SigmoidT

class brainstate.nn.SigmoidT(lower, upper)

Sigmoid transformation mapping unbounded values to a bounded interval.

This transformation uses the logistic sigmoid function to map any real value to a bounded interval [lower, upper]. It is particularly useful for constraining parameters that must lie within specific bounds, such as probabilities or correlation coefficients.

The transformation is defined by:

\[\text{forward}(x) = \text{lower} + (\text{upper} - \text{lower}) \cdot \sigma(x)\]

where \(\sigma(x) = \frac{1}{1 + e^{-x}}\) is the standard sigmoid function.

The inverse transformation is:

\[\text{inverse}(y) = \log\left(\frac{y - \text{lower}}{\text{upper} - y}\right)\]

Parameters:

• lower (Array | ndarray | bool | number | bool | int | float | complex | Quantity) – Lower bound of the target interval.

• upper (Array | ndarray | bool | number | bool | int | float | complex | Quantity) – Upper bound of the target interval.

lower

Lower bound of the interval.

Type:

array_like

width

Width of the interval (upper - lower).

Type:

array_like

unit

Physical unit of the bounds.

Type:

brainunit.Unit

Notes

The sigmoid function provides a smooth, differentiable mapping with asymptotes at the specified bounds. The transformation is bijective from ℝ to (lower, upper), though numerical precision may limit the effective range near the boundaries.

Examples

>>> # Map to probability range [0, 1]
>>> transform = SigmoidT(0.0, 1.0)
>>> x = jnp.array([-2.0, 0.0, 2.0])
>>> y = transform.forward(x)
>>> # y ≈ [0.12, 0.5, 0.88]

>>> # Map to correlation range [-1, 1]
>>> transform = SigmoidT(-1.0, 1.0)
>>> x = jnp.array([0.0])
>>> y = transform.forward(x)
>>> # y ≈ [0.0]

forward(x)

Transform unbounded input to bounded interval using sigmoid function.

Parameters:

x (Array | ndarray | bool | number | bool | int | float | complex | Quantity) – Input values in unbounded domain \((-\infty, \infty)\).

Returns:

Transformed values in interval [lower, upper].

Return type:

Array

Notes

Uses numerically stable exponential to prevent overflow for large |x|.

inverse(y)

Transform bounded input back to unbounded domain using logit function.

Parameters:

y (Array | ndarray | bool | number | bool | int | float | complex | Quantity) – Input values in bounded interval [lower, upper].

Returns:

Transformed values in unbounded domain (-∞, ∞).

Return type:

Array

Notes

For numerical stability, input should be strictly within (lower, upper). Values at the boundaries will result in infinite outputs.

log_abs_det_jacobian(x, y)

Compute log absolute determinant of the Jacobian.

For sigmoid: d/dx[lower + width * sigmoid(x)] = width * sigmoid(x) * (1 - sigmoid(x)) log|det J| = sum(log(width) + log(sigmoid(x)) + log(1 - sigmoid(x)))

Return type:

Array