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