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 importlib
import os
import time
import numpy as np
import torch
from matplotlib import pyplot as plt
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-5, 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: 100x0.000293s
100 NxN convolution, with N = 200: 100x0.001489s
100 NxN convolution, with N = 500: 100x0.009204s
10 NxN convolution, with N = 1000: 10x0.037353s
1 NxN convolution, with N = 2000: 1x0.158314s
1 NxN convolution, with N = 5000: 1x0.919527s
1 NxN convolution, with N = 10000: 1x3.404064s
1 NxN convolution, with N = 20000: 1x13.318650s
**
Too slow !
Backend : PyTorch, Device : cuda -------------
100 NxN convolution, with N = 100: 100x0.000208s
100 NxN convolution, with N = 200: 100x0.000208s
100 NxN convolution, with N = 500: 100x0.000208s
100 NxN convolution, with N = 1000: 100x0.000168s
100 NxN convolution, with N = 2000: 100x0.000394s
100 NxN convolution, with N = 5000: 100x0.002254s
10 NxN convolution, with N = 10000: 10x0.009712s
10 NxN convolution, with N = 20000: 10x0.038868s
**
Memory overflow !
Backend : KeOps, Device : cuda -------------
100 NxN convolution, with N = 100: 100x0.000108s
100 NxN convolution, with N = 200: 100x0.000110s
100 NxN convolution, with N = 500: 100x0.000121s
100 NxN convolution, with N = 1000: 100x0.000140s
100 NxN convolution, with N = 2000: 100x0.000162s
100 NxN convolution, with N = 5000: 100x0.000240s
100 NxN convolution, with N = 10000: 100x0.000373s
100 NxN convolution, with N = 20000: 100x0.001021s
100 NxN convolution, with N = 50000: 100x0.004758s
10 NxN convolution, with N = 100000: 10x0.018011s
10 NxN convolution, with N = 200000: 10x0.070430s
1 NxN convolution, with N = 500000: 1x0.426327s
1 NxN convolution, with N =1000000: 1x1.713124s
```

## 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: 100x0.000113s
100 NxN convolution, with N = 200: 100x0.000116s
100 NxN convolution, with N = 500: 100x0.000124s
100 NxN convolution, with N = 1000: 100x0.000140s
100 NxN convolution, with N = 2000: 100x0.000184s
100 NxN convolution, with N = 5000: 100x0.000265s
100 NxN convolution, with N = 10000: 100x0.000418s
100 NxN convolution, with N = 20000: 100x0.001039s
100 NxN convolution, with N = 50000: 100x0.004804s
10 NxN convolution, with N = 100000: 10x0.018154s
10 NxN convolution, with N = 200000: 10x0.071626s
1 NxN convolution, with N = 500000: 1x0.439303s
1 NxN convolution, with N =1000000: 1x1.740866s
Backend : KeOps (LazyTensor), Device : cuda -------------
100 NxN convolution, with N = 100: 100x0.000290s
100 NxN convolution, with N = 200: 100x0.000288s
100 NxN convolution, with N = 500: 100x0.000296s
100 NxN convolution, with N = 1000: 100x0.000311s
100 NxN convolution, with N = 2000: 100x0.000346s
100 NxN convolution, with N = 5000: 100x0.000435s
100 NxN convolution, with N = 10000: 100x0.000621s
100 NxN convolution, with N = 20000: 100x0.001351s
100 NxN convolution, with N = 50000: 100x0.005181s
10 NxN convolution, with N = 100000: 10x0.018410s
10 NxN convolution, with N = 200000: 10x0.072628s
1 NxN convolution, with N = 500000: 1x0.467757s
1 NxN convolution, with N =1000000: 1x2.785968s
Backend : KeOps (LazyTensor, batchsize=10), Device : cuda -------------
10x100 NxN convolution, with N = 100: 10x100x0.000069s
10x100 NxN convolution, with N = 200: 10x100x0.000070s
10x100 NxN convolution, with N = 500: 10x100x0.000079s
10x100 NxN convolution, with N = 1000: 10x100x0.000073s
10x100 NxN convolution, with N = 2000: 10x100x0.000081s
10x100 NxN convolution, with N = 5000: 10x100x0.000119s
10x100 NxN convolution, with N = 10000: 10x100x0.000307s
10x100 NxN convolution, with N = 20000: 10x100x0.001222s
10x100 NxN convolution, with N = 50000: 10x100x0.015417s
10x 10 NxN convolution, with N = 100000: 10x 10x0.056252s
10x 1 NxN convolution, with N = 200000: 10x 1x0.199319s
10x 1 NxN convolution, with N = 500000: 10x 1x2.374233s
10x 1 NxN convolution, with N =1000000: 10x 1x9.617984s
/home/bcharlier/keops/pykeops/benchmarks/plot_benchmarks_convolutions_3D.py:258: 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:** ( 5 minutes 7.280 seconds)