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1. Asymptotically Kasner-like singularities

   https://doi.org/10.1353/ajm.2023.a902957  

   Fournodavlos, Grigorios; Luk, Jonathan (August 2023, American Journal of Mathematics)

   abstract: We prove existence, uniqueness and regularity of solutions to the Einstein vacuum equations taking the form $$ {}^{(4)}g = -dt^2 + \sum_{i,j=1}^3 a_{ij}t^{2 p_{\max\{i,j\}}}\,{\rm d} x^i\,{\rm d} x^j $$ on $$(0,T]_t\times\Bbb{T}^3_x$$, where $$a_{ij}(t,x)$$ and $$p_i(x)$$ are regular functions without symmetry or analyticity assumptions. These metrics are singular and asymptotically Kasner-like as $$t\to 0^+$$. These solutions are expected to be highly non-generic, and our construction can be viewed as solving a singular initial value problem with Fuchsian-type analysis where the data are posed on the ``singular hypersurface'' $$\{t=0\}$$. This is the first such result without imposing symmetry or analyticity. To carry out the analysis, we study the problem in a synchronized coordinate system. In particular, we introduce a novel way to perform (weighted) energy estimates in such a coordinate system based on estimating the second fundamental forms of the constant-$$t$$ hypersurfaces. 

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2. Analysis of stochastic Lanczos quadrature for spectrum approximation

   Chen, Tyler; Trogdon, Thomas; Ubaru, Shashanka (January 2021, Proceedings of the 38th International Conference on Machine Learning)

   The cumulative empirical spectral measure (CESM) $$\Phi[\mathbf{A}] : \mathbb{R} \to [0,1]$$ of a $$n\times n$$ symmetric matrix $$\mathbf{A}$$ is defined as the fraction of eigenvalues of $$\mathbf{A}$$ less than a given threshold, i.e., $$\Phi[\mathbf{A}](x) := \sum_{i=1}^{n} \frac{1}{n} {\large\unicode{x1D7D9}}[ \lambda_i[\mathbf{A}]\leq x]$$. Spectral sums $$\operatorname{tr}(f[\mathbf{A}])$$ can be computed as the Riemann–Stieltjes integral of $$f$$ against $$\Phi[\mathbf{A}]$$, so the task of estimating CESM arises frequently in a number of applications, including machine learning. We present an error analysis for stochastic Lanczos quadrature (SLQ). We show that SLQ obtains an approximation to the CESM within a Wasserstein distance of $$t \: | \lambda_{\text{max}}[\mathbf{A}] - \lambda_{\text{min}}[\mathbf{A}] |$$ with probability at least $$1-\eta$$, by applying the Lanczos algorithm for $$\lceil 12 t^{-1} + \frac{1}{2} \rceil$$ iterations to $$\lceil 4 ( n+2 )^{-1}t^{-2} \ln(2n\eta^{-1}) \rceil$$ vectors sampled independently and uniformly from the unit sphere. We additionally provide (matrix-dependent) a posteriori error bounds for the Wasserstein and Kolmogorov–Smirnov distances between the output of this algorithm and the true CESM. The quality of our bounds is demonstrated using numerical experiments. 

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3. Subconvexity in Inhomogeneous Vinogradov Systems

   https://doi.org/10.1093/qmath/haac027  

   Wooley, Trevor D. (August 2022, The Quarterly Journal of Mathematics)

   Abstract When k and s are natural numbers and $${\mathbf h}\in {\mathbb Z}^k$$, denote by $$J_{s,k}(X;\,{\mathbf h})$$ the number of integral solutions of the system $$ \sum_{i=1}^s(x_i^j-y_i^j)=h_j\quad (1\leqslant j\leqslant k), $$ with $$1\leqslant x_i,y_i\leqslant X$$. When $$s\lt k(k+1)/2$$ and $$(h_1,\ldots ,h_{k-1})\ne {\mathbf 0}$$, Brandes and Hughes have shown that $$J_{s,k}(X;\,{\mathbf h})=o(X^s)$$. In this paper we improve on quantitative aspects of this result, and, subject to an extension of the main conjecture in Vinogradov’s mean value theorem, we obtain an asymptotic formula for $$J_{s,k}(X;\,{\mathbf h})$$ in the critical case $s=k(k+1)/2$. The latter requires minor arc estimates going beyond square-root cancellation. 

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4. Mean Field Approximations via Log-Concavity

   https://doi.org/10.1093/imrn/rnad302  

   Lacker, Daniel; Mukherjee, Sumit; Yeung, Lane Chun (December 2023, International Mathematics Research Notices)

   Abstract We propose a new approach to deriving quantitative mean field approximations for any probability measure $$P$$ on $$\mathbb {R}^{n}$$ with density proportional to $$e^{f(x)}$$, for $$f$$ strongly concave. We bound the mean field approximation for the log partition function $$\log \int e^{f(x)}dx$$ in terms of $$\sum _{i \neq j}\mathbb {E}_{Q^{*}}|\partial _{ij}f|^{2}$$, for a semi-explicit probability measure $$Q^{*}$$ characterized as the unique mean field optimizer, or equivalently as the minimizer of the relative entropy $$H(\cdot \,|\,P)$$ over product measures. This notably does not involve metric-entropy or gradient-complexity concepts which are common in prior work on nonlinear large deviations. Three implications are discussed, in the contexts of continuous Gibbs measures on large graphs, high-dimensional Bayesian linear regression, and the construction of decentralized near-optimizers in high-dimensional stochastic control problems. Our arguments are based primarily on functional inequalities and the notion of displacement convexity from optimal transport. 

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5. Outlier robust multivariate polynomial regression

   Arora, Vipul; Bhattacharyya, Arnab; Boban, Mathews; Guruswami, Venkatesan; Kelman, Esty (September 0024, Proceedings of the European Symposium on Algorithms (ESA),)

   We study the problem of robust multivariate polynomial regression: let p\colon\mathbb{R}^n\to\mathbb{R} be an unknown n-variate polynomial of degree at most d in each variable. We are given as input a set of random samples (\mathbf{x}_i,y_i) \in [-1,1]^n \times \mathbb{R} that are noisy versions of (\mathbf{x}_i,p(\mathbf{x}_i)). More precisely, each \mathbf{x}_i is sampled independently from some distribution \chi on [-1,1]^n, and for each i independently, y_i is arbitrary (i.e., an outlier) with probability at most \rho < 1/2, and otherwise satisfies |y_i-p(\mathbf{x}_i)|\leq\sigma. The goal is to output a polynomial \hat{p}, of degree at most d in each variable, within an \ell_\infty-distance of at most O(\sigma) from p. Kane, Karmalkar, and Price [FOCS'17] solved this problem for n=1. We generalize their results to the n-variate setting, showing an algorithm that achieves a sample complexity of O_n(d^n\log d), where the hidden constant depends on n, if \chi is the n-dimensional Chebyshev distribution. The sample complexity is O_n(d^{2n}\log d), if the samples are drawn from the uniform distribution instead. The approximation error is guaranteed to be at most O(\sigma), and the run-time depends on \log(1/\sigma). In the setting where each \mathbf{x}_i and y_i are known up to N bits of precision, the run-time's dependence on N is linear. We also show that our sample complexities are optimal in terms of d^n. Furthermore, we show that it is possible to have the run-time be independent of 1/\sigma, at the cost of a higher sample complexity. 

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