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: 100x0.001627s
100 NxN convolution, with N = 500: 100x0.007978s
10 NxN convolution, with N = 1000: 10x0.035831s
1 NxN convolution, with N = 2000: 1x0.143121s
1 NxN convolution, with N = 5000: 1x0.867392s
1 NxN convolution, with N = 10000: 1x3.352783s
1 NxN convolution, with N = 20000: 1x13.611010s
**
Too slow !
Backend : PyTorch, Device : cuda -------------
100 NxN convolution, with N = 100: 100x0.000241s
100 NxN convolution, with N = 200: 100x0.000229s
100 NxN convolution, with N = 500: 100x0.000229s
100 NxN convolution, with N = 1000: 100x0.000238s
100 NxN convolution, with N = 2000: 100x0.000423s
100 NxN convolution, with N = 5000: 100x0.002336s
10 NxN convolution, with N = 10000: 10x0.010855s
10 NxN convolution, with N = 20000: 10x0.039532s
**
Memory overflow !
Backend : KeOps, Device : cuda -------------
Compiling libKeOpstorch9451ba29f0 in /home/bcharlier/tmp/libkeops/pykeops/common/../build//build-libKeOpstorch9451ba29f0:
formula: Sum_Reduction(Exp(-SqDist(X,Y)) * B,0)
aliases: X = Vi(0,3); Y = Vj(1,3); B = Vj(2,1);
dtype : float32
... Done.
100 NxN convolution, with N = 100: 100x0.000166s
100 NxN convolution, with N = 200: 100x0.000138s
100 NxN convolution, with N = 500: 100x0.000150s
100 NxN convolution, with N = 1000: 100x0.000161s
100 NxN convolution, with N = 2000: 100x0.000195s
100 NxN convolution, with N = 5000: 100x0.000301s
100 NxN convolution, with N = 10000: 100x0.000469s
100 NxN convolution, with N = 20000: 100x0.001180s
100 NxN convolution, with N = 50000: 100x0.004697s
10 NxN convolution, with N = 100000: 10x0.017843s
10 NxN convolution, with N = 200000: 10x0.070646s
1 NxN convolution, with N = 500000: 1x0.428023s
1 NxN convolution, with N =1000000: 1x1.709376s
```

## 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: 100x0.000127s
100 NxN convolution, with N = 200: 100x0.000130s
100 NxN convolution, with N = 500: 100x0.000138s
100 NxN convolution, with N = 1000: 100x0.000150s
100 NxN convolution, with N = 2000: 100x0.000178s
100 NxN convolution, with N = 5000: 100x0.000268s
100 NxN convolution, with N = 10000: 100x0.000404s
100 NxN convolution, with N = 20000: 100x0.001037s
100 NxN convolution, with N = 50000: 100x0.004816s
10 NxN convolution, with N = 100000: 10x0.018006s
10 NxN convolution, with N = 200000: 10x0.071372s
1 NxN convolution, with N = 500000: 1x0.432676s
1 NxN convolution, with N =1000000: 1x1.729578s
Backend : KeOps (LazyTensor), Device : cuda -------------
Compiling libKeOpstorch4cd7af70c0 in /home/bcharlier/tmp/libkeops/pykeops/common/../build//build-libKeOpstorch4cd7af70c0:
formula: Sum_Reduction((Exp(Minus(Sum(Square((Var(0,3,0) - Var(1,3,1)))))) * Var(2,1,1)),0)
aliases: Var(0,3,0); Var(1,3,1); Var(2,1,1);
dtype : float32
... Done.
100 NxN convolution, with N = 100: 100x0.000331s
100 NxN convolution, with N = 200: 100x0.000314s
100 NxN convolution, with N = 500: 100x0.000316s
100 NxN convolution, with N = 1000: 100x0.000329s
100 NxN convolution, with N = 2000: 100x0.000363s
100 NxN convolution, with N = 5000: 100x0.000468s
100 NxN convolution, with N = 10000: 100x0.000640s
100 NxN convolution, with N = 20000: 100x0.001355s
100 NxN convolution, with N = 50000: 100x0.004976s
10 NxN convolution, with N = 100000: 10x0.018273s
10 NxN convolution, with N = 200000: 10x0.071754s
1 NxN convolution, with N = 500000: 1x0.436207s
1 NxN convolution, with N =1000000: 1x1.744026s
Backend : KeOps (LazyTensor, batchsize=10), Device : cuda -------------
10x100 NxN convolution, with N = 100: 10x100x0.000075s
10x100 NxN convolution, with N = 200: 10x100x0.000073s
10x100 NxN convolution, with N = 500: 10x100x0.000073s
10x100 NxN convolution, with N = 1000: 10x100x0.000075s
10x100 NxN convolution, with N = 2000: 10x100x0.000081s
10x100 NxN convolution, with N = 5000: 10x100x0.000119s
10x100 NxN convolution, with N = 10000: 10x100x0.000253s
10x100 NxN convolution, with N = 20000: 10x100x0.000789s
10x100 NxN convolution, with N = 50000: 10x100x0.004729s
10x 10 NxN convolution, with N = 100000: 10x 10x0.026386s
10x 1 NxN convolution, with N = 200000: 10x 1x0.140907s
10x 1 NxN convolution, with N = 500000: 10x 1x1.586358s
10x 1 NxN convolution, with N =1000000: 10x 1x8.246210s
/home/bcharlier/tmp/libkeops/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:** ( 4 minutes 50.098 seconds)