lsmtest10 0.0.1

Scaling MMD-MA.

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In MMD-MA (Liu et al. 2019, Jointly Embedding Multiple Single-Cell Omics Measurements), we project two sets of points, from two different spaces endowed with a positive definite kernel, to a shared Euclidean space of dimension low_dim. The mappings from high to low dimensional space are obtained using functions belonging to the respective RKHS. To obtain the mappings, we minimise a loss function that is composed of three terms:

• an MMD term between the low dimensional representations of the two views, which encourages them to have the same distribution.
• two non-collapsing penalty terms (corresponding to the pen_dual or pen_primal functions), one for each view. These terms ensure that the low dimensional representations are mutually orthogonal, preventing collapsing.
• two distortion penalties (corresponding to the dis_dual or dis_primal functions), one for each view. These terms encourage the low dimensional representation to obtain the same pairwise structure as the original views.

MMD-MA can be formulated using either the primal (when we use the linear kernel in the input spaces) or the dual problem. Each has advantages or disadvantages depending on the input data. In each view, when the number of features is larger than the number of samples p_featureX >> n_sampleX, then the dual formulation is beneficial in terms of runtime and memory, while if n_sampleX >> p_sampleX, the primal formulation is favorable.

Example

Work in progress. See main.py for usage.