ActPC-Geom: Neural-Symbolic Learning

• ActPC-Geom is a scalable neural-symbolic framework that integrates active predictive coding with geometric optimization using the Wasserstein metric.
• It employs measure-dependent natural gradient flows and compressed embeddings to improve convergence, stability, and real-time adaptation in complex AI systems.
• The architecture seamlessly combines continuous neural and discrete symbolic representations, enabling compositional reasoning and few-shot learning for advanced cognitive tasks.

ActPC-Geom is a scalable architecture for neural-symbolic learning that accelerates Active Predictive Coding (ActPC) by integrating information geometric principles—particularly those based on the Wasserstein metric—into learning dynamics, embedding methodology, and symbolic-subsymbolic integration. The design aims to enable robust, real-time, and compositional cognitive features for large-scale artificial intelligence systems by addressing the limitations of both standard backpropagation and classical predictive coding, while allowing seamless connection of continuous neural and discrete symbolic representations.

1. Foundational Principles of ActPC-Geom

ActPC-Geom enhances Active Predictive Coding (ActPC) by leveraging a rigorous information geometric framework. Standard ActPC implements biologically-inspired, brainlike local predictive error minimization and online adaptation, but may suffer from slow convergence or instability on complex, high-dimensional problems. By recasting learning as natural gradient flows on probabilistic manifolds endowed with the Wasserstein metric, ActPC-Geom aligns parameter updates with the intrinsic geometry of distributional prediction spaces. This design ensures updates are geometrically consistent, improving stability, convergence, and scalability for neural networks with complex architectures—such as those required for hybrid neural-symbolic inference, compositional reasoning, and online deliberative learning.

The use of Wasserstein geometry in ActPC-Geom directly replaces the Kullback-Leibler (KL) divergence with Wasserstein distance in both prediction error assessment and parameter update rules. This symmetry ensures the “outer” loss metric and the “inner” learning dynamics are harmonized, reducing mismatch between optimization objectives and actual update steps.

2. Information-Geometric Optimization and Wasserstein Measure

The core mathematical innovation in ActPC-Geom is the implementation of measure-dependent natural gradient flows using Wasserstein geometry. For model parameters $\boldsymbol{\xi}$ inducing a predictive distribution $p(\cdot|\boldsymbol{\xi})$ and loss functional $F$, the natural gradient flow is governed by

$$\frac{d\boldsymbol{\xi}}{dt} = - G(\boldsymbol{\xi})^{-1} \nabla_{\xi} F(p(\boldsymbol{\xi}))$$

Here, $G(\boldsymbol{\xi})$ denotes a metric tensor constructed from the Jacobian of the model’s probability map and the (pseudo-)inverse of the measure-dependent Laplacian $L(p(\boldsymbol{\xi}))^\dagger$. For discrete supports, the Laplacian takes the form $L(p)_{ij} = \omega_{ij}(p_i + p_j)$ for $i \neq j$.

The Wasserstein-2 ($W_2$) distance enables cost to be meaningfully computed even between distributions with disjoint supports, a property not shared by KL divergence. This property is essential for tasks requiring creativity, compositionality, or semantic interpolation, where the model must reason about novel or non-overlapping possibilities.

3. Neural Approximators, Embeddings, and Algebraic Structure

ActPC-Geom introduces computational strategies for efficient real-time operation at scale by employing:

• Neural approximators for $L(p)^\dagger$: Since exact inversion of the measure-dependent Laplacian is intractable in large systems, trainable neural modules are used to predict approximations of this inverse based on compressed state features.
• Approximate kernel PCA (kPCA) embeddings: To enable scalable and meaningful approximations, low-rank representations of $L(p)^\dagger$ are compressed via kPCA, with kernel choice guided by Wasserstein ground metrics and efficient computation using Nystrom or random feature methods.
• Compositional hypervector embeddings: High-dimensional hypervector representations (thousands of dimensions), supporting binding and bundling operations, encode both the kPCA geometry and compositional algebraic structure. Hypervectors facilitate symbolic-like manipulation in continuous vector space, supporting algebraic composition, property–value binding, and context-dependent representations.

Fuzzy Formal Concept Analysis (FCA) lattices are learned on top of these embeddings, yielding a data-driven, algebraically structured ontology for both neural and symbolic reasoning.

4. Symbolic-Subsymbolic Integration

A distinctive feature of ActPC-Geom is its capacity for deep integration between neural (continuous) and symbolic (discrete) representations. Both subsystems are unified via:

• Shared predictive coding dynamics: Continuous (e.g., transformer- or Hopfield-like) and discrete (e.g., ActPC-Chem for chemical rewrite rule evolution) architectures minimize predictive error using the same information-geometric gradient flow principles.
• Common probabilistic, concept-lattice, and hypervector models: Knowledge, uncertainty, and reasoning are represented within versatile embedding spaces accessible to both neural nets and symbolic rules.
• Neural-symbolic communication: Discrete events, rewrite rules, and algebraic concepts are all embedded into hypervector or metric representations, enabling joint reasoning and mutual influence without requiring prior manual ontologization.

This unified design supports “cognitive synergy,” consistent with frameworks such as OpenCog Hyperon, which blends MeTTa logical calculus, pattern mining, and neural learning under the ActPC-Geom paradigm.

5. Advanced Cognitive Mechanisms

ActPC-Geom incorporates multiple mechanisms to support higher-level cognition:

• Hopfield-net dynamics: Attractor networks with lateral connections provide associative memory, rapid pattern retrieval, sequence learning, and resilience against catastrophic forgetting.
• Transformer-style predictive coding: Local error propagation supports real-time, sublayer-wise online learning and instruction tuning. Self-attention distributions are interpreted as probability measures, ensuring information geometry is respected at every level.
• Few-shot learning and online weight adaptation: The system achieves a fine balance between fast, ephemeral adaptation (in activation space) and slower, persistent weight updates (in parameter space), enabling effective few-shot learning and continuous deliberation.

Online updates can be performed at a high activation-to-weight update ratio (e.g., 10:1), with both forms of adaptation guided by Wasserstein-gradient flows and embedding compression, yielding rapid task adaptation and incremental learning.

6. Hybrid Processing: Galois Connections and Specialized HPC Design

For hybrid neural-symbolic reasoning, ActPC-Geom adopts concurrent, expansion–shrinking processes managed by Galois connections. Candidate generation and pruning are coordinated across continuous and discrete representations, using abstract dynamic programming theorems to ensure convergence toward joint optimum.

A specialized high-performance computing (HPC) architecture is proposed:

• GPU subsystems manage high-throughput, real-valued neural activity and hypervector algebra.
• Multi-core CPU and parallel managers orchestrate symbolic rewriting, discrete candidate manipulation, and concept lattice updates.
• Focus-of-attention is managed dynamically, with local update cycles maximizing concurrency and minimizing global synchronization, mimicking biological cognition.
• The architecture is scalable, amenable to deployment on supercomputing and decentralized frameworks (e.g., via SingularityNET/NuNet).

7. Applications and Frameworks

ActPC-Geom is positioned for deployment in settings requiring advanced neuro-symbolic cognition, few-shot adaptation, and compositional reasoning:

• OpenCog Hyperon: Combines neural and symbolic modules, leveraging the ActPC-Geom embedding space for logical inference and pattern mining.
• ActPC-Chem: Facilitates algorithmic chemistry evolution as a discrete symbolic companion to standard ActPC, both linked by measure-based geometry and shared probabilistic concepts.
• Commonsense and Deliberative Reasoning: Hypervector algebra supports robust, interpretable representations suitable for planning, analogical reasoning, and high-level cognition.
• Real-time robotics and AI decision-making: The architecture’s focused attention and deliberation mechanisms are optimized for actionable decision-making within strict time constraints.

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Summary Table of Key Mathematical Components

Component	Formula / Definition	Purpose
Wasserstein Distance	$$W_2(p, q) = \inf_{\pi\in\Gamma(p, q)} \left(\int \|x-y\|^2 d\pi(x,y)\right)^{1/2}$$	Metric for distributional comparison and optimization
Measure-Dependent Laplacian	$L(p)_{ij} = \omega_{ij}(p_i + p_j)$	Governs geometry of parameter updates
Natural Gradient Update	$$\frac{d\boldsymbol{\xi}}{dt} = -G(\boldsymbol{\xi})^{-1} \nabla_{\xi} F(p(\boldsymbol{\xi}))$$	Geometric gradient flow on the probability manifold
Compositional Hypervector Embedding	Hypervector $\mathbf{v}(S)$ built via binding and bundling of concept memberships	Enables symbolic-algebraic manipulation in vector space

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ActPC-Geom represents an overview of biologically-inspired predictively-coding neural learning, modern information geometric optimization, and compositional neural-symbolic embedding techniques. It is designed for scalable, robust, and real-time cognitive systems capable of flexible reasoning, memory, and cross-level integration. This framework offers a mathematically grounded path forward for constructing large-scale artificial intelligence systems with advanced symbolic-subsymbolic competencies, compositional semantic reasoning, and focus-driven deliberation.