Beyond Optimization: Intelligence as Metric-Topology Factorization under Geometric Incompleteness

TL;DR

The paper reframes intelligence as the ability to manage geometry under topological incompleteness, arguing that fixed-metric optimization cannot globally funnel inference on semantically complex spaces. It introduces Metric-Topology Factorization (MTF), separating topological structure (task identity) from metric warps (local geometry), enabling topology-driven routing via the Topological Urysohn Machine (TUM) and fast metric contraction through Memory-Amortized Metric Inference (MAMI). Theoretical results based on Morse theory establish a fundamental geometric incompleteness: semantically complex manifolds necessitate saddles, making global funneling impossible without topology-aware switching. The proposed hybrid architecture uses spectral topological signatures to index task-specific metric warps, allowing rapid adaptation across permutations, reflections, and continual learning tasks, while mitigating catastrophic forgetting. Empirical demonstrations in Möbius-topology environments and permuted MNIST illustrate improved adaptation latency and stability, suggesting a shift from parameter consolidation to geometric control for robust learning.

Abstract

Contemporary ML often equates intelligence with optimization: searching for solutions within a fixed representational geometry. This works in static regimes but breaks under distributional shift, task permutation, and continual learning, where even mild topological changes can invalidate learned solutions and trigger catastrophic forgetting. We propose Metric-Topology Factorization (MTF) as a unifying geometric principle: intelligence is not navigation through a fixed maze, but the ability to reshape representational geometry so desired behaviors become stable attractors. Learning corresponds to metric contraction (a controlled deformation of Riemannian structure), while task identity and environmental variation are encoded topologically and stored separately in memory. We show any fixed metric is geometrically incomplete: for any local metric representation, some topological transformations make it singular or incoherent, implying an unavoidable stability-plasticity tradeoff for weight-based systems. MTF resolves this by factorizing stable topology from plastic metric warps, enabling rapid adaptation via geometric switching rather than re-optimization. Building on this, we introduce the Topological Urysohn Machine (TUM), implementing MTF through memory-amortized metric inference (MAMI): spectral task signatures index amortized metric transformations, letting a single learned geometry be reused across permuted, reflected, or parity-altered environments. This explains robustness to task reordering, resistance to catastrophic forgetting, and generalization across transformations that defeat conventional continual learning methods (e.g., EWC).

Figures (5)

• Figure 1: The Bowl-Maze Analogy: From path-finding to metric contraction.Left: The Maze (Topological Obstruction). Conventional learning systems, such as stochastic gradient descent (SGD) and elastic weight consolidation (EWC), treat intelligence as navigation within a fixed, complex geometry. In this "Maze"-based regime, topological features, such as holes, walls, or parity flips, manifest as local minima and dead-ends. Learning is a slow, procedural search for a valid path, which is easily invalidated by any change in the environment's topology. Right: The Bowl (MTF). Under the MTF framework, intelligence is redefined as the ability to shape the metric structure of the space. Instead of searching for paths, the agent performs topological indexing to switch into a coordinate system where the solution emerges as a stable, global attractor. By warping the maze into a "Bowl" (topological condensation followed by metric contraction), the agent replaces complex navigation with a simple downhill descent, ensuring that once the topology is identified, the solution is both reachable and consistent.
• Figure 2: Metric-Topology Factorization.Top: Unfactorized inference attempts to solve all tasks via optimization in a fixed metric space, requiring re-optimization under topological change. Bottom: MTF separates topological identification from metric inference. Once the correct topology is selected, metric descent proceeds within a specialized fiber and is invariant to global topological transformations.
• Figure 3: Empirical Validation of MTF on a Möbius State Space. (a) The Plain SGD learner (red) exhibits characteristic "sawtooth" spikes, indicating a failure to resolve the topological obstruction through a single global metric. The MTF Learner (blue) demonstrates near-instantaneous recovery after the initial learning of both parity signatures. (b) Absolute cosine similarity between task-specific metric warps $W_A$ and $W_B$ after training with the orthogonality penalty ($\alpha=0.5$). The vanishing overlap ($\approx 2.5 \times 10^{-4}$) provides empirical proof of the orthogonal routing required for representation stability.
• Figure 4: Impact of Topological Complexity on Learning Stability. (Left) Convergence time as a function of the first Betti number $\beta_1$. The Single-Metric Learner (red) exhibits a super-linear increase in training time, while the MTF Learner (blue) remains invariant to topological genus. (Right) A proxy for the Morse Index, measured via the density of negative Hessian eigenvalues. Higher topological complexity necessitates a more convoluted loss surface with a higher density of saddle points for unfactorized models.
• Figure 5: Continual Learning Performance on Permuted MNIST. Comparison of Average Accuracy over a sequence of 5 permuted tasks. While Elastic Weight Consolidation (EWC) (black) maintains higher stability than naive fine-tuning, it exhibits a characteristic "stiffness" as the Fisher constraints accumulate, leading to a performance plateau. The MAMI-MTF approach (red) achieves superior plasticity by treating the network as a fluid coordinate system, utilizing iterative Newton-Schulz whitening to amortize the metric warp across task boundaries.

Theorems & Definitions (16)

• Definition 1: Riemannian Manifold
• Definition 2: The Gradient Operator
• Lemma 3: Urysohn's Lemma urysohn1925zum- The Functional Form
• Lemma 4: Urysohn-Brouwer Lemma willard2012general - The Geometric Form
• Definition 5: Semantic Complexity
• Lemma 6: No-Saddle Funnels Force Simplicity
• Theorem 7: Metric-Topological Incompleteness
• Definition 8: Topological Representation
• Definition 9: Metric Representation
• Definition 10: Metric-Topology Factorization