True/False Binarization

• True/False binarization is the process of converting continuous or multi-level data into binary outputs using thresholding and optimized quantization schemes.
• It underpins applications from deep neural network quantization and document image preprocessing to efficient data structures like Bloom filters.
• Recent advances include adaptive thresholding, gradient approximations, and entropy-conserving methods that balance computational efficiency with semantic accuracy.

True/False binarization refers to the transformation of multi-level or continuous data into a binary representation, in which each element is discretized as either "true" (1, YES, +1) or "false" (0, NO, –1). The concept underpins a wide variety of domains, including deep neural network model compression, document image preprocessing, data structures (e.g., Bloom filters), omics data modeling (e.g., gene expression for Boolean GRNs), and universal data encoding for compression algorithms. Methodologies span hard-thresholding, learning-based classification, optimized quantization, entropy-conserving encoding, and regulation-based discretization. Precise binarization is often pivotal for both computational efficiency and semantic fidelity.

1. Mathematical Foundations and Canonical Algorithms

True/False binarization employs explicit functions to map continuous or discrete inputs to binary outputs. The most common primitive is thresholding:

• For a signal $x$, binarization is given by:

$$y = \begin{cases} 1 & x \geq \tau \ 0 & x < \tau \end{cases}$$

where $\tau$ is a fixed or adaptive threshold.

In neural networks, weights $w$ or activations $a$ are binarized via the sign function or shifted unit step:

• Weight binarization: $\operatorname{sgn}(w) = +1$ if $w \ge 0$, $-1$ otherwise.
• Activation binarization: $H(a-\tau)$, which outputs 1 if $a \ge \tau$, 0 otherwise.

For non-binary sources, entropy-conserving schemes (e.g., Srivastava's algorithm) structure the binarization into $m-1$ binary streams for an $m$-ary alphabet, preserving original entropy across streams via symbol ordering and sequential elimination (Srivastava, 2014).

In data structures such as Bloom filters, binarization applies a threshold $\Theta$ to counting vectors to obtain Boolean bitmaps, then membership is tested via dot-products against decision thresholds $T$ (Kleyko et al., 2017).

2. Applications in Deep Neural Network Quantization

Binarization is central for model compression in neural architectures (Xu et al., 2019, Shang et al., 2022, Lu et al., 27 Feb 2024). When network weights and activations are quantized to one bit, substantial reductions in memory footprint and computational complexity result, enabling fast integer or bitwise operations (e.g., XNOR-popcount convolutions).

Key advances for recovering accuracy lost by 1-bit quantization include:

• Trainable scaling factors for weights ($\alpha$) and activations ($\beta$), optimized via backpropagation alongside full-precision parameters.
• Tight gradient approximations for the discontinuous binarization functions, preserving nonzero gradient flow at the hard threshold boundaries to ensure effective learning.
• $L_2$ regularization of scaling factors to limit runaway magnitudes and improve generalization.

Contrastive Mutual Information Maximization (CMIM) pulls binary and FP activations together, maximizing shared information and mitigating quantization error (Shang et al., 2022).

Recent theoretical frameworks (ProxConnect++, BNN++) derive forward and backward quantizers from proximal operators with formal convergence proofs and empirical robustness in CNNs and vision transformers (Lu et al., 27 Feb 2024). The forward quantizer need not equal its backward gradient surrogate, as long as both correspond to coherent optimization steps.

3. Image and Document Binarization: Algorithms and Architectures

Document and image binarization is essential for distinguishing foreground (e.g., text, ink) from background in pre-processing for OCR and analysis (Nicolaou et al., 2016, Chan, 2019, Quattrini et al., 26 Apr 2024, Nnolim, 2021, Wu et al., 2015). Methods encompass:

• Adaptive local thresholding (e.g., Sauvola’s method): thresholds are dynamically computed from local mean and variance in a pixel window. Efficient implementations use sliding window stripe-sums and recursive updates for mean/variance, achieving true/false classification with $\Theta(HW)$ time and $\Theta(\min\{H,W\})$ space (Chan, 2019).
• PDE-based binarization: fractional/integer-order differential equations evolve a binarization potential $U(x,t)$ with source, edge, and diffusion terms. This approach robustly restores text and eliminates artifacts (Nnolim, 2021).
• Learning-based classifiers: High-dimensional features encode all binarization heuristics, then learned classifiers assign binary states. Recent approaches use balanced sampling, ExtraTrees classification, and novel features (e.g., Logarithm Intensity Percentile, Relative Darkness Index) to achieve contest-level performance (Wu et al., 2015).
• Frequency-aware neural architectures: Fast Fourier Convolutional U-Nets (FourBi) fuse spatial-local and global-spectral filtering, outperforming pure CNN or transformer-based designs in both accuracy and efficiency. Training employs the Charbonnier loss for stable binarization (Quattrini et al., 26 Apr 2024).

Hardware-specific binarization, such as real-time fingerprint image processing, involves pipeline structures chosen via block-factor optimization for block size and morphologically informed dilation stages (Kheiri et al., 2017).

4. Data Structure Binarization and Tradeoff Control

Binarization is critical in probabilistic data structures, notably Bloom filters. Autoscaling Bloom filters employ thresholded counting Bloom filters (CBF) and a secondary dot-product decision threshold to partition bitmaps into true/false outcomes, balancing accuracy vs. false positive rates analytically via binomial models (Kleyko et al., 2017). One can select binarization and decision thresholds $(\Theta, T)$ to achieve any desired TPR/FPR tradeoff without rebuilding the filter or changing hash functions. The approach generalizes fixed Bloom filters to dynamic contexts.

5. Regulation-Based True/False Binarization in Systems Biology

Boolean gene regulatory network (GRN) synthesis from omics data necessitates binarized gene expression states. Standard thresholding is biologically deficient due to gene-specific roles and data uncertainty. Regulation-based forward–backward binarization combines per-gene thresholding, forward propagation (activators/inhibitors), backward completion (attribute missing regulator states from observed targets using consistency scores), harmonization, and inconsistency re-initialization until fixpoint convergence, ensuring functional and regulatory consistency (Belgacem et al., 17 Oct 2025). The result is robust transformation of snapshot data into Boolean profiles, validated against ODE simulations and real biological datasets.

6. Theoretical Perspectives and Generalizations

Entropy-conserving binarization, as in Srivastava’s algorithm, guarantees that transforming $m$-ary data into binary streams retains the exact average entropy, regardless of the source probability distributions or ordering of symbol elimination. This property is essential for universal binary coding frontends in video/image compression (e.g., CABAC) and circumvents the suboptimality of classic binarization techniques (e.g., fixed-length or Huffman coding tied to specific distributions) (Srivastava, 2014).

In the context of LLMs, binarization of truth value (true/false statements) aligns with the identification of a linear direction in representation space discriminating factual truth. Simple difference-of-means probes sufficiently recover label separability, and mass-mean probes can causally control factual outputs by latent interventions. However, scope is limited to non-ambiguous statements, and transfer generalization may depend heavily on dataset diversity (Marks et al., 2023).

7. Limitations, Open Challenges, and Future Research

While binarization schemes span numerous domains and have achieved empirical and theoretical milestones, several challenges persist:

• In deep nets, activation binarization remains a bottleneck for closing the performance gap versus full precision; advanced surrogate gradient design and contrastive objectives have narrowed this but not fully closed it (Xu et al., 2019, Shang et al., 2022).
• Biological binarization is limited by snapshot data resolution and uncertainties; future work may integrate dynamic models and more sophisticated value imputation strategies (Belgacem et al., 17 Oct 2025).
• Compression applications face tradeoffs in stream overhead for large alphabets and contextual adaptation; joint optimization of order and context models is a candidate for further improvement (Srivastava, 2014).
• In LLMs, probe-based binarization of truth may not account for higher-moment structure or chain-of-thought complexity (Marks et al., 2023).

A plausible implication is that continued progress will rely on fully data-driven binarization pipelines, integration of multi-level context, and principled optimization frameworks—whether for neural architectures, data structures, biological networks, or compression frontends.