5 ms·
> For instance, we solve nontrivial nonlinear PDEs (one HJB equation and one Black-Scholes equation) in 100,000 dimensions in 6 hours on a single GPU using SDG
by SeanAnderson 3y ago
> For instance, we solve nontrivial nonlinear PDEs (one HJB equation and one Black-Scholes equation) in 100,000 dimensions in 6 hours on a single GPU using SDGD with PINNs
100,000 dimensions? I thought there were like... 11 tops? https://imagine.gsfc.nasa.gov/science/questions/superstring.html#:~:text=The%20only%20consistent%20framework%20to%20describe%20those%20strings%20implies%20a%2010%2D%20or%20even%20conceivably%20an%2011%2Ddimension%20world https://imagine.gsfc.nasa.gov/science/questions/superstring....
(edit: oops, I misunderstood the context of dimensions here. my bad. thanks)
- Legend2440 3y agoThe way ML people use "dimension", each free parameter is an extra dimension. A high-resolution 2D image is considered to have millions of dimensions - one for each pixel.
- atoav 3y agoAnd each pixel is what, a RGB-vector?
- dartos 3y agoIn the context of neural networks, kind of. It’s 3 numbers on the input layer, but that could influence N parameters in later layers.
- PartiallyTyped 3y agoWe usually do HxWxC, for height, width, and channels, so each pixel is addressed via the two first dims of the input, and then it has 3 channels. Of course, you can transpose the tensor to CxHxW or CxWxH. Different ordering behaves differently with respect to memory locality.
- polynomial 3y agoYes, each pixel is a vector of the rasterized vector space.
- hommelix 3y agoThanks for clarifying the dimensions.
- reedf1 3y agoThese are all expressions of the same concept. The confusion is that physicists are describing the dimensionality of specific systems (space, spacetime, superstring theory, supergravity) - that doesn't mean that this limits the dimensionality of other systems (which is often where confusion lies between laymen).
- tnecniv 3y agoThat’s not just ML, that’s linear algebra. Moreover, across fields in math, the dimension of an object is defined differently, but it’s always a very fundamental property of an object that captures loosely how complex an object is. Often, that complexity is related to how many numbers you need to describe said object.
- relyks 3y agoPhysics uses vectors, which are multi-dimensional and can represent a lot more than 11 dimensions... Machine learning typically uses feature vectors, which are basically just lists of numerical properties.
- lapinot 3y agoTo add to the other answers: dimensions also appear in higher-order PDE: you typically go from an nth order 1-dimensional equation (an equation mentioning the n-th derivative of your unknown function) to an n-dimensional 1st order equation. There is a general pattern of adding parameters to have several simple problems instead of one complicated.