相关论文: Fast linear algebra is stable
Many areas of machine learning and science involve large linear algebra problems, such as eigendecompositions, solving linear systems, computing matrix exponentials, and trace estimation. The matrices involved often have Kronecker,…
The complexity of matrix multiplication is measured in terms of $\omega$, the smallest real number such that two $n\times n$ matrices can be multiplied using $O(n^{\omega+\epsilon})$ field operations for all $\epsilon>0$; the best bound…
Given a hereditary family $\mathcal{G}$ of admissible graphs and a function $\lambda(G)$ that linearly depends on the statistics of order-$\kappa$ subgraphs in a graph $G$, we consider the extremal problem of determining…
We reconsider a recently published algorithm (Dalkilic et al.) for merging lists by way of the perfect shuffle. The original publication gave only experimental results which, although consistent with linear execution time on the samples…
In this paper we introduce a general framework for casting fully dynamic transitive closure into the problem of reevaluating polynomials over matrices. With this technique, we improve the best known bounds for fully dynamic transitive…
We present an algorithm computing the determinant of an integer matrix A. The algorithm is introspective in the sense that it uses several distinct algorithms that run in a concurrent manner. During the course of the algorithm partial…
Making new methods for quantum problems often relies on using basic operations in linear algebra. Often these routines are hidden behind well-known libraries that have been optimized over decades. Attempting to improve on those basic…
We survey classical and recent developments in numerical linear algebra, focusing on two issues: computational complexity, or arithmetic costs, and numerical stability, or performance under roundoff error. We present a brief account of the…
We observe that any $T(n)$ time algorithm (quantum or classical) for several central linear algebraic problems, such as computing $\det(A)$, $tr(A^3)$, or $tr(A^{-1})$ for an $n \times n$ integer matrix $A$, yields a $O(T(n)) + \tilde…
We present substantially generalized and improved quantum algorithms over prior work for inhomogeneous linear and nonlinear ordinary differential equations (ODE). Specifically, we show how the norm of the matrix exponential characterizes…
Stability analysis of Wilkinson's iterative refinement with a relaxation IR(omega) for solving linear systems is given. It extends existing results for omega=1, i.e., for Wilkinson's iterative refinement. We assume that all computations are…
Solving evolutionary equations in a parallel-in-time manner is an attractive topic and many algorithms are proposed in recent two decades. The algorithm based on the block $\alpha$-circulant preconditioning technique has shown promising…
Matrix multiplication is a cornerstone operation in a wide array of scientific fields, including machine learning and computer graphics. The standard algorithm for matrix multiplication has a complexity of $\mathcal{O}(n^3)$ for $n\times n$…
Motivated by Maulik-Okounkov stable maps associated to quiver varieties, we define and construct algebraic stable maps on tensor products of representations in the category O of the Borel subalgebra of an arbitrary untwisted quantum affine…
It is widely known that the lower bound for the algorithmic complexity of square matrix multiplication resorts to at least $n^2$ arithmetic operations. The justification builds upon the following reasoning: given that there are $2 n^2$…
Orthogonal systems in $\mathrm{L}_2(\mathbb{R})$, once implemented in spectral methods, enjoy a number of important advantages if their differentiation matrix is skew-symmetric and highly structured. Such systems, where the differentiation…
Analytical tools to $K$-theory; namely, self-stabilization of rapidly decreasing matrices, linearization of cyclic loops, and the contractibility of the pointed stable Toeplitz algebra are discussed in terms of concrete formulas. Adaptation…
Algebraic matrix multiplication algorithms are designed by bounding the rank of matrix multiplication tensors, and then using a recursive method. However, designing algorithms in this way quickly leads to large constant factors: if one…
We analyze the bit complexity of efficient algorithms for fundamental optimization problems, such as linear regression, $p$-norm regression, and linear programming (LP). State-of-the-art algorithms are iterative, and in terms of the number…
Nowhere dense classes of graphs are classes of sparse graphs with rich structural and algorithmic properties, however, they fail to capture even simple classes of dense graphs. Monadically stable classes, originating from model theory,…