Bit-Biased Embeddings

• Bit-biased embeddings are compact numerical representations using extremely limited bits (e.g., binary, ternary) to encode data efficiently.
• They enable fast, resource-optimized applications such as large-scale nearest neighbor retrieval, neural network compression, and privacy-sensitive computation.
• Theoretical guarantees like restricted isometry and quantization error bounds ensure that essential geometric structures and task-relevant properties are preserved under severe bit constraints.

Bit-biased embeddings are numerical representations in which the information is encoded using extremely limited numbers of bits—typically 1 bit (binary), a few bits per coordinate, or through quantization that introduces strong non-uniformity or explicit statistical bias into the bit patterns. These encoding schemes are designed for efficiency in storage, computation, or transmission, and are widely utilized in large-scale retrieval, neural network compression, and privacy-sensitive distributed settings. Bit-biased embeddings span both data structure–centric approaches (such as binary and ternary codes, circulant or structured random projections, and ultra-quantization) and bias-in-signal issues (statistical or social bias, as in language or multimodal embedding spaces). Their paper addresses the preservation of geometric or task-relevant structure subject to severe bit constraints, the mitigation or exploitation of statistical bias, and the impact of bit-level encoding decisions on downstream tasks.

1. Methodologies for Constructing Bit-Biased Embeddings

Bit-biased embedding construction encompasses several principled approaches engineered for efficiency and distortion control:

• Binary Embedding via Random Projection and Quantization: Linear projections into lower-dimensional spaces followed by quantization using the sign function are foundational techniques (Yu et al., 2015, Bilyk et al., 2015, Spencer, 2016, Dirksen et al., 2020). For x ∈ ℝᵈ, a projection h(x) = sign(Ax) yields a vector in {–1, +1}ᵏ or {0, 1}ᵏ. Structured random matrices (circulant, Hadamard, block-wise, etc.) are utilized to enable O(d log d) computation by virtue of the Fast Fourier Transform or other fast transforms (Yu et al., 2015, Dirksen et al., 2020).
• Data-Dependent and Learned Embeddings: Embedding parameters (projection vectors, codebooks) can be optimized using reconstruction or distortion objectives—often via alternating minimization in time and frequency domains or autoencoder architectures (Yu et al., 2015, Tissier et al., 2018).
• Bernoulli and Biased Coin Embeddings: For discrete binary codes, a Bernoulli probability is assigned to each bit, and embeddings are modeled as draws from independent (possibly learned and node- or entity-specific) biased Bernoulli variables. Optimization occurs via the expected loss over this distribution, using continuous relaxations and (potentially) reparameterization techniques (Misra et al., 2018).
• Multi-bit and Ternary Quantization: More aggressive quantization includes mapping to ternary codes (–1, 0, +1, yielding an average of 1.58 bits per value) (Connor et al., 31 May 2025), or fixed-bit quantization per embedding coordinate, as in 4-bit uniform or codebook-based quantization of embedding tables (Guan et al., 2019).
• Clustering and Semantic Binarization: In LLMs, embedding binarization is guided by clustering structures (e.g., k-means or autoencoders with bit-based bottlenecks), where semantic or distributional intersections are reflected in the bit patterns (Tissier et al., 2018, Ferrer-Aran et al., 2020).
• Representation Bias in Structural Sampling: Biased sampling in random walks (e.g., GlobalWalk) or explicit regularization in embedding optimization allows structural bias to be propagated or controlled in the embedding space (Xue et al., 2022).

2. Theoretical Guarantees and Geometric Preservation

Bit-biased embeddings are theoretically justified via multiple geometric and statistical properties:

• Restricted Isometry and Binary ε-Stable Embedding (BεSE): Binary embeddings derived from Gaussian projections and 1-bit quantization are shown to satisfy forms of the Restricted Isometry Property (RIP), ensuring near-isometric distortion of distances or angles for structured data such as sparse vectors or generative model outputs (Bilyk et al., 2015, Spencer, 2016, Liu et al., 2020). The expected Hamming distance between binary codes is an unbiased estimator of angular distance:

$$\mathbb{E}\left[ \frac{1}{2k} \| \Phi(x) - \Phi(y) \|_1 \right] = \frac{\theta}{\pi}$$

where θ is the angle between inputs (Yu et al., 2015).

• Sample Complexity: For 1-bit sensing and recovery of signals in the range of generative models, the number of measurements m required to ensure BεSE grows as $O((k/\epsilon) \log(L r/\epsilon^2))$ for a latent dimension k, Lipschitz constant L, and ball radius r (Liu et al., 2020). Lower bounds match up to logarithmic factors.
• Quantization Error Bounds: Uniform and codebook quantization approaches are benchmarked with respect to L₂ (squared error) and retrieval metrics; aggressive ultra-quantization (e.g., ternary encoding via equi-volume Voronoi polytopes) preserves pairwise distance rankings (Spearman ρ ~0.94–0.96) and k-NN recall at markedly lower resource footprints (Guan et al., 2019, Connor et al., 31 May 2025).
• Empirical Speedups: SIMD-friendly bit-encoded embeddings facilitate vector comparisons at >100× speedup over floating-point distance evaluations in large-scale retrieval tasks (Connor et al., 31 May 2025).

3. Applications and System-level Impact

Bit-biased embeddings are engineered for demanding applications requiring resource-optimized representations:

• Large-scale Nearest Neighbor Retrieval: Indexes supporting k-NN search over bit-encoded databases leverage extremely compact storage and bitwise distance calculations. Binarized word or image embeddings enable real-time, on-device search and matching (Tissier et al., 2018, Connor et al., 31 May 2025).
• Recommender Systems and Embedding Tables: 4-bit quantization of embedding tables for recommendation reduces model size to ~13.89% of the single-precision baseline, with neutral or improved performance on test-time metrics (Guan et al., 2019).
• Text, Graph, and Multimodal Embeddings: Binary and bit-biased codes are employed for graph node similarity, pre-ranking in information retrieval, efficient computation in NLP pipelines, and privacy-sensitive contexts where memory and compute must be minimized (Misra et al., 2018, Xue et al., 2022, Tissier et al., 2018).
• Neural Network Compression and Edge Deployment: Aggressive quantization strategies (including 1.58-bit ternary encodings) are directly applicable to neural parameter compression, federated or edge-based learning, and efficient matrix–vector multiply (Connor et al., 31 May 2025).
• Impact of Statistical and Social Bias: Representational bias in embedding spaces—especially in text-to-image diffusion pipelines—propagates into downstream generative and evaluative metrics, making unbiased bit allocation a necessary condition for representational fairness and auditability (Kuchlous et al., 15 Sep 2024).

4. Measurement and Mitigation of Bias in Embedding Spaces

In addition to the bit allocation strategies above, there is substantive work on the measurement and mitigation of societal or statistical bias in bit-based embedding schemes:

• Association Testing and Salience-based Clustering: Bias in word vectors is measured using association tests (e.g., WEAT, LIWC-domain scoring), cosine similarity to attribute centroids, and frequency-weighted salience measures. Clustering salient words into conceptual “biased concepts” aids in interpretability and audit (Sutton et al., 2018, Ferrer-Aran et al., 2020).
• Projection-based Debiasing: Identifying bias subspaces (e.g., via the he–she direction) and projecting vectors onto the orthogonal complement is shown to mitigate stereotypical associations, sometimes at the cost of one dimension in the embedding (Sutton et al., 2018, Dev et al., 2019).
• NLI-based Intrinsic Bias Detection: Inference-based metrics using large-scale template pairs in NLI tasks allow the direct identification of bias in embeddings as measured by invalid entailment/contradiction outputs, extending diagnosis beyond geometric proximity and into task-specific inference (Dev et al., 2019).
• Fairness in Diffusion and Multimodal Embeddings: Bias in text or multimodal prompt embeddings leads to representational imbalance in diffusion-generated samples. Statistical group fairness conditions—such as multiaccuracy and representational balance—are enforced via intrinsic alignment criteria and mitigation using subclass scoring and average-then-score strategies (Kuchlous et al., 15 Sep 2024).

5. Optimization, Trade-offs, and Future Lines of Inquiry

The engineering and theoretical challenges in bit-biased embedding construction and evaluation are multi-faceted:

• Optimization under Discreteness: Direct gradient propagation in training discrete (binary or ternary) embeddings is intractable due to non-differentiability. Continuous relaxations, surrogate losses, and expected-loss optimization (over coin-flip bias variables) enable effective learning (Misra et al., 2018).
• Expressivity–Compacity Trade-off: There exists a fundamental tension between aggressive quantization (compactness) and the preservation of geometric, semantic, or functional structure (expressivity). Overly aggressive encoding can lead to loss of information, while under-quantization undermines the efficiency objectives.
• Parameterization and Hyperparameter Sensitivity: The assignment of per-row or per-component codebooks, calibration of clipping thresholds, and scheduling of annealing parameters in structured sampling or biased walks strongly influence both retrieval accuracy and bias propagation (Guan et al., 2019, Xue et al., 2022).
• Robustness to Noise and Quantization Error: Bit-biased embeddings in compressive-sensing-type settings are robust to additive white noise only when distance metrics are adapted (using distortion metrics appropriate to the noise) (Spencer, 2016).
• Open Questions and Future Directions: Further areas include integrating semi-supervised or fairness constraints into bit allocation (Yu et al., 2015, Kuchlous et al., 15 Sep 2024), refining statistical bias mitigation under non-binary settings, extending ultra-quantization schemes to heterogeneous embedding spaces, and hardware-level co-design to exploit SIMD and bitwise arithmetic (Connor et al., 31 May 2025).

6. Tables: Core Methodologies and Properties

Approach/Class	Bit Encoding Mechanism	Notable Property/Guarantee
Circulant Binary Embedding (Yu et al., 2015)	sign(C(r)Dx), FFT-accelerated	O(d log d) compute, storage O(d); angle preservation
One-Bit Sensing/1-Bit CS (Bilyk et al., 2015, Spencer, 2016, Liu et al., 2020)	Random projections + sign quantization	δ-RIP and BεSE for sparse or generative-model signals
Autoencoder Binarization (Tissier et al., 2018)	Learned W, sign activation	Recall, semantic similarity, up to 97% size reduction
Bernoulli Coin Embedding (Misra et al., 2018)	Bitwise coin-flip with learned biases	End-to-end learned, avoids post-hoc quantization error
4-Bit Uniform/Codebook (Guan et al., 2019)	Per-row quantization, codebook	Model compression to 13.89%, neutral test quality
Ultra-Quantization EVP (Connor et al., 31 May 2025)	Ternary (–1,0,+1), equi-volume polytope	1.58 bits/coordinate, high Spearman ρ, SIMD speedup

7. Significance and Implications

Bit-biased embeddings enable scalable operations on high-dimensional data through order-of-magnitude reductions in memory and computational requirements, while preserving geometric and semantic fidelity to a degree sufficient for information retrieval, classification, and other machine learning tasks. Their use surfaces unique challenges, particularly in the propagation and amplification of statistical or societal bias, necessitating systematic evaluation, debiasing, and fairness criteria in both construction and application. Continued advances in quantization, optimization, and fair representation learning are expected to expand the applicability and reliability of bit-biased embeddings across data modalities and downstream domains.