Uploaded June 2021 | Updated September 2026, 1 day ago
Selecting a kernel has a huge impact on the types of dynamics you can analyze. Here we go over what happens when you use the exponential dot product kernel and the Gaussian RBF (and many other spaces) for Dynamic Mode Decompositions.
In the end, this comes down to products and quotients of power series yield... power series (provided the denominator is nonzero). We go over a couple of basic concepts in kernel spaces, and establish a couple of theorems.
Note that here we took the perspective of having a densely defined Liouivlle operator (i.e. Koopman generator). In the bounded case, the theorem is much easier to establish. Also, this style of argument applies to pretty much any function space you select, and yes, you must select a function space, since otherwise convergence of sequences of functions is meaningless.
NOTE: I did leave out one detail in the discussion. Here we are leveraging the universality of our kernels to ensure that there is at least one function that doesn't vanish at a selected point (for multiplication operators), and the universality of their gradients to ensure that for each x and v, there is at least one function such that grad g(x)*v doesn't vanish. This ensures that the respective kernel at the selected point is not the zero function, and that h_{x,v} is not the zero function either. Universality isn't the only way to achieve this, and many spaces will satisfy the conditions.
Music:
Come 2gether by Ooyy
Remove the Complexities by Peter Sandberg
Supine by Peter Sandberg
Guardians + Tek by Craig Hardgrove
Selecting a kernel has a huge impact on the types of dynamics you can analyze. Here we go over what happens when you use the exponential dot product kernel and the Gaussian RBF (and many other spaces) for Dynamic Mode Decompositions.
In the end, this comes down to products and quotients of power series yield... power series (provided the denominator is nonzero). We go over a couple of basic concepts in kernel spaces, and establish a couple of theorems.
Note that here we took the perspective of having a densely defined Liouivlle operator (i.e. Koopman generator). In the bounded case, the theorem is much easier to establish. Also, this style of argument applies to pretty much any function space you select, and yes, you must select a function space, since otherwise convergence of sequences of functions is meaningless.
NOTE: I did leave out one detail in the discussion. Here we are leveraging the universality of our kernels to ensure that there is at least one function that doesn't vanish at a selected point (for multiplication operators), and the universality of their gradients to ensure that for each x and v, there is at least one function such that grad g(x)*v doesn't vanish. This ensures that the respective kernel at the selected point is not the zero function, and that h_{x,v} is not the zero function either. Universality isn't the only way to achieve this, and many spaces will satisfy the conditions.
Music:
Come 2gether by Ooyy
Remove the Complexities by Peter Sandberg
Supine by Peter Sandberg
Guardians + Tek by Craig Hardgrove
