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What does calculus mean ?

odyssey7 2y ago
Time to schedule with a dentist
seanhunter 2y ago
Calculus is the branch of mathematics that deals with continuous change. Broadly speaking there are two parts to it - differential calculus, which deals with rates of change and integral calculus which deals with areas, volumes and that sort of thing. Pretty early on you learn that these are essentially two sides of the same coin. So in this particular instance since we are talking about matrix calculus it’s a type of multivariable calculus where you’re dealing with functions which take matrices as inputs and “matrix-valued functions” (ie functions which return matrices as an output). Calculus is used for a lot of things but for example if you have a continuous function you can use calculus to find maxima and minima, inflection points etc. Since the focus here is machine learning, one of the most important things you want to be able to do is gradient descent to optimise some cost function by tweaking the weights on your model. The gradient here is a vector field which at every point in the space of your model points in the direction of steepest ascent[1]. So if you want to go down you take the gradient at a particular point and go exactly in the opposite direction. That means you know exactly what model weight tweaks will cause the biggest decrease in your cost function the next time you do a training run. [1] To imagine the gradient, think of an old-school contour map like you’re going to do a hike or something. This is a “scalar field” (a map from spatial coordinates to a scalar value - altitude). The contour lines on the map link points of equal altitude. The gradient is a “vector field” which is a map from spatial coordinates to a vector. So imagine at every point on your contour map there was a little arrow that pointed in the direction of steepest ascent. Because this is matrix calculus you will be dealing with “matrix fields” (maps from spatial coordinates to matrices) as well. So for example say you did a measurement of stress in a steel beam. At each point you would have a “stress tensor” which says what the forces are at that point and in which direction they are pointing. This is a “tensor field” (map from spatial coordinates to a tensor) and a tensor is like a multidimensional matrix but with some additional rules about how it transforms.

Time to schedule with a dentist

Calculus is the branch of mathematics that deals with continuous change. Broadly speaking there are two parts to it - differential calculus, which deals with rates of change and integral calculus which deals with areas, volumes and that sort of thing. Pretty early on you learn that these are essentially two sides of the same coin. So in this particular instance since we are talking about matrix calculus it’s a type of multivariable calculus where you’re dealing with functions which take matrices as inputs and “matrix-valued functions” (ie functions which return matrices as an output). Calculus is used for a lot of things but for example if you have a continuous function you can use calculus to find maxima and minima, inflection points etc. Since the focus here is machine learning, one of the most important things you want to be able to do is gradient descent to optimise some cost function by tweaking the weights on your model. The gradient here is a vector field which at every point in the space of your model points in the direction of steepest ascent[1]. So if you want to go down you take the gradient at a particular point and go exactly in the opposite direction. That means you know exactly what model weight tweaks will cause the biggest decrease in your cost function the next time you do a training run. [1] To imagine the gradient, think of an old-school contour map like you’re going to do a hike or something. This is a “scalar field” (a map from spatial coordinates to a scalar value - altitude). The contour lines on the map link points of equal altitude. The gradient is a “vector field” which is a map from spatial coordinates to a vector. So imagine at every point on your contour map there was a little arrow that pointed in the direction of steepest ascent. Because this is matrix calculus you will be dealing with “matrix fields” (maps from spatial coordinates to matrices) as well. So for example say you did a measurement of stress in a steel beam. At each point you would have a “stress tensor” which says what the forces are at that point and in which direction they are pointing. This is a “tensor field” (map from spatial coordinates to a tensor) and a tensor is like a multidimensional matrix but with some additional rules about how it transforms.