5,769 research outputs found
A reduced complexity numerical method for optimal gate synthesis
Although quantum computers have the potential to efficiently solve certain
problems considered difficult by known classical approaches, the design of a
quantum circuit remains computationally difficult. It is known that the optimal
gate design problem is equivalent to the solution of an associated optimal
control problem, the solution to which is also computationally intensive.
Hence, in this article, we introduce the application of a class of numerical
methods (termed the max-plus curse of dimensionality free techniques) that
determine the optimal control thereby synthesizing the desired unitary gate.
The application of this technique to quantum systems has a growth in complexity
that depends on the cardinality of the control set approximation rather than
the much larger growth with respect to spatial dimensions in approaches based
on gridding of the space, used in previous literature. This technique is
demonstrated by obtaining an approximate solution for the gate synthesis on
- a problem that is computationally intractable by grid based
approaches.Comment: 8 pages, 4 figure
Curse of dimensionality reduction in max-plus based approximation methods: theoretical estimates and improved pruning algorithms
Max-plus based methods have been recently developed to approximate the value
function of possibly high dimensional optimal control problems. A critical step
of these methods consists in approximating a function by a supremum of a small
number of functions (max-plus "basis functions") taken from a prescribed
dictionary. We study several variants of this approximation problem, which we
show to be continuous versions of the facility location and -center
combinatorial optimization problems, in which the connection costs arise from a
Bregman distance. We give theoretical error estimates, quantifying the number
of basis functions needed to reach a prescribed accuracy. We derive from our
approach a refinement of the curse of dimensionality free method introduced
previously by McEneaney, with a higher accuracy for a comparable computational
cost.Comment: 8pages 5 figure
Communication Theoretic Data Analytics
Widespread use of the Internet and social networks invokes the generation of
big data, which is proving to be useful in a number of applications. To deal
with explosively growing amounts of data, data analytics has emerged as a
critical technology related to computing, signal processing, and information
networking. In this paper, a formalism is considered in which data is modeled
as a generalized social network and communication theory and information theory
are thereby extended to data analytics. First, the creation of an equalizer to
optimize information transfer between two data variables is considered, and
financial data is used to demonstrate the advantages. Then, an information
coupling approach based on information geometry is applied for dimensionality
reduction, with a pattern recognition example to illustrate the effectiveness.
These initial trials suggest the potential of communication theoretic data
analytics for a wide range of applications.Comment: Published in IEEE Journal on Selected Areas in Communications, Jan.
201
Patent Analytics Based on Feature Vector Space Model: A Case of IoT
The number of approved patents worldwide increases rapidly each year, which
requires new patent analytics to efficiently mine the valuable information
attached to these patents. Vector space model (VSM) represents documents as
high-dimensional vectors, where each dimension corresponds to a unique term.
While originally proposed for information retrieval systems, VSM has also seen
wide applications in patent analytics, and used as a fundamental tool to map
patent documents to structured data. However, VSM method suffers from several
limitations when applied to patent analysis tasks, such as loss of
sentence-level semantics and curse-of-dimensionality problems. In order to
address the above limitations, we propose a patent analytics based on feature
vector space model (FVSM), where the FVSM is constructed by mapping patent
documents to feature vectors extracted by convolutional neural networks (CNN).
The applications of FVSM for three typical patent analysis tasks, i.e., patents
similarity comparison, patent clustering, and patent map generation are
discussed. A case study using patents related to Internet of Things (IoT)
technology is illustrated to demonstrate the performance and effectiveness of
FVSM. The proposed FVSM can be adopted by other patent analysis studies to
replace VSM, based on which various big data learning tasks can be performed
Data complexity measured by principal graphs
How to measure the complexity of a finite set of vectors embedded in a
multidimensional space? This is a non-trivial question which can be approached
in many different ways. Here we suggest a set of data complexity measures using
universal approximators, principal cubic complexes. Principal cubic complexes
generalise the notion of principal manifolds for datasets with non-trivial
topologies. The type of the principal cubic complex is determined by its
dimension and a grammar of elementary graph transformations. The simplest
grammar produces principal trees.
We introduce three natural types of data complexity: 1) geometric (deviation
of the data's approximator from some "idealized" configuration, such as
deviation from harmonicity); 2) structural (how many elements of a principal
graph are needed to approximate the data), and 3) construction complexity (how
many applications of elementary graph transformations are needed to construct
the principal object starting from the simplest one).
We compute these measures for several simulated and real-life data
distributions and show them in the "accuracy-complexity" plots, helping to
optimize the accuracy/complexity ratio. We discuss various issues connected
with measuring data complexity. Software for computing data complexity measures
from principal cubic complexes is provided as well.Comment: Computers and Mathematics with Applications, in pres
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