Representation Group Flow

• Representation Group (RG) Flow is a categorical framework that generalizes the renormalization group for evolving data representations via functorial and dynamical systems.
• It employs neural ODEs and invertible, discretized layers to model continuous representation evolution and hierarchical disentanglement across scales.
• The framework finds applications in genomic analysis and image modeling, offering a modular approach for efficient generative and classification tasks.

A Representation Group (RG) Flow is a categorical and functional framework for the evolution of data representations, generalizing the renormalization group (RG) paradigm from theoretical physics into machine learning and data science contexts. In this approach, one describes the scale-dependent transformation of representations—typically as maps between categories of data objects and vector spaces—via flows governed by dynamical systems, neural ODEs, and learnable functorial structures. The RG flow structure is semigroup- or group-valued, highly compositional, and supports both hierarchical inference and generative modeling. Key features include categorical functoriality, invertible flow architectures, hyperbolic latent geometry, and an explicit correspondence with representation disentanglement at varying scales (Sheshmani et al., 2022).

1. Categorical Structure and Definition

• Categories of Data and Representations: The foundational objects are abstract categories $\mathcal{C}$ (data objects, e.g. genomic sequences, images) and $\mathcal{V}$ (linear spaces, typically $\mathbb{R}^n$). Morphisms in $\mathcal{C}$ correspond to permissible data transformations (e.g., subsequence embeddings, token reorderings).
• The Representation Functor: A representation is a functor $R: \mathcal{C} \rightarrow \mathcal{V}$ assigning vector spaces to data objects and linear maps to morphisms, thus encoding the semantics of data within a feature space.
• RG-flow as Natural Transformation/Endo-functor: The RG-flow operator is defined as a one-parameter family of endo-functors $\{\mathcal{R}_t\}_{t\geq0}$ acting on functor categories $[\mathcal{C},\mathcal{V}]$. Composition law $$\mathcal{R}_s \circ \mathcal{R}_t = \mathcal{R}_{s+t}$$ enforces semigroup properties. For learned invertible instances, $\mathcal{R}_t$ forms a group.

2. Evolution Equations and Neural ODE Realization

• Continuous Representation Flow: Representations evolve according to a neural ODE of the form

$\frac{dR_t}{dt} = F(R_t), \quad R_{t=0} = R_0$

where $F$ is a learnable library of vector fields on $[\mathcal{C},\mathcal{V}]$ (Sheshmani et al., 2022).

• Time-Ordered Exponential Solutions: The solution is given by the path-ordered exponential:

$$\mathcal{R}_t = \mathcal{T} \exp \left( \int_0^t F(R_s)\, ds \right)$$

• Discrete Steps and Bijective Layering: Practically, the RG flow is discretized: at each layer $k$, a bijective map splits the representation $\phi^{(k-1)}$ into a coarser “relevant” part $\phi^{(k)}$ and an “irrelevant” latent part $\zeta^{(k)}$. This decimation mimics the physical RG’s “integrate out” step while maintaining invertibility.

3. Architecture: RG-Flow Categorifier

• Multi-scale Layering: Each layer comprises a disentangler (Neural-ODE vector field integration) and decimator (splitting features into relevant/coarse and irrelevant/bulk), recursively assembling a hierarchical latent tree.
• Dual Usage—Generation and Classification: Generation involves sampling bulk latents $\zeta^{(k)}$ and backward-inverting the flow; classification uses coarse representations for semantic prediction.
• Losses and Training: The architecture trains via negative log-likelihood (KL divergence), Min-Bulk-Mutual-Information regularizers (enforcing independence/disentanglement of bulk variables), and supervised heads for classification.

4. Hyperbolic Latent Geometry and Holographic Analogy

• Bulk Index-Set Structure: The index set for “irrelevant” bulk variables $J^{(k)}$ forms a discrete hyperbolic lattice; exponential volume growth in scale realizes negative curvature, paralleling AdS/CFT intuitions.
• Jacobians and “Bulk Actions”: The log-Jacobian determinant of the neural ODE step is interpreted as a coupling action; recursion over layers mirrors Hamiltonian evolution along RG “time.”
• Hierarchical Disentanglement: At shallow depth (small $t$), representations encode local motifs; deeper layers capture long-range dependencies, with disentangled latent codes at every scale.

5. Empirical Applications and Representation Disentanglement

• Genomic Sequence Analysis: RG-flow categorifiers extract dominant features, symmetries, and clusters in biomedical sequence-to-function mapping. They enable likelihood estimation, classification, and generation of novel sequences consistent with learned grammar (Sheshmani et al., 2022).
• Multi-scale Image Models: The RG-Flow model in generative learning partitions information by scale, yielding interpretable latent variables for style, content, and object semantics (Hu et al., 2020). Hierarchical inpainting achieves complexity $O(\log L)$ for patch size $L$, outperforming $O(L^2)$ methods in scalability.
• Style Mixing and Semantic Control: Coarse/fine style mixing is formalized by recombination of latents at different scales; receptive-field analysis shows that features in latent space align with semantic attributes at hierarchical scales.

6. Theoretical and Mathematical Principles

• Functoriality and Algebraic Composition: The RG-flow framework supports systematic layer composition, facilitating modular design and theoretical analysis.
• Invertible, Continuous-depth Flows: Neural ODEs introduce continuous-depth, bijective transformations avoiding information loss and ensuring tractable likelihood evaluation.
• Parallelism with Classical RG: Where classical RG implements non-invertible semigroups on couplings via hand-crafted coarse-graining, the RG categorifier learns invertible flows through data, adapts scale transitions, and generalizes to non-physical settings.

7. Synthesis and Generalization

RG-flow as a representation group flow unifies the categorical, dynamical-systems, and information-theoretic approaches to scale-adaptive data analysis. It generalizes Wilsonian RG to arbitrary data domains, supports modular generative modeling, and achieves multi-scale disentanglement. The emergent hyperbolic geometry, coupled with invertible neural ODE layers and functorial semantics, positions RG-flow categorifiers as a rigorous, extensible framework for hierarchical representation learning, classification, and generation (Sheshmani et al., 2022, Hu et al., 2020).