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In the theory of linear switching systems with discrete time, as in other areas of mathematics, the problem of studying the growth rate of the norms of all possible matrix products $A_{\sigma_{n}}\cdots A_{\sigma_{0}}$ with factors from a set of matrices $\mathscr{A}$ arises. So far, only for a relatively small number of classes of matrices $\mathscr{A}$ has it been possible to accurately describe the sequences of matrices that guarantee the maximum rate of increase of the corresponding norms. Moreover, in almost all cases studied theoretically, the index sequences $\{\sigma_{n}\}$ of matrices maximizing the norms of the corresponding matrix products have been shown to be periodic or so-called Sturmian, which entails a whole set of "good" properties of the sequences $\{A_{\sigma_{n}}\}$, in particular the existence of a limiting frequency of occurrence of each matrix factor $A_{i}\in\mathscr{A}$ in them. In the paper it is shown that this is not always the case: a class of matrices is defined consisting of two $2\times 2$ matrices, similar to rotations in the plane, in which the sequence $\{A_{\sigma_{n}}\}$ maximizing the growth rate of the norms $\|A_{\sigma_{n}}\cdots A_{\sigma_{0}}\|$ is not Sturmian. All considerations are based on numerical modeling and cannot be considered mathematically rigorous in this part; rather, they should be interpreted as a set of questions for further comprehensive theoretical analysis.

We develop the theory of a metric, which we call the $\nu$-based Wasserstein metric and denote by $W_\nu$, on the set of probability measures $\mathcal P(X)$ on a domain $X \subseteq \mathbb{R}^m$. This metric is based on a slight refinement of the notion of generalized geodesics with respect to a base measure $\nu$ and is relevant in particular for the case when $\nu$ is singular with respect to $m$-dimensional Lebesgue measure; it is also closely related to the concept of linearized optimal transport. The $\nu$-based Wasserstein metric is defined in terms of an iterated variational problem involving optimal transport to $\nu$; we also characterize it in terms of integrations of classical Wasserstein distance between the conditional probabilities and through limits of certain multi-marginal optimal transport problems. As we vary the base measure $\nu$, the $\nu$-based Wasserstein metric interpolates between the usual quadratic Wasserstein distance and a metric associated with the uniquely defined generalized geodesics obtained when $\nu$ is sufficiently regular. When $\nu$ concentrates on a lower dimensional submanifold of $\mathbb{R}^m$, we prove that the variational problem in the definition of the $\nu$-based Wasserstein distance has a unique solution. We establish geodesic convexity of the usual class of functionals and of the set of source measures $\mu$ such that optimal transport between $\mu$ and $\nu$ satisfies a strengthening of the generalized nestedness condition introduced in \cite{McCannPass20}. We also present two applications of the ideas introduced here. First, our dual metric is used to prove convergence of an iterative scheme to solve a variational problem arising in game theory. We also use the multi-marginal formulation to characterize solutions to the multi-marginal problem by an ordinary differential equation, yielding a new numerical method for it.

The FEAST eigensolver is extended to the computation of the singular triplets of a large matrix $A$ with the singular values in a given interval. It is subspace iteration in nature applied to an approximate spectral projector associated with the cross-product matrix $A^TA$ and constructs approximate left and right singular subspaces corresponding to the desired singular values, onto which $A$ is projected to obtain approximations to the desired singular triplets. Approximate spectral projectors are constructed using the Chebyshev--Jackson series expansion other than contour integration and quadrature rules, and they are proven to be always symmetric positive semi-definite with the eigenvalues in $[0,1]$. Compact estimates are established for pointwise approximation errors of a specific step function that corresponds to the exact spectral projector, the accuracy of the approximate spectral projector, the number of desired singular triplets,the distance between the desired right singular subspace and the subspace generated each iteration, and the convergence of the FEAST SVDsolver. Practical selection strategies are proposed for the series degree and the subspace dimension. Numerical experiments illustrate that the FEAST SVDsolver is robust and efficient.

Sparse PCA is the optimization problem obtained from PCA by adding a sparsity constraint on the principal components. Sparse PCA is NP-hard and hard to approximate even in the single-component case. In this paper we settle the computational complexity of sparse PCA with respect to the rank of the covariance matrix. We show that, if the rank of the covariance matrix is a fixed value, then there is an algorithm that solves sparse PCA to global optimality, whose running time is polynomial in the number of features. We also prove a similar result for the version of sparse PCA which requires the principal components to have disjoint supports.

Domain decomposition methods are among the most efficient for solving sparse linear systems of equations. Their effectiveness relies on a judiciously chosen coarse space. Originally introduced and theoretically proved to be efficient for self-adjoint operators, spectral coarse spaces have been proposed in the past few years for indefinite and non-self-adjoint operators. This paper presents a new spectral coarse space that can be constructed in a fully-algebraic way unlike most existing spectral coarse spaces. We present theoretical convergence result for Hermitian positive definite diagonally dominant matrices. Numerical experiments and comparisons against state-of-the-art preconditioners in the multigrid community show that the resulting two-level Schwarz preconditioner is efficient especially for non-self-adjoint operators. Furthermore, in this case, our proposed preconditioner outperforms state-of-the-art preconditioners.

Inverse kinematics (IK) is the problem of finding robot joint configurations that satisfy constraints on the position or pose of one or more end-effectors. For robots with redundant degrees of freedom, there is often an infinite, nonconvex set of solutions. The IK problem is further complicated when collision avoidance constraints are imposed by obstacles in the workspace. In general, closed-form expressions yielding feasible configurations do not exist, motivating the use of numerical solution methods. However, these approaches rely on local optimization of nonconvex problems, often requiring an accurate initialization or numerous re-initializations to converge to a valid solution. In this work, we first formulate inverse kinematics with complex workspace constraints as a convex feasibility problem whose low-rank feasible points provide exact IK solutions. We then present \texttt{CIDGIK} (Convex Iteration for Distance-Geometric Inverse Kinematics), an algorithm that solves this feasibility problem with a sequence of semidefinite programs whose objectives are designed to encourage low-rank minimizers. Our problem formulation elegantly unifies the configuration space and workspace constraints of a robot: intrinsic robot geometry and obstacle avoidance are both expressed as simple linear matrix equations and inequalities. Our experimental results for a variety of popular manipulator models demonstrate faster and more accurate convergence than a conventional nonlinear optimization-based approach, especially in environments with many obstacles.

We show that the Wynn recurrence (the missing identity of Frobenius of the Pad\'{e} approximation theory) can be incorporated into the theory of integrable systems as a reduction of the discrete Schwarzian Kadomtsev-Petviashvili equation. This allows, in particular, to present the geometric meaning of the recurrence as a construction of the appropriately constrained quadrangular set of points. The interpretation is valid for a projective line over arbitrary skew field what motivates to consider non-commutative Pad\'{e} theory. We transfer the corresponding elements, including the Frobenius identities, to the non-commutative level using the quasideterminants. Using an example of the characteristic series of the Fibonacci language we present an application of the theory to the regular languages. We introduce the non-commutative version of the discrete-time Toda lattice equations together with their integrability structure. Finally, we discuss application of the Wynn recurrence in a different context of the geometric theory of discrete analytic functions.

In this paper, we show that the diagonal of a high-dimensional sample covariance matrix stemming from $n$ independent observations of a $p$-dimensional time series with finite fourth moments can be approximated in spectral norm by the diagonal of the population covariance matrix. We assume that $n,p\to \infty$ with $p/n$ tending to a constant which might be positive or zero. As applications, we provide an approximation of the sample correlation matrix ${\mathbf R}$ and derive a variety of results for its eigenvalues. We identify the limiting spectral distribution of ${\mathbf R}$ and construct an estimator for the population correlation matrix and its eigenvalues. Finally, the almost sure limits of the extreme eigenvalues of ${\mathbf R}$ in a generalized spiked correlation model are analyzed.

Several recent applications of optimal transport (OT) theory to machine learning have relied on regularization, notably entropy and the Sinkhorn algorithm. Because matrix-vector products are pervasive in the Sinkhorn algorithm, several works have proposed to \textit{approximate} kernel matrices appearing in its iterations using low-rank factors. Another route lies instead in imposing low-rank constraints on the feasible set of couplings considered in OT problems, with no approximations on cost nor kernel matrices. This route was first explored by Forrow et al., 2018, who proposed an algorithm tailored for the squared Euclidean ground cost, using a proxy objective that can be solved through the machinery of regularized 2-Wasserstein barycenters. Building on this, we introduce in this work a generic approach that aims at solving, in full generality, the OT problem under low-rank constraints with arbitrary costs. Our algorithm relies on an explicit factorization of low rank couplings as a product of \textit{sub-coupling} factors linked by a common marginal; similar to an NMF approach, we alternatively updates these factors. We prove the non-asymptotic stationary convergence of this algorithm and illustrate its efficiency on benchmark experiments.

Modeling and generating graphs is fundamental for studying networks in biology, engineering, and social sciences. However, modeling complex distributions over graphs and then efficiently sampling from these distributions is challenging due to the non-unique, high-dimensional nature of graphs and the complex, non-local dependencies that exist between edges in a given graph. Here we propose GraphRNN, a deep autoregressive model that addresses the above challenges and approximates any distribution of graphs with minimal assumptions about their structure. GraphRNN learns to generate graphs by training on a representative set of graphs and decomposes the graph generation process into a sequence of node and edge formations, conditioned on the graph structure generated so far. In order to quantitatively evaluate the performance of GraphRNN, we introduce a benchmark suite of datasets, baselines and novel evaluation metrics based on Maximum Mean Discrepancy, which measure distances between sets of graphs. Our experiments show that GraphRNN significantly outperforms all baselines, learning to generate diverse graphs that match the structural characteristics of a target set, while also scaling to graphs 50 times larger than previous deep models.