Fixed-Length Number Embedding Vectors

• Fixed-length number embedding vectors are transformations that map numbers into predetermined, compact vectors preserving numerical and algebraic structures.
• They employ methods such as interpolation, neural encoding, prototype averaging, and discrete coding to balance expressivity, efficiency, and scalability.
• These embeddings enable efficient classification, metric learning, and retrieval by preserving inherent numerical relationships and supporting approximate arithmetic operations.

A fixed-length number embedding vector is a transformation that maps numbers—real, integer, or otherwise—into real-valued or binary vectors of predetermined dimension, such that the numerical properties, relationships, and algebraic or geometric structure are preserved or approximately encoded. These vectors serve as compact, permutation-invariant representations for downstream models in tasks such as classification, metric learning, information retrieval, numerical reasoning, and efficient storage. Multiple methodologies exist, each making distinct trade-offs in expressivity, efficiency, and compatibility with target domains.

1. Foundational Approaches to Fixed-Length Number Embeddings

Methods for embedding numbers into fixed-length vectors include discretization/interpolation schemes, prototype-based averaging, learnable field-isomorphic encodings, compositional discrete codes, and random-projection–based binarization. Each has precise algorithmic and statistical properties suited for different modalities:

• Interpolated Discretized (ID) Embedding: For a real number $x$, choose $C+1$ ordered breakpoints $b_0 < b_1 < \dots < b_C$ forming $C$ bins; $x$ is encoded as a sparse vector $\phi(x) \in \mathbb{R}^C$ with only two adjacent non-zero entries, set via barycentric interpolation between breakpoints. The construction generalizes to $D$-dimensional vectors using $D$ grids and multilinear weights over $2^D$ corners, resulting in a sparse vector in $\mathbb{R}^{C^D}$ (Pele et al., 2016).
• Neural Isomorphic Fields (NIF): Rational numbers are encoded as digit sequences and mapped via a Transformer encoder into $\mathbb{R}^d$ ($d=512$ in experiments). Algebraic operations (addition, multiplication, comparison) are implemented via specialized neural operators designed to preserve field-theoretic structure as closely as possible, with the entire mapping trained via autoencoding and field-isomorphism losses (Sadeghi et al., 17 Jan 2026).
• Prototype-Based Numeral Embeddings: Numbers are expressed as weighted averages of learnable prototype vectors. Prototypes are induced via self-organizing maps (SOM) or Gaussian Mixture Models (GMM) over the observed numerical domain, and any number $n$ is embedded as $E(n) = \sum_{k=1}^K w_k(n) p_k$, where $p_k$ are prototype vectors and weights $w_k(n)$ are similarity-based (Jiang et al., 2019).
• K-way D-dimensional Discrete Codes (“KD Encoding”): Each discrete symbol (including numbers) $s_i$ receives a code $(c_i^1, ..., c_i^D)$ with components in $\{1, ..., K\}$. It is embedded via composition of $D$ trainable code-embedding matrices, either linearly projected or fed through a recurrent (e.g., LSTM) function, yielding a final embedding vector. This approach achieves superlinear compression relative to naive one-hot schemes (Chen et al., 2017).
• Binarized Johnson-Lindenstrauss Embeddings: Any finite set $D \subset \mathbb{R}^d$ is embedded by a random Gaussian projection into $\mathbb{R}^m$ ($m = O(\epsilon^{-2} \log N)$), followed by entrywise sign comparison with random thresholds—yielding $\{\pm1\}^m$ codes. Distances and inner products in the original space can be recovered from Hamming distances or structured inner products between the binary embeddings (Dirksen et al., 2020).
• Population Coding and Detector-Based Sparse Embedding: Numbers or tokens are assigned to fixed-length bit vectors or dense vectors by activating detectors—predefined or learned regions in code space—whose masks are merged via bitwise OR (for binary) or activation pooling (for real-valued). Stochastic layout methods are used to preserve long- and short-range similarities in vector space (Kashitsyn et al., 19 Jul 2025).

2. Algorithmic Workflows and Embedding Construction

Discretization and Interpolation

In ID embeddings, the workflow for a scalar involves locating the enclosing bin for $x$ and assigning weights to adjacent coordinates based on the linear position of $x$ within the bin. For vectors, the generalization entails a Cartesian product of univariate bin indices and multilinear interpolation, but the representation remains highly sparse—at most $2^D$ nonzero entries (Pele et al., 2016).

Prototype Induction and Averaging

SOM or GMM induction is applied to observed numbers, resulting in $K$ prototypes. For a number $n$, similarity kernels (power-law or posterior probability) define weights for constructing $E(n)$ as a convex combination of prototype vectors (Jiang et al., 2019). The resulting embedding is of fixed dimension $d$, shared across the domain.

Structured Neural Encoding

For NIF, digit-tokenized numbers are fed to a Transformer encoder. The output vector is then subject to algebraic operations implemented as commutative neural operators, and field consistency is enforced through carefully designed loss terms. Decoding uses an auto-regressive Transformer conditioned on the embedding (Sadeghi et al., 17 Jan 2026).

Discrete Code Composition

In KD encoding, each symbol’s code is discretely chosen to minimize reconstruction loss against pre-trained or downstream embedding targets. Composition functions—linear sums (with projection) or recurrent aggregations—allow substantial parameter reduction and efficient representation (Chen et al., 2017).

Random Projection and Quantization

The binarized JL embedding applies an oblivious random matrix and binarizes each coordinate via comparison to a uniform random threshold, supporting provable guarantees on distance or inner product approximation, with embedding length governed by dataset Gaussian complexity and covering numbers (Dirksen et al., 2020).

Population Coding and Sparse Detectors

Population coding assigns random or structured k-hot bitvector codes; geometric “detectors” aggregate local neighborhoods in the stochastic code layout, yielding real-valued or binary activations that provide fixed-length representations reflecting neighborhood structure (Kashitsyn et al., 19 Jul 2025).

3. Theoretical Properties, Complexity, and Trade-Offs

The expressiveness, computational cost, and optimality of fixed-length number embedding vectors are determined by algorithmic choices and application constraints:

• Sparsity and Dimensionality: ID embeddings yield highly sparse vectors ($O(2^D)$ nonzeros) even as ambient dimension $C^D$ explodes. Selection of bin count $C$ and dimension $D$ is a primary trade-off—larger $C$ reduces quantization error but increases embedding size exponentially in $D$ (Pele et al., 2016).
• Algebraic Structure Preservation: NIF embeddings approximate field structure—addition achieves near-perfect satisfaction of algebraic axioms (>95% for identity, closure, associativity), while multiplication is less reliable (53–73%) due to digit-length extrapolation effects (Sadeghi et al., 17 Jan 2026).
• Memory and Parameter Complexity: KD encoding reduces overall parameters from $O(Nd)$ to $O(Kd' \log N/ \log K)$, and can realize >90% compression with minimal degradation in downstream perplexity for language modeling tasks (Chen et al., 2017).
• Approximation Guarantees: Binarized JL embeddings present instance-optimal rates: to preserve pairwise distances within $\epsilon$, it suffices to use $O(\epsilon^{-2} \log N)$ bits for $N$ points. Inner products can be reconstructed within additive errors using pairs of independent random embeddings (Dirksen et al., 2020).
• Computational Complexity: For ID, embedding lookup and weight computation cost is $O(D \log C + 2^D D)$ per vector; for binarized JL, encoding is $O(dm)$ per input, and similarity queries are $O(m)$ (Pele et al., 2016, Dirksen et al., 2020). Prototype-based and population-coded approaches have similar linear or sublinear complexities, adaptable to batch or GPU computation (Kashitsyn et al., 19 Jul 2025).

4. Embedding Properties and Empirical Outcomes

Embedding approaches have been evaluated on a diverse set of tasks, revealing characteristic strengths and weaknesses:

• Number-Theoretic Structure Capture: Integer embeddings trained on mathematical sequences localize algebraic properties in single dimensions (e.g., evenness), render divisibility and primality linearly separable, and cluster number classes (primes, powers) without explicit labels (Ryskina et al., 2021).
• Downstream Compatibility: Prototype-based numeral embeddings outperform treat-as-token or digit-LSTM baselines on tasks of word similarity, magnitude classification, numeral prediction, and sequence labeling, including out-of-vocabulary settings (Jiang et al., 2019).
• Algebraic Manipulability: Neural field embeddings can directly implement arithmetic operations in vector space, supporting algebraic queries (addition, comparison) with high fidelity for addition—albeit with notable limitations for multiplication due to data range extrapolation (Sadeghi et al., 17 Jan 2026).
• Bit-Efficient Retrieval: Binarized Johnson-Lindenstrauss codes support efficient approximate nearest-neighbor search, inner product, or distance estimation using only bitwise operations, achieving information-theoretic optimality for embedding length (Dirksen et al., 2020).
• System-Level Advantages: Fixed-number multi-vector representations in retrieval systems (e.g., ConstBERT) enable constant-slot disk storage, improved OS paging, and predictable indexing, with empirical storage–effectiveness trade-offs precisely quantified (MacAvaney et al., 2 Apr 2025).

5. Application Domains and Utility

Fixed-length number embedding vectors are deployed or proposed in several distinct contexts:

• Metric Learning and Distance Approximation: ID embeddings enable efficient regression and classification over functionals of vectors, and universal approximation of general (including non-Euclidean) semimetrics (Pele et al., 2016).
• Numerical Reasoning in NLP: Sequence-trained integer embeddings directly support sequence completion, analogy, and class expansion tasks in mathematical and symbolic contexts, markedly outperforming canonical word embeddings (Ryskina et al., 2021).
• Numeral-Preserving Word Embedding: In NLP, prototype-based numeral embeddings and KD encodings provide numeracy and out-of-vocabulary robustness unattainable by standard "one-hot" or fixed-token approaches (Jiang et al., 2019, Chen et al., 2017).
• Field-Theoretic Computation: Neural Isomorphic Fields represent a step toward learnable algebraic computation at the embedding level, with the model acting as a neural field isomorphic (up to approximation error) to the rational numbers (Sadeghi et al., 17 Jan 2026).
• Efficient Indexing and Retrieval: Fixed-length binary or real-valued embeddings, whether from binarized random projections or sparse detector codes, enable scalable retrieval and similarity search across massive data corpora, with predictable storage footprint (Dirksen et al., 2020, MacAvaney et al., 2 Apr 2025).
• Interpretable, Modular Representation Learning: Population coding and geometric detectors afford interpretable, modifiable, and locally-editable representations suitable for continual learning, memory augmentation, and RAG (Retrieval-Augmented Generation) scenarios (Kashitsyn et al., 19 Jul 2025).

6. Current Limitations and Future Directions

While fixed-length number embedding vectors offer substantial utility, several technical limitations and open challenges remain:

• Expressivity in Arithmetic: Neural Isomorphic Fields achieve high fidelity for addition but underperform multiplication, especially for extrapolated ranges as encountered when multiplying numbers with aggregated digit length outside the training distribution. Suggested remedies include architecture specialization, corpus extension, and enhanced optimization (Sadeghi et al., 17 Jan 2026).
• Scalability: Exponential growth of representation dimension in interpolated discretization for high $D$ is mitigated only by maintaining extreme sparsity; grouping or partial Cartesian products are sometimes needed to keep complexity tractable (Pele et al., 2016).
• Nearest-Neighbor Precision: Binarized embeddings reconstruct only approximate distances or similarities; some tasks requiring high-precision recovery may require alternate encodings or increased bit budget (Dirksen et al., 2020).
• Domain Adaptation: Learned prototype and code compositions rely on representative sample distributions; performance degrades when distributions shift or for heavy-tailed domains unless prototypes are dynamically adapted or hierarchically structured (Jiang et al., 2019, Chen et al., 2017).
• Interpretable Structure Mapping: Detector-based approaches' mapping between semantically meaningful features and activated code dimensions is often emergent rather than explicitly designed, warranting further algorithmic stabilization for high-stakes interpretability (Kashitsyn et al., 19 Jul 2025).

Continued research targets deeper isomorphic encoding of algebraic structure, integration of additional algebraic operations (division, exponentiation), broader generalization to unseen number types, optimization of quantization trade-offs, and domain-specific adaptation procedures.