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Testing various ideas on SVD.

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#+TITLE: Testing Svd
#+title: Testing SVD
#+author: Vijay Gopal Chilkuri
#+email: vijay.gopal.c@gmail.com
* Generate a block random distribution of positions
Here's a small function to generate random points inside a 3D box of given length.
#+name: generate3Dpoints
#+begin_src python
def get_sphere_distribution(n, dmin, Ls, maxiter=1e4, allow_wall=True):
"""Get random points in a box with given dimensions and minimum separation.
Parameters:
- n: number of points
- dmin: minimum distance
- Ls: dimensions of box, shape (3,) array
- maxiter: maximum number of iterations.
- allow_wall: whether to allow points on wall;
(if False: points need to keep distance dmin/2 from the walls.)
Return:
- ps: array (n, 3) of point positions,
with 0 <= ps[:, i] < Ls[i]
- n_iter: number of iterations
- dratio: average nearest-neighbor distance, divided by dmin.
Note: with a fill density (sphere volume divided by box volume) above about
0.53, it takes very long. (Random close-packed spheres have a fill density
of 0.64).
Author: Han-Kwang Nienhuys (2020)
Copying: BSD, GPL, LGPL, CC-BY, CC-BY-SA
See Stackoverflow: https://stackoverflow.com/a/62895898/6228891
"""
Ls = np.array(Ls).reshape(3)
if not allow_wall:
Ls -= dmin
# filling factor; 0.64 is for random close-packed spheres
# This is an estimate because close packing is complicated near the walls.
# It doesn't work well for small L/dmin ratios.
sphere_vol = np.pi/6*dmin**3
box_vol = np.prod(Ls + 0.5*dmin)
fill_dens = n*sphere_vol/box_vol
if fill_dens > 0.64:
msg = f'Too many to fit in the volume, density {fill_dens:.3g}>0.64'
raise ValueError(msg)
# initial try
ps = np.random.uniform(size=(n, 3)) * Ls
# distance-squared matrix (diagonal is self-distance, don't count)
dsq = ((ps - ps.reshape(n, 1, 3))**2).sum(axis=2)
dsq[np.arange(n), np.arange(n)] = np.infty
for iter_no in range(int(maxiter)):
# find points that have too close neighbors
close_counts = np.sum(dsq < dmin**2, axis=1) # shape (n,)
n_close = np.count_nonzero(close_counts)
if n_close == 0:
break
# Move the one with the largest number of too-close neighbors
imv = np.argmax(close_counts)
# new positions
newp = np.random.uniform(size=3)*Ls
ps[imv]= newp
# update distance matrix
new_dsq_row = ((ps - newp.reshape(1, 3))**2).sum(axis=-1)
dsq[imv, :] = dsq[:, imv] = new_dsq_row
dsq[imv, imv] = np.inf
else:
raise RuntimeError(f'Failed after {iter_no+1} iterations.')
if not allow_wall:
ps += dmin/2
dratio = (np.sqrt(dsq.min(axis=1))/dmin).mean()
return ps, iter_no+1, dratio
#+end_src
A function that generates a block random set of points
#+name: generateBlocks
#+begin_src python :noweb yes
<<generate3Dpoints>>
def generateBlockRandomPointsAtShftApart(n,L1,dmin,shift):
"""
Parameters:
- n: number of points
- L1: dimensions of box, shape (3,) array
- dmin: minimum intra block distance
- shift: inter block distance
Return:
- r: array (n, 3) of point positions,
"""
Ls = np.array([L1,L1,L1]) # lengths of the box
# Quadrant +,+
dist11,a,b = get_sphere_distribution(n,dmin,Ls)
dist11 += np.array([shift,shift,0.0])
# Quadrant -,+
dist21,a,b = get_sphere_distribution(n,dmin,Ls)
dist21 += np.array([-shift,shift,0.0])
# Quadrant +,-
dist12,a,b = get_sphere_distribution(n,dmin,Ls)
dist12 += np.array([shift,-shift,0.0])
# Quadrant -,-
dist22,a,b = get_sphere_distribution(n,dmin,Ls)
dist22 += np.array([-shift,-shift,0.0])
xs = np.concatenate((dist11[:,0],dist12[:,0],dist21[:,0],dist22[:,0]))
ys = np.concatenate((dist11[:,1],dist12[:,1],dist21[:,1],dist22[:,1]))
zs = np.concatenate((dist11[:,2],dist12[:,2],dist21[:,2],dist22[:,2]))
return((np.array((xs,ys,zs))).T)
#+end_src
#+RESULTS: generateBlocks
#+begin_example
None
#+end_example
#+begin_src python :noweb yes :results file :exports results
# matplotlib related
import matplotlib.pyplot as plt
# linear algebra
import numpy as np
<<generate3Dpoints>>
shift = -1.0
# Quadrant +,+
L1 = 1.0
n = 50 # number of points
dmin = 0.1 # min dist between points
Ls = np.array([L1,L1,L1]) # lengths of the box
dist11,a,b = get_sphere_distribution(n,dmin,Ls)
dist11 += np.array([shift,shift,0.0])
# Quadrant -,+
L1 = 1.0
n = 50 # number of points
dmin = 0.1 # min dist between points
Ls = np.array([L1,L1,L1]) # lengths of the box
dist21,a,b = get_sphere_distribution(n,dmin,Ls)
dist21 += np.array([-shift,shift,0.0])
# Quadrant +,-
L1 = 1.0
n = 50 # number of points
dmin = 0.1 # min dist between points
Ls = np.array([L1,L1,L1]) # lengths of the box
dist12,a,b = get_sphere_distribution(n,dmin,Ls)
dist12 += np.array([shift,-shift,0.0])
# Quadrant -,-
L1 = 1.0
n = 50 # number of points
dmin = 0.1 # min dist between points
Ls = np.array([L1,L1,L1]) # lengths of the box
dist22,a,b = get_sphere_distribution(n,dmin,Ls)
dist22 += np.array([-shift,-shift,0.0])
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
xs = np.concatenate((dist11[:,0],dist12[:,0],dist21[:,0],dist22[:,0]))
ys = np.concatenate((dist11[:,1],dist12[:,1],dist21[:,1],dist22[:,1]))
zs = np.concatenate((dist11[:,2],dist12[:,2],dist21[:,2],dist22[:,2]))
ax.scatter(xs, ys, zs, marker='o')
fig.savefig('/tmp/test3.png')
#plt.show()
return '/tmp/test3.png'
#+end_src
#+RESULTS:
[[file:/tmp/test3.png]]
* Is the SVD decomposition of distances really useful ?
The problem is the following:
** Given a metric F(r1,r2) finding the best representatin of F(r1,r2)
*** Slater metric
The metric can be a scaled distance such as
\[
F(r_1,r_2) = \exp(-\kappa(|r_1 - r_2|))
\]
Where,\(\kappa\) is the rate of the breadth of the slater.
#+begin_src python :results file :exports results
# matplotlib related
import matplotlib
import matplotlib.pyplot as plt
# linear algebra
import numpy
# Data for plotting
kappa = 1.0/2.0
xstart = 0.0
xend = 2.0
xstep = 0.1
s = numpy.array(list(map(lambda x : numpy.exp(-x * numpy.arange(xstart,xend,xstep)), [100, 25, 10,5,2,1]))).T
#s = numpy.exp(-kappa * numpy.arange(0,1,0.1))
t = numpy.arange(xstart,xend,xstep)
fig, ax = plt.subplots()
ax.plot(t, s)
ax.set(xlabel=r'$r_{12}$', ylabel=r'$F(r_1,r_2)$',
title='Comparison of Kappa')
#ax.set_yscale('log')
ax.grid()
fig.savefig('/tmp/test1.png')
#plt.show()
return '/tmp/test1.png'
#+end_src
#+RESULTS:
[[file:/tmp/test1.png]]
*** Gaussian metric
The metric can be a scaled distance such as
\[
F(r_1,r_2) = \exp(-\kappa(|r_1 - r_2|^2))
\]
Where,\(\kappa\) is the rate of the breadth of the gaussian.
#+begin_src python :results file :exports results
# matplotlib related
import matplotlib
import matplotlib.pyplot as plt
# linear algebra
import numpy
# Data for plotting
kappa = 1.0/2.0
xstart = 0.0
xend = 2.0
xstep = 0.1
s = numpy.array(list(map(lambda x : numpy.exp(-x * numpy.power(numpy.arange(xstart,xend,xstep),2)), [100, 25, 10,5,2,1]))).T
#s = numpy.exp(-kappa * numpy.arange(0,1,0.1))
t = numpy.arange(xstart,xend,xstep)
fig, ax = plt.subplots()
ax.plot(t, s)
ax.set(xlabel=r'$r_{12}$', ylabel=r'$F(r_1,r_2)$',
title='Comparison of Kappa')
#ax.set_yscale('log')
ax.grid()
fig.savefig('/tmp/test2.png')
#plt.show()
return '/tmp/test2.png'
#+end_src
#+RESULTS:
[[file:/tmp/test2.png]]
* Generating a box of electrons centered around nucleii
** Generate randomly distributed nucleii
The nucleii will serve as anchors for the screening of distances for calculating the
screened \(\tilde{F}(r_1,r_2)\) metric.
* Calculating SVD of the distance vector
** Slater metric
Calculate the matrix of the \(F(r_1,r_2)\) metric i.e. the slater metric.
#+begin_src python :noweb yes :results file :exports results
import numpy as np
from functools import reduce
import matplotlib.pyplot as plt
<<generateBlocks>>
L1 = 1.0
n = 10 # number of points
dmin = 0.01 # min dist between points
Ls = np.array([L1,L1,L1]) # lengths of the box
shift = -1.0
kappa = 0.6
rlist = generateBlockRandomPointsAtShftApart(n,L1,dmin,shift)
print(rlist.shape)
rij = np.zeros(shape=(rlist.shape[0],rlist.shape[0]))
def funcF(x,y):
return(np.exp(-kappa * np.sqrt(np.abs(np.dot(x,y)))))
rij = np.array([[funcF(xval, yval) for yval in rlist] for xval in rlist])
u,d,vt = np.linalg.svd(rij)
#print(d)
#plt.imshow(rij)
#plt.colorbar()
#plt.show()
plt.plot(range(d.shape[0]),d)
plt.yscale('log')
plt.savefig('/tmp/plot3.png')
return '/tmp/plot3.png'
#+end_src
#+RESULTS:
[[file:/tmp/plot3.png]]
#+begin_src python :results output
import numpy
a = numpy.array([[1,2,3],[4,5,6],[7,8,9]])
b = numpy.array([[11,12,13],[14,15,16],[17,18,19]])
print(list(zip(a,b))[0][1])
#+end_src
#+RESULTS:
#+begin_example
[11 12 13]
#+end_example
** Gaussian metric
Calculate the matrix of the \(FG(r_1,r_2)\) metric i.e. the gaussian metric.
#+begin_src python :noweb yes :results file :exports results
import numpy as np
from functools import reduce
import matplotlib.pyplot as plt
<<generateBlocks>>
L1 = 1.0
n = 50 # number of points
dmin = 0.1 # min dist between points
Ls = np.array([L1,L1,L1]) # lengths of the box
shift = -10.0
kappa = 2.0
rlist = generateBlockRandomPointsAtShftApart(n,L1,dmin,shift)
print(rlist.shape)
rij = np.zeros(shape=(rlist.shape[0],rlist.shape[0]))
def funcF(x,y):
return(np.exp(-kappa * np.sqrt(np.abs(np.dot(x,y)))))
def funcFG(x,y):
return(np.exp(-kappa * np.abs(np.dot(x,y))))
rijSlater = np.array([[funcF(xval, yval) for yval in rlist] for xval in rlist])
rijGaussian = np.array([[funcFG(xval, yval) for yval in rlist] for xval in rlist])
u,dS,vt = np.linalg.svd(rijSlater)
u,dG,vt = np.linalg.svd(rijGaussian)
#print(d)
#plt.imshow(rij)
#plt.colorbar()
#plt.show()
plt.plot(range(dG.shape[0]),np.array([dS,dG]).T)
plt.yscale('log')
plt.savefig('/tmp/plot4.png')
return '/tmp/plot4.png'
#+end_src
#+RESULTS:
[[file:/tmp/plot4.png]]