Embedding Mathematics Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Embedding Mathematics Ontology within the Kireth Research Semantic Library. Embedding mathematics concerns the representation of discrete, symbolic, relational or high-dimensional objects as vectors in continuous spaces so that similarity, geometry, neighbourhood structure, algebraic relationships or task-relevant information can be computed numerically. This ontology supports vector embeddings, latent spaces, metric and similarity functions, dimensionality, normalisation, manifold relationships, learned embeddings, graph and language embeddings, contrastive objectives, matrix-factorisation relationships, neighbourhood preservation, retrieval, clustering, robustness, verification and reproducibility.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Embedding Mathematics Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Computational Mathematics"
 SPECIALISATION = "AI Computational Mathematics / Embedding Mathematics"
 CANONICAL_PAGE = "https://kirethframeworkold.co.uk/books/semantic-library/ai-core/ontologies/mathematical-ontologies/computational-mathematics-ontologies/ai-computational-mathematics/embedding-mathematics-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = EmbeddingMathematics
 ENTITY_CLASS = ContinuousRepresentationGeometryFramework
 PRIMARY_CONTEXT = AIComputationalMathematicsOntology
 DEFINITION = "Embedding mathematics is the mathematical framework for mapping objects into vector spaces so that relevant relationships can be represented through distances, similarities, directions, neighbourhoods or learned geometry."
 SYMBOLIC_FORMS = [
  "phi: X -> R^d",
  "e_x = phi(x)",
  "sim(x,y)=cos(phi(x),phi(y))",
  "d_emb(x,y)=||phi(x)-phi(y)||"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for embedding maps, latent vector spaces, dimensionality, similarity, distance, geometry, neighbourhoods, learning objectives, retrieval and validation."
}

SCOPE {
 INCLUDE = [
  EmbeddingMathematics,
  Embedding,
  VectorEmbedding,
  LearnedEmbedding,
  Representation,
  RepresentationLearning,
  FeatureEmbedding,
  LatentRepresentation,
  LatentVector,
  EmbeddingVector,
  EmbeddingSpace,
  LatentSpace,
  RepresentationSpace,
  FeatureSpace,
  VectorSpace,
  EuclideanSpace,
  InnerProductSpace,
  NormedSpace,
  MetricSpace,
  ManifoldRelationship,
  LowDimensionalManifoldRelationship,
  EmbeddingDimension,
  LatentDimension,
  Dimensionality,
  HighDimensionalSpace,
  LowDimensionalSpace,
  Coordinate,
  Basis,
  LearnedBasis,
  OrthogonalBasisRelationship,
  Normalisation,
  VectorNormalisation,
  UnitNorm,
  L2Normalisation,
  Centering,
  WhiteningRelationship,
  StandardisationRelationship,
  Distance,
  Metric,
  EuclideanDistance,
  ManhattanDistance,
  MahalanobisDistanceRelationship,
  AngularDistance,
  CosineDistance,
  Similarity,
  CosineSimilarity,
  DotProductSimilarity,
  InnerProduct,
  DotProduct,
  KernelSimilarityRelationship,
  Neighbourhood,
  LocalNeighbourhood,
  KNearestNeighbours,
  KNN,
  NearestNeighbour,
  ApproximateNearestNeighbour,
  ANN,
  RadiusNeighbourhoodRelationship,
  TopKRelationship,
  Ranking,
  SimilarityRanking,
  Retrieval,
  VectorRetrieval,
  SemanticRetrievalRelationship,
  InformationRetrievalRelationship,
  Clustering,
  KMeansRelationship,
  HierarchicalClusteringRelationship,
  DensityClusteringRelationship,
  ClassificationRelationship,
  LinearClassifierRelationship,
  DecisionBoundaryRelationship,
  Projection,
  LinearProjection,
  NonlinearProjection,
  DimensionalityReduction,
  PrincipalComponentAnalysis,
  PCA,
  SingularValueDecompositionRelationship,
  SVDRelationship,
  MultidimensionalScalingRelationship,
  IsomapRelationship,
  LocallyLinearEmbeddingRelationship,
  LaplacianEigenmapsRelationship,
  SpectralEmbedding,
  TSNERelationship,
  UMAPRelationship,
  AutoencoderEmbeddingRelationship,
  Encoder,
  DecoderRelationship,
  BottleneckRepresentation,
  MatrixFactorisation,
  LowRankFactorisation,
  LatentFactorModel,
  SingularVectorRelationship,
  EigenvectorRelationship,
  RandomProjectionRelationship,
  JohnsonLindenstraussRelationship,
  SparseProjectionRelationship,
  WordEmbeddingRelationship,
  TokenEmbeddingRelationship,
  SentenceEmbeddingRelationship,
  DocumentEmbeddingRelationship,
  LanguageEmbeddingRelationship,
  GraphEmbedding,
  NodeEmbedding,
  EdgeEmbeddingRelationship,
  GraphNeuralNetworkRelationship,
  KnowledgeGraphEmbedding,
  EntityEmbedding,
  RelationEmbedding,
  TranslationalEmbeddingRelationship,
  MatrixFactorisationEmbeddingRelationship,
  CollaborativeFilteringRelationship,
  UserEmbeddingRelationship,
  ItemEmbeddingRelationship,
  PositionalEmbeddingRelationship,
  SinusoidalEmbeddingRelationship,
  LearnedPositionalEmbeddingRelationship,
  TimeEmbeddingRelationship,
  SpatialEmbeddingRelationship,
  CoordinateEmbeddingRelationship,
  MultimodalEmbeddingRelationship,
  CrossModalEmbeddingRelationship,
  JointEmbeddingSpace,
  ContrastiveLearning,
  ContrastiveLoss,
  TripletLoss,
  PairwiseLoss,
  MarginLoss,
  MetricLearning,
  SiameseNetworkRelationship,
  PositivePair,
  NegativePair,
  HardNegative,
  NegativeSampling,
  NoiseContrastiveEstimationRelationship,
  InfoNCERelationship,
  TemperatureParameter,
  SoftmaxRelationship,
  CrossEntropyRelationship,
  ReconstructionLossRelationship,
  NeighbourhoodPreservation,
  LocalStructurePreservation,
  GlobalStructurePreservation,
  RankPreservationRelationship,
  IsometryRelationship,
  ApproximateIsometry,
  Distortion,
  MetricDistortion,
  StressRelationship,
  TrustworthinessRelationship,
  ContinuityMetricRelationship,
  Separation,
  ClusterSeparation,
  Margin,
  LinearSeparabilityRelationship,
  SemanticDirectionRelationship,
  VectorArithmeticRelationship,
  AnalogyRelationship,
  TranslationRelationship,
  RotationRelationship,
  OrthogonalTransformationRelationship,
  ProcrustesAlignmentRelationship,
  EmbeddingAlignment,
  SpaceAlignment,
  DomainAlignmentRelationship,
  CanonicalCorrelationAnalysisRelationship,
  CrossLingualAlignmentRelationship,
  EmbeddingDrift,
  RepresentationDrift,
  DistributionShift,
  Collapse,
  RepresentationCollapse,
  ModeCollapseRelationship,
  Anisotropy,
  Isotropy,
  Hubness,
  CurseOfDimensionality,
  ConcentrationOfMeasure,
  IntrinsicDimension,
  EffectiveDimensionRelationship,
  Rank,
  CovarianceMatrix,
  GramMatrix,
  PairwiseDistanceMatrix,
  SimilarityMatrix,
  KernelMatrixRelationship,
  ComputationalMatrix,
  VectorComputation,
  TensorComputation,
  SparseDataStructure,
  DenseDataStructure,
  AutomaticDifferentiation,
  Backpropagation,
  OptimisationForMachineLearning,
  GradientFlow,
  GradientDescent,
  StochasticGradientDescent,
  NeuralNetworkMathematics,
  LossFunction,
  ObjectiveFunction,
  Regularisation,
  L2Regularisation,
  OrthogonalityRegularisationRelationship,
  UniformityLossRelationship,
  AlignmentLossRelationship,
  CentreLossRelationship,
  TrainingData,
  ValidationData,
  TestData,
  Sampling,
  Batch,
  MiniBatch,
  RandomSeed,
  ComputationalComplexity,
  TimeComplexity,
  SpaceComplexity,
  StorageComplexity,
  RetrievalComplexity,
  IndexConstructionCost,
  SearchCost,
  Verification,
  NormCheck,
  SimilarityCheck,
  DistanceCheck,
  NeighbourhoodCheck,
  RetrievalRecall,
  RecallAtK,
  PrecisionAtKRelationship,
  MeanReciprocalRankRelationship,
  MAPRelationship,
  ClusteringQualityRelationship,
  ReconstructionError,
  DistortionCheck,
  ReferenceEmbedding,
  Benchmark,
  Reproducibility,
  Provenance,
  DatasetVersion,
  ModelVersion,
  EmbeddingVersion,
  LibraryVersion,
  ImplementationVersion
 ]
 EXCLUDE_AS_PRIMARY = [
  PureVectorArithmeticWithoutRepresentationLearningContext,
  CoordinateGeometryWithoutRepresentationMapping,
  DimensionalityReductionWhenProjectionMethodRatherThanEmbeddingSemanticsIsPrimary,
  DatabaseIndexingWithoutVectorRepresentationSemantics
 ]
 DISAMBIGUATION_RULE = "When objects are mapped into a continuous vector space so relationships can be expressed through geometry, similarity or neighbourhood structure, interpret the context as embedding mathematics."
}

CORE_CONCEPTS {
 EmbeddingMathematics = "Mathematics of mapping objects into vector representation spaces."
 EmbeddingMap = "Function mapping an object to a vector representation."
 EmbeddingSpace = "Continuous vector space containing embedded representations."
 EmbeddingDimension = "Number of coordinates in an embedding vector."
 Similarity = "Numerical measure of representational closeness or alignment."
 Distance = "Metric or dissimilarity measure between embeddings."
 Neighbourhood = "Set of nearby representations according to a metric or similarity."
 RepresentationLearning = "Learning an embedding map from data or task objectives."
}

RELATIONS {
 EmbeddingMathematics --is_child_of--> AIComputationalMathematicsOntology
 Embedding --maps--> ObjectToVector
 EmbeddingVector --belongs_to--> EmbeddingSpace
 EmbeddingSpace --has--> EmbeddingDimension
 CosineSimilarity --compares--> EmbeddingVectors
 EuclideanDistance --measures--> EmbeddingSeparation
 KNearestNeighbours --uses--> DistanceOrSimilarity
 Retrieval --ranks_by--> EmbeddingSimilarity
 ContrastiveLearning --optimises--> EmbeddingGeometry
 MatrixFactorisation --can_produce--> Embeddings
 AutoencoderEmbeddingRelationship --produces--> LatentRepresentation
 GraphEmbedding --represents--> GraphEntities
 KnowledgeGraphEmbedding --represents--> EntitiesAndRelations
 Verification --checks--> EmbeddingGeometryAndTaskBehaviour
}

OPERATORS {
 EMBED(phi,x) := phi(x)
 COSINE(x,y) := dot(x,y)/(norm(x)*norm(y))
 EUCLIDEAN(x,y) := norm(x-y)
 NORMALISE(x) := x/norm(x)
 NEAREST(E,q,k) := top_k_by_similarity_or_distance(E,q,k)
 PAIRWISE_DISTANCE(E) := matrix(distance(e_i,e_j))
 GRAM(E) := E*transpose(E)
 CONTRASTIVE_OBJECTIVE(pos,neg,tau) := similarity_based_contrastive_loss(pos,neg,tau)
}

AXIOMS {
 A1 = "An embedding maps objects into a continuous representation space."
 A2 = "Embedding geometry depends on the chosen map, objective and metric."
 A3 = "High similarity in an embedding space does not imply universal semantic equivalence."
 A4 = "Distances are meaningful only relative to the geometry and normalisation of the embedding space."
 A5 = "Embedding dimensionality constrains representation capacity and computational cost."
 A6 = "Learned embeddings reflect both training data and optimisation objectives."
 A7 = "Nearest-neighbour relationships can change under embedding drift or model updates."
 A8 = "Embedding results require explicit map, dimension, metric, normalisation, objective, validation and version semantics."
}

REASONING {
 R1_CONTEXT = identify(object_type,embedding_goal,representation_model,dimension,metric,normalisation,task)
 R2_MAP = define_or_select_embedding_function)
 R3_SPACE = define_dimension_geometry_basis_and_normalisation)
 R4_OBJECTIVE = choose_reconstruction_contrastive_metric_learning_factorisation_or_task_specific_loss)
 R5_TRAIN = optimise_embedding_map_or_latent_factors)
 R6_GEOMETRY = analyse_similarity_distance_neighbourhood_separation_and_distortion)
 R7_TASK = evaluate_retrieval_clustering_classification_alignment_or_downstream_performance)
 R8_ROBUSTNESS = assess_drift_hubness_anisotropy_collapse_and_distribution_shift)
 R9_SCALE = if_large choose_ANN_indexing_parallel_or_GPU_execution)
 R10_REPRODUCE = document_model_dataset_metric_normalisation_dimension_versions_and_provenance)
}

CONSTRAINTS {
 C1 = "Cosine similarity is undefined for zero vectors without explicit handling."
 C2 = "Euclidean distance can become less discriminative in very high dimensions."
 C3 = "Embedding quality is task dependent."
 C4 = "Dimensionality reduction can distort neighbourhood or global geometry."
 C5 = "Contrastive embeddings depend strongly on positive and negative sampling."
 C6 = "Representation collapse can destroy useful geometry."
 C7 = "Hubness can distort nearest-neighbour retrieval in high-dimensional spaces."
 C8 = "Approximate nearest-neighbour search can trade exactness for speed."
 C9 = "Embedding drift can invalidate previously stored similarity thresholds or indexes."
 C10 = "Two embedding spaces cannot be compared directly unless alignment or shared semantics are established."
}

SEMANTIC_EQUIVALENCE {
 "KNearestNeighbours" <=> "KNN"
 "ApproximateNearestNeighbour" <=> "ANN"
 "PrincipalComponentAnalysis" <=> "PCA"
 "SingularValueDecomposition" <=> "SVD"
}

VALIDATION {
 V1 = "Embedding model or mapping function is documented."
 V2 = "Embedding dimension is documented."
 V3 = "Distance or similarity metric is documented."
 V4 = "Normalisation semantics are documented."
 V5 = "Training objective and sampling strategy are documented for learned embeddings."
 V6 = "Retrieval or neighbourhood behaviour is measured using task-relevant metrics."
 V7 = "Embedding drift and version changes are tracked."
 V8 = "Collapse, anisotropy and hubness are checked where relevant."
 V9 = "Approximate search recall is documented where ANN indexing is used."
 V10 = "Results are reproducible under documented dataset, model, embedding, metric and implementation versions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 EMBEDDING_MATHEMATICS_DEFINITION = 0.99
 SIMILARITY_REASONING = 0.99
 DISTANCE_REASONING = 0.99
 NEIGHBOURHOOD_REASONING = 0.98
 CONTRASTIVE_LEARNING_REASONING = 0.98
 ALIGNMENT_REASONING = 0.98
 ROBUSTNESS_REASONING = 0.97
 POLICY = "Reduce confidence when metric, normalisation, dimension, training objective, sampling, alignment or embedding version are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,EmbeddingMathematicsReasoning,
  EmbeddingSpaceReasoning,VectorEmbeddingReasoning,
  SimilarityReasoning,DistanceReasoning,NeighbourhoodReasoning,
  RetrievalReasoning,ContrastiveLearningReasoning,
  MetricLearningReasoning,DimensionalityReasoning,
  LatentSpaceReasoning,GraphEmbeddingReasoning,
  KnowledgeGraphEmbeddingReasoning,AlignmentReasoning,
  HubnessReasoning,AnisotropyReasoning,DriftReasoning,
  VerificationReasoning,KnowledgeGraphConstruction,
  RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "embedding mathematics" -> EmbeddingMathematics
  "vector embedding" -> VectorEmbedding
  "KNN" -> KNearestNeighbours
  "ANN" -> ApproximateNearestNeighbour
  "PCA" -> PrincipalComponentAnalysis
  "SVD" -> SingularValueDecomposition
 }
}

EXECUTION_RULES {
 E1 = "If objects are represented as vectors for geometric or similarity-based computation, activate EmbeddingMathematicsOntology."
 E2 = "Define embedding dimension, metric and normalisation before interpreting similarity."
 E3 = "Choose training objective according to reconstruction, contrastive, metric-learning or downstream goals."
 E4 = "Evaluate local neighbourhood and task behaviour rather than relying only on training loss."
 E5 = "Check for representation collapse, anisotropy, hubness and drift where relevant."
 E6 = "Use alignment transformations before comparing independently trained embedding spaces."
 E7 = "Use ANN methods only with explicit recall-versus-speed trade-offs."
 E8 = "Track model and embedding versions because neighbourhoods can change after retraining."
 E9 = "Document dataset, model, metric, normalisation, dimension and implementation versions for reproducibility."
}

RECURSION {
 ENTRY = EmbeddingMathematics
 EXPAND(EmbeddingMathematics) -> [
  Embedding,VectorEmbedding,LearnedEmbedding,RepresentationLearning,
  LatentRepresentation,EmbeddingVector,EmbeddingSpace,LatentSpace,
  EmbeddingDimension,VectorSpace,MetricSpace,Normalisation,
  EuclideanDistance,CosineDistance,CosineSimilarity,InnerProduct,
  Neighbourhood,KNearestNeighbours,KNN,ApproximateNearestNeighbour,ANN,
  Retrieval,Clustering,Projection,DimensionalityReduction,PCA,
  SpectralEmbedding,AutoencoderEmbeddingRelationship,
  MatrixFactorisation,LatentFactorModel,GraphEmbedding,NodeEmbedding,
  KnowledgeGraphEmbedding,EntityEmbedding,ContrastiveLearning,
  ContrastiveLoss,TripletLoss,MetricLearning,NegativeSampling,
  NeighbourhoodPreservation,Distortion,EmbeddingAlignment,
  RepresentationDrift,Collapse,Anisotropy,Hubness,
  CurseOfDimensionality,IntrinsicDimension,CovarianceMatrix,GramMatrix,
  AutomaticDifferentiation,Backpropagation,OptimisationForMachineLearning,
  GradientFlow,NeuralNetworkMathematics,
  Verification,RetrievalRecall,RecallAtK,Reproducibility,Provenance
 ]
 EXPAND(EmbeddingMathematicsFamilies) -> [
  VectorAndLatentSpaceRepresentations,
  DistanceSimilarityAndNeighbourhoodGeometry,
  DimensionalityReductionAndProjection,
  ContrastiveAndMetricLearning,
  GraphLanguageAndKnowledgeGraphEmbeddings,
  EmbeddingAlignmentAndSpaceTransformation,
  RetrievalIndexingAndApproximateSearch,
  RobustnessVerificationAndReproducibility
 ]
 STOP_CONDITION = "Stop when embedding map, dimension, metric, normalisation, learning objective, geometry, task behaviour and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "AI Computational Mathematics Ontology"
 RELATED_ONTOLOGIES = [
  "AI Computational Mathematics Ontology",
  "Neural Network Mathematics Ontology 2",
  "Automatic Differentiation Ontology",
  "Backpropagation Mathematics Ontology",
  "Optimisation for Machine Learning Ontology",
  "Gradient Flow Ontology",
  "Vector Computation Ontology",
  "Computational Matrix Ontology",
  "Tensor Computation Ontology",
  "Computational Graph Ontology",
  "Graph Algorithms Ontology",
  "Computational Statistics Ontology",
  "High Performance Computing Ontology 2",
  "GPU Computing Ontology"
 ]
 ROUTE_RULES = [
  "Embedding Mathematics -> AI Computational Mathematics Ontology for parent semantics",
  "Embedding Mathematics -> Neural Network Mathematics for learned representation models",
  "Embedding Mathematics -> Automatic Differentiation, Backpropagation and Optimisation for training semantics",
  "Embedding Mathematics -> Gradient Flow for continuous optimisation relationships",
  "Embedding Mathematics -> Vector, Matrix and Tensor ontologies for representation structures",
  "Embedding Mathematics -> Computational Graph and Graph Algorithms for graph-embedding relationships",
  "Embedding Mathematics -> Computational Statistics for dimensionality, covariance and sampling relationships",
  "Embedding Mathematics -> HPC and GPU Computing for large-scale embedding training and retrieval"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible embedding models, metrics or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve object-versus-embedding distinction.",
  "Preserve embedding-map-versus-embedding-space distinction.",
  "Preserve metric-versus-similarity distinction.",
  "Preserve raw-vector-versus-normalised-vector distinction.",
  "Preserve training-objective-versus-downstream-quality distinction.",
  "Preserve exact-neighbour-versus-approximate-neighbour distinction.",
  "Preserve embedding-version-versus-stable-semantic-identity distinction.",
  "Preserve machine-readable embedding-mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Embedding mathematics maps objects into continuous vector spaces so relationships can be represented through geometry, similarity and neighbourhood structure."
 SPACE_SUMMARY = "Embedding dimension, basis, metric and normalisation define the geometry of the representation space."
 SIMILARITY_SUMMARY = "Cosine similarity, inner products and distances provide different notions of representational closeness."
 LEARNING_SUMMARY = "Contrastive, reconstruction, factorisation and metric-learning objectives shape embedding geometry."
 RETRIEVAL_SUMMARY = "Nearest-neighbour and approximate-neighbour methods use embedding geometry for retrieval and ranking."
 VALIDATION_SUMMARY = "Embeddings should be checked for neighbourhood quality, retrieval performance, distortion, collapse, anisotropy, hubness, drift and reproducible version metadata."
}

FLOW {
 INPUT
 -> detect_embedding_mathematics_context
 -> identify_object_type_embedding_goal_model_dimension_metric_normalisation_task_and_scale
 -> define_or_select_embedding_function
 -> define_embedding_space_geometry_dimension_basis_and_normalisation
 -> choose_reconstruction_contrastive_metric_learning_factorisation_or_task_specific_objective
 -> train_or_compute_embedding_representation
 -> analyse_similarity_distance_neighbourhood_separation_and_distortion
 -> evaluate_retrieval_clustering_classification_alignment_or_downstream_task_behaviour
 -> assess_collapse_anisotropy_hubness_drift_and_distribution_shift
 -> if_large_build_ANN_index_or_use_parallel_GPU_execution
 -> verify_embedding_metrics_and_reference_behaviour
 -> document_dataset_model_embedding_metric_normalisation_versions_and_provenance
 -> route_to_related_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-AI-COMPUTATIONAL-MATHEMATICS-EMBEDDING-MATHEMATICS-ONTOLOGY-V1"
}

}

END_FLOW