28,499 research outputs found
Range entropy: A bridge between signal complexity and self-similarity
Approximate entropy (ApEn) and sample entropy (SampEn) are widely used for
temporal complexity analysis of real-world phenomena. However, their
relationship with the Hurst exponent as a measure of self-similarity is not
widely studied. Additionally, ApEn and SampEn are susceptible to signal
amplitude changes. A common practice for addressing this issue is to correct
their input signal amplitude by its standard deviation. In this study, we first
show, using simulations, that ApEn and SampEn are related to the Hurst exponent
in their tolerance r and embedding dimension m parameters. We then propose a
modification to ApEn and SampEn called range entropy or RangeEn. We show that
RangeEn is more robust to nonstationary signal changes, and it has a more
linear relationship with the Hurst exponent, compared to ApEn and SampEn.
RangeEn is bounded in the tolerance r-plane between 0 (maximum entropy) and 1
(minimum entropy) and it has no need for signal amplitude correction. Finally,
we demonstrate the clinical usefulness of signal entropy measures for
characterisation of epileptic EEG data as a real-world example.Comment: This is the revised and published version in Entrop
Towards a Constructive Version of Banaszczyk's Vector Balancing Theorem
An important theorem of Banaszczyk (Random Structures & Algorithms `98)
states that for any sequence of vectors of norm at most and any
convex body of Gaussian measure in , there exists a
signed combination of these vectors which lands inside . A major open
problem is to devise a constructive version of Banaszczyk's vector balancing
theorem, i.e. to find an efficient algorithm which constructs the signed
combination.
We make progress towards this goal along several fronts. As our first
contribution, we show an equivalence between Banaszczyk's theorem and the
existence of -subgaussian distributions over signed combinations. For the
case of symmetric convex bodies, our equivalence implies the existence of a
universal signing algorithm (i.e. independent of the body), which simply
samples from the subgaussian sign distribution and checks to see if the
associated combination lands inside the body. For asymmetric convex bodies, we
provide a novel recentering procedure, which allows us to reduce to the case
where the body is symmetric.
As our second main contribution, we show that the above framework can be
efficiently implemented when the vectors have length ,
recovering Banaszczyk's results under this stronger assumption. More precisely,
we use random walk techniques to produce the required -subgaussian
signing distributions when the vectors have length , and
use a stochastic gradient ascent method to implement the recentering procedure
for asymmetric bodies
Minimum d-dimensional arrangement with fixed points
In the Minimum -Dimensional Arrangement Problem (d-dimAP) we are given a
graph with edge weights, and the goal is to find a 1-1 map of the vertices into
(for some fixed dimension ) minimizing the total
weighted stretch of the edges. This problem arises in VLSI placement and chip
design.
Motivated by these applications, we consider a generalization of d-dimAP,
where the positions of some of the vertices (pins) is fixed and specified as
part of the input. We are asked to extend this partial map to a map of all the
vertices, again minimizing the weighted stretch of edges. This generalization,
which we refer to as d-dimAP+, arises naturally in these application domains
(since it can capture blocked-off parts of the board, or the requirement of
power-carrying pins to be in certain locations, etc.). Perhaps surprisingly,
very little is known about this problem from an approximation viewpoint.
For dimension , we obtain an -approximation
algorithm, based on a strengthening of the spreading-metric LP for 2-dimAP. The
integrality gap for this LP is shown to be . We also show that
it is NP-hard to approximate 2-dimAP+ within a factor better than
\Omega(k^{1/4-\eps}). We also consider a (conceptually harder, but
practically even more interesting) variant of 2-dimAP+, where the target space
is the grid , instead of
the entire integer lattice . For this problem, we obtain a -approximation using the same LP relaxation. We complement
this upper bound by showing an integrality gap of , and an
\Omega(k^{1/2-\eps})-inapproximability result.
Our results naturally extend to the case of arbitrary fixed target dimension
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