2,004 research outputs found
On the complexity of computing Gr\"obner bases for weighted homogeneous systems
Solving polynomial systems arising from applications is frequently made
easier by the structure of the systems. Weighted homogeneity (or
quasi-homogeneity) is one example of such a structure: given a system of
weights , -homogeneous polynomials are polynomials
which are homogeneous w.r.t the weighted degree
. Gr\"obner bases for weighted homogeneous systems can be
computed by adapting existing algorithms for homogeneous systems to the
weighted homogeneous case. We show that in this case, the complexity estimate
for Algorithm~\F5 \left(\binom{n+\dmax-1}{\dmax}^{\omega}\right) can be
divided by a factor . For zero-dimensional
systems, the complexity of Algorithm~\FGLM (where is the
number of solutions of the system) can be divided by the same factor
. Under genericity assumptions, for
zero-dimensional weighted homogeneous systems of -degree
, these complexity estimates are polynomial in the
weighted B\'ezout bound .
Furthermore, the maximum degree reached in a run of Algorithm \F5 is bounded by
the weighted Macaulay bound , and this bound is
sharp if we can order the weights so that . For overdetermined
semi-regular systems, estimates from the homogeneous case can be adapted to the
weighted case. We provide some experimental results based on systems arising
from a cryptography problem and from polynomial inversion problems. They show
that taking advantage of the weighted homogeneous structure yields substantial
speed-ups, and allows us to solve systems which were otherwise out of reach
Explicit CM-theory for level 2-structures on abelian surfaces
For a complex abelian variety with endomorphism ring isomorphic to the
maximal order in a quartic CM-field , the Igusa invariants generate an abelian extension of the reflex field of . In
this paper we give an explicit description of the Galois action of the class
group of this reflex field on . We give a geometric
description which can be expressed by maps between various Siegel modular
varieties. We can explicitly compute this action for ideals of small norm, and
this allows us to improve the CRT method for computing Igusa class polynomials.
Furthermore, we find cycles in isogeny graphs for abelian surfaces, thereby
implying that the `isogeny volcano' algorithm to compute endomorphism rings of
ordinary elliptic curves over finite fields does not have a straightforward
generalization to computing endomorphism rings of abelian surfaces over finite
fields
Cram\'er-Rao Bounds for Polynomial Signal Estimation using Sensors with AR(1) Drift
We seek to characterize the estimation performance of a sensor network where
the individual sensors exhibit the phenomenon of drift, i.e., a gradual change
of the bias. Though estimation in the presence of random errors has been
extensively studied in the literature, the loss of estimation performance due
to systematic errors like drift have rarely been looked into. In this paper, we
derive closed-form Fisher Information matrix and subsequently Cram\'er-Rao
bounds (upto reasonable approximation) for the estimation accuracy of
drift-corrupted signals. We assume a polynomial time-series as the
representative signal and an autoregressive process model for the drift. When
the Markov parameter for drift \rho<1, we show that the first-order effect of
drift is asymptotically equivalent to scaling the measurement noise by an
appropriate factor. For \rho=1, i.e., when the drift is non-stationary, we show
that the constant part of a signal can only be estimated inconsistently
(non-zero asymptotic variance). Practical usage of the results are demonstrated
through the analysis of 1) networks with multiple sensors and 2) bandwidth
limited networks communicating only quantized observations.Comment: 14 pages, 6 figures, This paper will appear in the Oct/Nov 2012 issue
of IEEE Transactions on Signal Processin
Different quantum f-divergences and the reversibility of quantum operations
The concept of classical -divergences gives a unified framework to
construct and study measures of dissimilarity of probability distributions;
special cases include the relative entropy and the R\'enyi divergences. Various
quantum versions of this concept, and more narrowly, the concept of R\'enyi
divergences, have been introduced in the literature with applications in
quantum information theory; most notably Petz' quasi-entropies (standard
-divergences), Matsumoto's maximal -divergences, measured
-divergences, and sandwiched and --R\'enyi divergences.
In this paper we give a systematic overview of the various concepts of
quantum -divergences with a main focus on their monotonicity under quantum
operations, and the implications of the preservation of a quantum
-divergence by a quantum operation. In particular, we compare the standard
and the maximal -divergences regarding their ability to detect the
reversibility of quantum operations. We also show that these two quantum
-divergences are strictly different for non-commuting operators unless
is a polynomial, and obtain some analogous partial results for the relation
between the measured and the standard -divergences.
We also study the monotonicity of the --R\'enyi divergences under
the special class of bistochastic maps that leave one of the arguments of the
R\'enyi divergence invariant, and determine domains of the parameters
where monotonicity holds, and where the preservation of the
--R\'enyi divergence implies the reversibility of the quantum
operation.Comment: 70 pages. v4: New Proposition 3.8 and Appendix D on the continuity
properties of the standard f-divergences. The 2-positivity assumption removed
from Theorem 3.34. The achievability of the measured f-divergence is shown in
Proposition 4.17, and Theorem 4.18 is updated accordingl
Sampling and Recovery of Pulse Streams
Compressive Sensing (CS) is a new technique for the efficient acquisition of
signals, images, and other data that have a sparse representation in some
basis, frame, or dictionary. By sparse we mean that the N-dimensional basis
representation has just K<<N significant coefficients; in this case, the CS
theory maintains that just M = K log N random linear signal measurements will
both preserve all of the signal information and enable robust signal
reconstruction in polynomial time. In this paper, we extend the CS theory to
pulse stream data, which correspond to S-sparse signals/images that are
convolved with an unknown F-sparse pulse shape. Ignoring their convolutional
structure, a pulse stream signal is K=SF sparse. Such signals figure
prominently in a number of applications, from neuroscience to astronomy. Our
specific contributions are threefold. First, we propose a pulse stream signal
model and show that it is equivalent to an infinite union of subspaces. Second,
we derive a lower bound on the number of measurements M required to preserve
the essential information present in pulse streams. The bound is linear in the
total number of degrees of freedom S + F, which is significantly smaller than
the naive bound based on the total signal sparsity K=SF. Third, we develop an
efficient signal recovery algorithm that infers both the shape of the impulse
response as well as the locations and amplitudes of the pulses. The algorithm
alternatively estimates the pulse locations and the pulse shape in a manner
reminiscent of classical deconvolution algorithms. Numerical experiments on
synthetic and real data demonstrate the advantages of our approach over
standard CS
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