Gradients Sample Clauses

Gradients. ‌ > 1). We can see, that it could be numerically problematic to optimize the marginal likelihood, as it could be small and it only ranges positive values. Computers can only perform numerical computations, and are limited by the precision of the system. The precision of a computer system is bound by the number of bits of information being able to be stored in one number. This number is finite for computers. The most prominent problem is if there is multiple dimensions D 1, as it then is a product of potentially small values. In order to relieve this problem, we 1We let p(Y|X) ≡ p(Y|X, θ). employ the natural logarithm (log). The logarithm is a monotonic transformation, this means, all optima in the original space, will be optima in log space as well. The logarithm stretches the space from 0 to 1 into the negative values. Additionally, what used to be a product of values, becomes a sum in log-space. We sum up positive and negative values, instead of multiplying. Additionally, the logarithm reduces complexity in implementation of gradient based optimization in two ways. First, the products in the original space become independent parts in the sum. We can compute the partial gradients for each part and sum the results together to get the overall gradient. Second, the exponential of the Gaussian distribution gets cancelled out and simplifies the gradient, as the exponential would remain for gradient computations. The maximization is done over the hyper-parameters θ, maximizing the log marginal likelihood L ∶= log p(Y|X, θ, σ2): − 1 L = lo g ((D ((2π)N |K + σ2I |) 2 ) exp {− 1 tr Yt(K + σ2I )−1Y}) = − ND log 2π − D log |K + σ2I | − tr Yt(K + σ2I )−1Y (2.2)

Gradients. These are to be measured along the inside curve of corners and centreline of straight sections where applicable.  1 in 20 maximum for parking areas