prediction and anti-collapse requirements, and motivates a principled alternative to heuristic pretraining safeguards. It concludes by positioning the paper’s later sections as a first-principles derivation of a scalable JEPA solution.

2.1 Notations and Definitions

  • The framework uses independently and identically distributed datasets shaped as (N, V, D), without requiring a particular organization of views or tensor design.N denotes samples, V views, and D feature dimensions; views may represent augmentations or temporally ordered video frames.Source: §2.1 · p. 3
  • JEPAs predict one view’s embedding from another while preventing degenerate representations, enabling downstream tasks in zero or few shots.Prediction may directly compare embeddings or use a predictor when views are asymmetric, such as when conditioning on observed robotics actions [Khazatsky et al., 2024].Source: §2.1 · pp. 3–4

2.2 The Need for Reliable Pretraining

2.3 The Need for Actionable Theory

  • Unlike supervised and reconstruction learning, JEPA research lacks mature actionable theory, encouraging post-hoc explanations, scaling existing recipes, and data curation over principled innovation.The paper therefore aims to derive a JEPA from proved necessary optimality conditions while reconciling exploratory research, scalability, and state-of-the-art performance.Source: §2.3 · pp. 4–5

3 Latent Euclidean: Embeddings Should be Isotropic Gaussian

Section 3 proves that isotropic Gaussian embeddings uniquely minimize downstream prediction risk across both linear and nonlinear probes. The theory and empirical evidence establish isotropic Gaussian embeddings as LeJEPA’s design principle for minimizing worst-case downstream risk.

3 Latent Euclidean: Embeddings Should be Isotropic Gaussian

  • Isotropic Gaussian embeddings are the unique optimal distribution for minimizing downstream prediction risk under both linear and nonlinear probing.This establishes the design principle later implemented by LeJEPA.Source: §3 · p. 5, §3.2 · p. 6

3.1 Linear Probing

  • Anisotropy increases linear-probe bias under positive Tikhonov regularization and increases total estimator variance under ordinary least squares.For λ > 0, an anisotropic embedding can yield higher bias for some downstream task; with λ = 0, its estimator variance exceeds the isotropic case.Source: §3.1 · p. 5

3.3 Geometric and Practical Insights

  • Empirical evaluations show anisotropic embeddings degrade parameter alignment and produce higher estimator variance than isotropic embeddings across training samples.Figure 3 illustrates greater variability for anisotropic embeddings, while the reported bias degradation persists across sample sizes and regularization strengths.Source: Figure 3 · p. 6, §3.3 · p. 6

3.2 Nonlinear Probing

  • Under fixed total variance or scalar covariance constraints, isotropic Gaussian embeddings uniquely minimize nonlinear-probe integrated square bias.The analysis covers radius-based k-NN and kernel methods, subject to additional regularity conditions.Source: §3.2 · pp. 5–6

4 SIGReg: Reliable Isotropic Gaussian Regularization in High-Dimension

SIGReg turns isotropic-Gaussian matching into a statistically grounded, directional testing objective that is differentiable, scalable, provable, and interpretable. Its Epps–Pulley implementation provides stable linear-complexity regularization, while projection smoothness and resampling overcome high-dimensionality concerns.

4.1 Hypothesis Testing as a Judge

  • The method tests the null P_θ = Q using scalar statistics and critical values, with directional tests sufficient for the original multivariate hypothesis.The sufficiency result relies on the union-intersection principle and a modified Cramér-Wold theorem.Source: §4.1 · pp. 6–7

4.2 SIGReg: Sketching the Epps-Pulley Test is Stable and Scalable

  • SIGReg averages tests over sampled directions, recommending Epps–Pulley to avoid sparse gradients and support differentiable, distributed computation.The Epps–Pulley implementation has O(N) time and memory complexity and uses DDP-friendly all_reduce operations.Source: §4.2 · p. 7, §4.2.3 · pp. 9–10

4.2.1 Moments are Unstable and Insufficient

  • Moment tests cannot simultaneously provide identifiability and stable training: finite moments permit shortcuts, while higher-order moments produce exploding gradient norms and variance.The gradient norm scales as O(k), and Monte Carlo gradient variance grows as O(k^2m^2(k−1)) for the kth moment.Source: §4.2.1 · p. 8

4.2.2 Cumulative Density Functions are Impractical

  • CDF-based tests are unsuitable for LeJEPA because sorting costs O(Nlog(N)), disrupts multi-GPU parallelism, and introduces nondifferentiable operations or extra hyperparameters.Kolmogorov-Smirnov yields sparse gradients, while Shapiro-Wilk was unstable in practice.Source: §4.2.2 · p. 9

4.2.3 Characteristic Functions are Stable, Scalable and Identifiable

  • Epps–Pulley is preferred because characteristic functions are differentiable, distributed-friendly, and bounded, yielding stable gradients and curvature regardless of the input distribution.The resulting implementation is hyper-parameter free and has O(N) computational and memory complexity.Source: §4.2.3 · pp. 9–10

4.3 How SIGReg Beats the Curse of Dimensionality

Empirical Validation on Synthetic Data

  • In synthetic validation, SIGReg recovered two adversarial dimensions even in D = 512 with M = 16 randomly resampled directions.The experiment directly optimized embeddings and showed that SIGReg unfolded the degenerate dimensions toward the isotropic-Gaussian target.Source: Empirical Validation on Synthetic Data · p. 11

5 LeJEPA: Stable and Scalable Implementation

LeJEPA combines SIGReg with a multi-view prediction loss in a compact implementation using characteristic-function testing, while relating its slicing approach to prior distribution-matching methods and recovering VICReg under a degenerate test. The implementation uses trapezoidal quadrature, has minibatch bias of order O(1/n), and requires only one trade-off hyperparameter without prototypes, stop-gradients, or teacher-student networks.

5.1 LeJEPA: SIGReg + Prediction Loss

  • LeJEPA combines SIGReg and multi-view prediction loss in a few dozen PyTorch lines, using one trade-off hyperparameter and no prototypes, stop-gradients, or teacher-student networks.SIGReg uses the bounded, scalable Epps–Pulley test with trapezoidal quadrature; minibatch loss and gradient bias is O(1/n), reported as minimal even for batches of 16.Source: §5.1 · pp. 11–12

5.2 Relation to Prior Work

  • SIGReg’s slicing and distribution matching connect LeJEPA to Sliced Score Matching [Song et al., 2020], sliced Wasserstein distance [Bonneel et al., 2015, Nguyen and Ho, 2023], and kernel MMD [Sriperumbudur et al., 2010, Gretton et al., 2012, Chwialkowski et al., 2016].Exact integration recovers kernel MMD per slice, but with quadratic complexity.Source: §5.2 · p. 12
  • Using a particular statistical test, LeJEPA recovers VICReg in the large-slice limit, enforcing E[Z] = 0 and Cov(Z) = I_d but risking shortcut solutions.The paper advises against this degenerate setting based on theorem 3 and shortcut behavior observed in VICReg.Source: §5.2 · p. 12

6 LeJEPA: Empirical Validation

LeJEPA is validated across diverse datasets, architectures, hyperparameters, scales, and tasks, consistently producing stable, informative, and semantically rich representations. The experiments show strong in-domain transfer, scalability, and label-free model selection without collapse-prevention heuristics.

6.1 LeJEPA’s Stability Across Hyper-Parameters and Architectures

  • Across approximately 50 architectures from 8 families, LeJEPA achieves 91.5%–95% top-1 accuracy with frozen-backbone linear probing on ImageNet-10.Performance remains stable across standard hyperparameters, including views and loss weighting, while competitive ImageNet-1K results are obtained with batch sizes as small as 128.Source: §6.1 · pp. 13–14, Figure 9 · p. 14
  • SIGReg lets LeJEPA remove predictors and teacher-student architectures without collapse, while register tokens are unnecessary for stable performance.A teacher-student configuration can provide a small ViT performance boost, and SWA may optionally be applied with ViTs.Source: §6.1 · p. 14

6.2 LeJEPA’s Training Loss is Informative of Downstream Performance

  • LeJEPA’s training loss correlates about 85% with downstream accuracy across datasets and architectures, rising to nearly 99% with α≈0.4 for label-free model selection.Lower LeJEPA loss reliably indicates better downstream performance, enabling cross-validation without labeled data.Source: §6.2 · p. 15, Figure 11 · p. 15

6.3 In-Domain LeJEPA Outperforms Frontier Model Transfer Learning

  • LeJEPA substantially outperforms DINOv2 and DINOv3 on Galaxy10 with both frozen-backbone linear probing and full finetuning, using default hyperparameters.The in-domain setup uses 11,000 samples across 10 galaxy types and 400 pretraining epochs, with additional results reported on datasets including flowers102.Source: Figure 12 · p. 16, §6.3 · p. 17

6.4 LeJEPA Scales Across Data and Models

  • LeJEPA reaches 77.1% online ImageNet-1K linear-probe accuracy with ViT-Large and 78.5% with ConvNeXtV2-Huge, while consistently outperforming I-JEPA using smaller models and shorter schedules.The training curve is smooth and stable, indicating optimization that reduces the need for careful hyperparameter selection; scaling is also demonstrated with a 1.8B-parameter ViT-gigantic.Source: §6.4 · p. 17

6.5 Emergent Semantic Structure in LeJEPA Representations

  • LeJEPA’s learned attention maps reveal object boundaries and support unsupervised video segmentation with temporally coherent foreground masks.Thresholding [CLS]-token self-attention produces binary masks without segmentation labels, indicating spatial semantics and temporal structure in the representations.Source: Figure 13 · p. 16, Figure 14 · p. 17, §6.5 · p. 18

7 Conclusion

The paper establishes a principled JEPA framework by proving isotropic Gaussian embeddings minimize worst-case downstream risk and introducing SIGReg to enforce this distribution without ad-hoc anti-collapse mechanisms. LeJEPA validates this theory across domains and more than 60 architectures, matching state-of-the-art performance with fewer than 50 lines of core implementation.

Theoretical contributions

  • LeJEPA proves that isotropic Gaussian embeddings uniquely minimize worst-case downstream risk and uses SIGReg to enforce this distribution while preventing representational collapse by design.The approach avoids teacher-student networks, stop-gradients, and asymmetric architectures.Source: §7 · p. 18

Empirical validation

  • LeJEPA matches state-of-the-art performance across domains and over 60 architectures, including models with 1.8B parameters, while requiring fewer than 50 lines of core implementation.The results support a mathematically rigorous foundation that directly informs practical self-supervised learning design.Source: §7 · p. 18

LeJEPA … B Proofs

The appendix establishes that isotropic Gaussian embeddings uniquely optimize downstream nonlinear probing under stated regularity and covariance constraints. It proves this for both radial k-NN and kernel regression, covering bias and variance behavior.

A.1 kNN Probing

  • Radial k-NN probing uses sample-dependent neighborhood sizes, improving evaluation for non-uniform sample distributions in regression and classification.Classification is handled through the conditional probability target function under smoothness and boundedness assumptions.Source: §A.1 · p. 24
  • The k-NN analysis derives pointwise bias and integrated-bias results under isotropic-gradient and finite scalar covariance constraints.These conditions support the theorem identifying the unique isotropic Gaussian minimizer.Source: §A.1 · p. 24
  • Isotropic Gaussian embeddings uniquely minimize integrated squared bias for radial k-NN probing, yielding the optimal embedding density under finite covariance constraints.The result assumes regular embedding density and target-function conditions, with query points distributed according to the training density.Source: §A.1 · p. 25

A.2 Kernel Probing

  • Kernel regression is analyzed through the Nadaraya-Watson estimator, whose performance depends on the training-point distribution and requires additional light-tail assumptions.The derivation follows the k-NN regularity conditions and introduces conditional target variance.Source: §A.2 · p. 25
  • Under fixed mean and total-covariance constraints, isotropic Gaussian embeddings uniquely minimize kernel-regression bias and variance at every test point.The result is established for a restricted smoothness class of target functions and formalized through the kernel bias and variance theorem.Source: §A.2 · pp. 25–26
  • For the Nadaraya-Watson estimator, integrated variance is independent of the embedding density, while isotropic Gaussian density uniquely minimizes integrated squared bias under total-variance constraints.The proof applies among densities on R^d with constraints such as Tr(Σ) = c.Source: §A.2 · p. 26

B.1 Proof of lemma. 1 · B.2 Proof of lemma. 2 · B.3 Proof of lemma. 4

These appendices prove that anisotropic covariances can amplify ridge-regression bias, derive the corresponding unbiased estimator variance, and establish a ball-average surrogate through controlled local expansions.

B.1 Proof of lemma. 1

  • The argument begins with the closed-form ridge estimator and centered-label model Y = Xβtrue + ε, where E[ε] = 0.Bias is analyzed under Tikhonov-regularized least squares before comparing covariance structures.Source: §B.1 · p. 26
  • Because the smallest anisotropic eigenvalue satisfies λp < λ̄, strict inequality holds whenever the covariance is not isotropic.This eigenvalue comparison supplies the key inequality behind the increased-bias conclusion.Source: §B.1 · p. 26
  • Anisotropic covariance increases downstream estimator bias for tasks whose true parameter aligns unfavorably with low-variance directions.The proof compares isotropic and anisotropic covariances with equal total variance and uses the smallest-eigenvalue direction.Source: §B.1 · p. 26

B.2 Proof of lemma. 2

  • With weight decay set to zero, the estimator is unbiased, after which the proof computes its covariance matrix and total variance.The derivation uses the same estimator formula as B.1 and an eigendecomposition.Source: §B.2 · p. 27

B.3 Proof of lemma. 4

  • Under PPP, conditional expectations of bη(x) equal the normalized average over a ball, providing the surrogate used in subsequent calculations.The proof also invokes symmetry-based ball-integral identities.Source: §B.3 · p. 27
  • Assuming p ∈ C3 and η ∈ C2 with bounded derivatives, the proof applies second-order Taylor expansions with uniformly cubic remainders.The remainder bounds hold for local displacements within radius r0.Source: §B.3 · p. 27
  • Symmetry eliminates cubic terms, while quartic contributions are O(r0^(d+4)); the resulting denominator expansion contains a curvature term involving tr(Hp(x)).The expansion is obtained after simplifying the denominator and subtracting η(x)D(x) from the bias numerator.Source: §B.3 · p. 28

B.4 Proof of thm. 7 … B.7 Proof of thm. 8

These appendices establish that isotropic Gaussians uniquely minimize Fisher information under several covariance constraints, while supplying technical derivations for the bias, variance, and theorem bounds. The proofs reduce the results to Cramér–Rao, eigenvalue optimization, Taylor expansion, algebraic identities, and norm inequalities.

B.4 Proof of thm. 7

  • Theorem 7’s bias analysis isolates the score-gradient contribution as the leading p-dependent term, with cross terms vanishing under orthogonality or otherwise contributing only a finite constant.The score-gradient term recovers the Fisher-information functional scaled by τ^2, and the leading term remains τ^2 g J(p) without the orthogonality assumption.Source: §B.4 · p. 29
  • For fixed covariance Σ, Fisher information satisfies J(p) ≥ tr(Σ^-1), with equality if and only if p is the Gaussian N(0,Σ).The proof applies the matrix Cramér–Rao bound to the location family and characterizes equality through an affine score.Source: §B.4 · p. 30

Step 2: Optimizing over covariance shapes under scalar constraints

  • Under trace, determinant, Frobenius-norm, or spectral-radius constraints, minimizing tr(Σ^-1) forces all covariance eigenvalues equal, yielding Σ = sI_d.For example, the determinant constraint gives tr(Σ^-1) = dδ^-1/d at Σ = δ^1/d I_d, while the spectral-radius bound is attained at Σ = rI_d.Source: Step 2: Optimizing over covariance… · pp. 30–31

Conclusion: Isotropic Gaussian is optimal

  • Consequently, the isotropic Gaussian N(0,sI_d) uniquely minimizes J(p), up to null sets, for each stated scalar covariance constraint.The scale s is chosen to satisfy the selected constraint, and uniqueness follows from equality conditions in the Fisher-information bound.Source: Conclusion: Isotropic Gaussian is… · p. 32

B.5 Proof of lemma. 5

  • The proof of Lemma 5 computes kernel-weighted denominator and numerator expectations using isotropy, the product rule, and independence, then linearizes the ratio to obtain leading-order variance.The calculations recover the stated bias and variance expressions for the estimator.Source: §B.5 · p. 33

B.6 Proof of Equation (5) to Equation (6)

  • The proof of Equation (5) to Equation (6) verifies the identity by separately expanding both sides and matching the resulting terms.The appendix completes the algebraic equivalence through successive expansions.Source: §B.6 · p. 34

B.7 Proof of thm. 8

  • Theorem 8’s proof bounds integrated squared bias and variance by combining pointwise quadratic inequalities, the uniform bound |Δm(x)| ≤ B, Cauchy–Schwarz, and integration against p(x).The resulting bounds are combined to obtain the desired theorem statement for both integrated bias and variance.Source: §B.7 · p. 35

B.8 Proof of lemma. 3 · B.9 Proof of thm. 2 · B.10 Proof of thm. 5

The appendices prove the projection-based characterization, consistency, and spherical interpolation bounds underlying the paper’s theoretical results. These proofs rely on characteristic functions, dense directional coverage, and Sobolev-harmonic approximation on spheres.

B.8 Proof of lemma. 3

  • B.8 proves that matching all unit-direction projections implies equality of multivariate laws via characteristic-function uniqueness.Writing any t as s u transfers equality of one-dimensional projections to characteristic functions throughout R^d.Source: §B.8 · p. 35
  • B.8 also establishes convergence of every fixed linear projection by decomposing t into its norm and unit direction.The continuous mapping theorem yields convergence of the projected random variables and their characteristic functions.Source: §B.8 · pp. 35–36

B.9 Proof of thm. 2

  • B.9 requires population-level equivalence of laws, directional-set mesh Δ(A_n) → 0, and a neighborhood around a separating direction.Near that direction, the one-dimensional statistic exceeds the global null threshold with probability tending to one.Source: §B.9 · p. 36
  • B.9 proves level control under P = Q and consistency under P ≠ Q for tests scanning increasingly dense directional sets.Calibration gives lim sup Pr(Ψ_n = 1) ≤ α under the null, while a separating direction eventually appears in the scanned set.Source: §B.9 · p. 36

B.10 Proof of thm. 5

  • B.10 reduces approximation of characteristic functions, CDFs, and moments to Hölder-regular functions on the sphere.For fixed t or k, the quantity of interest belongs to H_α(S^{D−1}), enabling spherical interpolation analysis.Source: §B.10 · p. 36
  • B.10 uses spherical harmonics and Marcinkiewicz–Zygmund inequalities to bound interpolation error from M quasi-uniform points.For degree L proportional to M^(1/d), the sampled points reconstruct spherical harmonics and yield the stated L2 error bound.Source: §B.10 · pp. 36–37
  • B.10 obtains the theorem’s final bounds by integrating or summing pointwise interpolation errors, with weighted moment sums ensuring convergence.The weights may be chosen as w_k = 1/k!, while the explicit constant C(D, α) comes from spherical Sobolev theory.Source: §B.10 · p. 37

B.11 Proof of thm. 3 … F Multivariate Statistics

The merged sections establish theoretical properties of characteristic-function and moment-based regularization, recover zero-mean identity-covariance embeddings, and provide supporting statistical constructions, proofs, and implementation details. They also connect these tools to classical normality tests, regression, and low-discrepancy sampling.

B.11 Proof of thm. 3

  • Moment-based sample gradients are unbounded for k≥2, whereas empirical-characteristic-function gradients are uniformly bounded and Lipschitz under the stated assumptions.The moment gradient is polynomial in X_i with degree at most k−1 and grows like |X_i|^(k−1) when the top-weighted deviation is nonzero.Source: §B.11 · p. 38

B.13 Proof of thm. 6

  • Deterministic quadrature preserves sampling unbiasedness while adding only deterministic approximation error to the target characteristic-functional statistic.The construction uses a uniform grid and Gaussian window, with trapezoidal quadrature as an example.Source: §B.13 · pp. 39–40

B.14 Proof of VICReg’s Recovery

  • VICReg’s directional constraints imply E[X] = 0 and Cov(X) = I_d.The proof establishes unit diagonal covariance entries and zero off-diagonal entries.Source: §B.14 · pp. 40–41

C Background

  • The background introduces Gaussian-error linear regression and derives the ordinary least-squares estimator by minimizing squared residuals, assuming X^T X is invertible.The error vector has distribution N(0, σ^2I_n), with zero means, variance σ^2, and zero cross-covariances.Source: C Background · p. 41

D Details on Low-Discrepancy Sequences

  • Quasi-Monte Carlo points are transformed through the inverse standard-normal CDF and normalized, yielding empirically more uniform directions on the hypersphere.The method leverages rotational invariance of the multivariate normal distribution, although the nonlinear mapping does not strictly preserve low discrepancy.Source: D Details on Low-Discrepancy Sequences · pp. 42–43

E Shapiro-Wilk Test

  • The Shapiro-Wilk background defines normal order-statistic moments and covariance, then notes asymptotic and approximation-based variants including Shapiro-Francia and Rahman-Govindarajulu formulations.The cited variants replace or approximate the covariance and expected order statistics using established approximations.Source: E Shapiro-Wilk Test · p. 44

F Multivariate Statistics

  • Multivariate statistics compare empirical characteristic functions with target distributions and relate resulting tests to HZ, moment-generating-function, combined, derivative, and kurtosis-based criteria.For Gaussian data, the derivative statistic should be 0 and kurtosis should be d(d+2).Source: F Multivariate Statistics · pp. 44–46

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