# Kernel Density Estimation as Loss Function

I am working on a 3D point cloud co-registration problem. I have an observed lidar point cloud and a generative model with several stochastic parameters that simulates point clouds. I am interested in using a Kernel Density Estimate generated from the observed lidar point cloud to compute the log probability of the simulated points and then optimize the generative model’s parameters to minimize the negative log likelihood from the KDE.

The sklearn KDE score_samples method is essentially what I’d like to reproduce in PyTorch as a loss function so that it can capture gradients and not just the log probability. I don’t see a straightforward way to implement KDE in PyTorch beyond building one from scratch (e.g., a list-like object containing as many 3D-Normal distributions as there are observed lidar points, which would be looped through to calculate the log_pdf for every simulated point against every one of these KDE distributions and then summing them across all distributions for all simulated points). This seems like a lot of looping, and am not sure if there’s an effective way to broadcast this to be efficient.

Any ideas?

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Just wanted to provide a quick update that I have not found a good solution yet. This project has been on a back-burner, but will probably be picking up again in later this fall. Would still welcome any suggestions.

I’m trying to do something similar (still not quite sure, still learning)
Trying to figure out if this is about the same problem:

I came across this blog post mentioned in a comment to a question that has since been taken down from Stack Overflow.

The authors of the blog post seem like they’re attempting much more complicated modeling than I am, and are using TensorFlow, but it appears they are also looping through each point/kernel to calculate an overall density:

``````def density_output_layer(w, y, cluster_centers, sigma):
output_kernels = []
for center in cluster_centers:
output_kernels.append(tf.reshape(gaussian_kernel(y, center, sigma), [-1, 1]))
output_kernels = tf.concat(output_kernels, axis=1)
output_nodes = tf.multiply(w, output_kernels)
return tf.reduce_sum(output_nodes, axis=1) / total_weight
``````

Got something working for vectorized calculation of log probability. I have noticed this can gobble up memory fast if you have large numbers of points. Would welcome any more thoughts. Here is a class to implement it:

``````import torch
from torch.distributions import MultivariateNormal, Normal
from torch.distributions.distribution import Distribution

class GaussianKDE(Distribution):
def __init__(self, X, bw):
"""
X : tensor (n, d)
`n` points with `d` dimensions to which KDE will be fit
bw : numeric
bandwidth for Gaussian kernel
"""
self.X = X
self.bw = bw
self.dims = X.shape[-1]
self.n = X.shape[0]
self.mvn = MultivariateNormal(loc=torch.zeros(self.dims),
covariance_matrix=torch.eye(self.dims))

def sample(self, num_samples):
idxs = (np.random.uniform(0, 1, num_samples) * self.n).astype(int)
norm = Normal(loc=self.X[idxs], scale=self.bw)
return norm.sample()

def score_samples(self, Y, X=None):
"""Returns the kernel density estimates of each point in `Y`.

Parameters
----------
Y : tensor (m, d)
`m` points with `d` dimensions for which the probability density will
be calculated
X : tensor (n, d), optional
`n` points with `d` dimensions to which KDE will be fit. Provided to
allow batch calculations in `log_prob`. By default, `X` is None and
all points used to initialize KernelDensityEstimator are included.

Returns
-------
log_probs : tensor (m)
log probability densities for each of the queried points in `Y`
"""
if X == None:
X = self.X
log_probs = torch.log(
(self.bw**(-self.dims) *
torch.exp(self.mvn.log_prob(
(X.unsqueeze(1) - Y) / self.bw))).sum(dim=0) / self.n)

return log_probs

def log_prob(self, Y):
"""Returns the total log probability of one or more points, `Y`, using
a Multivariate Normal kernel fit to `X` and scaled using `bw`.

Parameters
----------
Y : tensor (m, d)
`m` points with `d` dimensions for which the probability density will
be calculated

Returns
-------
log_prob : numeric
total log probability density for the queried points, `Y`
"""

X_chunks = self.X.split(1000)
Y_chunks = Y.split(1000)

log_prob = 0

for x in X_chunks:
for y in Y_chunks:
log_prob += self.score_samples(y, x).sum(dim=0)

return log_prob
``````

Here’s a toy example working with gradients:

``````from sklearn.datasets import make_blobs
from torch.optim import LBFGS

# centers of blobs we'll try to find through optimization
# are at (0.1, 0.3) and (-0.2, -0.1)
X, y = make_blobs(5000, centers=[[0.1,0.3],[-0.2,-0.1]], cluster_std=0.1)
kde = GaussianKDE(X=torch.tensor(X, dtype=torch.float32), bw=0.03)
optimizer = LBFGS([test_pts])

for _ in range(10):  # take 10 optimization steps
def closure():