Matrix and Tensor Representations

Matrix and Tensor Representations are structured Behavioral Representations whose scientific meaning is distributed jointly across two or more explicitly declared logical axes. These axes include their coordinate domains, element semantics, units or value types, support mappings, masks, and structural constraints. It is essential to establish immediately that a matrix or tensor representation is not merely defined by being stored in a two- or N-dimensional programming array. Furthermore, the tensor order or tensor rank in this context is distinct from the matrix rank known in linear algebra. Here, the term tensor denotes a semantically indexed multidimensional representation rather than automatically asserting the coordinate-transformation properties associated with mathematical tensors in multilinear or differential geometry.

Meaning and Boundaries of Matrix and Tensor Representation

A Matrix or Tensor Behavioral Representation is a regular multidimensional arrangement of representation elements indexed jointly by two or more declared logical axes. The scientific meaning of each element arises from the full tuple of axis coordinates that index it. The representation's identity encompasses:

• Axis names and roles: What each axis semantically represents and its scientific function.
• Axis order: The fixed ordering of axes that defines the logical indexing scheme.
• Coordinate values or IDs: The explicit set of coordinate identifiers along each axis.
• Shape policy: The cardinality (size) of each axis and constraints on shape.
• Element semantics: The scientific meaning assigned to each element, including units or value type restrictions.
• Support mapping: The declared domain of validity or observed coordinates.
• Missingness and sparse/default semantics: How absent or default values are handled and interpreted.
• Structural symmetries: Any declared symmetry, antisymmetry, directionality, or diagonal properties.
• Representation version: The versioning or revision history of the representation definition.

Key Distinctions

Order, Shape, Named Axes, and Coordinates

A generic tensor-element abstraction for the ith representation instance is:

Here:

• i indexes the representation instance.
• Z_i is one matrix or tensor Representation Instance.
• k is the number of declared logical axes.
• j_r is an index along logical axis r for r = 1, …, k.
• 𝒱 is the declared element value domain.

The scientific meaning of the element requires the semantic identity and coordinate associated with every j_r, not merely its stored numeric value.

The tensor shape is defined as:

Where:

• n_r is the cardinality of logical axis r.
• k is the tensor order or number of logical axes.

Note that k is not matrix rank, intrinsic dimension, sample size, or information content. Equal shape does not establish semantic compatibility.

Named Axes

Named axes serve as stable semantic keys of matrix and tensor structure. Representative axes include:

• time
• descriptor
• channel
• sensor
• joint
• coordinate
• frequency
• scale
• entity
• source
• target
• modality
• participant
• trial
• condition
• component

Positional labels such as axis 0 or last dimension are implementation references and should never substitute for scientific axis identity.

Axis Coordinates

Axis coordinates are the values or IDs identifying locations along an axis. Examples include:

• Timestamps (for time axis).
• Frequencies (for frequency axis).
• Scale values (for scale axis).
• Descriptor IDs (for descriptor axis).
• Joint names (for joint axis).
• Coordinate labels such as x, y, z (for coordinate axis).
• Participant or modality IDs.
• Entity IDs.

Axis cardinality (length) alone does not reveal coordinate semantics. For example, an axis of length 17 could represent 17 joints, 17 frequencies, 17 participants, or 17 latent components.

Axis Role Comparison

Canonical Behavioral Matrix and Tensor Schemas

Representative matrix schemas include:

• time × descriptor: Common in time series feature analysis; rows correspond to time points, columns to features or descriptors.
• channel × time: Sensor signals recorded over time; rows are channels, columns are time samples.
• time × frequency: Time-frequency representations; rows are time points, columns frequency bins.
• joint × coordinate: Spatial pose data; rows are joints, columns spatial dimensions.
• entity × entity: Pairwise relationships or interactions between entities.

Each two-dimensional shape can encode fundamentally different semantics. The interpretation of rows and columns, their coordinate identities, units, support domain, and any diagonal or symmetry rules belong to the representation definition, not to shape alone.

Representative higher-order tensor schemas include:

• joint × coordinate × time: Pose trajectories over time.
• participant × time × descriptor: Experimental data across multiple participants.
• sensor × axis × time: Multichannel inertial or EEG signals with spatial axes.
• source × target × frequency: Pairwise interactions resolved over frequencies.
• scale × time × descriptor: Multiscale time series features.
• entity × time × component: Multivariate dynamic entity properties.

These examples illustrate axis composition rather than a fixed ontology. Adding an axis creates a new structural distinction only if the axis carries a defensible scientific role.

Mixed Semantic Axes and Unit Structure

A tensor can comprise axes with heterogeneous semantic and physical units. For example, one axis's coordinates might correspond to descriptors with different physical units (e.g., meters, seconds, volts), another axis may represent time, and another participants. Homogeneous numeric data type (dtype) does not imply homogeneous scientific units. Operations across mixed-unit axes require unit-aware or otherwise justified semantics to avoid invalid interpretations.

Canonical Schemas Comparison

Axis Permutation, Slicing, Stacking, and Alignment

Axis permutation (or transpose) refers to reordering the logical axes of a representation. A permutation preserves the same scientific representation only if every axis identity, coordinate mapping, and element correspondence is preserved. While permutation changes storage/index order, it does not necessarily alter scientific content. Losing the permutation map or incorrectly relabeling axes creates a semantically different or invalid representation, even if all numeric values remain intact.

Slicing and subsetting restrict the representation to declared coordinates or ranges along one or more axes. Slicing can preserve the representation schema over a smaller support or component set but changes support, available entities/components, boundary context, or comparability when the removed coordinates matter. A positional slice must be interpreted through coordinate identity rather than array position alone.

Stacking introduces a new logical axis that collects compatible matrix/tensor instances or components. Stacking requires a meaningful new coordinate, such as participant, trial, condition, or modality. Concatenation extends an existing axis, requiring compatible semantics on all unaffected axes and a defensible ordering or coordinate union on the extended axis.

Coordinate-aware alignment before combination is critical:

• Inner alignment retains only coordinates shared by all inputs.
• Outer alignment forms a union of all coordinates, potentially introducing missing entries.
• Left/right/reference alignment follows a declared coordinate set.

Same axis length does not prove coordinate identity. Silently aligning by position risks pairing different participants, timestamps, frequencies, descriptors, joints, or entities.

Structural Operations Comparison

Reshape, Flattening, Broadcasting, and Reduction Boundaries

Reshape changes the arrangement of stored values into another shape under an explicit index mapping. Reshape is scientifically semantics-preserving only when the new logical axes and the mapping between old and new index tuples are declared. Preserving element count and numeric order alone is insufficient when the new shape obscures which dimensions correspond to time, descriptors, joints, or entities.

Flattening merges two or more logical axes into one composite axis. A reversible flattening requires a composite-coordinate map that can recover the original axis identities and coordinates. Without such a map, values remain numerically intact but the representation loses semantic structure. For example, flattening joint × coordinate × time into one vector differs scientifically from retaining the original axes, even if an inverse array reshape is technically possible.

Singleton-axis insertion/removal and broadcasting:

• Inserting a size-one axis allows value reuse across another axis.
• Broadcasting replicates one value or vector across compatible shapes.

Shape compatibility is not semantic compatibility. Broadcasting participant-level context across time can be legitimate, but broadcasting one participant's value across other participant IDs manufactures unsupported evidence.

Reductions and contractions collapse an axis by summing, averaging, maximizing, integrating, or other operations. Reduction removes that axis and produces another representation or descriptor-like summary. The semantics depend on the operation and weighting. Axis reduction is not a neutral reshape and should not be treated as information preserving unless the removed distinction is intentionally irrelevant under the declared purpose.

Structural Operations Comparison

Dense, Sparse, Masked, and Block-Structured Forms

Dense matrix/tensor forms explicitly store every coordinate tuple in the logical domain.

Sparse forms store only selected entries plus explicit default or absence semantics. An unstored sparse entry can mean mathematical zero, no relation, unavailable value, unobserved coordinate, or another declared state. Sparse storage is not scientifically interpretable without its default semantics.

Masks and validity structures provide information separate from stored element values. They distinguish:

• Valid zero
• Missing
• Invalid
• Padded
• Structurally inapplicable
• Censored
• Unavailable
• Unobserved entries

A tensor filled with zeros and a mask can represent something fundamentally different from the same numeric tensor in which every zero is a genuine observation.

Block-structured matrices/tensors group subsets of coordinates into scientifically meaningful blocks such as modality blocks, body regions, sensor groups, descriptor families, within- versus between-entity regions, or multiscale bands. Block identity carries semantics not recoverable from shape alone. Reordering or combining blocks requires preserving membership and coordinate lineage.

Ragged and variable-cardinality boundaries arise when the regular tensor assumption of a rectangular Cartesian product of axis coordinate sets fails. Variable-length sequences, changing entity sets, variable landmark counts, or irregular nested structures may require padding/masks, ragged containers, sets, graphs, or composite representations rather than forcing every instance into the same dense tensor domain.

Storage Form Comparison

Repeated Axes, Symmetry, Directionality, and Diagonals

Repeated-domain axes occur in representations such as:

• entity × entity
• channel × channel
• joint × joint
• source × target

Two axes can draw coordinates from the same entity set but play different roles. It is critical to preserve source/target or row/column role identity. An asymmetric relation matrix can change meaning under transpose even when both axes contain identical labels.

Symmetric, antisymmetric, directed/asymmetric, triangular, and unconstrained matrix structures are declared representational constraints:

• Symmetry can encode undirected relations or mathematical properties of the represented quantity.
• Symmetry must not be inferred solely from equal axis labels.
• Storing only one triangle is semantics-preserving only when the omitted half is recoverable under a declared symmetry rule.

Diagonal semantics in repeated-axis matrices vary widely:

• Self-relation
• Variance or self-similarity
• Unity by normalization
• Zero by definition
• Unavailable self-comparison
• Other domain-specific quantities

Do not force diagonals to zero or one merely because a matrix resembles adjacency, distance, correlation, covariance, or similarity matrices.

A relation matrix/tensor is distinct from a graph representation:

• A matrix organizes pairwise values over repeated axes without declaring nodes, edges, topology, edge existence, graph attributes, or graph-theoretic semantics.
• Thresholding or interpreting nonzero entries as edges creates an additional relational/graph mapping rather than exposing an inherent graph in every pairwise matrix.

Repeated-Axis and Structural Constraint Comparison

Semantic Compatibility, Round Trips, and Provenance

Semantic compatibility between Matrix/Tensor Representation Instances requires:

• Compatible representation definitions and versions.
• Named axes with matching roles.
• Coordinate identities and units/value semantics aligned.
• Support and fitted state consistent.
• Missingness and default value rules compatible.
• Structural constraints (symmetry, directionality) matched.

Equal data type and shape are weak implementation checks and do not establish scientific compatibility.

Semantic round trips involve inverse operations such as transpose/inverse-transpose, flatten/unflatten, dense/sparse conversion, padding/unpadding, serialization/deserialization, and chunk/reassembly. A round trip is scientifically lossless only when values and all metadata needed to recover axis, component, and support semantics survive. Recovering the same byte or numeric sequence while losing coordinate identities is not a semantic round trip.

Logical axis order is distinct from physical storage and memory layout. Variations such as row-major vs. column-major layout, strides, chunk order, compression blocks, contiguous vs. noncontiguous views, device placement, and library-specific tensor layouts affect computation but do not define the scientific order or meaning of logical axes. Scientific serialization should preserve sufficient metadata to reconstruct logical structure independently of any particular library's memory conventions.

Integrated Worked Comparison: Walking Episode Representations

Consider a walking episode recorded with multiple sensor modalities and descriptors:

• time × descriptor matrix: A contour of behavioral features extracted at each time point.
• sensor × axis × time tensor: Multichannel inertial signals, where axis indexes spatial dimensions (e.g., x, y, z).
• joint × coordinate × time tensor: Pose landmarks capturing joint positions over time.
• time × frequency matrix: Time-frequency spectral representation of signals.
• source × target × frequency tensor: Pairwise relational descriptors (e.g., connectivity) across frequencies.

Demonstrations:

• Safe axis permutation: The pose tensor joint × coordinate × time permuted to time × joint × coordinate remains semantically equivalent only when an explicit permutation map preserves axis identities and coordinate correspondences.

• Same-shaped but semantically incompatible matrix: A time × descriptor matrix with identical shape but where rows encode sensor channels and columns frequency bins is semantically different despite shape equality.

• Coordinate-aware concatenation: Time-series matrices from multiple trials concatenated along a new trial axis with explicit trial IDs preserve semantic clarity.

• Outer alignment with missing entries: Combining sensor data from sessions with differing sensor sets requires outer alignment on the sensor axis, introducing missing entries for absent sensors.

• Reversible flattening with composite-index map: Flattening the joint × coordinate × time tensor into a vector using a composite coordinate map enables full unflattening and semantic recovery.

• Flattening losing semantic structure: Flattening without preserving the coordinate map leads to unrecoverable semantics, converting structured pose data into an opaque vector.

• Valid zero distinct from missing: A connectivity tensor where a zero indicates no detected connection is semantically different from a missing or unobserved entry, which must be masked or otherwise marked.

• Repeated-axis tensor with domain-specific diagonal: A source × target × frequency tensor where diagonal entries represent self-connection strength, a domain-specific quantity not defaulting to zero or one.

• Relation matrix is not automatically a graph: Thresholding the entity × entity relation matrix to form edges is an additional operation; the matrix itself lacks inherent graph semantics.

• Time-frequency matrix is a transform representation: Despite matrix shape, the time × frequency matrix represents a transform domain, not a descriptor or sample matrix.

Axis Permutation, Slicing, Stacking, and Alignment

Reshape, Flattening, Broadcasting, and Reduction Boundaries

Dense, Sparse, Masked, and Block-Structured Forms

Repeated Axes, Symmetry, Directionality, and Diagonals

Semantic Compatibility, Round Trips, and Provenance

Matrix and Tensor Representation provenance comprises the comprehensive information required to reconstruct and scientifically interpret each multidimensional structure. This includes, when material:

• Representation definition and version.
• Tensor order and shape policy.
• Named axes and their canonical order.
• Axis roles and coordinate domains/IDs including ordering.
• Element semantics, units, and value types.
• Source/support mapping and transformations used to assemble axes.
• Stack, concatenation, and alignment policies.
• Reshape and flattening maps.
• Masks and missingness semantics.
• Sparse default semantics.
• Block structure and coordinate groupings.
• Repeated-axis role identities.
• Symmetry, directionality, and diagonal constraints.
• Fitted-space or latent representation identity if applicable.
• Distinction between logical axis order and physical storage/memory layout.
• Implementation details and version.
• Comparability rules, uncertainty, and sensitivity information.

A defensible matrix or tensor representation is defined by the semantics of its axes and their joint indexing, not by dimensionality or storage shape alone. Preserving this metadata enables scientific reproducibility, valid combination, and meaningful interpretation.