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+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/_sources/book/papers/rff/rff.ipynb b/_sources/book/papers/rff/rff.ipynb index 52e59cd2..71882c8a 100644 --- a/_sources/book/papers/rff/rff.ipynb +++ b/_sources/book/papers/rff/rff.ipynb @@ -12,11 +12,11 @@ "source": [ "# Random Fourier features\n", "\n", - "\n", - "Star\n", + "Star\n", "Issue\n", "Watch\n", "Follow\n", + "\n", "\n", "One central difficulty with Gaussian Processes (GPs), and more generally all kernel methods such as Support Vector Machines (SVMs), is their computational cost. Exact GP regression scales cubically $\\mathcal{O}(N^3)$ in the number of datapoints $N,$ which is prohibitive for even modestly large datasets.\n", "Therefore we typically have to make approximations, of which there is a wealth of possible options. This page presents Random Fourier Features (RFF), {cite}`rahimi2007random` an approximation which is applicable to stationary kernels.\n", diff --git a/book/papers/rff/rff.html b/book/papers/rff/rff.html index 85a5e5e5..86be6a1f 100644 --- a/book/papers/rff/rff.html +++ b/book/papers/rff/rff.html @@ -429,11 +429,11 @@

Contents

Random Fourier features#

- -

Star +

Star Issue Watch Follow

+

One central difficulty with Gaussian Processes (GPs), and more generally all kernel methods such as Support Vector Machines (SVMs), is their computational cost. Exact GP regression scales cubically \(\mathcal{O}(N^3)\) in the number of datapoints \(N,\) which is prohibitive for even modestly large datasets. Therefore we typically have to make approximations, of which there is a wealth of possible options. This page presents Random Fourier Features (RFF), [Rahimi et al., 2007] an approximation which is applicable to stationary kernels.

RFF relies on the fact that kernels of stationary processes can be expressed as the Fourier transform of a probability density function. @@ -625,7 +625,7 @@

Sampling from the prior
-../../../_images/0d52c17090e987bca5119998b6cb17ec8c9f2e7de33a158f7aeb5a225e98f396.svg
+../../../_images/59dd7db9479748fbeb14212ef8202b0f319d95885cc60af672585d0491bfa309.svg

@@ -635,7 +635,7 @@

Regression with RFF features\(N = 5000,\) which would be quite slow to process with an exact GP model.

-../../../_images/049c3cdb224db146ecc6247782819ed3b83ad6c87e748b6a7cd91ce6750416f7.svg
+../../../_images/55b57407bf62e81b290fde85f49eabc285831da1c469fc151ee587d52a47cf0c.svg

The function rff_posterior below implements Bayesian linear regression with randomly sampled Fourier features. For more details on Bayesian linear regression see Chapter 3 of Bishop’s PRML book.[Bishop, 2006]

@@ -732,8 +732,8 @@

Regression with RFF features
Timing exact GP regression:
-
CPU times: user 10.6 s, sys: 1.33 s, total: 11.9 s
-Wall time: 4.29 s
+
CPU times: user 10.8 s, sys: 1.2 s, total: 12 s
+Wall time: 4.11 s
@@ -742,15 +742,15 @@

Regression with RFF features 
Timing exact approximate RFF regression:
 
-CPU times: user 237 ms, sys: 139 ms, total: 375 ms
-Wall time: 96.5 ms
+CPU times: user 202 ms, sys: 105 ms, total: 308 ms
+Wall time: 78.7 ms

-../../../_images/14fa6dfc78f73a6d7fe3e6e2a8e77d09802af8a9735bb8427c286a9e82a420ef.svg
+../../../_images/6ea44423180a290649f9077f103717c39cdd6d2c88f2d5bebc9dbf95b6cd6918.svg

RFF has produced sensible regressors in each case, significantly faster than exact GP regression. The approximate posteriors roughly match the exact posteriors, while being significantly quicker to compute.

@@ -759,11 +759,11 @@

Variance starvation
-../../../_images/2bd00cbb0a1b0264dee9444fa9b3c10869c02e059c4248c06618bb4433d3eabe.svg
+../../../_images/5f7d86f792b91e8bb21c7c15594e6da669b251375cb4cefe3b673912eaf19c3c.svg
-../../../_images/4712213dcac9a64a5adb0a08a3a8ac33b1df131a4d7897bf4b4451f43b370943.svg
+../../../_images/0d12a93b195058bc3b36ca70689970a2480a162fbdea91fc0457288013bc5385.svg

The variance of the approximate estimator in between the data is (in some cases), signiticantly smaller than that of the exact posterior. So the speedup that the RFF gives does not come without a cost. In certain cases, we can end up with approximate posteriors which are significantly overfitted. This can be alleviated by increasing the number of RFF features. However this increases the computational cost of performing regression and may defeat the purpose of using RFF features in the first place.

diff --git a/prf-prf.html b/prf-prf.html index d79d6232..633095c6 100644 --- a/prf-prf.html +++ b/prf-prf.html @@ -484,7 +484,7 @@

Proof Index

- definition-1 (book/toc/002-cfl) + definition-1 (book/papers/rff/rff) definition   @@ -908,7 +908,7 @@

Proof Index

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