Skeletonization using Proxies

In [1]:
import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as pt

import scipy.linalg.interpolative as sli

eps = 1e-7
In [2]:
sources = np.random.rand(2, 200)
targets = np.random.rand(2, 200) + 3

pt.plot(sources[0], sources[1], "go")
pt.plot(targets[0], targets[1], "ro")

pt.xlim([-1, 5])
pt.ylim([-1, 5])

pt.gca().set_aspect("equal")
In [3]:
def interaction_mat(t, s):
    all_distvecs = s.reshape(2, 1, -1) - t.reshape(2, -1, 1)
    dists = np.sqrt(np.sum(all_distvecs**2, axis=0))
    return np.log(dists)
In [4]:
def numerical_rank(A, eps):
    _, sigma, _ = la.svd(A)
    return np.sum(sigma >= eps)

Check the interaction rank:

In [5]:
numerical_rank(interaction_mat(targets, sources), eps)
Out[5]:
9

Idea:

  • Don't want to build whole matrix to find the few rows/columns that actually matter.
  • Introduces "proxies" that stand in for
    • all sources outside the targets or
    • all targets outside these sources

Target Skeletonization

In [38]:
nproxies = 25

angles = np.linspace(0, 2*np.pi, nproxies)
target_proxies = 3.5 + 1.5 * np.array([np.cos(angles), np.sin(angles)])
In [39]:
pt.plot(sources[0], sources[1], "go")
pt.plot(targets[0], targets[1], "ro")
pt.plot(target_proxies[0], target_proxies[1], "bo")

pt.xlim([-1, 5])
pt.ylim([-1, 5])

pt.gca().set_aspect("equal")

Construct the interaction matrix from the target proxies to the targets as target_proxy_mat.

A note on terminology: The target_proxies are near the targets but stand in for far-away sources.

In [61]:
target_proxy_mat = interaction_mat(targets, target_proxies)

Check its numerical rank and shape:

In [62]:
numerical_rank(target_proxy_mat, eps)
Out[62]:
24
In [42]:
target_proxy_mat.shape
Out[42]:
(200, 25)

Now compute an ID (row or column?):

In [43]:
idx, proj = sli.interp_decomp(target_proxy_mat.T, nproxies)

Find the target skeleton as target_skeleton, i.e. the indices of the targets from which the remaining values can be recovered:

In [44]:
target_skeleton = idx[:nproxies]

Check that the ID does what is promises:

In [45]:
P = np.hstack([np.eye(nproxies), proj])[:,np.argsort(idx)]
tpm_approx = P.T.dot(target_proxy_mat[target_skeleton])

la.norm(tpm_approx - target_proxy_mat, 2)
Out[45]:
5.7389031370755866e-15

Plot the chosen proxies:

In [46]:
pt.plot(sources[0], sources[1], "go")
pt.plot(targets[0], targets[1], "ro", alpha=0.05)
pt.plot(targets[0, target_skeleton], targets[1, target_skeleton], "ro")
pt.plot(target_proxies[0], target_proxies[1], "bo")

pt.xlim([-1, 5])
pt.ylim([-1, 5])

pt.gca().set_aspect("equal")

What does this mean?

  • We have now got a moral equivalent to a local expansion: The point values at the target skeleton points.
  • Is it a coincidence that the skeleton points sit at the boundary of the target region?
  • How many target proxies should we choose?
  • Can cheaply recompute potential at any target from those few points.
  • Have thus reduce LA-based evaluation cost to same as expansion-based cost.

Can we come up with an equivalent of a multipole expansion?

Check that this works for 'our' sources:

In [49]:
imat_error = (
    P.T.dot(interaction_mat(targets[:, target_skeleton], sources))
    -
    interaction_mat(targets, sources))

la.norm(imat_error, 2)
Out[49]:
6.1074574232342807e-09

Source Skeletonization

In [54]:
nproxies = 25

angles = np.linspace(0, 2*np.pi, nproxies)
source_proxies = 0.5 + 1.5 * np.array([np.cos(angles), np.sin(angles)])
In [56]:
pt.plot(sources[0], sources[1], "go")
pt.plot(targets[0], targets[1], "ro")
pt.plot(source_proxies[0], source_proxies[1], "bo")

pt.xlim([-1, 5])
pt.ylim([-1, 5])

pt.gca().set_aspect("equal")

Construct the interaction matrix from the sources to the source proxies as source_proxy_mat:

A note on terminology: The source_proxies are near the sources but stand in for far-away targets.

In [73]:
source_proxy_mat = interaction_mat(source_proxies, sources)
In [74]:
source_proxy_mat.shape
Out[74]:
(25, 200)

Now compute an ID (row or column?):

In [75]:
idx, proj = sli.interp_decomp(source_proxy_mat, nproxies)
In [76]:
source_skeleton = idx[:nproxies]
In [78]:
P = np.hstack([np.eye(nproxies), proj])[:,np.argsort(idx)]
tsm_approx = source_proxy_mat[:, source_skeleton].dot(P)

la.norm(tsm_approx - source_proxy_mat, 2)
Out[78]:
6.3367197566876269e-15

Plot the chosen proxies:

In [79]:
pt.plot(sources[0], sources[1], "go", alpha=0.05)
pt.plot(targets[0], targets[1], "ro")
pt.plot(sources[0, source_skeleton], sources[1, source_skeleton], "go")
pt.plot(source_proxies[0], source_proxies[1], "bo")

pt.xlim([-1, 5])
pt.ylim([-1, 5])

pt.gca().set_aspect("equal")

Check that it works for 'our' targets:

In [81]:
imat_error = (
    interaction_mat(targets, sources[:, source_skeleton]).dot(P)
    -
    interaction_mat(targets, sources))

la.norm(imat_error, 2)
Out[81]:
1.5507578112941745e-08
  • Sensibly, this is just the transpose of the target skeletonization process.
    • For a given point cluster, the same skeleton can serve for target and source skeletonization!
  • Computationally, starting from your original charges $x$, you accumulate 'new' charges $Px$ at the skeleton points and then only compute the interaction from the source skeleton to the targets.