Note

Click here to download the full example code

# Scaling up Gaussian convolutions on 3D point clouds¶

Let’s compare the performances of PyTorch and KeOps on simple Gaussian RBF kernel products, as the number of samples grows from 100 to 1,000,000.

Note

In this demo, we use exact **bruteforce** computations
(tensorized for PyTorch and online for KeOps), without leveraging any multiscale
or low-rank (multipole) decomposition of the Kernel matrix.
Please visit the documentation of the GeomLoss package
for a discussion of clever, scalable schemes.

## Setup¶

```
import os
import numpy as np
import time
from matplotlib import pyplot as plt
import importlib
import torch
use_cuda = torch.cuda.is_available()
```

Benchmark specifications:

```
D = 3 # Let's do this in 3D
MAXTIME = 10 if use_cuda else 1 # Max number of seconds before we break the loop
REDTIME = 2 if use_cuda else .2 # Decrease the number of runs if computations take longer than 2s...
# Number of samples that we'll loop upon
NS = [100, 200, 500,
1000, 2000, 5000,
10000, 20000, 50000,
100000, 200000, 500000,
1000000]
```

Synthetic dataset. Feel free to use a Stanford Bunny, or whatever!

```
def generate_samples(N, device, lang, batchsize=None):
"""Create point clouds sampled non-uniformly on a sphere of diameter 1."""
B = () if batchsize is None else (batchsize,)
if lang == 'torch':
if device == 'cuda':
torch.cuda.manual_seed_all(1234)
else:
torch.manual_seed(1234)
x = torch.randn( B + (N, D), device=device)
x[:,0] += 1
x = x / (2*x.norm(dim=1,keepdim=True))
y = torch.randn( B + (N, D), device=device)
y[:,1] += 2
y = y / (2*y.norm(dim=1,keepdim=True))
# Draw a random source signal:
b = torch.randn( B + (N, 1), device=device)
else:
np.random.seed(1234)
x = np.random.rand( *(B + (N, D)) ).astype('float32')
y = np.random.rand( *(B + (N, D))).astype('float32')
b = np.random.randn(*(B + (N,))).astype('float32')
return x, y, b
```

Define a simple Gaussian RBF product, using a **tensorized** implementation:

```
def gaussianconv_numpy(x, y, b):
K_xy = np.exp( - np.sum( (x[:,None,:] - y[None,:,:]) **2, axis=2) )
return K_xy@b
def gaussianconv_pytorch(x, y, b):
D_xx = (x*x).sum(-1).unsqueeze(1) # (N,1)
D_xy = torch.matmul( x, y.permute(1,0) ) # (N,D) @ (D,M) = (N,M)
D_yy = (y*y).sum(-1).unsqueeze(0) # (1,M)
D_xy = D_xx - 2*D_xy + D_yy
K_xy = (-D_xy).exp()
return K_xy@b
```

Define a simple Gaussian RBF product, using an **online** implementation:

```
from pykeops.torch import generic_sum
gaussianconv_keops = generic_sum("Exp(-SqDist(X,Y)) * B", # Formula
"A = Vi(1)", # Output
"X = Vi({})".format(D), # 1st argument
"Y = Vj({})".format(D), # 2nd argument
"B = Vj(1)" ) # 3rd argument
```

Finally, perform the same operation with our high-level `pykeops.torch.LazyTensor`

wrapper:

```
from pykeops.torch import LazyTensor
def gaussianconv_lazytensor(x, y, b):
nbatchdims = len(x.shape) - 2
x_i = LazyTensor( x.unsqueeze(-2) ) # (B, M, 1, D)
y_j = LazyTensor( y.unsqueeze(-3) ) # (B, 1, N, D)
D_ij = ((x_i - y_j) ** 2).sum(-1) # (B, M, N, 1)
K_ij = ( - D_ij).exp() # (B, M, N, 1)
S_ij = K_ij * b.unsqueeze(-3) # (B, M, N, 1) * (B, 1, N, 1)
return S_ij.sum(dim = nbatchdims+1)
```

## Benchmarking loops¶

```
def benchmark(routine_batchsize, dev, N, loops = 10, lang='torch') :
"""Times a convolution on an N-by-N problem."""
if isinstance(routine_batchsize, tuple):
Routine, B = routine_batchsize
else:
Routine, B = routine_batchsize, None
importlib.reload(torch) # In case we had a memory overflow just before...
device = torch.device(dev)
x, y, b = generate_samples(N, device, lang, batchsize=B)
# We simply benchmark a convolution
code = "a = Routine( x, y, b ) "
exec( code, locals() ) # Warmup run, to compile and load everything
t_0 = time.perf_counter() # Actual benchmark --------------------
if use_cuda: torch.cuda.synchronize()
for i in range(loops):
exec( code, locals() )
if use_cuda: torch.cuda.synchronize()
elapsed = time.perf_counter() - t_0 # ---------------------------
if B is None:
print("{:3} NxN convolution, with N ={:7}: {:3}x{:3.6f}s".format(loops, N, loops, elapsed / loops))
return elapsed / loops
else:
print("{:3}x{:3} NxN convolution, with N ={:7}: {:3}x{:3}x{:3.6f}s".format(
B, loops, N, B, loops, elapsed / (B*loops)))
return elapsed / (B * loops)
def bench_config(Routine, backend, dev, l) :
"""Times a convolution for an increasing number of samples."""
print("Backend : {}, Device : {} -------------".format(backend, dev))
times = []
try :
Nloops = [100, 10, 1]
nloops = Nloops.pop(0)
for n in NS :
elapsed = benchmark(Routine, dev, n, loops=nloops, lang=l)
times.append( elapsed )
if (nloops * elapsed > MAXTIME) \
or (nloops * elapsed > REDTIME/10 and len(Nloops) > 0 ) :
nloops = Nloops.pop(0)
except RuntimeError :
print("**\nMemory overflow !")
except IndexError :
print("**\nToo slow !")
return times + (len(NS)-len(times)) * [np.nan]
def full_bench(title, routines) :
"""Benchmarks the varied backends of a geometric loss function."""
backends = [ backend for (_, backend, _) in routines ]
print("Benchmarking : {} ===============================".format(title))
lines = [ NS ]
for routine, backend, lang in routines :
lines.append( bench_config(routine, backend, "cuda" if use_cuda else "cpu", lang) )
benches = np.array(lines).T
# Creates a pyplot figure:
plt.figure(figsize=(12,8))
linestyles = ["o-", "s-", "^-"]
for i, backend in enumerate(backends):
plt.plot( benches[:,0], benches[:,i+1], linestyles[i],
linewidth=2, label='backend = "{}"'.format(backend) )
for (j, val) in enumerate( benches[:,i+1] ):
if np.isnan(val) and j > 0:
x, y = benches[j-1,0], benches[j-1,i+1]
plt.annotate('Memory overflow!',
xy=(x, 1.05*y),
horizontalalignment='center',
verticalalignment='bottom')
break
plt.title('Runtimes for {} in dimension {}'.format(title, D))
plt.xlabel('Number of samples')
plt.ylabel('Seconds')
plt.yscale('log') ; plt.xscale('log')
plt.legend(loc='upper left')
plt.grid(True, which="major", linestyle="-")
plt.grid(True, which="minor", linestyle="dotted")
plt.axis([ NS[0], NS[-1], 1e-4, MAXTIME ])
plt.tight_layout()
# Save as a .csv to put a nice Tikz figure in the papers:
header = "Npoints " + " ".join(backends)
os.makedirs("output", exist_ok=True)
np.savetxt("output/benchmark_convolutions_3D.csv", benches,
fmt='%-9.5f', header=header, comments='')
```

## NumPy vs. PyTorch vs. KeOps¶

```
routines = [ (gaussianconv_numpy, "Numpy", "numpy"),
(gaussianconv_pytorch, "PyTorch", "torch"),
(gaussianconv_keops, "KeOps", "torch"),]
full_bench( "Gaussian Matrix-Vector products", routines )
```

Out:

```
Benchmarking : Gaussian Matrix-Vector products ===============================
Backend : Numpy, Device : cuda -------------
100 NxN convolution, with N = 100: 100x0.000421s
100 NxN convolution, with N = 200: 100x0.001759s
100 NxN convolution, with N = 500: 100x0.010887s
10 NxN convolution, with N = 1000: 10x0.037291s
1 NxN convolution, with N = 2000: 1x0.151025s
1 NxN convolution, with N = 5000: 1x0.841828s
1 NxN convolution, with N = 10000: 1x3.242485s
1 NxN convolution, with N = 20000: 1x12.794234s
**
Too slow !
Backend : PyTorch, Device : cuda -------------
100 NxN convolution, with N = 100: 100x0.000159s
100 NxN convolution, with N = 200: 100x0.000157s
100 NxN convolution, with N = 500: 100x0.000159s
100 NxN convolution, with N = 1000: 100x0.000170s
100 NxN convolution, with N = 2000: 100x0.000424s
100 NxN convolution, with N = 5000: 100x0.002323s
10 NxN convolution, with N = 10000: 10x0.010904s
10 NxN convolution, with N = 20000: 10x0.039553s
**
Memory overflow !
Backend : KeOps, Device : cuda -------------
100 NxN convolution, with N = 100: 100x0.000125s
100 NxN convolution, with N = 200: 100x0.000125s
100 NxN convolution, with N = 500: 100x0.000133s
100 NxN convolution, with N = 1000: 100x0.000146s
100 NxN convolution, with N = 2000: 100x0.000173s
100 NxN convolution, with N = 5000: 100x0.000258s
100 NxN convolution, with N = 10000: 100x0.000417s
100 NxN convolution, with N = 20000: 100x0.001067s
100 NxN convolution, with N = 50000: 100x0.004768s
10 NxN convolution, with N = 100000: 10x0.017990s
10 NxN convolution, with N = 200000: 10x0.071256s
1 NxN convolution, with N = 500000: 1x0.432217s
1 NxN convolution, with N =1000000: 1x1.731090s
```

## Genred vs. LazyTensor vs. batched LazyTensor¶

```
routines = [ (gaussianconv_keops, "KeOps (Genred)", "torch"),
(gaussianconv_lazytensor, "KeOps (LazyTensor)", "torch"),
((gaussianconv_lazytensor, 10), "KeOps (LazyTensor, batchsize=10)", "torch"),]
full_bench( "Gaussian Matrix-Vector products", routines )
plt.show()
```

Out:

```
Benchmarking : Gaussian Matrix-Vector products ===============================
Backend : KeOps (Genred), Device : cuda -------------
100 NxN convolution, with N = 100: 100x0.000119s
100 NxN convolution, with N = 200: 100x0.000124s
100 NxN convolution, with N = 500: 100x0.000134s
100 NxN convolution, with N = 1000: 100x0.000148s
100 NxN convolution, with N = 2000: 100x0.000179s
100 NxN convolution, with N = 5000: 100x0.000280s
100 NxN convolution, with N = 10000: 100x0.000454s
100 NxN convolution, with N = 20000: 100x0.001131s
100 NxN convolution, with N = 50000: 100x0.004985s
10 NxN convolution, with N = 100000: 10x0.018807s
10 NxN convolution, with N = 200000: 10x0.072771s
1 NxN convolution, with N = 500000: 1x0.442033s
1 NxN convolution, with N =1000000: 1x2.343521s
Backend : KeOps (LazyTensor), Device : cuda -------------
100 NxN convolution, with N = 100: 100x0.000298s
100 NxN convolution, with N = 200: 100x0.000301s
100 NxN convolution, with N = 500: 100x0.000310s
100 NxN convolution, with N = 1000: 100x0.000322s
100 NxN convolution, with N = 2000: 100x0.000361s
100 NxN convolution, with N = 5000: 100x0.000471s
100 NxN convolution, with N = 10000: 100x0.000658s
100 NxN convolution, with N = 20000: 100x0.001436s
100 NxN convolution, with N = 50000: 100x0.007684s
10 NxN convolution, with N = 100000: 10x0.034021s
1 NxN convolution, with N = 200000: 1x0.133892s
1 NxN convolution, with N = 500000: 1x1.165402s
1 NxN convolution, with N =1000000: 1x12.377574s
**
Too slow !
Backend : KeOps (LazyTensor, batchsize=10), Device : cuda -------------
10x100 NxN convolution, with N = 100: 10x100x0.000074s
10x100 NxN convolution, with N = 200: 10x100x0.000078s
10x100 NxN convolution, with N = 500: 10x100x0.000085s
10x100 NxN convolution, with N = 1000: 10x100x0.000098s
10x100 NxN convolution, with N = 2000: 10x100x0.000148s
10x100 NxN convolution, with N = 5000: 10x100x0.000420s
10x100 NxN convolution, with N = 10000: 10x100x0.001404s
10x100 NxN convolution, with N = 20000: 10x100x0.005176s
10x 10 NxN convolution, with N = 50000: 10x 10x0.030826s
10x 1 NxN convolution, with N = 100000: 10x 1x0.121361s
10x 1 NxN convolution, with N = 200000: 10x 1x0.483594s
10x 1 NxN convolution, with N = 500000: 10x 1x3.012165s
10x 1 NxN convolution, with N =1000000: 10x 1x12.030773s
**
Too slow !
/home/bcharlier/keops/pykeops/benchmarks/plot_benchmarks_convolutions_3D.py:259: UserWarning: Matplotlib is currently using agg, which is a non-GUI backend, so cannot show the figure.
plt.show()
```

**Total running time of the script:** ( 6 minutes 37.333 seconds)