Parallel MDM

Mapped distance matrix

The Mapped Distance Matrix (MDM) of two sets $\mathcal{X}, \mathcal{Y}$ of n-dimensional points is an algebraic structure which is defined in general as follows, given a mapping $f$:

$$\mathbf{M}(\mathcal{X}, \mathcal{Y}, f)_{i,j} := f(\Vert \mathcal{X}_i - \mathcal{Y}_j\Vert)$$

where $\Vert \cdot \Vert$ is an appropriate distance notion on the space of definition of $\mathcal{X}$ and $\mathcal{Y}$.

The problem might be augmented by weighting the contributions with a matrix of weights $\mathbf{W}$; the updated definition is then:

$$\mathbf{M}(\mathcal{X}, \mathcal{Y}, f)_{i,j} := \mathbf{W}_{i,j} f(\Vert \mathcal{X}_{i} - \mathcal{Y}_{j}\Vert)$$

A particularly popular form of the problem (which is also what we treat in this repository) occurs when weights are defined individually for the members of $\mathcal{Y}$ (i.e. the columns of $\mathbf{W}$ are taken constants):

$$\mathbf{M}(\mathcal{X}, \mathcal{Y}, f)_{i,j} := \mathbf{W}_{j} f(\Vert \mathcal{X}_{i} - \mathcal{Y}_{j}\Vert)$$

A notable case: uniform grid

In general $\mathcal{X}, \mathcal{Y}$ identify two general sets of points. A few applications allow more assumptions on the two sets. For instance, $\mathcal{X}$ might be taken to be an uniform grid. In this case a few interesting optimizization can be taken into account for the computation of the matrix.

More assumptions

Practical applications usually require huge sets of points, which causes memory errors on commonly used devices. This is why it's preferrable to compute the vector $\tilde{\mathbf{M}}$ defined below instead of $\mathbf{M}$:

$$\tilde{\mathbf{M}}_{i} := \sum_{j} \mathbf{M}_{i,j}$$

For most use cases this is enough.

Roadmap

• Algorithms
  • Uniform grid algorithm
  • Scattered points algorithm
  • Fourier-transfor based algorithm
• Backends
  • NumPy/Numba
  • PyTorch
  • JAX(?)
• Parallelization
  • Multithreading/Multiprocessing
  • GPU w/ PyTorch
  • GPU w/ JAX
  • CUDA kernels(?)
• Tests
• Documentation
• Benchmark (+comparison with competitors)
  • CPU
  • GPU
  • Several different bin sizes
  • pts_per_future
• Future
  • Periodicity
  • More general about distance definitions

Authors

This repository is maintained by:

• Francesco Andreuzzi (CERN, SISSA)