54,352 research outputs found
Solving Inverse Problems with Reinforcement Learning
In this paper, we formally introduce, with rigorous derivations, the use of
reinforcement learning to the field of inverse problems by designing an
iterative algorithm, called REINFORCE-IP, for solving a general type of
non-linear inverse problem. By choosing specific probability models for the
action-selection rule, we connect our approach to the conventional
regularization methods of Tikhonov regularization and iterative regularization.
For the numerical implementation of our approach, we parameterize the
solution-searching rule with the help of neural networks and iteratively
improve the parameter using a reinforcement-learning algorithm~-- REINFORCE.
Under standard assumptions we prove the almost sure convergence of the
parameter to a locally optimal value. Our work provides two typical examples
(non-linear integral equations and parameter-identification problems in partial
differential equations) of how reinforcement learning can be applied in solving
non-linear inverse problems. Our numerical experiments show that REINFORCE-IP
is an efficient algorithm that can escape from local minimums and identify
multi-solutions for inverse problems with non-uniqueness.Comment: 33 pages, 10 figure
Monodromy Transform Approach to Solution of Some Field Equations in General Relativity and String Theory
A monodromy transform approach, presented in this communication, provides a
general base for solution of space-time symmetry reductions of Einstein
equations in all known integrable cases, which include vacuum, electrovacuum,
massless Weyl spinor field and stiff matter fluid, as well as some string
theory induced gravity models. It was found a special finite set of functional
parameters, defined as the monodromy data for the fundamental solution of
associated spectral problem. Similarly to the scattering data in the inverse
scattering transform, these monodromy data can be used for characterization of
any local solution of the field equations. A "direct" and "inverse" problems of
such monodromy transform admit unambiguous solutions. For the linear singular
integral equation with a scalar (i.e. non-matrix) kernel, which solves the
inverse problem of this monodromy transform, an equivalent regularization -- a
Fredholm linear integral equation of the second kind is constrcuted in several
convenient forms. For arbitrary choice of the monodromy data a simple iterative
method leads to an effective construction of the solution in terms of
homogeneously convergent functional series.Comment: 7 pages, 1 figure, LaTeX2e; as submitted to the Proceedings of the
Workshop "Nonlinearity Integrability and All That. Twenty Years After
NEEDS'79", Gallipoli (Lecce), Italy - July 1 - July 10, 1999
An Alternative Lattice Field Theory Formulation Inspired by Lattice Supersymmetry
We propose an unconventional formulation of lattice field theories which is
quite general, although originally motivated by the quest of exact lattice
supersymmetry. Two long standing problems have a solution in this context: 1)
Each degree of freedom on the lattice corresponds to degrees of freedom
in the continuum, but all these doublers have (in the case of fermions) the
same chirality and can be either identified, thus removing the degeneracy, or,
in some theories with extended supersymmetry, identified with different members
of the same supermultiplet. 2) The derivative operator, defined on the lattice
as a suitable periodic function of the lattice momentum, is an addittive and
conserved quantity, thus assuring that the Leibnitz rule is satisfied. This
implies that the product of two fields on the lattice is replaced by a
non-local "star product" which is however in general non-associative.
Associativity of the "star product" poses strong restrictions on the form of
the lattice derivative operator (which becomes the inverse gudermannian
function of the lattice momentum) and has the consequence that the degrees of
freedom of the lattice theory and of the continuum theory are in one-to-one
correspondence, so that the two theories are eventually equivalent.
Regularization of the ultraviolet divergences on the lattice is not associated
to the lattice spacing, which does not act as a regulator, but may be obtained
by a one parameter deformation of the lattice derivative, thus preserving the
lattice structure even in the limit of infinite momentum cutoff. However this
regularization breaks gauge invariance and a gauge invariant regularization
within the lattice formulation is still lacking.Comment: 68 pages, 7 figure
Inverse Problems and Self-similarity in Imaging
This thesis examines the concept of image self-similarity and provides solutions to various associated inverse problems such as resolution enhancement and missing fractal codes.
In general, many real-world inverse problems are ill-posed, mainly because of the lack of existence of a unique solution. The procedure of providing acceptable unique solutions to such problems is known as regularization. The concept of image prior, which has been of crucial importance in image modelling and processing, has also been important in solving inverse problems since it algebraically translates to the regularization procedure.
Indeed, much recent progress in imaging has been due to advances in the formulation and practice of regularization. This, coupled with progress in optimization and numerical analysis, has yielded much improvement in computational methods of solving inverse imaging problems.
Historically, the idea of self-similarity was important in the development of fractal image coding. Here we show that the self-similarity properties of natural images may be used to construct image priors for the purpose of addressing certain inverse problems. Indeed, new trends in the area of non-local image processing have provided a rejuvenated appreciation of image self-similarity and opportunities to explore novel self-similarity-based priors.
We first revisit the concept of fractal-based methods and address some open theoretical problems in the area. This includes formulating a necessary and sufficient condition for the contractivity of the block fractal transform operator. We shall also provide some more generalized formulations of fractal-based self-similarity constraints of an image. These formulations can be developed algebraically and also in terms of the set-based method of Projection Onto Convex Sets (POCS).
We then revisit the traditional inverse problems of single frame image zooming and multi-frame resolution enhancement, also known as super-resolution. Some ideas will be borrowed from newly developed non-local denoising algorithms in order to formulate self-similarity priors. Understanding the role of scale and choice of examples/samples is also important in these proposed models. For this purpose, we perform an extensive series of numerical experiments and analyze the results. These ideas naturally lead to the method of self-examples, which relies on the regularity properties of natural images at different scales, as a means of solving the single-frame image zooming problem.
Furthermore, we propose and investigate a multi-frame super-resolution counterpart which does not require explicit motion estimation among video sequences
On a continuation approach in Tikhonov regularization and its application in piecewise-constant parameter identification
We present a new approach to convexification of the Tikhonov regularization
using a continuation method strategy. We embed the original minimization
problem into a one-parameter family of minimization problems. Both the penalty
term and the minimizer of the Tikhonov functional become dependent on a
continuation parameter.
In this way we can independently treat two main roles of the regularization
term, which are stabilization of the ill-posed problem and introduction of the
a priori knowledge. For zero continuation parameter we solve a relaxed
regularization problem, which stabilizes the ill-posed problem in a weaker
sense. The problem is recast to the original minimization by the continuation
method and so the a priori knowledge is enforced.
We apply this approach in the context of topology-to-shape geometry
identification, where it allows to avoid the convergence of gradient-based
methods to a local minima. We present illustrative results for magnetic
induction tomography which is an example of PDE constrained inverse problem
Optimization Methods for Inverse Problems
Optimization plays an important role in solving many inverse problems.
Indeed, the task of inversion often either involves or is fully cast as a
solution of an optimization problem. In this light, the mere non-linear,
non-convex, and large-scale nature of many of these inversions gives rise to
some very challenging optimization problems. The inverse problem community has
long been developing various techniques for solving such optimization tasks.
However, other, seemingly disjoint communities, such as that of machine
learning, have developed, almost in parallel, interesting alternative methods
which might have stayed under the radar of the inverse problem community. In
this survey, we aim to change that. In doing so, we first discuss current
state-of-the-art optimization methods widely used in inverse problems. We then
survey recent related advances in addressing similar challenges in problems
faced by the machine learning community, and discuss their potential advantages
for solving inverse problems. By highlighting the similarities among the
optimization challenges faced by the inverse problem and the machine learning
communities, we hope that this survey can serve as a bridge in bringing
together these two communities and encourage cross fertilization of ideas.Comment: 13 page
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