arXiv subjects

Christopher Kennedy

Publications and source records attributed to Christopher Kennedy.

7 recordsLinked to original sources

arXiv2025

AI for a Planet Under Pressure

Artificial intelligence (AI) is already driving scientific breakthroughs in a variety of research fields, ranging from the life sciences to mathematics. This raises a critical question: can AI be applied both responsibly and effectively to address complex and interconnected sustainability challenges? This report is the result of a collaboration between the Stockholm resilience Centre (Stockholm University), the Potsdam Institute for Climate Impact Research (PIK), and Google DeepMind. Our work explores the potential and limitations of using AI as a research method to help tackle eight broad sustainability challenges. The results build on iterated expert dialogues and assessments, a systematic AI-supported literature overview including over 8,500 academic publications, and expert deep-dives into eight specific issue areas. The report also includes recommendations to sustainability scientists, research funders, the private sector, and philanthropies.

cs.CY↗

arXiv2024

Interaction between long internal waves and free surface waves in deep water

We consider a density-stratified fluid composed of two immiscible layers separated by a sharp interface. We study the regime of long internal waves interacting with modulated surface wave packets and describe their resonant interaction by a system of equations where the internal wave solves a high-order Benjamin-Ono (BO) equation coupled to a linear Schr\"{o}dinger equation for the envelope of the free surface. The perturbation methods are based on the Hamiltonian formulation for the original system of irrotational Euler's equations as described in Benjamin-Bridges [J. Fluid Mech. 333, 1997] and Craig-Guyenne-Kalisch [Comm. Pure Appl. Math. 58, 2005]. We also establish a local wellposedness result for a reduced BO-Schr\"{o}dinger system using an approach developed by Linares-Ponce-Pilod [J. Diff. Eqs. 250, 2011].

math.AP↗

arXiv2023

Tracing two decades of carbon emissions using a network approach

Carbon emissions are currently attributed to producers although a consumption-aware accounting is advocated. After constructing the Carbon Trade Network, we trace the flow of emissions over the past two decades. Our analysis reveals the presence of an unexpected, positive feedback: despite individual exchanges have become less carbon-intensive, the increase in trading activity has ultimately risen the amount of emissions directed from `net exporters' towards `net importers'. Adopting a consumption-aware accounting would re-distribute responsibility between the two groups, possibly reducing disparities.

physics.soc-ph↗

arXiv2019

A Bochner Formula on Path Space for the Ricci Flow

We generalize the classical Bochner formula for the heat flow on evolving manifolds $(M,g_{t})_{t \in [0,T]}$ to an infinite-dimensional Bochner formula for martingales on parabolic path space $P\mathcal{M}$ of space-time $\mathcal{M} = M \times [0,T]$. Our new Bochner formula and the inequalities that follow from it are strong enough to characterize solutions of the Ricci flow. Specifically, we obtain characterizations of the Ricci flow in terms of Bochner inequalities on parabolic path space. We also obtain gradient and Hessian estimates for martingales on parabolic path space, as well as condensed proofs of the prior characterizations of the Ricci flow from Haslhofer-Naber \cite{HN18a}. Our results are parabolic counterparts of the recent results in the elliptic setting from \cite{HN18b}.

math.DG↗

arXiv2018

Greedy Variance Estimation for the LASSO

Recent results have proven the minimax optimality of LASSO and related algorithms for noisy linear regression. However, these results tend to rely on variance estimators that are inefficient or optimizations that are slower than LASSO itself. We propose an efficient estimator for the noise variance in high dimensional linear regression that is faster than LASSO, only requiring $p$ matrix-vector multiplications. We prove this estimator is consistent with a good rate of convergence, under the condition that the design matrix satisfies the Restricted Isometry Property (RIP). In practice, our estimator scales incredibly well into high dimensions, is highly parallelizable, and only incurs a modest bias.

math.ST↗

arXiv2016

Fast Cross-Polytope Locality-Sensitive Hashing

We provide a variant of cross-polytope locality sensitive hashing with respect to angular distance which is provably optimal in asymptotic sensitivity and enjoys $\mathcal{O}(d \ln d )$ hash computation time. Building on a recent result (by Andoni, Indyk, Laarhoven, Razenshteyn, Schmidt, 2015), we show that optimal asymptotic sensitivity for cross-polytope LSH is retained even when the dense Gaussian matrix is replaced by a fast Johnson-Lindenstrauss transform followed by discrete pseudo-rotation, reducing the hash computation time from $\mathcal{O}(d^2)$ to $\mathcal{O}(d \ln d )$. Moreover, our scheme achieves the optimal rate of convergence for sensitivity. By incorporating a low-randomness Johnson-Lindenstrauss transform, our scheme can be modified to require only $\mathcal{O}(\ln^9(d))$ random bits

cs.DS↗

arXiv2015

Approximating the Little Grothendieck Problem over the Orthogonal and Unitary Groups

The little Grothendieck problem consists of maximizing $\sum_{ij}C_{ij}x_ix_j$ over binary variables $x_i\in\{\pm1\}$, where C is a positive semidefinite matrix. In this paper we focus on a natural generalization of this problem, the little Grothendieck problem over the orthogonal group. Given C a dn x dn positive semidefinite matrix, the objective is to maximize $\sum_{ij}Tr (C_{ij}^TO_iO_j^T)$ restricting $O_i$ to take values in the group of orthogonal matrices, where $C_{ij}$ denotes the (ij)-th d x d block of C. We propose an approximation algorithm, which we refer to as Orthogonal-Cut, to solve this problem and show a constant approximation ratio. Our method is based on semidefinite programming. For a given $d\geq 1$, we show a constant approximation ratio of $α_{R}(d)^2$, where $α_{R}(d)$ is the expected average singular value of a d x d matrix with random Gaussian $N(0,1/d)$ i.i.d. entries. For d=1 we recover the known $α_{R}(1)^2=2/π$ approximation guarantee for the classical little Grothendieck problem. Our algorithm and analysis naturally extends to the complex valued case also providing a constant approximation ratio for the analogous problem over the Unitary Group. Orthogonal-Cut also serves as an approximation algorithm for several applications, including the Procrustes problem where it improves over the best previously known approximation ratio of~$\frac1{2\sqrt{2}}$. The little Grothendieck problem falls under the class of problems approximated by a recent algorithm proposed in the context of the non-commutative Grothendieck inequality. Nonetheless, our approach is simpler and it provides a more efficient algorithm with better approximation ratios and matching integrality gaps. Finally, we also provide an improved approximation algorithm for the more general little Grothendieck problem over the orthogonal (or unitary) group with rank constraints.

cs.DS↗