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 [27]:
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 [28]:
def numerical_rank(A, eps):
_, sigma, _ = la.svd(A)
return np.sum(sigma >= eps)
Check the interaction rank:
In [29]:
numerical_rank(interaction_mat(targets, sources), eps)
Out[29]:
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 [30]:
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 [31]:
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 [32]:
target_proxy_mat = interaction_mat(targets, target_proxies)
Check its numerical rank and shape:
In [33]:
numerical_rank(target_proxy_mat, eps)
Out[33]:
24
In [34]:
target_proxy_mat.shape
Out[34]:
(200, 25)
Now compute an ID (row or column?):
In [35]:
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 [36]:
target_skeleton = idx[:nproxies]
Check that the ID does what is promises:
In [37]:
P = np.hstack([np.eye(nproxies), proj])[:,np.argsort(idx)]
tpm_approx = P.T @ target_proxy_mat[target_skeleton]
la.norm(tpm_approx - target_proxy_mat, 2)
Out[37]:
6.738945834835623e-15
Plot the chosen "skeleton" and the proxies:
In [38]:
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 [16]:
imat_error = (
P.T.dot(interaction_mat(targets[:, target_skeleton], sources))
-
interaction_mat(targets, sources))
la.norm(imat_error, 2)
Out[16]:
1.8776453951593488e-09
Source Skeletonization¶
In [17]:
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 [18]:
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 [19]:
source_proxy_mat = interaction_mat(source_proxies, sources)
In [20]:
source_proxy_mat.shape
Out[20]:
(25, 200)
Now compute an ID (row or column?):
In [21]:
idx, proj = sli.interp_decomp(source_proxy_mat, nproxies)
In [22]:
source_skeleton = idx[:nproxies]
In [23]:
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[23]:
5.2523961564651685e-15
Plot the chosen skeleton as well as the proxies:
In [24]:
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 [25]:
imat_error = (
interaction_mat(targets, sources[:, source_skeleton]) @ P
-
interaction_mat(targets, sources))
la.norm(imat_error, 2)
Out[25]:
5.966880531704515e-09
- 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.
Hierarchical Skeletonization¶
In [18]:
gathered_skeletons = np.concatenate(
[targets[:, target_skeleton],
targets[:, target_skeleton] + np.array([0, 1]).reshape(-1, 1),
targets[:, target_skeleton] + np.array([1, 0]).reshape(-1, 1),
targets[:, target_skeleton] + np.array([1, 1]).reshape(-1, 1),
], axis=1)
pt.plot(gathered_skeletons[0], gathered_skeletons[1], "ro", alpha=0.05)
target_proxies = 4 + 2 * np.array([np.cos(angles), np.sin(angles)])
pt.plot(target_proxies[0], target_proxies[1], "bo")
parent_proxy_mat = interaction_mat(gathered_skeletons, target_proxies)
idx, proj = sli.interp_decomp(parent_proxy_mat.T, nproxies)
parent_skeleton = idx[:nproxies]
pt.plot(gathered_skeletons[0, parent_skeleton], gathered_skeletons[1, parent_skeleton], "ro")
Out[18]:
[<matplotlib.lines.Line2D at 0x7fb821315fd0>]
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