This project is an investigation into the effect of different methods of data scaling in quantum machine learning models. The objectives are:
- O1: Code a simple QML model with 1 dimensional data input and show that unconstrained optimisation of a data scaling parameter leads to overfitting.
- O2: Develop a method to scale the data that does not lead to overfitting, and improves generalisation error with respect to models with fixed scaling.
- O3: Generalise the method in O2 to input data of arbitrary dimension and investigate performance on some data sets.
- O4: (if time) Observe a double descent curve in a simple QML model, and show that this phenomenon relies on a good choice of data scaling.
##Some intial ideas There are a number of studies in classical machine learning that investigate generalisation in 'fourier models', which share the same basic mathematical structure as QML models. We might be able to get some ideas from these papers, although I have not found anything that adresses the scaling of data explicity.
- Two models of double descent for weak features
Here they show a double descent model in a fourier model. - Occam's Razor
Something about scaling in a fourier model in bayesian setting, not sure if that relevant.
Another thing we could look at is somehow using fourier analysis to judiciously chose the data scaling, so that there is a large overlap between the frequencies avaliable in the quantum model and the highly weighted frequencies in the data. This will probably involve estimating the discrete fourier transform of the data. The issue here is that the data is generally not evenly spaced and there will be missing data, so we need techniques to estimate the DFT in this setting. I imagine there is a lot of literature that could help us here, see e.g. here.
Our dataset (data/custom_dataset.txt) consists of 1D noisy points sampled from the following function:
np.sin((2 * np.pi / 10) * x - np.pi)
All the testing code that is not relevant to the project can be moved inside a dummy/ folder, and git will ignore it.