Sampling-Free Uniformity

• Sampling-Free Uniformity is defined by deterministic objectives, such as geometric penalties and moment-matching, to enforce uniformity without negative sampling.
• It counteracts issues like mode collapse and sampling bias in contrastive learning and point selection by using global and structured loss formulations.
• Applications span graph representation, metric sampling, and robotic control, showing empirically higher efficiency and improved performance.

Sampling-free uniformity is a paradigm and set of methodologies for enforcing uniformity or diversity constraints in representation learning, point selection, and control systems without requiring negative sampling or explicit pairwise sampling strategies. In contrast to classical techniques that rely on stochastic exclusion or statistical comparison over randomly selected pairs or subsets, sampling-free uniformity leverages deterministic or global objectives—typically based on geometric, moment-matching, or coverage principles—to guarantee dispersion, uniform coverage, or entropy maximization directly. This approach spans graph representation learning, metric space sampling theory, robotic control, and contrastive learning, and addresses critical drawbacks of sampling-based methods such as sampling bias, computational inefficiency, mode collapse, and variance in uniformity measures.

1. Principles of Sampling-Free Uniformity

Sampling-free uniformity is defined by the imposition of explicit objectives, regularizers, or probabilistic constraints that enforce a uniform distribution or maximal dispersion in a given space, without reliance on random or adversarial negative sampling. Its core mechanisms are:

• Global or aggregate geometric penalties: Global moment-based losses (e.g., penalizing the squared empirical mean of normalized embeddings on a hypersphere), pairwise exponential repulsion, and gap ratio minimization over a point set (Chen et al., 30 Dec 2025, Bishnu et al., 2014).
• Deterministic surrogates for negatives: Use of structured “prototypes” or centroids as anchors for contrast rather than sampled instances, eliminating sampling-induced bias (Ou et al., 2024).
• Exact uniformity in distributional marginals: In robotic planning, constructing policies whose induced trajectory or state distributions match uniformity criteria at each step, measured by divergence from the uniform law on free configuration sets (Cao et al., 19 Oct 2025).
• Metric-based coverage guarantees: Uniformity measures such as gap ratio avoid volumetric or range-based discrepancy, seeking instead to optimize the worst-case covering vs. packing radii—a fundamentally sampling-free principle (Bishnu et al., 2014).

Sampling-free uniformity approaches are technically distinguished by being (a) operationally “sampling-free” in implementation, and (b) theoretically sampling-independent in their definition or optimization targets.

2. Sampling-Free Uniformity in Representation Learning

In high-dimensional representation learning, especially for graphs and contrastive frameworks, sampling-free uniformity objectives have been developed to counter instability and collapse associated with negative-sampling-based losses such as InfoNCE. The predominant formulations include:

• Hyperspherical mean regularization: HyperGRL penalizes the squared empirical mean of $N$ $\ell_2$-normalized embeddings $$\{\mathbf{z}_i\in\mathbb{R}^d: \|\mathbf{z}_i\|_2=1\}$$ by

$$\mathcal{L}_{\mathrm{unif}} = \Bigl\|\frac{1}{N}\sum_{i=1}^N \mathbf{z}_i\Bigr\|_2^2$$

This term is minimized uniquely when the $\mathbf{z}_i$ are uniformly distributed on the hypersphere, enforcing maximal dispersion and preventing representational collapse (Chen et al., 30 Dec 2025).

• Interpretation via MMD and pairwise distances: The uniformity loss is equivalent to matching the first moment of the empirical embedding distribution to that of the uniform law ($$\mathbb{E}_{\mathbf{z}\sim\text{Unif}(\mathbb{S}^{d-1})}[\mathbf{z}]=0$$), and algebraically, maximizing the mean squared pairwise distances among all embeddings.
• Contrast with sampling-based losses: Traditional methods (e.g., InfoNCE) scale as $O(N_{\text{pos}}\times N_{\text{neg}})$, depend on negative sampling, and require sensitive hyperparameters (e.g., temperature). Sampling-free uniformity attains global coverage in a parameter-free fashion, with provable avoidance of collapse and maximal entropy configuration on the sphere (Chen et al., 30 Dec 2025).
• Coupling with alignment objectives: In adversarial regularization (e.g., HyperGRL), an entropy-guided schedule dynamically balances local neighbor alignment against global uniformity, quantitatively regulating collapse and dispersion based on information-theoretic proxy metrics.

3. Uniformity without Sampling in Metric Spaces

In metric geometry, the notion of sampling-free uniformity is formalized via deterministic coverage measures, principally the gap ratio, which requires no random selection or sampling of candidate points:

• Gap ratio definition: For a finite point set $P\subset\mathcal{S}$ in metric space $(\mathcal{S},d)$, the gap ratio is

$\phi(P) = \frac{R_P}{r_P}$

where $r_P$ is the packing radius (half the minimum pairwise distance) and $R_P$ the covering radius (worst-case distance from any $x\in\mathcal{S}$ to $P$). Uniformity corresponds to achieving $\phi(P)\to 1$ (Bishnu et al., 2014).

• Relationship to discrepancy: Unlike discrepancy, which depends on counting over range spaces and volumes, gap ratio is purely metric, making it a “sampling-free” uniformity criterion.
• Algorithmic consequence: Farthest-point insertion yields a $2$-approximation for minimum gap ratio in any metric space, with explicit coreset constructions allowing $(1+\varepsilon)$-approximation for large-scale data (static and streaming settings). No random sampling of pairs or points is necessary.
• Complexity and bounds: Achievable uniformity is governed by the geometry of $\mathcal{S}$, with lower bounds on $\phi(P)$ established for various classes (path-connected spaces, $[0,1]^2$, graphs), and NP-hardness results for computing or approximating the optimal uniform sample (Bishnu et al., 2014).

4. Sampling-Free Uniformity in Contrastive Learning Frameworks

Sampling-free mechanisms have been adapted to contrastive learning settings with the aim of mitigating bias and collapse inherent to negative sampling:

• Prototype-based contrast: ProtoAU replaces sampled negatives with a fixed bank of $K$ learnable prototypes (${C^U},{C^I}$ for users/items). Each instance is contrasted against all prototypes, completely eliminating negative sampling. The prototypes serve as both anchors and negatives in the contrastive objective (Ou et al., 2024).
• Explicit uniformity penalty: To prevent dimensional collapse (i.e., all prototypes converging to a single point), the uniformity loss

$$l_{\text{uniform}} = \mathbb{E}_{(t,s)\sim\mathcal{P}_{\text{proto}}} [\exp(-\beta \|\mathbf{c}_t-\mathbf{c}_s\|_2^2)]$$

penalizes prototypes that are too close, driving their configuration toward an approximately spherical arrangement over the unit ball.

• Relation to collapse-avoidance: Without uniformity, the only minima for the contrastive prototype objective are degenerate. Adding sampling-free uniformity ensures that the representations remain diverse and informative, as confirmed by both empirical ablation and theoretical arguments grounded in entropy maximization on the sphere (Ou et al., 2024).
• Training procedure: All parameters (prototypes and instance encoders) are updated with first-order methods; loss terms are computed over the entire batch without sub-sampling negatives or positives.

5. Sampling-Free Uniformity in Robotics and Control: C-Free-Uniform Trajectory Sampling

In robotic planning and control, sampling-free uniformity achieves efficient and robust exploration of the configuration space by obviating the need for ad-hoc rejection or random sampling of valid states:

• C-Free-Uniform objective: The sampler $\pi_\theta(u \mid x_t, M_{\text{local}})$ is trained so that the marginal distribution at each time $t$ over states matches the uniform law on the $t$-th safe level set, $L_\text{safe}(x_0, t, M)$. For any measurable $A\subset L_\text{safe}$,

$$P_t^*(A|M) = \frac{\mu(A)}{\mu(L_\text{safe}(x_0,t,M))} \quad \text{if } \mu(L_\text{safe}) > 0$$

The policy is updated to minimize $$\sum_{t=0}^T \mathrm{D}_{KL}(P_t^* \,\|\, P_{\theta,t})$$ (Cao et al., 19 Oct 2025).

• Supervision via max-flow and reachability: Algorithmically, an expert policy is derived by solving max-flows over layered reachability graphs defined on the discretized state space, guaranteeing equal-probability coverage. This produces supervision for training a neural sampler that is explicitly map-conditioned and adaptively uniform.
• Quantitative uniformity metrics: Uniformity is assessed using average KL divergence to the uniform law, entropy ratio, and collision-free sample ratios. C-Free-Uniform achieves practical and statistically significant uniformity advantages, with empirical collision-free ratio improvements by a factor of $4$–$8$ over alternatives.
• Planning integration and success rates: Integration into MPPI (CFU-MPPI) yields superior success rates and path efficiency compared to classical and log-probability-based alternatives, attributed directly to the uniformity properties of the induced trajectories.
• Avoidance of rejection sampling: Since the action policy directly enforces uniform marginal distributions, expensive rejection or correction steps are unnecessary. The learned sampler generalizes across environments and can be deployed in diverse planning frameworks (Cao et al., 19 Oct 2025).

6. Comparative Overview and Theoretical Implications

Sampling-free uniformity is characterized by several system-wide benefits and theoretical insights:

• Elimination of sampling bias: By replacing local, stochastic sample-based criteria with global or deterministic ones, the risk of ambiguous or semantically overlapping negatives (or failure to explore configuration space) is largely mitigated (Ou et al., 2024, Chen et al., 30 Dec 2025, Cao et al., 19 Oct 2025).
• Prevention of mode collapse and degeneracy: Theoretically, minimization of uniformity losses (e.g., exponential pairwise repulsion, squared mean on the sphere, gap ratio) provably prevent the collapse of parameterized point sets—whether embeddings or prototypes—by maximizing entropy and dispersion.
• Computational scalability: Sampling-free uniformity typically requires only simple aggregate statistics or structured loss computation, scaling more favorably and stably than negative-pair-dependent methods.
• Hardness and approximation: While achieving optimal uniformity can be computationally challenging (NP-hard for gap ratio in Euclidean or graph settings), efficient constant-factor approximations and coreset reductions are available for practical data sizes (Bishnu et al., 2014).

A plausible implication is that as model and data sizes increase, the inefficiencies and pathologies of sampling-based uniformity regularization become more severe, further motivating adoption of sampling-free formulations across machine learning and robotics.

7. Applications and Empirical Validation

Sampling-free uniformity powers advances across several domains:

Domain	Sampling-Free Uniformity Mechanism	Empirical Impact
Graph Representation	Hyperspherical mean; entropy-guided balancing (Chen et al., 30 Dec 2025)	$+1.49\%$ node classification over baselines, robust to density
Recommendation	Prototypical uniformity with exponential repulsion (Ou et al., 2024)	$5.1\%$ Rec@20 uplift, collapse-resistance, better t-SNE spread
Robotic Planning/Control	Map-conditioned trajectory sampler; KL-uniform marginals (Cao et al., 19 Oct 2025)	$4$–$8\times$ higher collision-free ratios; up to $100\%$ real-robot navigation success
Metric Sampling	Gap ratio minimization, metric coresets (Bishnu et al., 2014)	$(1+\varepsilon)$-approximation via fast streaming/static algorithms

In all cases, empirical evidence supports that sampling-free uniformity leads to higher-quality, less biased, and more robust representations, trajectories, and samplings than sampling-based approaches. In settings susceptible to representational collapse or limited exploration, explicit uniformity objectives provide a strong safeguard.

• (Bishnu et al., 2014) Uniformity of point samples in metric spaces using gap ratio
• (Ou et al., 2024) Prototypical Contrastive Learning through Alignment and Uniformity for Recommendation
• (Cao et al., 19 Oct 2025) C-Free-Uniform: A Map-Conditioned Trajectory Sampler for Model Predictive Path Integral Control
• (Chen et al., 30 Dec 2025) Hyperspherical Graph Representation Learning via Adaptive Neighbor-Mean Alignment and Uniformity