Implemented mechanisms
This page lists the implemented coordinate transforms, value transforms, and composition modes. The primitive base-function ID table is in Primitive base function IDs. YAML parsers normalize case, spaces, hyphens, and underscores before matching names.
Base functions
Primitive functions are exposed through BasicFunctionId. Numeric IDs and
names can both be used in YAML. See Primitive base function IDs for the
complete table.
Coordinate transforms
Let \(x\) be the parent point, \(a\) be the assigned component optimum
stored in assigned_xopt, and \(t\) be the internally resolved child
optimum. The transform maps parent coordinates into child coordinates.
noneFull-dimensional shift:
\[T(x) = t + (x-a).\]rotationFull-dimensional shifted rotation with orthogonal matrix \(R\):
\[T(x) = t + R(x-a).\]affineFull-dimensional shifted affine transform with matrix \(A\). The generated matrix starts from a random rotation and scales each output row independently by a geometric factor between
1and100:\[T(x) = t + A(x-a).\]subspace-rotationSubspace transform. If
selected_indicesdefines projection \(P\), then \(x_{\mathrm{sub}} = Px\) and\[T(x) = t + R(x_{\mathrm{sub}} - a).\]The child function sees only
output_dimensionvariables.
Value transforms
Let \(u \ge 0\) be the shifted component value before the value transform.
none- \[\phi(u) = u.\]
powerParameters are
[alpha, p]:\[\phi(u) = \alpha u^p.\]oscillatoryParameters are
[epsilon, alpha]:\[\phi(u) = u\left(1 + \epsilon\sin(\alpha u)\right).\]cosine-zeroParameter is
[alpha]:\[\phi(u) = 1 - \cos(\alpha u).\]huberParameter is
[delta]:\[\begin{split}\phi(u) = \begin{cases} u^2/(2\delta), & u \le \delta,\\ u-\delta/2, & u > \delta. \end{cases}\end{split}\]logParameter is
[alpha]:\[\phi(u) = \log(1+\alpha u)/\alpha.\]softplus-thresholdParameters are
[tau, alpha]. With \(\operatorname{sp}(v)=\log(1+\exp(v))\),\[\phi(u) = \frac{\operatorname{sp}(\alpha(u-\tau)) -\operatorname{sp}(-\alpha\tau)}{\alpha}.\]dead-zoneParameters are
[tau, p]:\[\phi(u) = \max(0,u-\tau)^p.\]saturatingParameters are
[cap, c]:\[\phi(u) = \frac{\mathrm{cap}\,u}{u+c}.\]piecewise-powerParameters are
[tau, p1, p2]:\[\begin{split}\phi(u) = \begin{cases} u^{p_1}, & u \le \tau,\\ \tau^{p_1} + (u-\tau)^{p_2}, & u > \tau. \end{cases}\end{split}\]noisy-smoothParameters are
[epsilon, alpha]:\[\phi(u) = u\left(1+\epsilon\sin(\alpha u) \sin(0.371\alpha u+1.2345)\right).\]
Trigonometric value transforms reduce the phase modulo \(2\pi\) internally for more stable cross-platform numerical behavior.
Composition modes
Let \(z_i\) be transformed component values.
noneSingle-component identity:
\[\psi(z_1) = z_1.\]cpm-wsumCommon-point weighted sum:
\[\psi(z) = \sum_i w_i z_i.\]cpm-power-meanParameter is
[p]:\[\psi(z) = \left(\sum_i w_i z_i^p\right)^{1/p}.\]cpm-level-wellParameters are
[epsilon, alpha]. Let \(s = \sum_i w_i z_i\):\[\psi(z) = s\left(1 + \epsilon\sin(\alpha s)\right).\]cpm-maxMaximum component value:
\[\psi(z)=\max_i z_i.\]cpm-smoothmaxParameter is
[beta]. With \(z_{\max}=\max_i z_i\),\[\psi(z)=z_{\max}+ \frac{\log\sum_i \exp(\beta(z_i-z_{\max}))-\log m}{\beta}.\]cpm-constraint-penaltyParameters are
[rho, p]:\[\psi(z)=z_1+\rho\sum_{i=2}^m z_i^p.\]cpm-lexicographicParameter is
[decay]:\[\psi(z)=\sum_{i=1}^m \mathrm{decay}^{i-1}z_i.\]cpm-productParameter is
[alpha]:\[\psi(z)=\frac{\prod_i(1+\alpha z_i)-1}{\alpha}.\]cpm-max-plus-meanParameter is
[lambda]:\[\psi(z)=\lambda\max_i z_i+ (1-\lambda)\frac{1}{m}\sum_i z_i.\]cpm-cvarParameter is
[quantile]. Let \(z_{(1)}\ge\cdots\ge z_{(m)}\) and \(k=\lceil \mathrm{quantile}\,m\rceil\):\[\psi(z)=\frac{1}{k}\sum_{i=1}^k z_{(i)}.\]dpm-softmaxParameter is
[sharpness]. Let \(c_i\) be full-dimensional DPM centers, \(b_i\) be DPM biases, and \(\gamma\) be sharpness:\[q_i(x) = \exp(-\gamma\|x-c_i\|^2 - M), \qquad M = \max_j -\gamma\|x-c_j\|^2.\]Non-global centers are masked near the global center:
\[m_0(x)=1,\qquad m_i(x)=1-\exp(-\|x-c_0\|^2),\quad i>0.\]Then
\[\psi(x,z) = \frac{\sum_i q_i(x)m_i(x)(z_i+b_i)} {\sum_i q_i(x)m_i(x)}.\]dpm-bgsoftmaxParameters are
[sharpness, background_strength, background_sharpness]. It adds a smooth background term\[\beta(x) = \rho\left(1-\exp(-\eta \min_i\|x-c_i\|)\right)\]and computes
\[\psi(x,z) = \frac{\sum_i (q_i(x)m_i(x)+\beta(x))(z_i+b_i)} {\sum_i (q_i(x)m_i(x)+\beta(x))}.\]
DPM center 0 is the assigned global optimum. Other centers are deceptive locations. DPM biases are composition parameters, not component value transforms.
Name aliases
Examples of accepted aliases:
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