91,845 research outputs found
Computational Complexity and Phase Transitions
Phase transitions in combinatorial problems have recently been shown to be
useful in locating "hard" instances of combinatorial problems. The connection
between computational complexity and the existence of phase transitions has
been addressed in Statistical Mechanics and Artificial Intelligence, but not
studied rigorously.
We take a step in this direction by investigating the existence of sharp
thresholds for the class of generalized satisfiability problems defined by
Schaefer. In the case when all constraints are clauses we give a complete
characterization of such problems that have a sharp threshold.
While NP-completeness does not imply (even in this restricted case) the
existence of a sharp threshold, it "almost implies" this, since clausal
generalized satisfiability problems that lack a sharp threshold are either
1. polynomial time solvable, or
2. predicted, with success probability lower bounded by some positive
constant by across all the probability range, by a single, trivial procedure.Comment: A (slightly) revised version of the paper submitted to the 15th IEEE
Conference on Computational Complexit
Computational Complexity versus Statistical Performance on Sparse Recovery Problems
We show that several classical quantities controlling compressed sensing
performance directly match classical parameters controlling algorithmic
complexity. We first describe linearly convergent restart schemes on
first-order methods solving a broad range of compressed sensing problems, where
sharpness at the optimum controls convergence speed. We show that for sparse
recovery problems, this sharpness can be written as a condition number, given
by the ratio between true signal sparsity and the largest signal size that can
be recovered by the observation matrix. In a similar vein, Renegar's condition
number is a data-driven complexity measure for convex programs, generalizing
classical condition numbers for linear systems. We show that for a broad class
of compressed sensing problems, the worst case value of this algorithmic
complexity measure taken over all signals matches the restricted singular value
of the observation matrix which controls robust recovery performance. Overall,
this means in both cases that, in compressed sensing problems, a single
parameter directly controls both computational complexity and recovery
performance. Numerical experiments illustrate these points using several
classical algorithms.Comment: Final version, to appear in information and Inferenc
Error-Bounded and Feature Preserving Surface Remeshing with Minimal Angle Improvement
The typical goal of surface remeshing consists in finding a mesh that is (1)
geometrically faithful to the original geometry, (2) as coarse as possible to
obtain a low-complexity representation and (3) free of bad elements that would
hamper the desired application. In this paper, we design an algorithm to
address all three optimization goals simultaneously. The user specifies desired
bounds on approximation error {\delta}, minimal interior angle {\theta} and
maximum mesh complexity N (number of vertices). Since such a desired mesh might
not even exist, our optimization framework treats only the approximation error
bound {\delta} as a hard constraint and the other two criteria as optimization
goals. More specifically, we iteratively perform carefully prioritized local
operators, whenever they do not violate the approximation error bound and
improve the mesh otherwise. In this way our optimization framework greedily
searches for the coarsest mesh with minimal interior angle above {\theta} and
approximation error bounded by {\delta}. Fast runtime is enabled by a local
approximation error estimation, while implicit feature preservation is obtained
by specifically designed vertex relocation operators. Experiments show that our
approach delivers high-quality meshes with implicitly preserved features and
better balances between geometric fidelity, mesh complexity and element quality
than the state-of-the-art.Comment: 14 pages, 20 figures. Submitted to IEEE Transactions on Visualization
and Computer Graphic
Many Hard Examples in Exact Phase Transitions with Application to Generating Hard Satisfiable Instances
This paper first analyzes the resolution complexity of two random CSP models
(i.e. Model RB/RD) for which we can establish the existence of phase
transitions and identify the threshold points exactly. By encoding CSPs into
CNF formulas, it is proved that almost all instances of Model RB/RD have no
tree-like resolution proofs of less than exponential size. Thus, we not only
introduce new families of CNF formulas hard for resolution, which is a central
task of Proof-Complexity theory, but also propose models with both many hard
instances and exact phase transitions. Then, the implications of such models
are addressed. It is shown both theoretically and experimentally that an
application of Model RB/RD might be in the generation of hard satisfiable
instances, which is not only of practical importance but also related to some
open problems in cryptography such as generating one-way functions.
Subsequently, a further theoretical support for the generation method is shown
by establishing exponential lower bounds on the complexity of solving random
satisfiable and forced satisfiable instances of RB/RD near the threshold.
Finally, conclusions are presented, as well as a detailed comparison of Model
RB/RD with the Hamiltonian cycle problem and random 3-SAT, which, respectively,
exhibit three different kinds of phase transition behavior in NP-complete
problems.Comment: 19 pages, corrected mistakes in Theorems 5 and
Online Tool Condition Monitoring Based on Parsimonious Ensemble+
Accurate diagnosis of tool wear in metal turning process remains an open
challenge for both scientists and industrial practitioners because of
inhomogeneities in workpiece material, nonstationary machining settings to suit
production requirements, and nonlinear relations between measured variables and
tool wear. Common methodologies for tool condition monitoring still rely on
batch approaches which cannot cope with a fast sampling rate of metal cutting
process. Furthermore they require a retraining process to be completed from
scratch when dealing with a new set of machining parameters. This paper
presents an online tool condition monitoring approach based on Parsimonious
Ensemble+, pENsemble+. The unique feature of pENsemble+ lies in its highly
flexible principle where both ensemble structure and base-classifier structure
can automatically grow and shrink on the fly based on the characteristics of
data streams. Moreover, the online feature selection scenario is integrated to
actively sample relevant input attributes. The paper presents advancement of a
newly developed ensemble learning algorithm, pENsemble+, where online active
learning scenario is incorporated to reduce operator labelling effort. The
ensemble merging scenario is proposed which allows reduction of ensemble
complexity while retaining its diversity. Experimental studies utilising
real-world manufacturing data streams and comparisons with well known
algorithms were carried out. Furthermore, the efficacy of pENsemble was
examined using benchmark concept drift data streams. It has been found that
pENsemble+ incurs low structural complexity and results in a significant
reduction of operator labelling effort.Comment: this paper has been published by IEEE Transactions on Cybernetic
Mesh-based video coding for low bit-rate communications
In this paper, a new method for low bit-rate content-adaptive mesh-based video coding is proposed. Intra-frame coding of this method employs feature map extraction for node distribution at specific threshold levels to achieve higher density placement of initial nodes for regions that contain high frequency features and conversely sparse placement of initial nodes for smooth regions. Insignificant nodes are largely removed using a subsequent node elimination scheme. The Hilbert scan is then applied before quantization and entropy coding to reduce amount of transmitted information. For moving images, both node position and color parameters of only a subset of nodes may change from frame to frame. It is sufficient to transmit only these changed parameters. The proposed method is well-suited for video coding at very low bit rates, as processing results demonstrate that it provides good subjective and objective image quality at a lower number of required bits
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