The hierarchical matrix ($\mathcal{H}^{2}$-matrix) formalism provides a way to reinterpret the Fast Multipole Method and related fast summation schemes in linear algebraic terms. The idea is to tessellate a matrix into blocks in such as way that each block is either small or of numerically low rank; this enables the storage of the matrix and the application of it to a vector in linear or close to linear complexity. A key motivation for the reformulation is to extend the range of dense matrices that can be represented. Additionally, $\mathcal{H}^{2}$-matrices in principle also extend the range of operations that can be executed to include matrix inversion and factorization. While such algorithms can be highly efficient for certain specialized formats (such as HBS/HSS matrices based on ``weak admissibility''), inversion algorithms for general $\mathcal{H}^{2}$-matrices tend to be based on nested recursions and recompressions, making them challenging to implement efficiently. An exception is the \textit{strong recursive skeletonization (SRS)} algorithm by Minden, Ho, Damle, and Ying, which involves a simpler algorithmic flow. However, SRS greatly increases the number of blocks of the matrix that need to be stored explicitly, leading to high memory requirements. This manuscript presents the \textit{randomized strong recursive skeletonization (RSRS)} algorithm, which is a reformulation of SRS that incorporates the randomized SVD (RSVD) to simultaneously compress and factorize an $\mathcal{H}^{2}$-matrix. RSRS is a ``black box'' algorithm that interacts with the matrix to be compressed only via its action on vectors; this extends the range of the SRS algorithm (which relied on the ``proxy source'' compression technique) to include dense matrices that arise in sparse direct solvers.
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Online RL in Linearly $q^π$-Realizable MDPs Is as Easy as in Linear MDPs If You Learn What to Ignore
We consider online reinforcement learning (RL) in episodic Markov decision processes (MDPs) under the linear $q^\pi$-realizability assumption, where it is assumed that the action-values of all policies can be expressed as linear functions of state-action features. This class is known to be more general than linear MDPs, where the transition kernel and the reward function are assumed to be linear functions of the feature vectors. As our first contribution, we show that the difference between the two classes is the presence of states in linearly $q^\pi$-realizable MDPs where for any policy, all the actions have approximately equal values, and skipping over these states by following an arbitrarily fixed policy in those states transforms the problem to a linear MDP. Based on this observation, we derive a novel (computationally inefficient) learning algorithm for linearly $q^\pi$-realizable MDPs that simultaneously learns what states should be skipped over and runs another learning algorithm on the linear MDP hidden in the problem. The method returns an $\epsilon$-optimal policy after $\text{polylog}(H, d)/\epsilon^2$ interactions with the MDP, where $H$ is the time horizon and $d$ is the dimension of the feature vectors, giving the first polynomial-sample-complexity online RL algorithm for this setting. The results are proved for the misspecified case, where the sample complexity is shown to degrade gracefully with the misspecification error.
In the semi-streaming model for processing massive graphs, an algorithm makes multiple passes over the edges of a given $n$-vertex graph and is tasked with computing the solution to a problem using $O(n \cdot \text{polylog}(n))$ space. Semi-streaming algorithms for Maximal Independent Set (MIS) that run in $O(\log\log{n})$ passes have been known for almost a decade, however, the best lower bounds can only rule out single-pass algorithms. We close this large gap by proving that the current algorithms are optimal: Any semi-streaming algorithm for finding an MIS with constant probability of success requires $\Omega(\log\log{n})$ passes. This settles the complexity of this fundamental problem in the semi-streaming model, and constitutes one of the first optimal multi-pass lower bounds in this model. We establish our result by proving an optimal round vs communication tradeoff for the (multi-party) communication complexity of MIS. The key ingredient of this result is a new technique, called hierarchical embedding, for performing round elimination: we show how to pack many but small hard $(r-1)$-round instances of the problem into a single $r$-round instance, in a way that enforces any $r$-round protocol to effectively solve all these $(r-1)$-round instances also. These embeddings are obtained via a novel application of results from extremal graph theory -- in particular dense graphs with many disjoint unique shortest paths -- together with a newly designed graph product, and are analyzed via information-theoretic tools such as direct-sum and message compression arguments.
In this paper, we revisit the bilevel optimization problem, in which the upper-level objective function is generally nonconvex and the lower-level objective function is strongly convex. Although this type of problem has been studied extensively, it still remains an open question how to achieve an ${O}(\epsilon^{-1.5})$ sample complexity in Hessian/Jacobian-free stochastic bilevel optimization without any second-order derivative computation. To fill this gap, we propose a novel Hessian/Jacobian-free bilevel optimizer named FdeHBO, which features a simple fully single-loop structure, a projection-aided finite-difference Hessian/Jacobian-vector approximation, and momentum-based updates. Theoretically, we show that FdeHBO requires ${O}(\epsilon^{-1.5})$ iterations (each using ${O}(1)$ samples and only first-order gradient information) to find an $\epsilon$-accurate stationary point. As far as we know, this is the first Hessian/Jacobian-free method with an ${O}(\epsilon^{-1.5})$ sample complexity for nonconvex-strongly-convex stochastic bilevel optimization.
A novel H3N3-2$_\sigma$ interpolation approximation for the Caputo fractional derivative of order $\alpha\in(1,2)$ is derived in this paper, which improves the popular L2C formula with (3-$\alpha$)-order accuracy. By an interpolation technique, the second-order accuracy of the truncation error is skillfully estimated. Based on this formula, a finite difference scheme with second-order accuracy both in time and in space is constructed for the initial-boundary value problem of the time fractional hyperbolic equation. It is well known that the coefficient properties of discrete fractional derivatives are fundamental to the numerical stability of time fractional differential models. We prove the related properties of the coefficients of the H3N3-2$_\sigma$ approximate formula. With these properties, the numerical stability and convergence of the difference scheme is derived immediately by the energy method in the sense of $H^1$-norm. Considering the weak regularity of the solution to the problem at the starting time, a finite difference scheme on the graded meshes based on H3N3-2$_\sigma$ formula is also presented. The numerical simulations are performed to show the effectiveness of the derived finite difference schemes, in which the fast algorithms are employed to speed up the numerical computation.
Originating in Girard's Linear logic, Ehrhard and Regnier's Taylor expansion of $\lambda$-terms has been broadly used as a tool to approximate the terms of several variants of the $\lambda$-calculus. Many results arise from a Commutation theorem relating the normal form of the Taylor expansion of a term to its B\"ohm tree. This led us to consider extending this formalism to the infinitary $\lambda$-calculus, since the $\Lambda_{\infty}^{001}$ version of this calculus has B\"ohm trees as normal forms and seems to be the ideal framework to reformulate the Commutation theorem. We give a (co-)inductive presentation of $\Lambda_{\infty}^{001}$. We define a Taylor expansion on this calculus, and state that the infinitary $\beta$-reduction can be simulated through this Taylor expansion. The target language is the usual resource calculus, and in particular the resource reduction remains finite, confluent and terminating. Finally, we state the generalised Commutation theorem and use our results to provide simple proofs of some normalisation and confluence properties in the infinitary $\lambda$-calculus.
We show that the cohomology of the Regge complex in three dimensions is isomorphic to $\mathcal{H}^{{\scriptscriptstyle \bullet}}_{dR}(\Omega)\otimes\mathcal{RM}$, the infinitesimal-rigid-body-motion-valued de~Rham cohomology. Based on an observation that the twisted de~Rham complex extends the elasticity (Riemannian deformation) complex to the linearized version of coframes, connection 1-forms, curvature and Cartan's torsion, we construct a discrete version of linearized Riemann-Cartan geometry on any triangulation and determine its cohomology.
Partial differential equations (PDEs) with uncertain or random inputs have been considered in many studies of uncertainty quantification. In forward uncertainty quantification, one is interested in analyzing the stochastic response of the PDE subject to input uncertainty, which usually involves solving high-dimensional integrals of the PDE output over a sequence of stochastic variables. In practical computations, one typically needs to discretize the problem in several ways: approximating an infinite-dimensional input random field with a finite-dimensional random field, spatial discretization of the PDE using, e.g., finite elements, and approximating high-dimensional integrals using cubatures such as quasi-Monte Carlo methods. In this paper, we focus on the error resulting from dimension truncation of an input random field. We show how Taylor series can be used to derive theoretical dimension truncation rates for a wide class of problems and we provide a simple checklist of conditions that a parametric mathematical model needs to satisfy in order for our dimension truncation error bound to hold. Some of the novel features of our approach include that our results are applicable to non-affine parametric operator equations, dimensionally-truncated conforming finite element discretized solutions of parametric PDEs, and even compositions of PDE solutions with smooth nonlinear quantities of interest. As a specific application of our method, we derive an improved dimension truncation error bound for elliptic PDEs with lognormally parameterized diffusion coefficients. Numerical examples support our theoretical findings.
By incorporating a new matrix splitting and the momentum acceleration into the relaxed-based matrix splitting (RMS) method \cite{soso2023}, a generalization of the RMS (GRMS) iterative method for solving the generalized absolute value equations (GAVEs) is proposed. Unlike some existing methods, by using the Cauchy's convergence principle, we give some sufficient conditions for the existence and uniqueness of the solution to the GAVEs and prove that our method can converge to the unique solution of the GAVEs. Moreover, we can obtain a few new and weaker convergence conditions for some existing methods. Preliminary numerical experiments show that the proposed method is efficient.
We propose a threshold-type algorithm to the $L^2$-gradient flow of the Canham-Helfrich functional generalized to $\mathbb{R}^N$. The algorithm to the Willmore flow is derived as a special case in $\mathbb{R}^2$ or $\mathbb{R}^3$. This algorithm is constructed based on an asymptotic expansion of the solution to the initial value problem for a fourth order linear parabolic partial differential equation whose initial data is the indicator function on the compact set $\Omega_0$. The crucial points are to prove that the boundary $\partial\Omega_1$ of the new set $\Omega_1$ generated by our algorithm is included in $O(t)$-neighborhood from $\partial\Omega_0$ for small time $t>0$ and to show that the derivative of the threshold function in the normal direction for $\partial\Omega_0$ is far from zero in the small time interval. Finally, numerical examples of planar curves governed by the Willmore flow are provided by using our threshold-type algorithm.
We present algorithms for the computation of $\varepsilon$-coresets for $k$-median clustering of point sequences in $\mathbb{R}^d$ under the $p$-dynamic time warping (DTW) distance. Coresets under DTW have not been investigated before, and the analysis is not directly accessible to existing methods as DTW is not a metric. The three main ingredients that allow our construction of coresets are the adaptation of the $\varepsilon$-coreset framework of sensitivity sampling, bounds on the VC dimension of approximations to the range spaces of balls under DTW, and new approximation algorithms for the $k$-median problem under DTW. We achieve our results by investigating approximations of DTW that provide a trade-off between the provided accuracy and amenability to known techniques. In particular, we observe that given $n$ curves under DTW, one can directly construct a metric that approximates DTW on this set, permitting the use of the wealth of results on metric spaces for clustering purposes. The resulting approximations are the first with polynomial running time and achieve a very similar approximation factor as state-of-the-art techniques. We apply our results to produce a practical algorithm approximating $(k,\ell)$-median clustering under DTW.
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