Foundations of Computational Mathematics最新文献

We propose a novel non-parametric learning paradigm for the identification of drift and diffusion coefficients of multi-dimensional non-linear stochastic differential equations, which relies upon discrete-time observations of the state. The key idea essentially consists of fitting a RKHS-based approximation of the corresponding Fokker–Planck equation to such observations, yielding theoretical estimates of non-asymptotic learning rates which, unlike previous works, become increasingly tighter when the regularity of the unknown drift and diffusion coefficients becomes higher. Our method being kernel-based, offline pre-processing may be profitably leveraged to enable efficient numerical implementation, offering excellent balance between precision and computational complexity.

This paper introduces and develops the algebraic framework of moment polynomials, which are polynomial expressions in commuting variables and their formal mixed moments. Their positivity and optimization over probability measures supported on semialgebraic sets and subject to moment polynomial constraints is investigated. On the one hand, a positive solution to Hilbert’s 17th problem for pseudo-moments is given. On the other hand, moment polynomials positive on actual measures are shown to be sums of squares and formal moments of squares up to arbitrarily small perturbation of their coefficients. When only measures supported on a bounded semialgebraic set are considered, a stronger algebraic certificate for moment polynomial positivity is derived. This result gives rise to a converging hierarchy of semidefinite programs for moment polynomial optimization. Finally, as an application, two open nonlinear Bell inequalities from quantum physics are settled.

This article is concerned with the construction and analysis of new time discretizations for the KdV equation on a torus for low-regularity solutions below (H^1). New harmonic analysis tools, including averaging approximations to the exponential phase functions and trilinear estimates of the KdV operator, are established for the construction and analysis of time discretizations with higher convergence orders under low-regularity conditions. In addition, new perturbation techniques are introduced to establish stability estimates of time discretizations under low-regularity conditions without using filters when the energy techniques fail. The proposed method is proved to be convergent with order (gamma ) (up to a logarithmic factor) in (L^2) under the regularity condition (uin C([0,T];H^gamma )) for (gamma in (0,1]).

Sharpness is an almost generic assumption in continuous optimization that bounds the distance from minima by objective function suboptimality. It facilitates the acceleration of first-order methods through restarts. However, sharpness involves problem-specific constants that are typically unknown, and restart schemes typically reduce convergence rates. Moreover, these schemes are challenging to apply in the presence of noise or with approximate model classes (e.g., in compressive imaging or learning problems), and they generally assume that the first-order method used produces feasible iterates. We consider the assumption of approximate sharpness, a generalization of sharpness that incorporates an unknown constant perturbation to the objective function error. This constant offers greater robustness (e.g., with respect to noise or relaxation of model classes) for finding approximate minimizers. By employing a new type of search over the unknown constants, we design a restart scheme that applies to general first-order methods and does not require the first-order method to produce feasible iterates. Our scheme maintains the same convergence rate as when the constants are known. The convergence rates we achieve for various first-order methods match the optimal rates or improve on previously established rates for a wide range of problems. We showcase our restart scheme in several examples and highlight potential future applications and developments of our framework and theory.

We count singular vector tuples of a system of tensors assigned to the edges of a directed hypergraph. To do so, we study the generalisation of quivers to directed hypergraphs. Assigning vector spaces to the nodes of a hypergraph and multilinear maps to its hyperedges gives a hyperquiver representation. Hyperquiver representations generalise quiver representations (where all hyperedges are edges) and tensors (where there is only one multilinear map). The singular vectors of a hyperquiver representation are a compatible assignment of vectors to the nodes. We compute the dimension and degree of the variety of singular vectors of a sufficiently generic hyperquiver representation. Our formula specialises to known results that count the singular vectors and eigenvectors of a generic tensor. Lastly, we study a hypergraph generalisation of the inverse tensor eigenvalue problem and solve it algorithmically.

We introduce an algorithm to decompose matrix representations of the symmetric group over the reals into irreducible representations, which as a by-product also computes the multiplicities of the irreducible representations. The algorithm applied to a d-dimensional representation of (S_n) is shown to have a complexity of ({mathcal {O}}(n^2 d^3)) operations for determining which irreducible representations are present and their corresponding multiplicities and a further ({mathcal {O}}(n d^4)) operations to fully decompose representations with non-trivial multiplicities. These complexity bounds are pessimistic and in a practical implementation using floating point arithmetic and exploiting sparsity we observe better complexity. We demonstrate this algorithm on the problem of computing multiplicities of two tensor products of irreducible representations (the Kronecker coefficients problem) as well as higher order tensor products. For hook and hook-like irreducible representations the algorithm has polynomial complexity as n increases. We also demonstrate an application to constructing a basis of homogeneous polynomials so that applying a permutation of variables induces an irreducible representation.

A fundamental challenge in learning an unknown dynamical system is to reduce model uncertainty by making measurements while maintaining safety. In this work, we formulate a mathematical definition of what it means to safely learn a dynamical system by sequentially deciding where to initialize the next trajectory. In our framework, the state of the system is required to stay within a safety region for a horizon of T time steps under the action of all dynamical systems that (i) belong to a given initial uncertainty set, and (ii) are consistent with the information gathered so far. For our first set of results, we consider the setting of safely learning a linear dynamical system involving n states. For the case (T=1), we present a linear programming-based algorithm that either safely recovers the true dynamics from at most n trajectories, or certifies that safe learning is impossible. For (T=2), we give a semidefinite representation of the set of safe initial conditions and show that (lceil n/2 rceil ) trajectories generically suffice for safe learning. For (T = infty ), we provide semidefinite representable inner approximations of the set of safe initial conditions and show that one trajectory generically suffices for safe learning. Finally, we extend a number of our results to the cases where the initial uncertainty set contains sparse, low-rank, or permutation matrices, or when the dynamical system involves a control input. Our second set of results concerns the problem of safely learning a general class of nonlinear dynamical systems. For the case (T=1), we give a second-order cone programming based representation of the set of safe initial conditions. For (T=infty ), we provide semidefinite representable inner approximations to the set of safe initial conditions. We show how one can safely collect trajectories and fit a polynomial model of the nonlinear dynamics that is consistent with the initial uncertainty set and best agrees with the observations. We also present extensions of some of our results to the cases where the measurements are noisy or the dynamical system involves disturbances.

While decomposition of one-parameter persistence modules behaves nicely, as demonstrated by the algebraic stability theorem, decomposition of multiparameter modules is known to be unstable in a certain precise sense. Until now, it has not been clear that there is any way to get around this and build a meaningful stability theory for multiparameter module decomposition. We introduce new tools, in particular (epsilon )-refinements and (epsilon )-erosion neighborhoods, to start building such a theory. We then define the (epsilon )-pruning of a module, which is a new invariant acting like a “refined barcode” that shows great promise to extract features from a module by approximately decomposing it. Our main theorem can be interpreted as a generalization of the algebraic stability theorem to multiparameter modules up to a factor of 2r, where r is the maximal pointwise dimension of one of the modules. Furthermore, we show that the factor 2r is close to optimal. Finally, we discuss the possibility of strengthening the stability theorem for modules that decompose into pointwise low-dimensional summands, and pose a conjecture phrased purely in terms of basic linear algebra and graph theory that seems to capture the difficulty of doing this. We also show that this conjecture is relevant for other areas of multipersistence, like the computational complexity of approximating the interleaving distance, and recent applications of relative homological algebra to multipersistence.

In optimization-based approaches to inverse problems and to statistical estimation, it is common to augment criteria that enforce data fidelity with a regularizer that promotes desired structural properties in the solution. The choice of a suitable regularizer is typically driven by a combination of prior domain information and computational considerations. Convex regularizers are attractive computationally but they are limited in the types of structure they can promote. On the other hand, nonconvex regularizers are more flexible in the forms of structure they can promote and they have showcased strong empirical performance in some applications, but they come with the computational challenge of solving the associated optimization problems. In this paper, we seek a systematic understanding of the power and the limitations of convex regularization by investigating the following questions: Given a distribution, what is the optimal regularizer for data drawn from the distribution? What properties of a data source govern whether the optimal regularizer is convex? We address these questions for the class of regularizers specified by functionals that are continuous, positively homogeneous, and positive away from the origin. We say that a regularizer is optimal for a data distribution if the Gibbs density with energy given by the regularizer maximizes the population likelihood (or equivalently, minimizes cross-entropy loss) over all regularizer-induced Gibbs densities. As the regularizers we consider are in one-to-one correspondence with star bodies, we leverage dual Brunn-Minkowski theory to show that a radial function derived from a data distribution is akin to a “computational sufficient statistic” as it is the key quantity for identifying optimal regularizers and for assessing the amenability of a data source to convex regularization. Using tools such as (Gamma )-convergence from variational analysis, we show that our results are robust in the sense that the optimal regularizers for a sample drawn from a distribution converge to their population counterparts as the sample size grows large. Finally, we give generalization guarantees for various families of star bodies that recover previous results for polyhedral regularizers (i.e., dictionary learning) and lead to new ones for a variety of classes of star bodies.

We obtain essentially matching upper and lower bounds for the expected max-sliced 1-Wasserstein distance between a probability measure on a separable Hilbert space and its empirical distribution from n samples. By proving a Banach space version of this result, we also obtain an upper bound, that is sharp up to a log factor, for the expected max-sliced 2-Wasserstein distance between a symmetric probability measure (mu ) on a Euclidean space and its symmetrized empirical distribution in terms of the operator norm of the covariance matrix of (mu ) and the diameter of the support of (mu ).