Embedding Field Definition

• Embedding Field Definition is a mathematical construct that maps typed multi-graph elements to high-dimensional vectors, integrating symbolic and statistical data.
• It employs metrics such as Euclidean, cosine, and Bhattacharyya distances to quantify similarity and support noise-tolerant computations.
• This framework enables hybrid models by unifying schema-driven logic with continuous embeddings, crucial for applications in computer vision, NLP, and multi-modal AI.

An embedding field is a mathematical structure that serves as the ambient metric tensor space for mapping the elements (edges and possibly vertices) of a typed, tensor-valued multi-graph into a continuous, typically high-dimensional, vector or tensor space. This construct enables a unified data structure where symbolic/logical (categorical) and statistical (Bayesian) relationships cohabit, supporting direct, cross-domain computation of similarity and distance across heterogeneous data types such as visual, linguistic, or auditory representations. The embedding field formalism provides a rigorous scaffolding for integrating discrete relational data and continuous latent representations within a single framework, with direct implications for machine learning models in computer vision, NLP, and related domains.

1. Formal Definition of the Embedding Field

Let $G = (V, E, \tau_V, \tau_E, s, t, K_V, K_E)$ denote a directed, typed, tensor-valued multi-graph, where $V$ is a set of vertices, $E$ a set of directed edges, $\tau_V : V\rightarrow T_V$ and $\tau_E : E\rightarrow T_E$ are type assignments from finite sets, $s, t : E \rightarrow V$ are source and target maps, and $K_V$, $K_E$ specify, for each node and edge, the allowed dictionaries of attribute keys (with associated domains $D_k$).

An embedding field $F$ then consists of

$F = (X, \{\|\cdot\|_i\}_{i\in I}, d_F)$

where:

• $X = \bigsqcup_{i\in I} \mathbb{R}^{d_i}$ is a disjoint union of tensor spaces (or manifolds) of varying shapes $d_i$,
• Each $\|\cdot\|_i$ is a norm on $\mathbb{R}^{d_i}$, defining the metric $d_i(x,y) = \|x-y\|_i$,
• $d_F$ extends to all of $X$ by setting the distance between points in different components as infinite (or sufficiently large).

Immersion (embedding) of $G$ into $F$ is specified by:

• $\phi_V : V \rightarrow X_V$ (often $X_V = X$ or trivial),
• $\phi_E = \eta: E \rightarrow X_E \subset X$, where for each edge $e\in E$, $\eta(e)$ assigns a vector or tensor in $X$ of appropriate shape.

In the uniform scenario (all embeddings in $\mathbb{R}^d$):

$$\eta: E \longrightarrow \mathbb{R}^d,\qquad \phi_E(e) = \eta(e) \in \mathbb{R}^d$$

with optional vertex map $\phi_V: V \rightarrow \mathbb{R}^{d'}$.

2. Metric Structure and Similarity on the Embedding Field

The embedding field endows $G$ with a metric $d_F$ on the space of edge embeddings. For $e_i, e_j \in E$:

• $$d_F(\phi(e_i),\phi(e_j)) = \|\eta(e_i) - \eta(e_j)\|_p$$, particularly Euclidean ($p=2$):

$$d_F(\eta(e_i), \eta(e_j)) = \sqrt{\sum_{k=1}^d (\eta(e_i)_k - \eta(e_j)_k)^2}$$

• Cosine similarity:

$$\mathrm{sim}_F(\eta(e_i),\eta(e_j)) = \frac{\langle \eta(e_i), \eta(e_j)\rangle}{\|\eta(e_i)\|\;\|\eta(e_j)\|}$$

• For histogram-valued embeddings $p,q$ over $X$, Bhattacharyya distance:

$$BC(p,q) = \sum_{x\in X} \sqrt{p(x)q(x)},\qquad D_B(p,q) = -\ln(BC(p,q))$$

These induced metrics allow one to define continuous notions of affinity, similarity, or noise-tolerant relational distance between edges (and consequently, the facts or relations they encode).

3. Integration of Logical/Categorical and Statistical/Bayesian Structures

Embedding fields bridge the gap between categorical (symbolic/logical) data representations and continuous (statistical/Bayesian) representations:

• The categorical/logical structure is retained by the typing functions $\tau_V$, $\tau_E$ and by the explicit schema and predicate types enforced at the graph level (e.g., relations such as $\mathrm{HAPPENS\_BEFORE}$, $\mathrm{SPATIALLY\_CONTAINS}$).
• The statistical/Bayesian side is implemented by attributing edge (or vertex) embeddings as continuous-valued vectors/tensors; similarity and distance metrics then support probabilistic, soft, or noise-tolerant computations (e.g., likelihood of two relations being “the same under noise”).
• The construction admits a functorial viewpoint: there is a covariant functor from the category of typed graphs to the category of metric spaces, sending $G$ to $F$ and immersing $G$ into $F$ via $\phi$.

Paths (compositions of edges) in $G$ may be mapped to composed embeddings in $F$ (via sum, concatenation, or path-based kernels), allowing complex symbolic relationships to acquire continuous analogues.

4. Construction in Applied Settings: Video Analytics Example

An instantiation in video analytics illustrates the approach:

• Nodes $V_1, V_2, V_3$ model detections (e.g., faces/objects across frames), with each node augmented by a latent-space attribute $f_i(\mathrm{image}) \in \mathbb{R}^D$ provided by a CNN.
• Edges of type $\mathrm{HAPPENS\_BEFORE}$ represent temporal linkage, which can be embedded as scalar time-deltas ($\eta(e) = \Delta t$) or as one-hot vectors.
• Edges of type $\mathrm{IS\_SIMILAR\_AS\_COSINE\_ON\_FEATURES}$ between face nodes are embedded as difference-of-CNN-features vectors:

$$\eta(e) = f(\mathrm{face}_i) - f(\mathrm{face}_j) \in \mathbb{R}^D$$

and their statistical affinity is given by the Euclidean norm or cosine similarity of these embeddings.

The graph schema ensures only valid relations are constructed (schema-driven constraints), while statistical affinity enables downstream probabilistic operations such as Bayesian clustering.

5. Unification: Hybrid Relational–Statistical Models

The embedding field framework allows the construction of hybrid models possessing the following properties simultaneously:

• Exact, schema-driven type correctness and logical constraints (as in categorical databases or first-order relational logic).
• Smooth, differentiable, and noise-tolerant affinity measures based on continuous embeddings, supporting vector-space statistical methods and machine-learning objectives.
• Functorial compositionality, mapping relational compositions in the original graph to algebraic or analytical compositions in embedding space.

This duality permits the development of data architectures and algorithms where hard symbolic rules coexist with soft statistical relations, facilitating cross-domain reasoning and complex pattern extraction.

6. Mathematical Summary Table

Concept	Notation / Definition	Comments
Multi-Graph	$G = (V, E, \tau_V, \tau_E, s, t, K_V, K_E)$	Typed, tensor-valued
Embedding Field	$F= (X, \{\|\cdot\|_i\}_{i\in I}, d_F )$	Metric tensor space
Immersion Map	$$\eta: E \to \bigsqcup_{i\in I}\mathbb{R}^{d_i} \subset X$$	Edge embedding to appropriate space
Similarity / Distance	$$d_F(\eta(e_i),\eta(e_j)),\; \mathrm{sim}_F(\cdot,\cdot)$$	Euclidean/cosine/Bhattacharyya distances
Logical structure	Encoded by $(\tau_V, \tau_E)$	Predicate/type schema
Statistical structure	Encoded by $\eta$, $d_F$	Probabilistic/statistical affinities

7. Significance and Applications

Embedding fields are foundational in machine learning pipelines where hybrid data types and complex relational constraints must be jointly exploited. By endowing multi-graph data structures with geometric, metric-driven embeddings, one enables direct definition and computation of similarity across modalities, supports unified architectural data layers, and bridges the divide between logical reasoning and statistical inference. The generality and functorial formalism accommodate diverse data sources (e.g., vision, language, audio) and allow both theoretical expressiveness and practical tractability in designing modern AI systems (Bocse et al., 2020).