Hidden Manifold

Data for the benchmarking of machine learning models taken from Better than classical? The subtle art of benchmarking quantum machine learning models. Proposed by Goldt et al. (2021) [1], this classification task mimics the idea that data is labeled on a “hidden” manifold that is embedded into a space of different dimensionality. A machine learning model has to find the manifold to solve the problem. It is conjectured that the size of the manifold controls the difficulty of the problem.

Description of the dataset

Input vectors of m dimensions are sampled from a normal distribution in a low-dimensional space and labeled by a single-layer neural network initialised at random. The inputs are then projected to the final d-dimensional space.

There are two different dataset collections in this task:

• hidden_manifold varies only the dimension of the input vectors between $d=2,\ldots,20$ and keeps the dimension of the hidden manifold constant at $m=6$.
• `hidden_manifold_diff`` keeps the dimension constant at $d=10$ and varies the dimensionality $m$ of the manifold between $m=2,\ldots,20$

Additional details

• The class labels are defined as -1, 1.
• For each space, 240 labeled points are provided for training and 60 for testing.
• The datasets are balanced, meaning that they contain the same number of samples for each class.
• Please see the Source code tab to check how the data was generated.

Example usage

[ds] = qml.data.load("other", name="hidden-manifold")

ds.train['4']['inputs'] # points in 4-dimensional space
ds.train['4']['labels'] # labels for the points above

ds.diff_train['5']['inputs'] # points in 10-dimensional space, projected from a 5-dimensional manifold
ds.diff_train['5']['labels'] # labels for the points above

[1] S. Goldt, M. Mezard, F. Krzakala, and L. Zdeborova, Modeling the influence of data structure on learning in neural networks: The hidden manifold model, Physical Review X 10, 041044 (2020).

version 0.1 : initial public release