How does the the third-order approximation of a multivariable function work? Is like a 3D matrix of third-order partials?
motoole2
Exactly! This would start involving tensors and tensor notation. An m-tensor is effectively a m-dimensional structure, e.g., T \in R^{n x n x ... x n}. You are already familiar with 1-tensors (vectors) and 2-tensors (matrices). Working with m-tensors gets a little hairy from a notation perspective, but would be required if you want to work with higher-order approximations of multivariable functions.
wanshenl
Do high-dimensional tensors come up in graphics often? What are some examples? (It is difficult to search anything with the word tensor nowadays, thanks to TensorFlow...)
motoole2
Sure! Here's a project on Tensor Displays, a display where you can experience 3D content without the need for special glasses. The paper is a little technical, but the process of creating these 3D images involves working with high-dimensional tensors.
How does the the third-order approximation of a multivariable function work? Is like a 3D matrix of third-order partials?
Exactly! This would start involving tensors and tensor notation. An m-tensor is effectively a m-dimensional structure, e.g.,
T \in R^{n x n x ... x n}. You are already familiar with 1-tensors (vectors) and 2-tensors (matrices). Working with m-tensors gets a little hairy from a notation perspective, but would be required if you want to work with higher-order approximations of multivariable functions.Do high-dimensional tensors come up in graphics often? What are some examples? (It is difficult to search anything with the word tensor nowadays, thanks to TensorFlow...)
Sure! Here's a project on Tensor Displays, a display where you can experience 3D content without the need for special glasses. The paper is a little technical, but the process of creating these 3D images involves working with high-dimensional tensors.