1,049 research outputs found
A weakly stable algorithm for general Toeplitz systems
We show that a fast algorithm for the QR factorization of a Toeplitz or
Hankel matrix A is weakly stable in the sense that R^T.R is close to A^T.A.
Thus, when the algorithm is used to solve the semi-normal equations R^T.Rx =
A^Tb, we obtain a weakly stable method for the solution of a nonsingular
Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the
solution of the full-rank Toeplitz or Hankel least squares problem.Comment: 17 pages. An old Technical Report with postscript added. For further
details, see http://wwwmaths.anu.edu.au/~brent/pub/pub143.htm
Linear and Range Counting under Metric-based Local Differential Privacy
Local differential privacy (LDP) enables private data sharing and analytics
without the need for a trusted data collector. Error-optimal primitives (for,
e.g., estimating means and item frequencies) under LDP have been well studied.
For analytical tasks such as range queries, however, the best known error bound
is dependent on the domain size of private data, which is potentially
prohibitive. This deficiency is inherent as LDP protects the same level of
indistinguishability between any pair of private data values for each data
downer.
In this paper, we utilize an extension of -LDP called Metric-LDP or
-LDP, where a metric defines heterogeneous privacy guarantees for
different pairs of private data values and thus provides a more flexible knob
than does to relax LDP and tune utility-privacy trade-offs. We show
that, under such privacy relaxations, for analytical workloads such as linear
counting, multi-dimensional range counting queries, and quantile queries, we
can achieve significant gains in utility. In particular, for range queries
under -LDP where the metric is the -distance function scaled by
, we design mechanisms with errors independent on the domain sizes;
instead, their errors depend on the metric , which specifies in what
granularity the private data is protected. We believe that the primitives we
design for -LDP will be useful in developing mechanisms for other analytical
tasks, and encourage the adoption of LDP in practice
On maximum volume submatrices and cross approximation for symmetric semidefinite and diagonally dominant matrices
The problem of finding a submatrix of maximum volume of a matrix
is of interest in a variety of applications. For example, it yields a
quasi-best low-rank approximation constructed from the rows and columns of .
We show that such a submatrix can always be chosen to be a principal submatrix
if is symmetric semidefinite or diagonally dominant. Then we analyze the
low-rank approximation error returned by a greedy method for volume
maximization, cross approximation with complete pivoting. Our bound for general
matrices extends an existing result for symmetric semidefinite matrices and
yields new error estimates for diagonally dominant matrices. In particular, for
doubly diagonally dominant matrices the error is shown to remain within a
modest factor of the best approximation error. We also illustrate how the
application of our results to cross approximation for functions leads to new
and better convergence results
- …