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Convolutional kernel computation (CKN, NTK)

Package for computing exact kernel evaluations for some convolutional kernels, such as those from convolutional kernel networks (CKNs, see here, here, here, here) and neural tangent kernels for convolutional networks (NTK or CNTK, see here, here, here). The main code is in C++ and Eigen, with Cython bindings.

Compilation

The compilation requires Cython and Eigen (the latter will be downloaded automatically upon installation). You can compile the python package with the following command:

python setup.py build_ext -if

Examples

Here is a commented example for a two-layer CKN with exponential kernels on patches.

import numpy as np
import ckn_kernel

# data (e.g. MNIST digits)
X = np.random.rand(5, 28, 28, 1).astype(np.float64)

# define the model:
#   2 convolutional layers
#   3x3 patches at both layers
#   pooling/downsampling by a factor 2 at the first layer, 5 at the second
#   patch kernels are exp((u - 1) / sigma^2)
model = [{'npatch': 3, 'subsampling': 2, 'pooling': 'gaussian', 'kernel': 'exp', 'sigma': 0.65},
         {'npatch': 3, 'subsampling': 5, 'pooling': 'gaussian', 'kernel': 'exp', 'sigma': 0.65}]

# compute (symmetric) kernel matrix
K = ckn_kernel.compute_kernel_matrix(X, model=model)

# for computing non-symmetric blocks, e.g. train/test, with test data Xtest,
# or simply for parallelizing different block computations, use:
# K_train_test = ckn_kernel.compute_kernel_matrix(X, Xtest, model=model)

The NTK can be used as follows (only the ReLU / arc-cosine kernel is supported for now):

# model: same architecture as above, but with arc-cosine kernels (i.e. the dual of the ReLU activation)
model = [{'npatch': 3, 'subsampling': 2, 'pooling': 'gaussian', 'kernel': 'relu'},
         {'npatch': 3, 'subsampling': 5, 'pooling': 'gaussian', 'kernel': 'relu'}]

# compute NTK kernel matrix
K = ckn_kernel.compute_kernel_matrix(X, model=model, ntk=True)

Setting ntk=False in this last command would give the kernel corresponding to training only the last layer of a ReLU CNN at infinite width, which then corresponds to a more basic CKN.

For pooling, the options are gaussian, average or strided, where the latter corresponds to strided convolutions (i.e. downsampling with no blurring) and can be much faster, though it may induce aliasing.

Papers

The papers to cite are the following (first and third for CKN, second one for NTK):

A. Bietti, J. Mairal. Group Invariance, Stability to Deformations, and Complexity of Deep Convolutional Representations. JMLR, 2019.

A. Bietti, J. Mairal. On the Inductive Bias of Neural Tangent Kernels. NeurIPS, 2019.

A. Bietti. On Approximation in Deep Convolutional Networks: a Kernel Perspective. arXiv, 2021.

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