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The results combine techniques from information theory, affine differential geometry, convex analysis and many other fields. One of the most perspective information geometry approaches find applications in machine learning. For example, the developing of information-geometric optimization methods (mirror descent [6] and natural gradient descent ...
The properties of gradient descent depend on the properties of the objective function and the variant of gradient descent used (for example, if a line search step is used). The assumptions made affect the convergence rate, and other properties, that can be proven for gradient descent. [ 33 ]
For any smooth function f on a Riemannian manifold (M, g), the gradient of f is the vector field ∇f such that for any vector field X, (,) =, that is, ((),) = (), where g x ( , ) denotes the inner product of tangent vectors at x defined by the metric g and ∂ X f is the function that takes any point x ∈ M to the directional derivative of f ...
A Riemannian manifold is a smooth manifold together with a Riemannian metric. The techniques of differential and integral calculus are used to pull geometric data out of the Riemannian metric. For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel transport.
The adjoint state method is a numerical method for efficiently computing the gradient of a function or operator in a numerical optimization problem. [1] It has applications in geophysics, seismic imaging, photonics and more recently in neural networks. [2] The adjoint state space is chosen to simplify the physical interpretation of equation ...
The manifold hypothesis is related to the effectiveness of nonlinear dimensionality reduction techniques in machine learning. Many techniques of dimensional reduction make the assumption that data lies along a low-dimensional submanifold, such as manifold sculpting , manifold alignment , and manifold regularization .
Due to its resulting linear memory requirement, the L-BFGS method is particularly well suited for optimization problems with many variables. Instead of the inverse Hessian H k, L-BFGS maintains a history of the past m updates of the position x and gradient ∇f(x), where generally the history size m can be small (often <).
In machine learning, the vanishing gradient problem is the problem of greatly diverging gradient magnitudes between earlier and later layers encountered when training neural networks with backpropagation. In such methods, neural network weights are updated proportional to their partial derivative of the loss function. [1]