387 research outputs found
Linear Convergence of ISTA and FISTA
In this paper, we revisit the class of iterative shrinkage-thresholding
algorithms (ISTA) for solving the linear inverse problem with sparse
representation, which arises in signal and image processing. It is shown in the
numerical experiment to deblur an image that the convergence behavior in the
logarithmic-scale ordinate tends to be linear instead of logarithmic,
approximating to be flat. Making meticulous observations, we find that the
previous assumption for the smooth part to be convex weakens the least-square
model. Specifically, assuming the smooth part to be strongly convex is more
reasonable for the least-square model, even though the image matrix is probably
ill-conditioned. Furthermore, we improve the pivotal inequality tighter for
composite optimization with the smooth part to be strongly convex instead of
general convex, which is first found in [Li et al., 2022]. Based on this
pivotal inequality, we generalize the linear convergence to composite
optimization in both the objective value and the squared proximal subgradient
norm. Meanwhile, we set a simple ill-conditioned matrix which is easy to
compute the singular values instead of the original blur matrix. The new
numerical experiment shows the proximal generalization of Nesterov's
accelerated gradient descent (NAG) for the strongly convex function has a
faster linear convergence rate than ISTA. Based on the tighter pivotal
inequality, we also generalize the faster linear convergence rate to composite
optimization, in both the objective value and the squared proximal subgradient
norm, by taking advantage of the well-constructed Lyapunov function with a
slight modification and the phase-space representation based on the
high-resolution differential equation framework from the implicit-velocity
scheme.Comment: 16 pages, 4 figure
Gradient Norm Minimization of Nesterov Acceleration:
In the history of first-order algorithms, Nesterov's accelerated gradient
descent (NAG) is one of the milestones. However, the cause of the acceleration
has been a mystery for a long time. It has not been revealed with the existence
of gradient correction until the high-resolution differential equation
framework proposed in [Shi et al., 2021]. In this paper, we continue to
investigate the acceleration phenomenon. First, we provide a significantly
simplified proof based on precise observation and a tighter inequality for
-smooth functions. Then, a new implicit-velocity high-resolution
differential equation framework, as well as the corresponding implicit-velocity
version of phase-space representation and Lyapunov function, is proposed to
investigate the convergence behavior of the iterative sequence
of NAG. Furthermore, from two kinds of phase-space
representations, we find that the role played by gradient correction is
equivalent to that by velocity included implicitly in the gradient, where the
only difference comes from the iterative sequence
replaced by . Finally, for the open question of whether
the gradient norm minimization of NAG has a faster rate , we figure
out a positive answer with its proof. Meanwhile, a faster rate of objective
value minimization is shown for the case .Comment: 16 page
Proximal Subgradient Norm Minimization of ISTA and FISTA
For first-order smooth optimization, the research on the acceleration
phenomenon has a long-time history. Until recently, the mechanism leading to
acceleration was not successfully uncovered by the gradient correction term and
its equivalent implicit-velocity form. Furthermore, based on the
high-resolution differential equation framework with the corresponding emerging
techniques, phase-space representation and Lyapunov function, the squared
gradient norm of Nesterov's accelerated gradient descent (\texttt{NAG}) method
at an inverse cubic rate is discovered. However, this result cannot be directly
generalized to composite optimization widely used in practice, e.g., the linear
inverse problem with sparse representation. In this paper, we meticulously
observe a pivotal inequality used in composite optimization about the step size
and the Lipschitz constant and find that it can be improved tighter. We
apply the tighter inequality discovered in the well-constructed Lyapunov
function and then obtain the proximal subgradient norm minimization by the
phase-space representation, regardless of gradient-correction or
implicit-velocity. Furthermore, we demonstrate that the squared proximal
subgradient norm for the class of iterative shrinkage-thresholding algorithms
(ISTA) converges at an inverse square rate, and the squared proximal
subgradient norm for the class of faster iterative shrinkage-thresholding
algorithms (FISTA) is accelerated to convergence at an inverse cubic rate.Comment: 17 pages, 4 figure
The thermal evolution of nuclear matter at zero temperature and definite baryon number density in chiral perturbation theory
The thermal properties of cold dense nuclear matter are investigated with
chiral perturbation theory.
The evolution curves for the baryon number density, baryon number
susceptibility, pressure and the equation of state are obtained.
The chiral condensate is calculated and our result shows that when the baryon
chemical potential goes beyond , the absolute value of the
quark condensate decreases rapidly, which indicates a tendency of chiral
restoration.Comment: 17 pages, 9 figures, revtex
2,2,2-Trifluoroethyl 4-methylbenzenesulfonate
In the crystal structure of the title compound, C9H9F3O3S, intermolecular C—H⋯O hydrogen bonds link the molecules along the c-axis direction. Also present are slipped π–π stacking interactions between phenylene rings, with perpendicular interplanar distances of 3.55 (2) Å and centroid–centroid distances of 3.851 (2) Å
Methyl 2-amino-5-chlorobenzoate
The title compound, C8H8ClNO2, is almost planar, with an r.m.s. deviation of 0.0410 Å from the plane through the non-hydrogen atoms. In the crystal structure, intermolecular N—H⋯O hydrogen bonds link the molecules into chains along the b axis. An intramolecular N—H⋯O hydrogen bond results in the formation of a six-membered ring
Ethyl 2-(2-hydroxy-5-nitrophenyl)acetate
In the crystal structure of the title compound, C10H11NO5, intermolecular O—H⋯O hydrogen bonds link the molecules into chains along the b-axis direction. Weak C—H.·O hydrogen bonds also occur
Methyl 5-chloro-2-[N-(3-ethoxycarbonylpropyl)-4-methylbenzenesulfonamido]benzoate
In the title compound, C21H24ClNO6S, the benzene rings are oriented at a dihedral angles of 41.6 (2)°. In the crystal structure, weak intermolecular C—H⋯O interactions link the molecules
Methyl 5-chloro-2-(4-methylbenzenesulfonamido)benzoate
In the title compound, C15H14ClNO4S, the benzene rings are oriented at a dihedral angle of 85.42 (1)°. An intramolecular N—H⋯O hydrogen bond results in the formation of a five-membered ring and an intramolecular C—H⋯O interaction also occurs
2-Methyl-4-(2-methylbenzamido)benzoic acid
In the crystal structure of the title compound, C16H15NO3, intermolecular N—H⋯O hydrogen bonds link the molecules into chains parallel to the b axis and pairs of intermolecular O—H⋯O hydrogen bonds between inversion-related carboxylic acid groups link the molecules into dimers. The dihedral angle between the two benzene rings is 82.4 (2)°
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