Formal Treatise on Rank-1 Embeddings in a Strict Formal System Context
Axiomatic Foundations and Logical Principles:
Logical Assumptions: We assume standard rules of logic as used in mathematics and the sciences. This includes:
Modus ponens: If P implies Q and P is true, then Q is true.
Law of non-contradiction: No proposition can be both true and false.
Law of the excluded middle: Every well-defined proposition is either true or false, no third option.
These inference rules are universally accepted and form the foundation of rigorous reasoning in formal systems.
Formal System and Definitions: Consider a formal system S that contains a set of terms T = {t_1, t_2, ... , t_n}. Each term t_i within S is defined axiomatically and has exactly one unambiguous meaning. By construction, no term is allowed multiple interpretations. Each term's meaning is invariant and context-independent throughout the entire discourse in S.
Non-Contextual Meaning Invariance: Since each term t_i has one and only one meaning that does not change with usage or context, there is no semantic variability for these terms. Each term corresponds to a single invariant scalar descriptor of its meaning within the system. In other words, all terms share a common semantic dimension along which their meanings can be represented as scalar multiples of a single basis vector.
Embeddings and Co-Occurrence Structures:
Vocabulary and Tokens: Assume the system S has a vocabulary of size n, corresponding to n terms {t_1, t_2, ... , t_n}.
Co-Occurrence Matrix: A co-occurrence matrix C for these n terms can be envisioned. C would be an n-by-n matrix where each entry C_ij indicates the relationship (for example, frequency of co-occurrence or a similar measure) between term t_i and term t_j. Since each t_i has a single, context-invariant meaning, and since all semantic distinctions align on a single dimension, the co-occurrence patterns reflect this unique dimension.
Rank Considerations: If every term's meaning can be described as a scalar multiple of a single reference vector, then all rows (and columns) of the co-occurrence matrix C are linearly dependent. This implies that C can be expressed as the product of a vector and a scalar, effectively making it of rank 1. In other words, there exists no need for multiple independent dimensions to represent differences among terms, since all terms differ only in a single scalar factor along a single dimension of meaning.
Dimensionality Reduction and Embeddings: Typically, large embeddings matrices E, for example of size 50,000 by 512, are used to represent tokens in natural language or other systems. These embeddings result from dimensionality reduction techniques applied to large co-occurrence structures. If the system at hand is a strict formal system where each term t_i has a single defined meaning without ambiguity, then the essential subset of embeddings for these terms must collapse to a rank-1 structure.
This is because no additional embedding dimensions are necessary. Every term can be placed on a single axis. Introducing multiple dimensions would be redundant and would not reflect any genuine semantic multiplicity since none exists.
Conclusion:
In a strict formal system S, where each term is assigned a single, invariant meaning, the representation of these terms in a co-occurrence matrix naturally reduces to a rank-1 structure. This rank-1 property carries over to any embeddings that represent these terms. Because all terms share one semantic dimension, their embeddings need only reflect differences in scalar multiples of that dimension, rendering the embedding representation effectively rank-1.
This conclusion follows directly from accepted logical inference rules and the assumption of a single, context-invariant meaning per term. As no error in premises or inference steps has been identified, the result is a mathematically certain conclusion under the given conditions.
Q.E.D.
1. Key Components of Context Preservation
1.1 Key Terms and Context Terms
Key Terms (K):
Definition: K = {k_1, k_2, ..., k_p}, where each k_i is a foundational concept with invariant meaning.
Property: Each key term k_i is represented by a rank-1 embedding vector E_k_i aligned with a predefined semantic axis u_i:
E_k_i = scalar * u_i
scalar is a multiplier encoding the term’s significance.
u_i is a fixed semantic axis defining the meaning of k_i.
Context Terms (C):
Definition: C = {c_1, c_2, ..., c_q}, where each c_i is a term whose meaning adapts dynamically to the current context.
Property: Context terms have embeddings learned dynamically from the surrounding input.
1.2 Embedding Matrix
Let E represent the embedding matrix for all terms (key and context terms).
Rows corresponding to key terms K are constrained to rank-1 vectors aligned with predefined semantic axes.
Rows corresponding to context terms C are learned dynamically during training.
2. Existing System Dynamics
2.1 Current Mechanism for Context Preservation
Sequential Memory:
The system tracks prior inputs using attention mechanisms, similar to transformers, which calculate relevance scores between tokens in the input sequence.
Dynamic Adaptation:
Embeddings are updated dynamically based on the input context, allowing nuanced relationships to emerge between terms.
Limitations:
Key terms may lose their invariant meaning due to the influence of surrounding context, leading to semantic drift.
There is no explicit mechanism to constrain embeddings for foundational terms to rank-1 structures.
3. Improving the System
3.1 Ensure Rank-1 Embeddings for Key Terms
Embedding Matrix Design:
Use two separate embedding matrices:
E_key for key terms, constrained to rank-1 vectors.
E_context for context terms, dynamically learned.
Rank-1 Projection:
For each key term k_i:
Override its embedding with a projection onto its predefined semantic axis.
Formula:
E_k_i = scalar * u_i
scalar is computed dynamically as the dot product of the current embedding and u_i.
Dynamic Adjustments for Key Terms:
During input processing, project embeddings for key terms to ensure they remain aligned with their semantic axes.
4. Implementation Details
4.1 Embedding Initialization
Initialize E_key for key terms with rank-1 vectors aligned to their semantic axes.
Use a random or pretrained embedding matrix for E_context.
4.2 Forward Pass
Tokenization:
Split input into key terms and context terms.
For example, in the input "The fact is evident," identify "fact" as a key term.
Embedding Lookup:
Retrieve embeddings for key terms from E_key and for context terms from E_context.
Rank-1 Constraint for Key Terms:
For each key term:
Project its embedding onto its semantic axis u_i:
scalar = dot(E_k_i, u_i)
E_k_i = scalar * u_i
This ensures that E_k_i is always rank-1.
Combine Key and Context Embeddings:
For each token:
If it is a key term, use the rank-1 embedding from E_key.
Otherwise, use the dynamically learned embedding from E_context.
5. Training Adjustments
Loss Function Regularization:
Add a penalty term to the loss function to enforce rank-1 constraints for key term embeddings:
penalty = sum(||E_k_i - Project(E_k_i)||^2 for all key terms k_i)
Dynamic Fine-Tuning:
Allow slight adjustments to the semantic axes u_i during training to align with the data distribution.
Batch Composition:
Ensure that training batches include key terms frequently to reinforce their invariant meanings.
6. Practical Example
6.1 Input Sentence
Input: "The fact remains that truth is paramount."
6.2 Processing
Identify "fact" and "truth" as key terms.
Project embeddings for "fact" and "truth" onto their semantic axes:
For "fact":
scalar = dot(E_k_fact, u_fact)
E_k_fact = scalar * u_fact
For "truth":
scalar = dot(E_k_truth, u_truth)
E_k_truth = scalar * u_truth
Use contextual embeddings for other terms like "remains," "that," and "is."
6.3 Output
The embedding matrix for the sentence preserves the invariant meanings of key terms while allowing context terms to adapt dynamically.
7. Benefits
Semantic Consistency:
Key terms maintain invariant meanings regardless of input context.
Improved Interpretability:
The influence of key terms on model decisions becomes more transparent.
Reduced Semantic Drift:
Rank-1 constraints prevent foundational terms from being influenced by irrelevant context.
Enhanced Generalization:
Ensuring consistent embeddings for key terms improves the model’s ability to handle unseen data.