Feature Space Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Feature Space Ontology within the Kireth Research Semantic Library. Feature-space mathematics studies the mathematical representation of data as points, vectors, tensors or structured objects in spaces used for machine learning and artificial intelligence. This ontology supports features, coordinates, embeddings, basis transformations, scaling, normalisation, distance, similarity, kernels, latent spaces, manifolds, dimensionality reduction, sparsity, separability, metric structure and relationships with AI mathematics, neural networks, statistical learning, linear algebra, geometry and information theory.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Feature Space Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Artificial Intelligence Mathematics / Feature Space"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/artificial-intelligence-mathematics/feature-space-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = FeatureSpace
 ENTITY_CLASS = MathematicalRepresentationSpaceFramework
 PRIMARY_CONTEXT = AIMathematics
 DEFINITION = "A feature space is a mathematical space in which observations are represented by coordinates, vectors, tensors or structured embeddings corresponding to measurable or learned features."
 SYMBOLIC_FORMS = [
  "x in R^d",
  "phi:X->F",
  "z=W*x+b",
  "d(x,y)=||x-y||"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for feature vectors, coordinates, embeddings, similarity, distance, scaling, dimensionality, latent representations, kernels and geometry used by AI systems."
}

SCOPE {
 INCLUDE = [
  FeatureSpace,
  Feature,
  FeatureVector,
  Coordinate,
  Dimension,
  Dimensionality,
  InputSpace,
  RepresentationSpace,
  EmbeddingSpace,
  LatentSpace,
  VectorSpace,
  InnerProductSpace,
  NormedSpace,
  MetricSpace,
  HilbertSpace,
  EuclideanSpace,
  Manifold,
  FeatureMap,
  EmbeddingFunction,
  CoordinateSystem,
  Basis,
  BasisVector,
  Orthogonality,
  Projection,
  LinearTransformation,
  AffineTransformation,
  Standardisation,
  Normalisation,
  MinMaxScaling,
  Whitening,
  Centering,
  Distance,
  EuclideanDistance,
  ManhattanDistance,
  MinkowskiDistance,
  MahalanobisDistance,
  CosineSimilarity,
  InnerProduct,
  SimilarityMeasure,
  Kernel,
  KernelFunction,
  KernelSpace,
  ReproducingKernelHilbertSpace,
  RKHS,
  SparseFeatureVector,
  DenseFeatureVector,
  Sparsity,
  FeatureSelection,
  FeatureExtraction,
  FeatureEngineering,
  LearnedFeature,
  HandcraftedFeature,
  RepresentationLearning,
  DimensionalityReduction,
  PrincipalComponentAnalysis,
  PCA,
  SingularValueDecomposition,
  SVD,
  LinearDiscriminantAnalysis,
  LDA,
  AutoencoderEmbedding,
  ManifoldLearning,
  Isomap,
  LocallyLinearEmbedding,
  tSNE,
  UMAP,
  ClusterStructure,
  ClassSeparability,
  DecisionBoundary,
  Margin,
  CovarianceStructure,
  CorrelationStructure,
  FeatureDistribution,
  FeatureScaling,
  FeatureInteraction,
  PolynomialFeature,
  NonlinearFeature,
  HighDimensionalSpace,
  CurseOfDimensionality,
  ConcentrationOfMeasure,
  IntrinsicDimension,
  ExtrinsicDimension,
  FeatureRedundancy,
  FeatureRelevance,
  FeatureImportance,
  FeatureNoise,
  DistributionShift,
  CovariateShift,
  DomainShift
 ]
 EXCLUDE_AS_PRIMARY = [
  RawDataWithoutRepresentationSemantics,
  GeneralVectorSpaceWithoutFeatureInterpretation,
  NeuralArchitectureWithoutRepresentationFocus,
  PureDatabaseSchemaWithoutMathematicalFeatureSpace
 ]
 DISAMBIGUATION_RULE = "When observations are mathematically represented as coordinates, vectors, tensors or embeddings for learning, similarity, classification or geometric analysis, interpret the context as feature-space mathematics."
}

CORE_CONCEPTS {
 FeatureSpace = "Mathematical space used to represent observations by features."
 FeatureVector = "Ordered numerical or symbolic representation of an observation."
 FeatureMap = "Function mapping raw objects into a feature space."
 EmbeddingSpace = "Representation space in which objects are encoded as learned vectors or structured coordinates."
 Distance = "Function quantifying separation between feature representations."
 SimilarityMeasure = "Function quantifying likeness between representations."
 LatentSpace = "Internal representation space of hidden variables or learned factors."
 DimensionalityReduction = "Transformation from a higher-dimensional representation to a lower-dimensional one while preserving selected structure."
 Sparsity = "Property that many feature coordinates are zero or inactive."
 ClassSeparability = "Degree to which classes occupy distinguishable regions of feature space."
}

RELATIONS {
 FeatureSpace --is_child_of--> AIMathematics
 FeatureSpace --contains--> FeatureVector
 FeatureVector --contains--> Coordinate
 FeatureMap --maps--> RawObjectToFeatureSpace
 EmbeddingFunction --maps--> ObjectToEmbeddingSpace
 Basis --spans--> VectorSpace
 Projection --maps--> HigherToLowerSubspace
 Standardisation --transforms--> FeatureDistribution
 Whitening --decorrelates--> Features
 Distance --operates_on--> FeatureVector
 CosineSimilarity --uses--> InnerProduct
 MahalanobisDistance --uses--> CovarianceStructure
 KernelFunction --induces--> KernelSpace
 KernelSpace --may_be--> ReproducingKernelHilbertSpace
 FeatureSelection --reduces--> Dimensionality
 FeatureExtraction --constructs--> NewFeatures
 RepresentationLearning --learns--> EmbeddingSpace
 PCA --uses--> CovarianceStructure
 PCA --produces--> PrincipalSubspace
 SVD --supports--> LowRankRepresentation
 ManifoldLearning --models--> NonlinearLowDimensionalStructure
 ClassSeparability --affects--> ClassificationDifficulty
 Margin --measures--> SeparationFromDecisionBoundary
 CurseOfDimensionality --affects--> HighDimensionalSpace
 DistributionShift --changes--> FeatureDistribution
}

OPERATORS {
 FEATURE_MAP(phi,x) := z=phi(x)
 STANDARDISE(x,mu,sigma) := z=(x-mu)/sigma
 MINMAX_SCALE(x,a,b,min_x,max_x) := z=a+(x-min_x)*(b-a)/(max_x-min_x)
 CENTER(x,mu) := z=x-mu
 EUCLIDEAN_DISTANCE(x,y) := sqrt(sum_i (x_i-y_i)^2)
 MANHATTAN_DISTANCE(x,y) := sum_i abs(x_i-y_i)
 MINKOWSKI_DISTANCE(x,y,p) := (sum_i abs(x_i-y_i)^p)^(1/p)
 MAHALANOBIS_DISTANCE(x,y,Sigma) := sqrt((x-y)^T*Sigma^(-1)*(x-y))
 COSINE_SIMILARITY(x,y) := (x^T*y)/(||x||*||y||)
 LINEAR_PROJECTION(W,x) := z=W*x
 PCA_PROJECTION(U_k,x,mu) := z=U_k^T*(x-mu)
 KERNEL_MAP(k,x,y) := k(x,y)=<phi(x),phi(y)>
 SPARSITY(x) := count_i[x_i=0]/dimension(x)
 INTRINSIC_DIMENSION_APPROX(X) := estimate_minimal_dimension_preserving_selected_structure
}

AXIOMS {
 A1 = "Feature-space semantics require explicit definition of features or representation coordinates."
 A2 = "Distance and similarity depend on the geometry imposed on the feature space."
 A3 = "Feature scaling can materially alter geometry and optimisation."
 A4 = "Linear separability depends on the chosen representation."
 A5 = "High-dimensional feature spaces can exhibit concentration and sparsity effects."
 A6 = "A learned embedding is representation-dependent and objective-dependent."
 A7 = "Dimensionality reduction preserves only selected aspects of structure."
 A8 = "Feature-space analysis requires validation of geometry, scaling, distribution and task relevance."
}

REASONING {
 R1_CONTEXT = identify(objects,raw_variables,feature_representation,task,metric)
 R2_SPACE_CLASS = classify(Euclidean,inner_product,metric,kernel,latent,manifold,sparse)
 R3_PREPROCESS = select(centering,scaling,standardisation,normalisation,whitening)
 R4_GEOMETRY = analyse(distance,similarity,norm,angles,covariance)
 R5_REPRESENTATION = construct(handcrafted_features,learned_features,embeddings)
 R6_DIMENSION = analyse(extrinsic_dimension,intrinsic_dimension,redundancy,sparsity)
 R7_REDUCTION = select(PCA,SVD,LDA,autoencoder,manifold_learning_or_feature_selection)
 R8_STRUCTURE = analyse(clusters,separability,margins,neighbourhoods,manifolds)
 R9_SHIFT = assess(covariate_shift,domain_shift,feature_distribution_change)
 R10_VALIDATION = check(scale,metric_choice,stability,task_alignment,information_loss)
}

CONSTRAINTS {
 C1 = "Euclidean distance can become less informative in very high dimensions."
 C2 = "Feature scaling can change nearest neighbours, margins and optimisation trajectories."
 C3 = "Cosine similarity ignores vector magnitude."
 C4 = "Mahalanobis distance requires a suitable covariance estimate."
 C5 = "PCA preserves variance, not necessarily predictive information."
 C6 = "Nonlinear embeddings can distort global distances."
 C7 = "t-SNE and related visual embeddings should not automatically be treated as faithful global geometry."
 C8 = "Sparse and dense representations can require different computational methods."
 C9 = "Feature importance depends on model and representation context."
 C10 = "Distribution shift can invalidate feature-space relationships learned during training."
}

SEMANTIC_EQUIVALENCE {
 "FeatureSpace" <=> "RepresentationSpace"
 "EmbeddingSpace" <=> "VectorEmbeddingSpace"
 "FeatureMap" <=> "RepresentationMap"
 "Standardisation" <=> "ZScoreScaling"
 "ReproducingKernelHilbertSpace" <=> "RKHS"
 "PrincipalComponentAnalysis" <=> "PCA"
 "SingularValueDecomposition" <=> "SVD"
}

VALIDATION {
 V1 = "Feature definitions and dimensions are explicit."
 V2 = "Scaling and normalisation conventions are documented."
 V3 = "Distance or similarity function matches the feature semantics."
 V4 = "Covariance estimates are checked before Mahalanobis-type geometry."
 V5 = "Dimensionality-reduction outputs are assessed for information loss."
 V6 = "Embedding geometry is evaluated for task relevance."
 V7 = "Sparsity and computational representation are documented."
 V8 = "Class separability is evaluated independently of visual impression."
 V9 = "Distribution shift is assessed where deployment data differ from training data."
 V10 = "Feature-space transformations are reproducible."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 FEATURE_SPACE_DEFINITION = 0.99
 GEOMETRY_REASONING = 0.99
 SCALING_REASONING = 0.99
 EMBEDDING_REASONING = 0.98
 DIMENSIONALITY_REASONING = 0.98
 KERNEL_REASONING = 0.98
 MANIFOLD_REASONING = 0.97
 DISTRIBUTION_SHIFT_REASONING = 0.98
 POLICY = "Reduce confidence when feature definitions, scaling, distance metric, covariance, latent-space objective, dimensionality assumptions or distribution-shift conditions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  FeatureSpaceReasoning,
  FeatureVectorReasoning,
  FeatureMapReasoning,
  EmbeddingReasoning,
  DistanceReasoning,
  SimilarityReasoning,
  ScalingReasoning,
  NormalisationReasoning,
  KernelReasoning,
  SparseRepresentationReasoning,
  FeatureSelectionReasoning,
  FeatureExtractionReasoning,
  DimensionalityReductionReasoning,
  PCAReasoning,
  SVDReasoning,
  ManifoldLearningReasoning,
  ClassSeparabilityReasoning,
  DistributionShiftReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]
 NORMALISE {
  "feature space" -> FeatureSpace
  "representation space" -> FeatureSpace
  "feature vector" -> FeatureVector
  "embedding space" -> EmbeddingSpace
  "latent space" -> LatentSpace
  "cosine similarity" -> CosineSimilarity
  "Mahalanobis distance" -> MahalanobisDistance
  "PCA" -> PrincipalComponentAnalysis
  "SVD" -> SingularValueDecomposition
  "RKHS" -> ReproducingKernelHilbertSpace
 }
}

EXECUTION_RULES {
 E1 = "If observations are represented mathematically by features or embeddings, activate FeatureSpaceOntology."
 E2 = "If feature magnitudes differ materially, assess scaling before geometric analysis."
 E3 = "If similarity is angular rather than magnitude-based, consider cosine similarity."
 E4 = "If covariance geometry matters, consider Mahalanobis distance or whitening."
 E5 = "If dimensionality is excessive, assess PCA, SVD, feature selection or learned embeddings."
 E6 = "If nonlinear low-dimensional structure is suspected, consider manifold-learning methods."
 E7 = "If features are sparse, preserve sparse computational semantics where possible."
 E8 = "If representation changes between training and deployment, assess distribution shift."
 E9 = "Any feature transformation should be validated for information preservation and task relevance."
}

RECURSION {
 ENTRY = FeatureSpace
 EXPAND(FeatureSpace) -> [
  Feature, FeatureVector, Coordinate, Dimension, Dimensionality, InputSpace,
  RepresentationSpace, EmbeddingSpace, LatentSpace, VectorSpace,
  InnerProductSpace, NormedSpace, MetricSpace, HilbertSpace, EuclideanSpace,
  Manifold, FeatureMap, EmbeddingFunction, CoordinateSystem, Basis,
  BasisVector, Orthogonality, Projection, LinearTransformation,
  AffineTransformation, Standardisation, Normalisation, MinMaxScaling,
  Whitening, Centering, Distance, EuclideanDistance, ManhattanDistance,
  MinkowskiDistance, MahalanobisDistance, CosineSimilarity, InnerProduct,
  SimilarityMeasure, Kernel, KernelFunction, KernelSpace,
  ReproducingKernelHilbertSpace, SparseFeatureVector, DenseFeatureVector,
  Sparsity, FeatureSelection, FeatureExtraction, FeatureEngineering,
  LearnedFeature, HandcraftedFeature, RepresentationLearning,
  DimensionalityReduction, PrincipalComponentAnalysis,
  SingularValueDecomposition, LinearDiscriminantAnalysis,
  AutoencoderEmbedding, ManifoldLearning, Isomap, LocallyLinearEmbedding,
  tSNE, UMAP, ClusterStructure, ClassSeparability, DecisionBoundary,
  Margin, CovarianceStructure, CorrelationStructure, FeatureDistribution,
  FeatureScaling, FeatureInteraction, PolynomialFeature, NonlinearFeature,
  HighDimensionalSpace, CurseOfDimensionality, ConcentrationOfMeasure,
  IntrinsicDimension, ExtrinsicDimension, FeatureRedundancy,
  FeatureRelevance, FeatureImportance, FeatureNoise, DistributionShift,
  CovariateShift, DomainShift
 ]
 EXPAND(GeometryFamilies) -> [
  EuclideanGeometry,
  InnerProductGeometry,
  MetricGeometry,
  KernelGeometry,
  ManifoldGeometry,
  LatentGeometry
 ]
 STOP_CONDITION = "Stop when feature definition, space geometry, scaling, dimensionality, representation, similarity structure and distribution stability are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "AI Mathematics Ontology"
 IMMEDIATE_CONTEXTS = [
  "AI Mathematics Ontology",
  "Neural Network Mathematics Ontology",
  "Linear Algebra Ontologies",
  "Geometry Ontologies"
 ]
 RELATED_ONTOLOGIES = [
  "AI Mathematics Ontology",
  "Artificial Intelligence Mathematics",
  "Neural Network Mathematics Ontology",
  "Loss Function Ontology",
  "Linear Algebra Ontologies",
  "Geometry Ontologies",
  "Probability & Statistics Ontologies",
  "Information Theory Ontologies",
  "Kernel Method Ontologies",
  "Dimensionality Reduction Ontologies",
  "Graph Learning Ontologies",
  "Numerical Linear Algebra Ontology"
 ]
 ROUTE_RULES = [
  "Feature Space -> AI Mathematics Ontology for parent semantics",
  "Feature Space -> Neural Network Mathematics for learned representations",
  "Feature Space -> Linear Algebra for vector and matrix representation",
  "Feature Space -> Geometry for metric, manifold and angular structure",
  "Feature Space -> Probability & Statistics for covariance and feature distributions",
  "Feature Space -> Information Theory for representation quality",
  "Feature Space -> Kernel Methods for implicit high-dimensional feature maps",
  "Feature Space -> Dimensionality Reduction for lower-dimensional structure",
  "Feature Space -> Graph Learning for structured feature spaces",
  "Feature Space -> Numerical Linear Algebra for PCA, SVD and whitening"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible feature geometries, embedding methods or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve feature-coordinate semantics.",
  "Preserve input-versus-representation space distinction.",
  "Preserve distance-versus-similarity distinction.",
  "Preserve raw-versus-learned feature distinction.",
  "Preserve extrinsic-versus-intrinsic dimension distinction.",
  "Preserve metric-versus-kernel geometry distinction.",
  "Preserve scaling and distribution-shift semantics.",
  "Preserve machine-readable feature-space semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "A feature space is the mathematical representation space in which observations are encoded as feature vectors, tensors or embeddings."
 GEOMETRY_SUMMARY = "Distances, norms, similarities and kernels define the geometry used to compare feature representations."
 SCALING_SUMMARY = "Centering, standardisation, normalisation and whitening alter feature geometry and numerical behaviour."
 EMBEDDING_SUMMARY = "Embeddings map objects into learned representation spaces whose geometry reflects the training objective."
 DIMENSION_SUMMARY = "Dimensionality reduction and feature selection remove redundancy or compress structure while attempting to preserve task-relevant information."
 HIGH_DIMENSION_SUMMARY = "High-dimensional spaces can exhibit sparsity, concentration of measure and degraded distance discrimination."
 SHIFT_SUMMARY = "Feature distributions can change between training and deployment, creating covariate or domain shift."
 VALIDATION_SUMMARY = "Feature spaces should be checked for meaningful coordinates, scaling, metric choice, dimensionality, information preservation and distribution stability."
}

FLOW {
 INPUT
 -> detect_feature_space_context
 -> identify_objects_features_dimensions_representation_task_and_metric
 -> classify_Euclidean_inner_product_metric_kernel_latent_manifold_or_sparse_space
 -> apply_centering_scaling_normalisation_or_whitening_if_required
 -> construct_or_learn_feature_map
 -> analyse_distance_similarity_covariance_and_local_geometry
 -> assess_dimension_sparsity_redundancy_and_intrinsic_structure
 -> if_reduction_required_select_feature_selection_PCA_SVD_LDA_autoencoder_or_manifold_learning
 -> analyse_clusters_class_separability_margins_and_neighbourhoods
 -> assess_feature_distribution_shift_and_noise
 -> validate_geometry_information_preservation_numerical_stability_and_task_alignment
 -> route_to_AI_mathematics_neural_network_mathematics_linear_algebra_geometry_probability_statistics_information_theory_kernel_methods_dimensionality_reduction_graph_learning_or_numerical_linear_algebra_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Feature Space Ontology defines mathematical representation spaces for artificial intelligence and machine learning. It supports features and feature vectors, coordinates and dimensions, input, representation, embedding and latent spaces, vector, normed, metric, Hilbert and Euclidean spaces, manifolds, feature maps and embeddings, bases and projections, centering, standardisation, normalisation and whitening, Euclidean, Manhattan, Minkowski and Mahalanobis distances, cosine similarity and inner products, kernels and reproducing-kernel Hilbert spaces, sparse and dense representations, feature selection and extraction, representation learning, PCA, SVD, LDA, autoencoder embeddings and manifold learning, cluster structure, class separability and margins, covariance and correlation, high-dimensional geometry, curse of dimensionality, concentration of measure, intrinsic dimension, feature relevance and redundancy, noise, covariate shift and domain shift."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-FEATURE-SPACE-ONTOLOGY-V1"
}

}

END_FLOW