361,593 research outputs found
New bounds for circulant Johnson-Lindenstrauss embeddings
This paper analyzes circulant Johnson-Lindenstrauss (JL) embeddings which, as
an important class of structured random JL embeddings, are formed by
randomizing the column signs of a circulant matrix generated by a random
vector. With the help of recent decoupling techniques and matrix-valued
Bernstein inequalities, we obtain a new bound
for Gaussian circulant JL embeddings.
Moreover, by using the Laplace transform technique (also called Bernstein's
trick), we extend the result to subgaussian case. The bounds in this paper
offer a small improvement over the current best bounds for Gaussian circulant
JL embeddings for certain parameter regimes and are derived using more direct
methods.Comment: 11 pages; accepted by Communications in Mathematical Science
A Geometric Approach to Covariance Matrix Estimation and its Applications to Radar Problems
A new class of disturbance covariance matrix estimators for radar signal
processing applications is introduced following a geometric paradigm. Each
estimator is associated with a given unitary invariant norm and performs the
sample covariance matrix projection into a specific set of structured
covariance matrices. Regardless of the considered norm, an efficient solution
technique to handle the resulting constrained optimization problem is
developed. Specifically, it is shown that the new family of distribution-free
estimators shares a shrinkagetype form; besides, the eigenvalues estimate just
requires the solution of a one-dimensional convex problem whose objective
function depends on the considered unitary norm. For the two most common norm
instances, i.e., Frobenius and spectral, very efficient algorithms are
developed to solve the aforementioned one-dimensional optimization leading to
almost closed form covariance estimates. At the analysis stage, the performance
of the new estimators is assessed in terms of achievable Signal to Interference
plus Noise Ratio (SINR) both for a spatial and a Doppler processing assuming
different data statistical characterizations. The results show that interesting
SINR improvements with respect to some counterparts available in the open
literature can be achieved especially in training starved regimes.Comment: submitted for journal publicatio
Maximum of the resolvent over matrices with given spectrum
In numerical analysis it is often necessary to estimate the condition number
and the norm of the resolvent
of a given matrix . We derive new
spectral estimates for these quantities and compute explicit matrices that
achieve our bounds. We recover the well-known fact that the supremum of
over all matrices with and minimal absolute eigenvalue
is the Kronecker bound .
This result is subsequently generalized by computing the corresponding supremum
of for any . We find that the supremum
is attained by a triangular Toeplitz matrix. This provides a simple class of
structured matrices on which condition numbers and resolvent norm bounds can be
studied numerically. The occuring Toeplitz matrices are so-called model
matrices, i.e. matrix representations of the compressed backward shift operator
on the Hardy space to a finite-dimensional invariant subspace
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