The Three Walls of Pattern Recognition: Fundamental Limits and Structured Approximations in Discrete Bayesian Learning

Abstract

Machine learning is often advanced by scaling models, data, and computation, yet scaling encounters basic limits of learning. A model may require more memory than a machine can provide, more data than can realistically be collected, or a representation that does not match the structure of the problem. These difficulties recur throughout the history of pattern recognition in different forms, indicating persistent constraints of learning across technological regimes. This dissertation studies these constraints through discrete Bayesian classifiers, full-joint probability distribution memory models, and N-tuple subspace methods, formalizing and extending the author's prior research on structured N-tuple extensions published in IEEE Transactions on Systems, Man, and Cybernetics: Systems (2021). These systems provide a transparent setting in which learning can be examined directly, because training is based on stored statistical evidence and inference is based on retrieval and comparison. Within this setting, the dissertation introduces the Three Walls theory as a unified conceptual and operational framework for recurring limits of learning. The theory identifies limits imposed by physical capacity, finite data, and representational structure, while treating noise as a separate performance ceiling. The dissertation shows that the full-joint discrete model provides a precise reference for identifying when learning becomes physically infeasible because memory requirements grow too rapidly, statistically unreliable because available training data are too sparse relative to the discrete state space, and structurally limited when the chosen discrete representation does not adequately capture the organization of the data. The dissertation further shows that class-labeled observations possess exploitable statistical and relational structure, so lower-order subspace models can preserve important dependencies while substantially reducing storage and execution demands. Feasibility-constrained optimization can then systematically improve these structured subspace models and reduce the empirical gap to the full-joint reference. To resolve the repeated-evidence multiplicity inherent in overlapping subspaces, this dissertation introduces the V-Tuple architecture–an exact graph-induced clique–separator factorization of the discrete N-tuple network. By regulating shared coordinates through a running-intersection ordering, the V-Tuple lifts the classical subspace approximation into a decomposable graphical model. The empirical profiles confirm the theoretical guarantees: on controlled high-entropy and arithmetic regimes where uncorrected aggregation collapses to baseline guessing, the V-Tuple reconstructs the required dependence structure. In these regimes, the V-Tuple restores test performance from random-guessing baselines to the Full-Joint Bayesian reference level, reaching 1.0000 on Modulo Sum and matching the Full-Joint reference mean on XOR Parity. This closes the structural approximation gap in these regimes while explicitly quantifying the realized statistical support and hardware storage burdens required by the clique–separator factorization. The same separator-corrected probability head is also evaluated on quantized Transformer attention-head summaries. In an undertrained XOR Parity encoder regime, the V-Tuple head raises Transformer linear-readout test accuracy from 0.6703 to 0.9612, reduces negative log-likelihood from 0.5500 to 0.0829, and reduces expected calibration error from 0.0939 to 0.0231, with paired sign-flip values p < 0.0001. Taken together, these results show that learning feasibility is governed jointly by physical capacity, finite data, representational structure, and noise. Discrete Bayesian learning admits a hierarchy of structured alternatives under fixed resource constraints: the full-joint reference for identifying fundamental limits, lower-order subspace models for feasible execution, feasibility-constrained optimization for systematic performance improvement, the V-Tuple architecture for exact overlap-consistent dependency retention, and separator-corrected V-Tuple probability heads for intermediate neural representations.