94 research outputs found
Krylov-aware stochastic trace estimation
We introduce an algorithm for estimating the trace of a matrix function
using implicit products with a symmetric matrix .
Existing methods for implicit trace estimation of a matrix function tend to
treat matrix-vector products with as a black-box to be computed
by a Krylov subspace method. Like other recent algorithms for implicit trace
estimation, our approach is based on a combination of deflation and stochastic
trace estimation. However, we take a closer look at how products with
are integrated into these approaches which enables several
efficiencies not present in previously studied methods. In particular, we
describe a Krylov subspace method for computing a low-rank approximation of a
matrix function by a computationally efficient projection onto Krylov subspace.Comment: Figure 5.1 differs somewhat from the published version due to a
clerical error made when uploading the images to the journa
Scalable iterative methods for sampling from massive Gaussian random vectors
Sampling from Gaussian Markov random fields (GMRFs), that is multivariate
Gaussian ran- dom vectors that are parameterised by the inverse of their
covariance matrix, is a fundamental problem in computational statistics. In
this paper, we show how we can exploit arbitrarily accu- rate approximations to
a GMRF to speed up Krylov subspace sampling methods. We also show that these
methods can be used when computing the normalising constant of a large
multivariate Gaussian distribution, which is needed for both any
likelihood-based inference method. The method we derive is also applicable to
other structured Gaussian random vectors and, in particu- lar, we show that
when the precision matrix is a perturbation of a (block) circulant matrix, it
is still possible to derive O(n log n) sampling schemes.Comment: 17 Pages, 4 Figure
Suboptimal subspace construction for log-determinant approximation
Variance reduction is a crucial idea for Monte Carlo simulation and the
stochastic Lanczos quadrature method is a dedicated method to approximate the
trace of a matrix function. Inspired by their advantages, we combine these two
techniques to approximate the log-determinant of large-scale symmetric positive
definite matrices. Key questions to be answered for such a method are how to
construct or choose an appropriate projection subspace and derive guaranteed
theoretical analysis. This paper applies some probabilistic approaches
including the projection-cost-preserving sketch and matrix concentration
inequalities to construct a suboptimal subspace. Furthermore, we provide some
insights on choosing design parameters in the underlying algorithm by deriving
corresponding approximation error and probabilistic error estimations.
Numerical experiments demonstrate our method's effectiveness and illustrate the
quality of the derived error bounds
Faster randomized partial trace estimation
We develop randomized matrix-free algorithms for estimating partial traces.
Our algorithm improves on the typicality-based approach used in [T. Chen and
Y-C. Cheng, Numerical computation of the equilibrium-reduced density matrix for
strongly coupled open quantum systems, J. Chem. Phys. 157, 064106 (2022)] by
deflating important subspaces (e.g. corresponding to the low-energy
eigenstates) explicitly. This results in a significant variance reduction for
matrices with quickly decaying singular values. We then apply our algorithm to
study the thermodynamics of several Heisenberg spin systems, particularly the
entanglement spectrum and ergotropy
Randomized matrix-free quadrature for spectrum and spectral sum approximation
We study randomized matrix-free quadrature algorithms for spectrum and
spectral sum approximation. The algorithms studied are characterized by the use
of a Krylov subspace method to approximate independent and identically
distributed samples of , where
is an isotropic random vector, is a Hermitian matrix,
and is a matrix function. This class of algorithms includes the
kernel polynomial method and stochastic Lanczos quadrature, two widely used
methods for approximating spectra and spectral sums. Our analysis, discussion,
and numerical examples provide a unified framework for understanding randomized
matrix-free quadrature and shed light on the commonalities and tradeoffs
between them. Moreover, this framework provides new insights into the practical
implementation and use of these algorithms, particularly with regards to
parameter selection in the kernel polynomial method
Randomized low-rank approximation of monotone matrix functions
This work is concerned with computing low-rank approximations of a matrix
function for a large symmetric positive semi-definite matrix , a task
that arises in, e.g., statistical learning and inverse problems. The
application of popular randomized methods, such as the randomized singular
value decomposition or the Nystr\"om approximation, to requires
multiplying with a few random vectors. A significant disadvantage of
such an approach, matrix-vector products with are considerably more
expensive than matrix-vector products with , even when carried out only
approximately via, e.g., the Lanczos method. In this work, we present and
analyze funNystr\"om, a simple and inexpensive method that constructs a
low-rank approximation of directly from a Nystr\"om approximation of
, completely bypassing the need for matrix-vector products with . It
is sensible to use funNystr\"om whenever is monotone and satisfies . Under the stronger assumption that is operator monotone, which includes
the matrix square root and the matrix logarithm , we
derive probabilistic bounds for the error in the Frobenius, nuclear, and
operator norms. These bounds confirm the numerical observation that
funNystr\"om tends to return an approximation that compares well with the best
low-rank approximation of . Our method is also of interest when
estimating quantities associated with , such as the trace or the diagonal
entries of . In particular, we propose and analyze funNystr\"om++, a
combination of funNystr\"om with the recently developed Hutch++ method for
trace estimation
Accuracy of the finite-temperature Lanczos method compared to simple typicality-based estimates
We study trace estimators for equilibrium thermodynamic observables that rely on the idea of typicality and derivatives thereof such as the finite-temperature Lanczos method (FTLM). As numerical examples quantum spin systems are studied. Our initial aim was to identify pathological examples or circumstances, such as strong frustration or unusual densities of states, where these methods could fail. It turned out that all investigated systems allow such approximations. Only at temperatures of the order of the lowest energy gap is the convergence somewhat slower in the number of random vectors over which observables are averaged
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