Gradients

Gradients are the multi-dimensional equivalent to differential calculus in differential calculus. Gradients are vector-valued functions, as opposed to derivatives, which are scalar-valued.

Like derivatives, gradients represents the slope of the tangent of the graph of a function. The gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction.

Gradients are important for Machine Learning algorithms such as Stochastic Gradient Descent and Histogram of Oriented Gradients.

The example below depicts the gradients as arrows below the curved grid depiction of a function of two variables:

$\\ f\left ( x,y \right )= -\left (\cos ^{2}x+\cos ^{2}y\right )^{2}$

https://commons.wikimedia.org/wiki/File:Gradient99.png#/media/File:Gradient99.png

References

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