GRSL: Generation-Representation Scaling Law

• GRSL is a principle linking increased generative capacity with improved representation power, yielding predictable performance gains in complex systems.
• It is validated by empirical scaling laws in domains such as neural architectures, recommendation systems, and quantum-probabilistic models.
• The framework guides optimal resource allocation and model design by balancing system growth with representation richness.

The Generation-Representation Scaling Law (GRSL) describes principled relationships by which generative capacity and representational richness co-vary as system resources or complexity are increased, yielding predictable improvements in aggregate activity, model accuracy, or representation power in domains ranging from complex networks and urban systems to large-scale neural architectures and multimodal foundation models. GRSL articulates that as the generative aspects of a system scale—e.g., via increased population, compute, model size, or vocabulary—the corresponding representations become increasingly expressive and effective for downstream analysis, transfer, or task performance. This principle is supported by diverse empirical and mathematical analyses and embodies a unifying motif for resource-optimal model construction and theoretical understanding in modern large-scale learning, recommendation, motion synthesis, quantum-probabilistic modeling, and compressive training regimes.

1. Super-linear Growth and GRSL in Complex Systems

The paper "Growing Random Geometric Graph Models of Super-linear Scaling Law" (Zhang, 2012) formulates a foundational version of GRSL through the concept of super-linear growth in activities as a function of system size. Here, the scaling law is expressed as $X \propto P^\gamma$ for $\gamma > 1$, capturing the phenomenon where the total activity $X$ (such as GDP, patent creation, or communications in online communities) grows faster than linearly with population size $P$. The authors relate this to generation (network growth driven by new participants) and representation (the structure of connections shaped by spatial or similarity embedding):

• The key modeling element is a growing random geometric graph in $\mathbb{R}^d$, where new nodes are assigned locations and can connect only if they fall within an interaction radius of existing nodes; the number of resultant edges increases super-linearly due to local density.
• Central findings include that the super-linear exponent $\gamma$ is primarily controlled by the dimension $d$ of the geometric space, with fractal structure and clustering coefficient invariance arising as emergent network representations.
• Applied outcomes encompass area–population scaling ($A \propto P^\beta$ with $\beta<1$), diversity–population scaling, and scale-free degree distributions (when edge radii are heterogeneous).

GRSL here quantifies the tradeoff between the generative process (node aggregation by placement) and the richness of emergent network representations, with scaling exponents directly tied to spatial dimensionality, thus providing parsimonious, physically grounded explanatory power for complex system growth.

2. Empirical Power-Law Scaling in Generative Modeling

Scaling laws empirically verified in large autoregressive generative models (Henighan et al., 2020) reveal a power-law plus constant form for cross-entropy loss across diverse modalities, including images, video, multimodal text-image systems, and mathematical problem solving. The generative loss obeys:

$L(x) = L_\infty + (x_0/x)^{\alpha_x}$

where $x$ is model size, compute, or dataset size, $L_\infty$ is the irreducible loss (true data entropy), and the power-law term describes the reducible KL divergence between true and model distributions. These trends, nearly universal in exponent values, systematize the relationship between generative scaling and internal representation quality.

• The optimal model size at fixed compute follows $N_{\text{opt}} \propto C^{0.7}$.
• Mutual information between modalities (e.g., image-caption pairs) scales logarithmically with model size, directly linking representation information content to generative scaling.
• Extrapolation in out-of-distribution mathematical tasks and downstream fine-tuning (e.g., classification after generative modeling) both inherit the scaling improvement, demonstrating that finer representation structure emerges with reduced generative loss.

GRSL in this context formalizes the prediction that improvements in generative fidelity (lower KL divergence) translate into monotonic increases in representational utility, supporting resource-forecasting and informed scaling of architectures.

3. Scaling Laws in User Representation and Recommendation Systems

The Contrastive Learning User Encoder (CLUE) framework (Shin et al., 2021) and studies of large sequential recommendation models (Zhang et al., 2023) extend GRSL to user-centric and ID-based representation learning. The loss in these models diminishes as a power-law with total computation (measured in PF-days or model capacity):

• Pretraining loss: $\text{Loss} \propto (\text{Computation})^{-\alpha}$, where computation aggregates model parameters, batch size, sequence length, and training steps. Empirically, $\alpha$ captures diminishing returns for increased resources.
• Experiments confirm that scaling representation factors (model capacity, sequence length, batch size, and data) concurrently is required for optimal downstream performance, particularly for cold-start users or sparsely populated domains.
• For sequential recommenders, the observed test loss obeys $L(N) = E_N + (N_0 / N)^{\alpha_N}$, with larger exponents than typical NLP scaling, suggesting that representation scaling yields rapid improvements in recommendation performance, even in data-constrained regimes.

GRSL manifests as the predictable enhancement of behavioral representations for generalized and transferable recommendation tasks as system generation (in the form of resource scaling) is increased.

4. Mathematical Formulations in Linear and Quantum-Probabilistic Models

Recent analyses of scaling laws in linear regression (Lin et al., 12 Jun 2024) and quantum-probabilistic learning (Bai et al., 13 Oct 2024) derive precise mathematical decompositions for test error and representation quality under scalable training regimes:

• Linear regression: Test error decomposes into irreducible, approximation, and bias errors, with the excess risk bounded as $\Theta(M^{-(a-1)} + N^{-(a-1)/a})$ (where $M$ is model size and $N$ is data size, with $a$ the decay exponent of the covariance spectrum). Implicit regularization via one-pass SGD suppresses variance error, ensuring polynomial improvements as either data or representation scale.
• Quantum-probabilistic models: The negative log-likelihood $L$ for generative tensor networks initially grows linearly with the number of features $M$ due to the "catastrophe of orthogonality"—as $M$ increases, mutual state fidelity decays exponentially. Training introduces a negative quadratic correction, $L \simeq \beta M - \alpha M^2 + \text{const}$, where $\alpha$ and $\beta$ encode representation and generalization capacities and exhibit logarithmic scaling in network complexity (virtual dimension $\chi$).

GRSL here is underpinned by the theoretical axis linking data generation and representation scaling—via explicit error terms in linear models and via quantum mechanical alignment in tensor networks—clarifying the mechanisms by which increasing effective resources yields improved representational accuracy.

5. Scaling Laws in Architectures, Compression, and Motion Generation

Unified scaling laws encompassing both dense and sparse models (Hossain et al., 8 Aug 2025), compressed representations (Panferov et al., 2 Jun 2025), and motion generative systems (Lu et al., 19 Dec 2024) extend GRSL to practical model design and resource allocation:

• The generalized scaling law for LLMs accommodates both dense and sparsified architectures:

$$L(N, D, S) = e \cdot (1-S)^\gamma + [a \cdot (1-S)^\alpha + c \cdot S] / N^\alpha + b/D^\beta$$

where $S$ is the sparsity, smoothly interpolating representational resource effects.

• In compressed regimes, the law corrects parameter count by a representation-dependent "capacity" $\rho(R)$, derived from the ability to fit random Gaussian data, enabling direct comparison and prediction of loss across quantized, sparsified, and vector-coded formats.
• Motion generation models (ScaMo) establish a dual scaling principle: normalized test loss scales logarithmically with compute budget, while vocabulary and model sizes exhibit power-law dependencies with respect to compute, supporting the prediction of optimal configurations for a given resource envelope.

GRSL unifies these findings: for any generative system, representation richness (whether hidden activations, compressed weights, or discrete vocabularies) must be scaled in tandem with generation mechanisms for predictable performance improvement.

6. Interplay with Redundancy, Universality, and Multimodal Alignment

The mathematical origins of scaling exponents are formally explained as redundancy laws (Bi et al., 25 Sep 2025). In kernel regression, excess risk scales as:

$\alpha = \frac{2s}{2s + 1/\beta}$

with $\beta$ describing the polynomial tail of the representation spectrum (redundancy) and $s$ the target smoothness. A steeper spectrum (lower redundancy) accelerates performance scaling. The invariance of scaling law exponents under bounded representations and transformations demonstrates universality across domains and architectures, including multi-modal mixtures and Transformers.

In omnimodal representation learning with MLLMs (Xiao et al., 13 Oct 2025), GRSL is explicitly characterized: the generative quality (e.g., decoder alignment, mutual information between modalities) provides an upper bound on post-contrastive representation performance, supported by a formal PAC–Bayesian analysis linking lowered generative loss to improved downstream risk. Empirical studies confirm that enhanced generative pretraining improves representation power after lightweight contrastive refinement.

GRSL is thus both a theoretical and empirical principle: improvements in generative capacity directly bound, and often drive, advances in the system’s ability to generate expressive, discriminative, or transferable representations, with scaling laws providing the predictive bridge.

7. Implications and Outlook

GRSL offers a predictive and composable framework for building resource-optimal models in diverse domains—network science, generative modeling, recommendation, motion synthesis, quantum-probabilistic ML, and multimodal representation learning. By embodying scaling behaviors in rigorous empirical and mathematical relationships, GRSL guides the allocation of compute, data, model capacity, and representation structure for maximal gains in system activity, accuracy, and transferability.

Future work proceeds along refining scaling exponents, exploring joint generation-representation scaling in increasingly multimodal and agentic AI systems, and developing algorithms that adaptively tune representational scaling with generative advances. GRSL stands as a central theme for understanding and engineering complex, scalable learning systems.