irpjast/testing_svd.org

Testing Svd Testing SVD

• Generate a block random distribution of positions
• Is the SVD decomposition of distances really useful ?
  • Given a metric F(r1,r2) finding the best representatin of F(r1,r2)
    • Slater metric
    • Gaussian metric
• Generating a box of electrons centered around nucleii
  • Generate randomly distributed nucleii
• Calculating SVD of the distance vector
  • Slater metric
  • Gaussian metric
  • Palying around
  • Testing SVD for custom matrices

Generate a block random distribution of positions

Here's a small function to generate random points inside a 3D box of given length.

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

A function that generates a block random set of points

<<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)

None

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.

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.

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.

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])
print(numpy.square(a[:,0]))

def stepExp(a):
    def myexp(x):
        if numpy.abs(x) > 1e+0:
            return numpy.zeros_like(x)
        else:
            return numpy.exp(x)

    res = numpy.array([[myexp(x) for x in y] for y in a])
    return(res)

print(numpy.exp(a))
print(stepExp(a))

[11 12 13]
[ 1 16 49]
[[2.71828183e+00 7.38905610e+00 2.00855369e+01]
 [5.45981500e+01 1.48413159e+02 4.03428793e+02]
 [1.09663316e+03 2.98095799e+03 8.10308393e+03]]
[[2.71828183 0.         0.        ]
 [0.         0.         0.        ]
 [0.         0.         0.        ]]

Gaussian metric

Calculate the matrix of the \(FG(r_1,r_2)\) metric i.e. the gaussian metric.

Palying around

Calculate the matrix of the \(FG(r_1,r_2)\) metric i.e. the gaussian metric.

Testing SVD for custom matrices

import numpy
import matplotlib.pyplot as plt

a = numpy.array([[0,100,200],[100,0,200],[100,200,0]])
b = numpy.exp(-a)
print("Matrix A")
print(a)
print("Matrix Exp(A)")
print(numpy.around(b,10))
u,d,vt = numpy.linalg.svd(a)
d = d/numpy.linalg.norm(d)
print("Singular values of A")
print(numpy.around(d,3))
print("Singular vectors of A")
print(numpy.around(u,3))
u,d,vt = numpy.linalg.svd(b)
d = d/numpy.linalg.norm(d)
print("Singular values of Exp(A)")
print(numpy.around(d,3))
print("Singular vectors of Exp(A)")
print(numpy.around(u,3))

Matrix A
[[  0 100 200]
 [100   0 200]
 [100 200   0]]
Matrix Exp(A)
[[1. 0. 0.]
 [0. 1. 0.]
 [0. 0. 1.]]
Singular values of A
[0.813 0.53  0.24 ]
Singular vectors of A
[[-0.67   0.142  0.728]
 [-0.626  0.42  -0.657]
 [-0.399 -0.896 -0.193]]
Singular values of Exp(A)
[0.577 0.577 0.577]
Singular vectors of Exp(A)
[[-1.     0.    -0.   ]
 [-0.    -0.894 -0.447]
 [-0.    -0.447  0.894]]