March Boedihardjo, Thomas Strohmer, Roman Vershynin
We show how randomized rounding based on Grothendieck's identity can be used to prove a nearly tight bound on the covariance loss--the amount of covariance that is lost by taking conditional expectation. This result yields a new type of weak Szemeredi regularity lemma for positive semidefinite matrices and kernels. Moreover, it can be used to construct differentially private synthetic data.
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We prove a variant of the abstract probabilistic version of Szemerédi's regularity lemma, due to Tao, which applies to a number of structures (including graphs, hypergraphs, hypercubes, graphons, and many more) and works for random variables in $L_p$ for any $p>1$. Our approach is based on martingale difference sequences.
The covariance matrix is well-known for its following properties: affine equivariance, additivity, independence property and full affine equivariance. Generalizing the first one leads into the study of scatter functionals, commonly used as plug-in estimators to replace the covariance matrix in robust statistics. However, if the application requires also some of the other properties of the covariance matrix listed earlier, the success of the plug-in depends on whether the candidate scatter functional possesses these. In this short note we show that under natural regularity conditions the covari…
Sample Path Regularity of Gaussian Processes from the Covariance Kernel
本文研究了高斯过程样本路径的正则性，通过协方差核提出必要且充分条件，以Hölder正则性为核心，为机器学习中常用GPs的正则性提供精确刻画。
We consider a family of gradient Gaussian vector fields on the torus $(\mathbb{Z}/L^N\mathbb{Z})^d$. Adams, Kotecký, Müller and independently Bauerschmidt established the existence of a uniform finite range decomposition of the corresponding covariance operators, i.e., the covariance can be written as a sum of covariance operators supported on increasing cubes with diameter $L^k$. We improve this result and show that the decay behaviour of the kernels in Fourier space can be controlled. Then we show the regularity of the integration map that convolves functionals with the partial measures of t…
Complementary regularity between the integrand and integrator is a well known condition for the integral $\int_0^T f(r) \, \mathrm{d} g(r)$ to exist in the Riemann-Stieltjes sense. This condition also applies to the multi-dimensional case, in particular the 2D integral $\int_{[0, T]^2} f(s,t) \, \mathrm{d} g(s,t)$. In the paper, we give a new condition for the existence of the integral under the assumption that the integrator $g$ is a Volterra covariance function. We introduce the notion of strong Hölder bi-continuity, and show that if the integrand possess this property, the assumption on com…
We consider a family of gradient Gaussian vector fields on $\Z^d$, where the covariance operator is not translation invariant. A uniform finite range decomposition of the corresponding covariance operators is proven, i.e., the covariance operator can be written as a sum of covariance operators whose kernels are supported within cubes of increasing diameter. An optimal regularity bound for the subcovariance operators is proven. We also obtain regularity bounds as we vary the coefficients defining the gradient Gaussian measures. This extends a result of S. Adams, R. Kotecký and S. Müller \cite{1…
In this article, we fully characterize the measurable Gaussian processes $(U(x))_{x\in\mathcal{D}}$ whose sample paths lie in the Sobolev space of integer order $W^{m,p}(\mathcal{D}),\ m\in\mathbb{N}_0,\ 1 <p<+\infty$, where $\mathcal{D}$ is an arbitrary open set. The result is phrased in terms of a form of Sobolev regularity of the covariance function on the diagonal. This is then linked to the existence of suitable Mercer or otherwise nuclear decompositions of the integral operators associated to the covariance function and its cross-derivatives. In the Hilbert case $p=2$, additional links a…
Let a family of gradient Gaussian vector fields on $ \mathbb{Z}^d $ be given. We show the existence of a uniform finite range decomposition of the corresponding covariance operators, that is, the covariance operator can be written as a sum of covariance operators whose kernels are supported within cubes of diameters $ \sim L^k $. In addition we prove natural regularity for the subcovariance operators and we obtain regularity bounds as we vary within the given family of gradient Gaussian measures.
It is a well-known fact that finite rho-variation of the covariance (in 2D sense) of a general Gaussian process implies finite rho-variation of Cameron-Martin paths. In the special case of fractional Brownian motion (think: 2H=1/rho), in the rougher than Brownian regime, a sharper result holds thanks to a Besov-type embedding [Friz-Victoir, JFA, 2006]. In the present note we give a general result which closes this gap. We comment on the importance of this result for various applications.
Stochastic Partial Differential Equations Associated with Pseudo-Differential Operators and Hilbert Space-Valued Gaussian Processes
本文研究了Hilbert空间中由Gaussian过程驱动、与伪微分算子相关的随机偏微分方程，证明了其解的唯一存在性并建立了$L^{p}$-正则性框架，主要贡献包括建立$p$-阶矩极大不等式与Littlewood-Paley型不等式，并改进了Fourier乘子充分条件。
Isotropic Gaussian random fields on the sphere: Regularity, fast simulation and stochastic partial differential equations
研究球面上各向同性高斯随机场的正则性、快速模拟方法及其在环境科学模型中的应用，探讨其与随机偏微分方程的关系。
Existence, uniqueness, and regularity of a strong solution are obtained for stochastic PDEs with a colored noise $F$ and its super-linear diffusion coefficient: $$ du=(a^{ij}u_{x^ix^j}+b^iu_{x^i}+cu)dt+ξ|u|^{1+λ}dF, \quad (t,x)\in(0,\infty)\times\mathbb{R}^d, $$ where $λ\geq 0$ and the coefficients depend on $(ω,t,x)$. The strategy of handling nonlinearity of the diffusion coefficient is to find a sharp estimation for a general Lipschitz case, and apply it to the super-linear case. Moreover, investigation for the estimate provides a range of $λ$, a sufficient condition for the unique solvabili…
We study the sample path regularity of the solutions of a class of spde's which are second order in time and that includes the stochastic wave equation. Non-integer powers of the spatial Laplacian are allowed. The driving noise is white in time and spatially homogeneous. Continuing with the work initiated in Dalang and Mueller (2003), we prove that the solutions belong to a fractional $L^2$-Sobolev space. We also prove Hölder continuity in time and therefore, we obtain joint Hölder continuity in the time and space variables. Our conclusions rely on a precise analysis of the properties of the s…
We consider biased random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. Axelsson-Fisk and Häggström established for this model a phase transition for the asymptotic linear speed $\overline{\mathrm{v}}$ of the walk. Namely, there exists some critical value $λ_{\mathrm{c}}>0$ such that $\overline{\mathrm{v}}>0$ if $λ\in (0,λ_{\mathrm{c}})$ and $\overline{\mathrm{v}}=0$ if $λ>λ_{\mathrm{c}}$. We show that the speed $\overline{\mathrm{v}}$ is continuous in $λ$ on the interval $(0,λ_{\mathrm{c}})$ and differentiable on $(0,λ_{\mathrm{c}}/2)$. Mo…
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