Low-rank structures play important role in recent advances of many problems in image science and data science. As a natural extension of low-rank structures for data with nonlinear structures, the concept of the low-dimensional manifold structure has been considered in many data processing problems. Inspired by this concept, we consider a manifold based low-rank regularization as a linear appro...

one of the main steps within the geoid computation methodology without applying the stokes formula is downward continuation of the harmonic residual observables from the surface of the earth down to the surface of the reference ellipsoid. this downward continuation is done via the abel-poisson integral and its derivatives. this integral in which the unknowns, i.e. harmonic residual potential va...

Probabilistic topic modeling of text collections is a powerful tool for statistical text analysis. Determining the optimal number of topics remains a challenging problem in topic modeling. We propose a simple entropy regularization for topic selection in terms of Additive Regularization of Topic Models (ARTM), a multicriteria approach for combining regularizers. The entropy regularization gradu...

Regularization is a general method for solving ill-posed and ill-conditioned problems. Traditionally, ill-conditioning in system identiication problems is usually approached using regularization methods such as ridge regression and principal component regression. In this work it is argued that the Tikhonov regularization method is a powerful alternative for regulariza-tion of non-linear system ...

A key aspect of the recently proposed Hamiltonian Particle-Mesh (HPM) method is its time-staggered discretization combined with a regularization of the continuous governing equations. In this paper, the time discretization aspect of the HPM method is analysed for the linearized, rotating, shallow-water equations with orography and the combined effect of time-staggering and regularization is com...

In this paper, we study the manifold regularization for the Sliced Inverse Regression (SIR). The manifold regularization improves the standard SIR in two aspects: 1) it encodes the local geometry for SIR and 2) it enables SIR to deal with transductive and semi-supervised learning problems. We prove that the proposed graph Laplacian based regularization is convergent at rate root-n. The projecti...

We propose in this work to regularize the output of a svm classifier on pixels in order to promote smoothness in the predicted image. The learning problem can be cast as a semi-supervised SVM with a particular structure encoding pixel neighborhood in the regularization graph. We provide several optimization schemes in order to solve the problem for linear SVM with `2 or `1 regularization and sh...

We propose squared-loss mutual information regularization (SMIR) for multi-class probabilistic classification, following the information maximization principle. SMIR is convex under mild conditions and thus improves the nonconvexity of mutual information regularization. It offers all of the following four abilities to semi-supervised algorithms: Analytical solution, out-of-sample/multi-class cl...

We present a general framework of spectral methods for semi-supervised dimensionality reduction. Applying an approach called manifold regularization, our framework naturally generalizes existent supervised frameworks. Furthermore, by our two semi-supervised versions of the representer theorem, our framework can be kernelized as well. Using our framework, we give three examples of semi-supervise...