This work shows a random feature map that can be computed in near-linear time $O((d+m)\log d)$ and yields near-subgaussian concentration.
Abstract
Random Fourier features approximate a shift-invariant kernel in $\mathbb{R}^d$, such as the Gaussian kernel $\exp(-\tfrac12\|x-y\|_2^2)$, using independent samples from its spectral measure to construct an unbiased estimator for the kernel value with subgaussian concentration. However, generating $m$ such features requires $O(md)$ time. A long line of work on structured random features has accelerated this computation for the Gaussian kernel and other radial kernels, albeit with weaker concentration guarantees. We close this gap by showing a random feature map that can be computed in near-linear time $O((d+m)\log d)$ and yields near-subgaussian concentration. We obtain it through a new and tighter analysis of the Fastfood feature map of Le, Sarl\'{o}s and Smola (ICML 2013), using Brascamp-Lieb inequalities and the Schoenberg representation of radial kernels.
The bound depends only logarithmically on the ambient dimension and on manifold parameters such as volume and reach, while retaining the $1/\varepsilon^2$ Euclidean rate.
We establish strong Gaussian approximations for centered stationary random fields on $\mathbb Z^d$, $d\ge2$, represented as measurable functions of i.i.d.\ innovations. Under finite $p$th moments, $p>2$, and sufficiently fast decay of spatial dependence, we construct a single coupling of the field with an i.i.d.\ Gauss...
In this article, we study the spatial fluctuations of the Skorohod solution to a stochastic heat equation on $\mathbb R^d$ for $d<4$, driven by multiplicative Gaussian noise with the non-separable covariance kernel $p_{|t-s|}(x-y)$. We prove that the solution is strictly stationary and spatially ergodic at every fixed...
Given a random probability distribution \(F\), whose realizations are defined on \(\mathbb R^d\), and a kernel \(k\), our aim is to analyze the properties of the kernel mean embedding, \(\mu_F\), of \(F\) as a random element of the corresponding reproducing kernel Hilbert space \(\mathcal H\). This establishes a rigoro...
We study the problem of constructing highly nonlinear vectorial maps $F: \mathbb F_2^n \to \mathbb F_2^m$. Concretely, we want an $F$ and an $A = A(m,n)>0$ as small as possible, so that for every affine map $L: \mathbb F_2^n \to \mathbb F_2^m$ (of the form $L(x) = M x + b $) we have: $$\mathrm{agree}(F, L) := |\{ x \in...
S. Kopparty, Rishabh Kothary, Shanthanu S. Rai· Electron. Colloquium Comput....· 0 citations
Let $E(X)=\#\{(a,b)\in\Z^2:a^2+b^2\le X\}-\pi X$. We prove that $E(X)=O_\varepsilon(X^{209/670+\varepsilon})$ for every $\varepsilon>0$, uniformly for real $X\ge4$. This breaks the $5/16$ threshold suggested by Huxley as a possible limit of the Bombieri--Iwaniec method at the time. The proof introduces two refinements:...
Xi Cen, Xi-Tao Gao, Cheng Zhang et al.· 0 citations
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