相关论文: Improved variants of the Hutch++ algorithm for tra…
We present a method to reduce the variance of stochastic trace estimators used in quantum typicality (QT) methods via a randomized low-rank approximation of the finite-temperature density matrix $e^{-\beta H}$. The trace can be evaluated…
The multiplication of matrices is an important arithmetic operation in computational mathematics. In the context of hierarchical matrices, this operation can be realized by the multiplication of structured block-wise low-rank matrices,…
Calculating the trace of the product of $m$ $n$-qubit density matrices (multivariate trace) is a crucial subroutine in quantum error mitigation and information measures estimation. We propose an unified multivariate trace estimation (UMT)…
In recent years, various subspace algorithms have been developed to handle large-scale optimization problems. Although existing subspace Newton methods require fewer iterations to converge in practice, the matrix operations and full…
In this paper we propose and analyze an algorithm for identifying spectral gaps of a real symmetric matrix $A$ by simultaneously approximating the traces of spectral projectors associated with multiple different spectral slices. Our method…
Matrices with low-rank structure are ubiquitous in scientific computing. Choosing an appropriate rank is a key step in many computational algorithms that exploit low-rank structure. However, estimating the rank has been done largely in an…
In recent years, random subspace methods have been actively studied for large-dimensional nonconvex problems. Recent subspace methods have improved theoretical guarantees such as iteration complexity and local convergence rate while…
This work is concerned with computing low-rank approximations of a matrix function $f(A)$ for a large symmetric positive semi-definite matrix $A$, a task that arises in, e.g., statistical learning and inverse problems. The application of…
We present randomized algorithms for estimating the log-determinant of regularized symmetric positive semi-definite matrices. The algorithms access the matrix only through matrix vector products, and are based on the introduction of a…
We describe a randomized algorithm for producing a near-optimal hierarchical off-diagonal low-rank (HODLR) approximation to an $n\times n$ matrix $\mathbf{A}$, accessible only though matrix-vector products with $\mathbf{A}$ and…
Given an implicit $n\times n$ matrix $A$ with oracle access $x^TA x$ for any $x\in \mathbb{R}^n$, we study the query complexity of randomized algorithms for estimating the trace of the matrix. This problem has many applications in quantum…
Many kernel methods suffer from high time and space complexities and are thus prohibitive in big-data applications. To tackle the computational challenge, the Nystr\"om method has been extensively used to reduce time and space complexities…
We present a new heuristic point-to-point routing algorithm based on contraction hierarchies (CH). Given an epsilon >= 0, we can prove that the length of the path computed by our algorithm is at most (1+epsilon) times the length of the…
The trace of a matrix function f(A), most notably of the matrix inverse, can be estimated stochastically using samples< x,f(A)x> if the components of the random vectors x obey an appropriate probability distribution. However such a…
Optimizing smooth convex functions in stochastic settings, where only noisy estimates of gradients and Hessians are available, is a fundamental problem in optimization. While first-order methods possess a low per-iteration cost, their…
We present a novel statistical inference framework for convex empirical risk minimization, using approximate stochastic Newton steps. The proposed algorithm is based on the notion of finite differences and allows the approximation of a…
Hamiltonian Monte Carlo (HMC) is a Markov chain Monte Carlo (MCMC) algorithm that avoids the random walk behavior and sensitivity to correlated parameters that plague many MCMC methods by taking a series of steps informed by first-order…
Many relevant machine learning and scientific computing tasks involve high-dimensional linear operators accessible only via costly matrix-vector products. In this context, recent advances in sketched methods have enabled the construction of…
In quantum information, trace distance is a basic metric of distinguishability between quantum states. However, there is no known efficient approach to estimate the value of trace distance in general. In this paper, we propose efficient…
The Matrix-based Renyi's entropy enables us to directly measure information quantities from given data without the costly probability density estimation of underlying distributions, thus has been widely adopted in numerous statistical…