Gaussian similarity kernel

Read the equation

The kernel is one for identical inputs and decreases with their squared distance. The positive bandwidth σ sets the scale: a distance σ gives exp(−1/2). It is a similarity function, not by itself a normalized probability density.

Terms & definitions

How it works

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Original context

    }

    // Φ = 1/(1 + |x - 1|) - Consciousness constraint
    inline double phi_constraint(double x) {
        return 1.0 / (1.0 + std::abs(x - 1.0));
    }

    // Gaussian kernel: K(x,y) = exp(-||x-y||²/2σ²)
    inline double gaussian_kernel(double x, double y, double sigma = 1.0) {
        double diff = x - y;
        return std::exp(-diff * diff / (2.0 * sigma * sigma));
    }

    // Shannon entropy: H(p) = -p·log(p) - (1-p)·log(1-p)
    inline double binary_entropy(double p) {
        if (p <= 0.001 || p >= 0.999) return 0.0;
        return -(p * std::log(p) + (1.0 - p) * std::log(1.0 - p)) / LN2;
    }

Gaussian kernel: K(x,y) = exp(-||x-y||²/2σ²)

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