Chandan Singh | kernels

kernels

An introduction to kernels and recent research.

# kernel basics

## ch 4 from support vector machines book

• 4.1 - what is a valid kernel
• in general, most dot-product like things constitute valid kernels
• a function is a kernel iff it is a symmetric, positive definite function
• this refers to the $n$ x $n$ matrix with entries $f(x_{row}-x_{col})$ being a psd matrix
• a given kernel can have many feature spaces (can construct different feature spaces that yield the same inner products)
• 4.2 reproducing kernel hilbert space (RKHS) of a kernel
• hilbert space - abstract vector space with (1) an inner product and (2) is complete (i.e. enough limits so calculus works)
• the RKHS is the smallest feature space of a kernel, and can serve as a canonical feature space
• RKHS - a $\mathbb K$-hilbert space that consists of functions mapping from X to $\mathbb K$
• every RKHS has a unique reproducing kernel
• every kernel has a unique RKHS
• sums/products of kernels also work

# kernels in deep learning

• To understand deep learning we need to understand kernel learning - overfitted kernel classifiers can still fit the data well
• original kernels (neal 1994) + (lee et al. 2018) + (matthews et al. 2018)
• infinitely wide nets and only top layer is trained
• corresponds to kernel $\text{ker}(x, x’) = \mathbb E_{\theta \sim W}[f(\theta, x) \cdot f(\theta, x’)]$, where $W$ is an intialization distr. over $\theta$
• neural tangent kernel (jacot et al. 2018)
• $\text{ker}(x, x’) = \mathbb E_{\theta \sim W}[\left < \frac{f(\theta, x)}{\partial \theta} \cdot \frac{f(\theta, x’)}{\partial \theta} \right> ]$ - evolution of weights over time follows this kernel
• with very large width, this kernel is the NTK at initialization
• stays stable during training (since weights don’t change much)
• at initialization, artificial neural networks (ANNs) are equivalent to Gaussian processes in the infinite-width limit
• evolution of an ANN during training can also be described by a kernel (kernel gradient descent)
• different types of kernels impose different things on a function (e.g. want more / less low frequencies)
• gradient descent in kernel space can be convex if kernel is PD (even if nonconvex in the parameter space)
• understanding the neural tangent kernel (arora et al. 2019)

• method to compute the kernel quickly on a gpu
• Scaling description of generalization with number of parameters in deep learning (geiger et al. 2019)
• number of params = N
• above 0 training err, larger number of params reduces variance but doesn’t actually help
• ensembling with smaller N fixes problem
• the improvement of generalization performance with N in this classification task originates from reduced variance of fN when N gets large, as recently observed for mean-square regression
• On the Inductive Bias of Neural Tangent Kernels (bietti & mairal 2019)
• Kernel and Deep Regimes in Overparametrized Models (Woodworth…Srebro 2019)

• transition between kernel and deep regimes

# kernel papers

• data spectroscopy paper (shi et al. 2009)
• kernel matrix $K_n$
• Laplacian matrix $L_n = D_n - K_n$
• $D_n$ is diagonal matrix, with entries = column sums
• block-diagonal kernel matrix would imply a cluster
• eigenvalues of these kernel matrices can identify clusters (and are invariant to permutation)
• want to look for data points corresponding to same/similar eigenvectors
• hard to know what kernel to use, how many eigenvectors / groups look at
• here, look at population point of view - realted dependence of spectrum of $K_n$ on the data density function: $K_Pf(x) = \int K(x, y) f(y) dP(y)$

# spectral clustering

• interested in top eigenvectors of $K_n$ and bottom eigenvectors of $L_n$
• scott and longuet-higgins - embed data in space of top eigenvectors, normalize in that space, and investigate block structure
• perona and freeman - 2 clusters by thresholding top eigenvector
• shi & malik - normalized cut: threshold second smallest generalize eigenvector of $L_n$
• similarly we have kernel PCA, spectral dimensionality reduction, and SVMs (which can be viewed as fitting a linear classifier in the eigenspace of $K_n$)