The Geometry of Meaning in Neural Manifolds

The strongest aspect is the elegant simplicity of the foundational method, relative representations, which projects embeddings onto cosine similarities with shared anchors to produce invariant semantic coordinates. As the author notes, this requires "no optimization, no learned parameters, no iterative refinement—just matrix multiplication computing cosine similarities." The empirical scope is also notable: the same principle is shown to support angular, metric, spectral, sparse, and distributional alignments across CNNs, Transformers, graph networks, and even fMRI recordings. The consistent operator taxonomy provides a useful conceptual scaffold for what would otherwise appear as a disparate collection of tricks.

The central limitation is the lack of theoretical grounding for when and why intrinsic geometries converge. The author states plainly that "we cannot predict a priori whether two given models will have sufficiently similar intrinsic geometry for correspondence to enable useful transformation." Without sufficiency conditions, the framework risks being unfalsifiable: if alignment fails, one can always blame insufficient correspondence or violated geometric assumptions, while success is taken as evidence of a "fundamental primitive." Furthermore, claims that the principle extends to biological neural systems rely on a single fMRI application and conflate very different measurement modalities and learning regimes. The heavy reliance on the author’s own prior publications as evidence also raises the risk of confirmation bias, since negative results across unrelated architectures may not be reported.

The thesis situates its contribution fairly against transfer learning, Procrustes analysis, model stitching, and kernel similarity measures, correctly noting that "Zero-shot transformation across independently trained models with arbitrary architectures remained an open challenge." The evidence draws extensively on the representation similarity puzzle—e.g., "different optimization trajectories reach functionally similar solutions in distinct regions of parameter space"—to motivate the need for coordinate-invariant methods. However, the comparison is largely narrative rather than quantitative head-to-head; benchmarks against contemporary baselines are distributed across prior publications rather than aggregated in a single reproducible protocol. The claim that the framework accommodates "arbitrary architectures" is empirically supported mainly by pairwise demonstrations rather than by a systematic ablation over architecture divergence.

Reproducibility is mixed. The thesis benefits from precise mathematical notation and explicit definitions of operators such as $T_{\Gamma}:\mathcal{X}\to\mathcal{Y}$. Some methods link to public code; for example, relative representations are accompanied by "Code: https://github.com/lucmos/relreps." Yet because the thesis synthesizes fourteen distinct publications, replicating the full sweep would require reconstructing disparate codebases, datasets, and hyperparameters. Key details such as exact anchor counts, training budgets, and random seeds are often deferred to appendices of prior papers. The author advocates consistent empirical methodology, stating that each method chapter follows "stating correspondence assumptions explicitly, validating those assumptions on real problems with authentic experimental results," but a unified benchmark suite or standard training protocol is absent, which would be essential for independent verification.