We prove the method of power iteration for matrices with at most finite entries from the Levi-Civita field $\mathcal C$ under the assumption that there exists an eigenvalue with the strictly largest in absolute value complex part. In this case the weak convergence of a start vector to the eigenvector, that corresponds to the largest eigenvalue, is proven. Further, we prove that the Rayleigh quotient of the largest eigenvector also converges weakly to the corresponding eigenvalue. As a corollary, the same holds for matrices and polynomials over the Puiseux series field. In addition to that, we deliver an implementation of our method in Python.
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The mutual-visibility problem in a graph $G$ asks for the cardinality of a largest set of vertices $S\subseteq V(G)$ so that for any two vertices $x,y\in S$ there is a shortest $x,y$-path $P$ so that all internal vertices of $P$ are not in $S$. This is also said as $x,y$ are visible with respect to $S$, or $S$-visible for short. Variations of this problem are known, based on the extension of the visibility property of vertices that are in and/or outside $S$. Such variations are called total, outer and dual mutual-visibility problems. This work is focused on studying the corresponding four visibility parameters in graphs of diameter two, throughout showing bounds and/or closed formulae for these parameters. The mutual-visibility problem in the Cartesian product of two complete graphs is equivalent to (an instance of) the celebrated Zarankievicz's problem. Here we study the dual and outer mutual-visibility problem for the Cartesian product of two complete graphs and all the mutual-visibility problems for the direct product of such graphs as well. We also study all the mutual-visibility problems for the line graphs of complete and complete bipartite graphs. As a consequence of this study, we present several relationships between the mentioned problems and some instances of the classical Tur\'an problem. Moreover, we study the visibility problems for cographs and several non-trivial diameter-two graphs of minimum size.
There is a folkloric belief that a depth-$\Theta(m)$ quantum circuit is needed to estimate the trace of the product of $m$ density matrices (i.e., a multivariate trace), a subroutine crucial to applications in condensed matter and quantum information science. We prove that this belief is overly conservative by constructing a constant quantum-depth circuit for the task, inspired by the method of Shor error correction. Furthermore, our circuit demands only local gates in a two dimensional circuit -- we show how to implement it in a highly parallelized way on an architecture similar to that of Google's Sycamore processor. With these features, our algorithm brings the central task of multivariate trace estimation closer to the capabilities of near-term quantum processors. We instantiate the latter application with a theorem on estimating nonlinear functions of quantum states with "well-behaved" polynomial approximations.
In this contribution we deal with Gaussian quadrature rules based on orthogonal polynomials associated with a weight function $w(x)= x^{\alpha} e^{-x}$ supported on an interval $(0,z)$, $z>0.$ The modified Chebyshev algorithm is used in order to test the accuracy in the computation of the coefficients of the three-term recurrence relation, the zeros and weights, as well as the dependence on the parameter $z.$
A simple and effective method for the alignment of generative models is the best-of-$n$ policy, where $n$ samples are drawn from a base policy, and ranked based on a reward function, and the highest ranking one is selected. A commonly used analytical expression in the literature claims that the KL divergence between the best-of-$n$ policy and the base policy is equal to $\log (n) - (n-1)/n.$ We disprove the validity of this claim, and show that it is an upper bound on the actual KL divergence. We also explore the tightness of this upper bound in different regimes. Finally, we propose a new estimator for the KL divergence and empirically show that it provides a tight approximation through a few examples.
In this article, we explore the spectral properties of general random kernel matrices $[K(U_i,U_j)]_{1\leq i\neq j\leq n}$ from a Lipschitz kernel $K$ with $n$ independent random variables $U_1,U_2,\ldots, U_n$ distributed uniformly over $[0,1]$. In particular we identify a dichotomy in the extreme eigenvalue of the kernel matrix, where, if the kernel $K$ is degenerate, the largest eigenvalue of the kernel matrix (after proper normalization) converges weakly to a weighted sum of independent chi-squared random variables. In contrast, for non-degenerate kernels, it converges to a normal distribution extending and reinforcing earlier results from Koltchinskii and Gin\'e (2000). Further, we apply this result to show a dichotomy in the asymptotic behavior of extreme eigenvalues of $W$-random graphs, which are pivotal in modeling complex networks and analyzing large-scale graph behavior. These graphs are generated using a kernel $W$, termed as graphon, by connecting vertices $i$ and $j$ with probability $W(U_i, U_j)$. Our results show that for a Lipschitz graphon $W$, if the degree function is constant, the fluctuation of the largest eigenvalue (after proper normalization) converges to the weighted sum of independent chi-squared random variables and an independent normal distribution. Otherwise, it converges to a normal distribution.
We propose a method for estimating a log-concave density on $\mathbb R^d$ from samples, under the assumption that there exists an orthogonal transformation that makes the components of the random vector independent. While log-concave density estimation is hard both computationally and statistically, the independent components assumption alleviates both issues, while still maintaining a large non-parametric class. We prove that under mild conditions, at most $\tilde{\mathcal{O}}(\epsilon^{-4})$ samples (suppressing constants and log factors) suffice for our proposed estimator to be within $\epsilon$ of the original density in squared Hellinger distance. On the computational front, while the usual log-concave maximum likelihood estimate can be obtained via a finite-dimensional convex program, it is slow to compute -- especially in higher dimensions. We demonstrate through numerical experiments that our estimator can be computed efficiently, making it more practical to use.
The increasing availability of temporal data poses a challenge to time-series and signal-processing domains due to its high numerosity and complexity. Symbolic representation outperforms raw data in a variety of engineering applications due to its storage efficiency, reduced numerosity, and noise reduction. The most recent symbolic aggregate approximation technique called ABBA demonstrates outstanding performance in preserving essential shape information of time series and enhancing the downstream applications. However, ABBA cannot handle multiple time series with consistent symbols, i.e., the same symbols from distinct time series are not identical. Also, working with appropriate ABBA digitization involves the tedious task of tuning the hyperparameters, such as the number of symbols or tolerance. Therefore, we present a joint symbolic aggregate approximation that has symbolic consistency, and show how the hyperparameter of digitization can itself be optimized alongside the compression tolerance ahead of time. Besides, we propose a novel computing paradigm that enables parallel computing of symbolic approximation. The extensive experiments demonstrate its superb performance and outstanding speed regarding symbolic approximation and reconstruction.
A tuple (Z_1,...,Z_p) of matrices of size r is said to be a commuting extension of a tuple (A_1,...,A_p) of matrices of size n <r if the Z_i pairwise commute and each A_i sits in the upper left corner of a block decomposition of Z_i. This notion was discovered and rediscovered in several contexts including algebraic complexity theory (in Strassen's work on tensor rank), in numerical analysis for the construction of cubature formulas and in quantum mechanics for the study of computational methods and the study of the so-called "quantum Zeno dynamics." Commuting extensions have also attracted the attention of the linear algebra community. In this paper we present 3 types of results: (i) Theorems on the uniqueness of commuting extensions for three matrices or more. (ii) Algorithms for the computation of commuting extensions of minimal size. These algorithms work under the same assumptions as our uniqueness theorems. They are applicable up to r=4n/3, and are apparently the first provably efficient algorithms for this problem applicable beyond r=n+1. (iii) A genericity theorem showing that our algorithms and uniqueness theorems can be applied to a wide range of input matrices.
We compute explicitly the MTW tensor (or cross curvature) for the optimal transport problem on $\mathbb{R}^n$ with a cost function of form $\mathsf{c}(x, y) = \mathsf{u}(x^{\mathfrak{t}}y)$, where $\mathsf{u}$ is a scalar function with inverse $\mathsf{s}$, $x^{\ft}y$ is a nondegenerate bilinear pairing of vectors $x, y$ belonging to an open subset of $\mathbb{R}^n$. The condition that the MTW-tensor vanishes on null vectors under the Kim-McCann metric is a fourth-order nonlinear ODE, which could be reduced to a linear ODE of the form $\mathsf{s}^{(2)} - S\mathsf{s}^{(1)} + P\mathsf{s} = 0$ with constant coefficients $P$ and $S$. The resulting inverse functions include {\it Lambert} and {\it generalized inverse hyperbolic\slash trigonometric} functions. The square Euclidean metric and $\log$-type costs are equivalent to instances of these solutions. The optimal map for the family is also explicit. For cost functions of a similar form on a hyperboloid model of the hyperbolic space and unit sphere, we also express this tensor in terms of algebraic expressions in derivatives of $\mathsf{s}$ using the Gauss-Codazzi equation, obtaining new families of strictly regular costs for these manifolds, including new families of {\it power function costs}. We analyze the $\sinh$-type hyperbolic cost, providing examples of $\mathsf{c}$-convex functions and divergence.
We improve the best known upper bound for the bracketing number of $d$-dimensional axis-parallel boxes anchored in $0$ (or, put differently, of lower left orthants intersected with the $d$-dimensional unit cube $[0,1]^d$). More precisely, we provide a better upper bound for the cardinality of an algorithmic bracketing cover construction due to Eric Thi\'emard, which forms the core of his algorithm to approximate the star discrepancy of arbitrary point sets from [E. Thi\'emard, An algorithm to compute bounds for the star discrepancy, J.~Complexity 17 (2001), 850 -- 880]. Moreover, the new upper bound for the bracketing number of anchored axis-parallel boxes yields an improved upper bound for the bracketing number of arbitrary axis-parallel boxes in $[0,1]^d$. In our upper bounds all constants are fully explicit.
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