139 research outputs found
Resonant nonlinear periodic problems with the scalar p-Laplacian and a nonsmooth potential
We study periodic problems driven by the scalar p-Laplacian
with a nonsmooth potential. Using the nonsmooth critical point theory for
locally Lipsctiz functions,we prove two existence theorems under conditions
of resonance at infinity with respect to the first two eigenvalues of the
negative scalar p-Laplacian with periodic boundary conditions.Universidade de Aveir
Partial Sum Minimization of Singular Values in Robust PCA: Algorithm and Applications
Robust Principal Component Analysis (RPCA) via rank minimization is a
powerful tool for recovering underlying low-rank structure of clean data
corrupted with sparse noise/outliers. In many low-level vision problems, not
only it is known that the underlying structure of clean data is low-rank, but
the exact rank of clean data is also known. Yet, when applying conventional
rank minimization for those problems, the objective function is formulated in a
way that does not fully utilize a priori target rank information about the
problems. This observation motivates us to investigate whether there is a
better alternative solution when using rank minimization. In this paper,
instead of minimizing the nuclear norm, we propose to minimize the partial sum
of singular values, which implicitly encourages the target rank constraint. Our
experimental analyses show that, when the number of samples is deficient, our
approach leads to a higher success rate than conventional rank minimization,
while the solutions obtained by the two approaches are almost identical when
the number of samples is more than sufficient. We apply our approach to various
low-level vision problems, e.g. high dynamic range imaging, motion edge
detection, photometric stereo, image alignment and recovery, and show that our
results outperform those obtained by the conventional nuclear norm rank
minimization method.Comment: Accepted in Transactions on Pattern Analysis and Machine Intelligence
(TPAMI). To appea
Gateaux differentiability revisited
We revisit some basic concepts and ideas of the classical differential calculus and convex analysis extending them to a broader frame. We reformulate and generalize the notion of Gateaux differentiability and propose new notions of generalized derivative and generalized subdifferential in an arbitrary topological vector space. Meaningful examples preserving the key properties of the original notion of derivative are provided. Ā© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature
Three nontrivial solutions for nonlocal anisotropic inclusions under nonresonance
In this article, we study a pseudo-differential inclusion driven by a nonlocal anisotropic operator and a Clarke generalized subdifferential of a nonsmooth potential, which satisfies nonresonance conditions both at the origin and at infinity. We prove the existence of three nontrivial solutions: one positive, one negative and one of unknown sign, using variational methods based on nosmooth critical point theory, more precisely applying the second deformation theorem and spectral theory. Here, a nosmooth anisotropic version of the Hƶlder versus Sobolev minimizers relation play an important role
Generalized monotonicity and convexity ofĀ non-differentiable functions
AbstractThe relationships between (strict, strong) convexity of non-differentiable functions and (strict, strong) monotonicity of set-valued mappings, and (strict, strong, sharp) pseudo convexity of non-differentiable functions and (strict, strong) pseudo monotonicity of set-valued mappings, as well as quasi convexity of non-differentiable functions and quasi monotonicity of set-valued mappings are studied in this paper. In addition, the relations between generalized convexity of non-differentiable functions and generalized co-coerciveness of set-valued mappings are also analyzed
- ā¦