SPHERE-JEPA: Hyperspherical Self-Supervision

• The paper presents the SPHERE-JEPA framework which replaces Euclidean Gaussian optimality with hyperspherical uniformity to minimize worst-case risk over estimators.
• It integrates self-supervised learning with statistical tests such as MMD, KSD, and KDE-based KL divergence to enforce a uniform embedding geometry on the hypersphere.
• Empirical results show improved k-NN and linear-probe accuracies, faster convergence, and enhanced instance separation compared to traditional SUSReg approaches.

Searching arXiv for SPHERE-JEPA and closely related work to ground the article in the cited literature. SPHERE-JEPA is a self-supervised learning framework for spherical prediction with homogeneous embeddings. It learns unit-norm representations on the hypersphere $\mathbb S^{d-1}$ and regularizes their empirical distribution toward $\mathrm{Unif}(\mathbb S^{d-1})$, motivated by a minimax analysis on Riemannian manifolds in which uniform distributions are optimal for $k$-nearest neighbors and, on the sphere, for kernel ridge regression with both the exponential dot-product kernel and the linear kernel (Nicollier et al., 26 May 2026). A subsequent extension shows that the original sliced regularizer can be analytically integrated into a deterministic hyperspherical Maximum Mean Discrepancy and generalizes the framework to a family of full-dimensional statistical tests on the sphere, including MMD, KSD, and KDE-based KL divergence (Nicollier et al., 16 Jun 2026).

1. Theoretical setting and problem formulation

SPHERE-JEPA arises from the question of what representation geometry should be preferred when the downstream task is unknown. Following the minimax viewpoint adopted for LeJEPA, the representation distribution $p$ is chosen to minimize the worst-case risk of a family of estimators over target functions $g$. On a Riemannian manifold $\mathcal M$, the relevant quantity is the integrated squared bias

$$\mathrm{ISB}(g,p) = \int_{\mathcal M} \bigl(\mathbb E[\widehat g(z)]-g(z)\bigr)^2\, p(z)\,\mathrm{dvol}(z),$$

and the representation design problem is

$$p^\star = \arg\min_p \sup_{g\in\mathcal G} \mathrm{ISB}(g,p).$$

The central claim is that extending the Euclidean minimax analysis to smooth distributions supported on lower-dimensional manifolds changes the optimal solution. In Euclidean space, isotropic Gaussian embeddings had been identified as optimal for linear ridge and $k$-NN. On a manifold, however, curvature and volume measure enter through the Laplace–Beltrami operator $\Delta_{\mathcal M}$ and the Riemannian volume form. SPHERE-JEPA therefore replaces Gaussian optimality with hyperspherical uniformity as the organizing principle for representation design (Nicollier et al., 26 May 2026).

This theoretical shift is not merely a change of prior. The argument presented for SPHERE-JEPA is that non-uniform density on the sphere induces anisotropic $\mathrm{Unif}(\mathbb S^{d-1})$0-NN neighborhoods and therefore biases downstream estimators. A plausible implication is that the representation law itself becomes part of the estimator design problem: the geometry of the learned embedding space is not auxiliary, but structurally tied to downstream prediction risk.

2. Minimax optimality of hyperspherical uniformity

For $\mathrm{Unif}(\mathbb S^{d-1})$1-nearest neighbors on an $\mathrm{Unif}(\mathbb S^{d-1})$2-dimensional manifold, SPHERE-JEPA considers the function class

$\mathrm{Unif}(\mathbb S^{d-1})$3

and uses the leading bias expansion

$\mathrm{Unif}(\mathbb S^{d-1})$4

where $\mathrm{Unif}(\mathbb S^{d-1})$5 is the random $\mathrm{Unif}(\mathbb S^{d-1})$6-NN radius. Integrating and maximizing over $\mathrm{Unif}(\mathbb S^{d-1})$7 yields

$\mathrm{Unif}(\mathbb S^{d-1})$8

Within that formulation, the unique minimax solution is the uniform distribution on $\mathrm{Unif}(\mathbb S^{d-1})$9 (Nicollier et al., 26 May 2026).

For kernel ridge regression, the population covariance operator is

$k$0

and the worst-case integrated bias over

$k$1

reduces to

$k$2

where $k$3 are the eigenvalues of $k$4. Minimizing $k$5 is therefore equivalent to minimizing the top eigenvalue $k$6. For the exponential dot-product kernel $k$7 and the linear kernel $k$8, symmetry arguments and a Jensen-type convexity argument imply that $k$9 is smallest when $p$0 is rotationally invariant with zero radial variance, namely when $p$1 is uniform on $p$2.

Combining the $p$3-NN result with the linear-ridge and exponential-kernel-ridge criteria gives Theorem 4.4 in the SPHERE-JEPA paper: the unique minimax representation distribution is $p$4. A common misconception is that isotropic Gaussian optimality in Euclidean space should transfer unchanged to hyperspherical representations; the stated result is presented precisely as a counterexample to that transfer.

3. Training objective and the original SUSReg mechanism

In multi-view self-supervision, an encoder $p$5 maps each augmented view $p$6 to

$p$7

SPHERE-JEPA combines an invariance term with a hyperspherical uniformity regularizer. In the formulation given for the expanded framework, the loss is

$p$8

where $p$9 is the mean of the global views of sample $g$0 and $g$1 measures the discrepancy between the empirical distribution of normalized embeddings and the uniform law on $g$2 (Nicollier et al., 16 Jun 2026).

The practical form introduced in the original SPHERE-JEPA paper uses an alignment loss

$g$3

together with Sketched Uniform Spherical Regularization, or SUSReg. The regularizer is based on a Cramér–Wold characterization on the sphere: if $g$4, then for every direction $g$5, the scalar projection $g$6 has density

$g$7

Accordingly, one samples a random set of directions $g$8 and penalizes the mismatch between empirical projections and $g$9 through the Epps–Pulley test: $\mathcal M$0 The overall objective is

$\mathcal M$1

The operational recipe is correspondingly lightweight: normalize embeddings to unit length, sample random directions on $\mathcal M$2, project to one dimension, run the Epps–Pulley test toward $\mathcal M$3, and back-propagate. The limitation identified later is that the Monte Carlo approximation introduces projection variance into the training gradients, which appears as noisy gradient estimates, slower convergence, and less stable optimization (Nicollier et al., 26 May 2026).

4. Deterministic hyperspherical regularizers and spectral kernels

The extension "Expanding SPHERE-JEPA: A Family of Statistical Regularizers for the Hypersphere" reframes SUSReg as a sliced proxy for a full-dimensional discrepancy on the sphere. By Bochner’s theorem and Fubini’s theorem, the expected SUSReg statistic over random directions is exactly a Maximum Mean Discrepancy between the empirical embedding distribution $\mathcal M$4 and the uniform law $\mathcal M$5: $\mathcal M$6 Because the kernel is rotationally invariant, the latter two expectations can be precomputed analytically. The deterministic batch loss is

$\mathcal M$7

with constants $\mathcal M$8 and $\mathcal M$9. The paper also notes an equivalent shift and rescaling under which a collapsed batch has loss exactly $$\mathrm{ISB}(g,p) = \int_{\mathcal M} \bigl(\mathbb E[\widehat g(z)]-g(z)\bigr)^2\, p(z)\,\mathrm{dvol}(z),$$0 (Nicollier et al., 16 Jun 2026).

The induced zonal kernel can be written as

$$\mathrm{ISB}(g,p) = \int_{\mathcal M} \bigl(\mathbb E[\widehat g(z)]-g(z)\bigr)^2\, p(z)\,\mathrm{dvol}(z),$$1

In implementation, the integral is approximated with a small Gauss–Jacobi rule and the kernel is evaluated as $$\mathrm{ISB}(g,p) = \int_{\mathcal M} \bigl(\mathbb E[\widehat g(z)]-g(z)\bigr)^2\, p(z)\,\mathrm{dvol}(z),$$2.

The same work constructs rotationally invariant kernels through Schoenberg’s theorem: $$\mathrm{ISB}(g,p) = \int_{\mathcal M} \bigl(\mathbb E[\widehat g(z)]-g(z)\bigr)^2\, p(z)\,\mathrm{dvol}(z),$$3 Two canonical spectral filters are studied. The Heat kernel uses

$$\mathrm{ISB}(g,p) = \int_{\mathcal M} \bigl(\mathbb E[\widehat g(z)]-g(z)\bigr)^2\, p(z)\,\mathrm{dvol}(z),$$4

which gives smooth exponential decay of high-frequency modes. The Bandlimited kernel uses

$$\mathrm{ISB}(g,p) = \int_{\mathcal M} \bigl(\mathbb E[\widehat g(z)]-g(z)\bigr)^2\, p(z)\,\mathrm{dvol}(z),$$5

which imposes a strict low-pass cutoff. Since these kernels depend only on $$\mathrm{ISB}(g,p) = \int_{\mathcal M} \bigl(\mathbb E[\widehat g(z)]-g(z)\bigr)^2\, p(z)\,\mathrm{dvol}(z),$$6, they are fully rotationally invariant and therefore avoid spatial bias. The normalization condition

$$\mathrm{ISB}(g,p) = \int_{\mathcal M} \bigl(\mathbb E[\widehat g(z)]-g(z)\bigr)^2\, p(z)\,\mathrm{dvol}(z),$$7

sets unit scale.

Beyond MMD, the paper formulates two additional full-dimensional objectives on $$\mathrm{ISB}(g,p) = \int_{\mathcal M} \bigl(\mathbb E[\widehat g(z)]-g(z)\bigr)^2\, p(z)\,\mathrm{dvol}(z),$$8. For Kernel Stein Discrepancy, with the uniform density $$\mathrm{ISB}(g,p) = \int_{\mathcal M} \bigl(\mathbb E[\widehat g(z)]-g(z)\bigr)^2\, p(z)\,\mathrm{dvol}(z),$$9, a score-free Stein operator is constructed from divergence-free Killing vector fields. Writing $$p^\star = \arg\min_p \sup_{g\in\mathcal G} \mathrm{ISB}(g,p).$$0 for the Lie-algebra basis of skew-symmetric matrices,

$$p^\star = \arg\min_p \sup_{g\in\mathcal G} \mathrm{ISB}(g,p).$$1

and

$$p^\star = \arg\min_p \sup_{g\in\mathcal G} \mathrm{ISB}(g,p).$$2

For a zonal kernel $$p^\star = \arg\min_p \sup_{g\in\mathcal G} \mathrm{ISB}(g,p).$$3, the Stein kernel has the closed form

$$p^\star = \arg\min_p \sup_{g\in\mathcal G} \mathrm{ISB}(g,p).$$4

With $$p^\star = \arg\min_p \sup_{g\in\mathcal G} \mathrm{ISB}(g,p).$$5, the normalized loss is chosen so that total collapse yields loss $$p^\star = \arg\min_p \sup_{g\in\mathcal G} \mathrm{ISB}(g,p).$$6.

For KL divergence to the uniform law,

$$p^\star = \arg\min_p \sup_{g\in\mathcal G} \mathrm{ISB}(g,p).$$7

the empirical distribution is smoothed by leave-one-out KDE: $$p^\star = \arg\min_p \sup_{g\in\mathcal G} \mathrm{ISB}(g,p).$$8 leading to the estimator

$$p^\star = \arg\min_p \sup_{g\in\mathcal G} \mathrm{ISB}(g,p).$$9

The stated consequence is geometric: logarithmic repulsion through the KDE density encourages very fine-grained instance separation.

5. Empirical behavior across benchmarks

The original SPHERE-JEPA results are reported as consistent with the manifold minimax analysis. On a controlled texture-retrieval task, the framework improves mean average precision by over $k$0 points in the raw embedding space, from $k$1 to $k$2. On ImageNet-1K with ViT-B/14, it matches LeJEPA’s $k$3-NN accuracy at $k$4 while increasing linear-probe accuracy from $k$5 to $k$6. Across transfer tasks including DTD, Aircraft, CIFAR10/100, and Flowers, the linear-probing average rises from $k$7 to $k$8. Ablations on the regularizer weight $k$9 indicate that neither pure invariance ($\Delta_{\mathcal M}$0) nor pure SUSReg ($\Delta_{\mathcal M}$1) matches the combined objective, which is presented as evidence that global geometry and local alignment must be optimized jointly (Nicollier et al., 26 May 2026).

The expanded framework evaluates deterministic regularizers on ImageNet-100 and Galaxy10 with ResNet-18/50, $\Delta_{\mathcal M}$2 epochs, batch size $\Delta_{\mathcal M}$3, and identical augmentations. Replacing SUSReg with the induced MMD kernel removes projection-induced noise, and the reported effect is that deterministic MMD converges in roughly half the epochs of SUSReg and exhibits near-zero variance in the regularization gradient. On ImageNet-100 linear probes, MMD/Heat/KSD outperform SUSReg by up to $\Delta_{\mathcal M}$4 pct, from $\Delta_{\mathcal M}$5 to $\Delta_{\mathcal M}$6, and by $\Delta_{\mathcal M}$7 pct in $\Delta_{\mathcal M}$8-NN. On Galaxy10, the gains reach $\Delta_{\mathcal M}$9 pct linear and $\mathrm{Unif}(\mathbb S^{d-1})$00 pct $\mathrm{Unif}(\mathbb S^{d-1})$01-NN. Among kernels, Bandlimited with low-pass $\mathrm{Unif}(\mathbb S^{d-1})$02 is described as marginally best for classification (Nicollier et al., 16 Jun 2026).

On the non-clustered procedural texture retrieval benchmark Cloud, Disk, Flake, and Wood, the relative behavior is different. Recall@1 increases from $\mathrm{Unif}(\mathbb S^{d-1})$03 pct with SUSReg to $\mathrm{Unif}(\mathbb S^{d-1})$04 pct with KL-Heat, a $\mathrm{Unif}(\mathbb S^{d-1})$05 pct absolute gain. MMD and KSD achieve about $\mathrm{Unif}(\mathbb S^{d-1})$06 to $\mathrm{Unif}(\mathbb S^{d-1})$07 pct gains over SUSReg, but remain below KL. The explicit interpretation given in the paper is that the statistical test shapes the learned geometry: MMD and KSD favor locally clustered organization suitable for object-centric domains, whereas the continuous KDE-based KL divergence promotes fine-grained instance separation and performs best for unclustered procedural texture retrieval.

6. Interpretation, misconceptions, and nomenclature

The principal conceptual contribution of SPHERE-JEPA is the replacement of Euclidean Gaussian optimality with hyperspherical uniformity as the target geometry for normalized representations. The framework is therefore not simply a spherical variant of an existing JEPA objective. It is built around a distinct minimax claim: for the specified downstream estimators and function classes, uniformity on the manifold, and in particular on $\mathrm{Unif}(\mathbb S^{d-1})$08, is the unique minimax solution.

A second misconception is to regard SUSReg and the later full-dimensional regularizers as aiming at different distributional targets. The expansion paper argues instead that SUSReg is a Monte Carlo one-dimensional proxy to a closed-form hyperspherical MMD. This suggests that the main distinction is not the target law, which remains the uniform distribution on the sphere, but the variance properties and geometric biases of the estimator used to enforce that law.

A third point concerns latent-space organization. Uniformity does not imply a single universal arrangement of semantic structure. The empirical results explicitly separate object-centric domains from instance-level retrieval domains: MMD and KSD, both integral metrics, tolerate and even encourage intra-class clustering, whereas KL, through KDE-based entropy maximization, enforces stronger per-instance repulsion. A plausible implication is that SPHERE-JEPA is better viewed as a family of hyperspherical regularization principles than as a single fixed geometry.

The name should also not be conflated with "SPHERES," the neutron backscattering spectrometer at FRM II. That instrument is a third-generation neutron backscattering spectrometer operated by the Jülich Centre for Neutron Science and is unrelated to JEPA-based self-supervised learning (Wuttke et al., 2012).