Thresholding

📐 Definition

For threshold $\tau$ and scalar $x$ ,

T_{\tau}(x) = \begin{cases} 0, & x < \tau,\\[4pt] x, & x \ge \tau \end{cases}

Variants include:

hard thresholding to a constant level (e.g. $x\mapsto \tau\,\mathbf{1}_{x\ge \tau}$ ),
soft (shrinkage) thresholding $S_{\tau}(x) = \operatorname{sgn}(x)\max(|x|-\tau,0)$ .

Domain and Codomain

Domain: real numbers (extendable componentwise). For the definition above, the codomain is $\{0\}\cup[\tau,\infty)$ .

⚙️ Key Properties

Idempotent. For $\tau\ne 0$ it is discontinuous at $x=\tau$ (a jump from $0$ to $\tau$ ); for $\tau=0$ it reduces to the ReLU map and is continuous but not differentiable at the origin.

Soft-thresholding $S_\tau$ is continuous and $1$ -Lipschitz (non-expansive), and is commonly used when stability under perturbations is desired.

As $\tau \to -\infty$ , thresholding becomes the identity; as $\tau \to \infty$ , it maps all values to $0$ .

🎯 Special Cases and Limits

$\tau\to-\infty$ approaches identity.
$\tau\to\infty$ maps everything to $0$ .

Clamping enforces two-sided bounds; gain control can be applied after thresholding to rescale retained components.

Usage in Oakfield

Oakfield uses thresholding mainly as a “gate” on features and diagnostics:

Phase feature extraction (phase_feature operator) suppresses samples with magnitude below threshold before writing features.
Phase coherence diagnostics ignore samples below a magnitude gate derived from configured absolute/relative thresholds before computing coherence order parameters.
Operator logic commonly uses small epsilons/thresholds to avoid division by zero (e.g. when forming unit directions like $z/|z|$ ).

Historical Foundations

📜 Hard Thresholds and Sparsity

Thresholding is the simplest nonlinear sparsification primitive and appears in denoising and compressed representations as a hard “keep/drop” rule.

🌍 Modern Perspective

It is used when exact sparsity or strict cutoffs are desired; smooth surrogates are often used when differentiability is required.

📚 References

Mallat, A Wavelet Tour of Signal Processing