509 research outputs found
Monotone matrix functions of successive orders
This paper extends a result obtained by Wigner and von Neumann. We prove that a non-constant real-valued function, f(x), in C^3(I) where I is an interval of the real line, is a monotone matrix function of order n+1 on I if and only if a related, modified function gx0 (x) is a monotone matrix function of order n for every value of x0 in I, assuming that f' is strictly positive on I
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
On Searching a Table Consistent with Division Poset
Suppose is a partially ordered set with the partial order
defined by divisibility, that is, for any two distinct elements
satisfying divides , . A table of
distinct real numbers is said to be \emph{consistent} with , provided for
any two distinct elements satisfying divides ,
. Given an real number , we want to determine whether ,
by comparing with as few entries of as possible. In this paper we
investigate the complexity , measured in the number of comparisons, of
the above search problem. We present a search
algorithm for and prove a lower bound on
by using an adversary argument.Comment: 16 pages, no figure; same results, representation improved, add
reference
- …