HSAS and Geometric Deep Learning

 HSAS and Geometric Deep Learning

Table of Contents

  1. Geometric Deep Learning as Semantic Constraint Engineering
    1.1 High-dimensional learning and the failure of generic approximation
    1.2 Structure as search-space contraction
    1.3 Domain \(\Omega\), signal \(X(\Omega,C)\), representation and semantic state
    1.4 GDL geometry versus FSAS semantic access
    1.5 Representation \(\neq\) semantics \(\neq\) source reality
    1.6 Protected-consequence contracts for learned representations

  2. Symmetry, Invariance and Admissible Transformations
    2.1 Groups, actions and representations
    2.2 Invariance as protected consequence
    2.3 Equivariance as structured semantic transport
    2.4 Automorphism versus isomorphism
    2.5 Structural equivalence versus semantic identity
    2.6 Symmetry subgroup selection as preservation of distinctions
    2.7 When invariance destroys information

  3. Local Geometry, Deformation and Semantic Curvature
    3.1 Exact global symmetry as limiting case
    3.2 Local and approximate invariance
    3.3 Deformation stability
    3.4 Local curvature versus global averaging
    3.5 Context-dependent admissibility
    3.6 Fracture: transformations that cross semantic boundaries
    3.7 Residue generated by imperfect equivalence
    The original text already moves from exact group symmetry toward local, inexact deformation stability, with \(||f(\rho(\tau)x)-f(x)||\le Cc(\tau)||x||\). 2104.13478v2

  4. Scale Separation and Controlled Collapse
    4.1 Locality and receptive fields
    4.2 Pooling and coarsening
    4.3 Multiscale representations
    4.4 Information preserved under collapse
    4.5 Information irreversibly discarded
    4.6 Residue accounting across scales
    4.7 Recoverability and semantic aliasing

  5. The FSAS–GDL Blueprint
    5.1 Linear equivariant operators
    5.2 Pointwise nonlinearities
    5.3 Local pooling
    5.4 Global invariant readout
    5.5 Typed composition of layers
    5.6 Admissibility validators between layers
    5.7 Consequence-relative equivalence
    5.8 Failure-triggered reopening and retyping

  6. Graphs and Relational Domains
    6.1 Sets as permutation domains
    6.2 Graph topology and node signals
    6.3 Graph isomorphism and automorphism
    6.4 Message passing as local semantic access
    6.5 Oversmoothing as destructive collapse
    6.6 Long-range dependency and higher-order structure
    6.7 Deep Sets, Transformers and latent graphs
    6.8 Large language models as dynamically induced relational geometry

  7. Grids and Ordered Domains
    7.1 Euclidean grids
    7.2 Translation symmetry
    7.3 Convolution
    7.4 Resolution and aliasing
    7.5 Temporal grids
    7.6 Recurrent networks and LSTMs
    7.7 Time warping and deformation stability

  8. Groups and Homogeneous Spaces
    8.1 Group convolution
    8.2 Representation spaces
    8.3 Homogeneous domains
    8.4 Spherical and rotational equivariance
    8.5 TacticAI as constrained geometric reasoning

  9. Geodesics, Manifolds and Geometric Graphs
    9.1 Riemannian metrics
    9.2 Geodesic distance
    9.3 Intrinsic versus extrinsic representation
    9.4 Manifold and mesh CNNs
    9.5 Equivariant message passing in physical space
    9.6 AlphaFold-style geometric reasoning

  10. Gauges, Bundles and Local Reference Frames
    10.1 Tangent spaces and bundles
    10.2 Gauge choice
    10.3 Gauge transformations
    10.4 Parallel transport
    10.5 Gauge equivariance
    10.6 Local frame choice versus physical invariant
    10.7 Holonomy as evidence of curvature

  11. Beyond Groups: FSAS Horizons
    11.1 Why fixed global groups are insufficient
    11.2 State-dependent symmetries
    11.3 Partial transformations and local operators
    11.4 Translation between nonidentical semantic domains
    11.5 Closed translation loops and residue
    11.6 Higher-order nodes and nonfactorizable structure
    11.7 Recursive restructuring of the representation space

  12. Domain-Aware Constraint Validation
    12.1 Geometric validity
    12.2 Semantic validity
    12.3 Consequence validity
    12.4 Destructive substitution tests
    12.5 Local closure versus forbidden global closure
    12.6 Failure as information
    12.7 Geometry as an access structure rather than an ontology 

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