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Ordination or gradient analysis, in multivariate analysis, is a method complementary to data clustering, and used mainly in exploratory data analysis (rather than in hypothesis testing). In contrast to cluster analysis, ordination orders quantities in a (usually lower-dimensional) latent space.
In this way, it is possible to backpropagate the gradient without involving stochastic variable during the update. The scheme of a variational autoencoder after the reparameterization trick. In Variational Autoencoders (VAEs), the VAE objective function, known as the Evidence Lower Bound (ELBO), is given by:
The gradient of the function f(x,y) = −(cos 2 x + cos 2 y) 2 depicted as a projected vector field on the bottom plane. The gradient (or gradient vector field) of a scalar function f(x 1, x 2, x 3, …, x n) is denoted ∇f or ∇ → f where ∇ denotes the vector differential operator, del.
In Cartesian coordinates, the divergence of a continuously differentiable vector field = + + is the scalar-valued function: = = (, , ) (, , ) = + +.. As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.
An adjoint state equation is introduced, including a new unknown variable. The adjoint method formulates the gradient of a function towards its parameters in a constraint optimization form. By using the dual form of this constraint optimization problem, it can be used to calculate the gradient very fast.
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.
Quasi-Newton methods for optimization are based on Newton's method to find the stationary points of a function, points where the gradient is 0. Newton's method assumes that the function can be locally approximated as a quadratic in the region around the optimum, and uses the first and second derivatives to find the stationary point.
The gradient theorem states that if the vector field F is the gradient of some scalar-valued function (i.e., if F is conservative), then F is a path-independent vector field (i.e., the integral of F over some piecewise-differentiable curve is dependent only on end points). This theorem has a powerful converse: