Kernel Methods, part 1 - Arthur Gretton - MLSS 2020, Tübingen

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0:00:00 Introduction
0:02:10 Representing and comparing probabilities with kernels: Part 1
0:02:56 A motivation: comparing two samples
0:03:34 A real-life example: two-sample tests
0:04:34 Training generative models
0:06:04 A second task: testing goodness of fit
0:07:18 A third task: testing independence
0:08:37 Outline: these slides
0:10:16 Outline: next slides
0:10:43 Reproducing Kernel Hilbert Spaces
0:10:48 Kernels and feature space (1): XOR example
0:12:30 Kernels and feature space (2): smoothing
0:13:43 Hilbert space
0:17:06 New kernels from old: sums, transformations
0:18:03 New kernels from old: products
0:22:19 Sums and products =polynomials
0:22:40 Infinite sequences
0:23:21 The reproducing property
0:24:42 Infinite sequences (proof)
0:25:15 A famous infinite feature space kernel
0:27:38 Positive definite functions
0:29:43 Kernels are positive definite
0:31:48 Sum of kernels is a kernel
0:32:34 Functions of infinitely many features
0:33:22 Functions of finitely many features
0:35:07 Functions of infinitely many features
0:35:55 Expressing the functions with kernels
0:38:42 The feature map (x) is also a function
0:41:00 The reproducing property
0:41:45 Understanding smoothness in the RKHS
0:42:30 Constructing an infinite feature space: fourier series
0:47:27 Fourier series for top hat function
0:49:37 Fourier series for kernel function
0:51:49 Fourier series for Gaussian-spectrum kernel
0:53:00 RKHS via fourier series
0:59:23 Roughness penalty explained
1:03:38 Feature map and reproducing property
1:07:29 Q&A
1:09:28 Link back to original RKHS function definition
1:12:31 Main message
1:14:33 Q&A

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