Laplacian Term Research Articles

Smoothing effect and exponential stability of discrete-time Schrödinger equation with fractional regularization term

This paper investigates the discrete-time problem of the regularized Schrödinger equation with fractional Laplacian term in bounded domains. We proved the exponential stability and the Kato smoothing effect of the discrete-time Crank–Nicolson scheme. The proofs are based on uniform estimates of the semi-discrete resolvent obtained by the Z-Transformation. We constructed a well-posed approximation scheme and carried out numerical simulations to verify numerically the exponential stability and the smoothing effect.

Spacetime modulation in floating thin elastic plates

We consider the propagation of flexural-gravity waves in thin elastic plates floating atop nonviscous fluids, e.g., seawater, which are governed by a partial differential equation with Laplacian and tri-Laplacian terms. We investigate the effect of time modulation as well as spacetime modulation on thin floating elastic plates and show the peculiarity of the phenomena of the $k$-band gap and the rotated $k$-band gap in the context of flexural-gravity waves. This makes possible floating plates with nonreciprocal features and behaving as elastodynamic analogs of luminal electromagnetic metamaterials, with exotic applications in enhanced control of ocean waves, such as filtering devices, unidirectional acoustic propagation, and isolation effects and energy harvesting in maritime engineering.

Rational Krylov methods for fractional diffusion problems on graphs

In this paper we propose a method to compute the solution to the fractional diffusion equation on directed networks, which can be expressed in terms of the graph Laplacian L as a product f(L^T) varvec{b}, where f is a non-analytic function involving fractional powers and varvec{b} is a given vector. The graph Laplacian is a singular matrix, causing Krylov methods for f(L^T) varvec{b} to converge more slowly. In order to overcome this difficulty and achieve faster convergence, we use rational Krylov methods applied to a desingularized version of the graph Laplacian, obtained with either a rank-one shift or a projection on a subspace.

A Transfer Model Based on Supervised Multi-Layer Dictionary Learning for Brain Tumor MRI Image Recognition.

Artificial intelligence (AI) is an effective technology for automatic brain tumor MRI image recognition. The training of an AI model requires a large number of labeled data, but medical data needs to be labeled by professional clinicians, which makes data collection complex and expensive. Moreover, a traditional AI model requires that the training data and test data must follow the independent and identically distributed. To solve this problem, we propose a transfer model based on supervised multi-layer dictionary learning (TSMDL) for brain tumor MRI image recognition in this paper. With the help of the knowledge learned from related domains, the goal of this model is to solve the task of transfer learning where the target domain has only a small number of labeled samples. Based on the framework of multi-layer dictionary learning, the proposed model learns the common shared dictionary of source and target domains in each layer to explore the intrinsic connections and shared information between different domains. At the same time, by making full use of the label information of samples, the Laplacian regularization term is introduced to make the dictionary coding of similar samples as close as possible and the dictionary coding of different class samples as different as possible. The recognition experiments on brain MRI image datasets REMBRANDT and Figshare show that the model performs better than competitive state of-the-art methods.

Adams’ inequality with logarithmic weights in ℝ⁴

Trudinger-Moser inequality with logarithmic weight was first established by Calanchi and Ruf [J. Differential Equations 258 (2015), pp. 1967–1989]. The aim of this paper is to address the higher order version; more precisely, we show the following inequality sup u ∈ W 0 , r a d 2 , 2 ( B , ω ) , ‖ Δ u ‖ ω ≤ 1 ∫ B exp ⁡ ( α | u | 2 1 − β ) d x > + ∞ \begin{equation*} \sup _{u \in W_{0,rad}^{2,2}(B,\omega ),{{\left \| {\Delta u} \right \|}_\omega } \le 1} \int _B {\exp \left ( {\alpha {{\left | u \right |}^{\frac {2}{{1 - \beta }}}}} \right )} dx > + \infty \end{equation*} holds if and only if \[ α ≤ α β = 4 [ 8 π 2 ( 1 − β ) ] 1 1 − β , \alpha \le {\alpha _\beta } = 4{\left [ {8{\pi ^2}\left ( {1 - \beta } \right )} \right ]^{\frac {1}{{1 - \beta }}}}, \] where B B denotes the unit ball in R 4 \mathbb {R}^{4} , β ∈ ( 0 , 1 ) \beta \in \left ( {0,1} \right ) , ω ( x ) = ( log ⁡ 1 | x | ) β \omega \left ( x \right ) = {\left ( {\log \frac {1}{{\left | x \right |}}} \right )^\beta } or ( log ⁡ e | x | ) β {\left ( {\log \frac {e}{{\left | x \right |}}} \right )^\beta } , and W 0 , r a d 2 , 2 ( B , ω ) W_{0,rad}^{2,2}(B,\omega ) is the weighted Sobolev spaces. Our proof is based on a suitable change of variable that allows us to represent the laplacian of u u in terms of the second derivatives with respect to the new variable; this method was first used by Tarsi [Potential Anal. 37 (2012), pp. 353–385].

Kink–antikink interaction forces and bound states in a ϕ 4 model with quadratic and quartic dispersion

We consider the interaction of solitary waves in a model involving the well-known ϕ 4 Klein–Gordon theory, but now bearing both Laplacian and biharmonic terms with different prefactors. As a result of the competition of the respective linear operators, we obtain three distinct cases as we vary the model parameters. In the first the biharmonic effect dominates, yielding an oscillatory inter-wave interaction; in the third the harmonic effect prevails yielding exponential interactions, while we find an intriguing linearly modulated exponential effect in the critical second case, separating the above two regimes. For each case, we calculate the force between the kink and antikink when initially separated with sufficient distance. Being able to write the acceleration as a function of the separation distance, and its corresponding ordinary differential equation, we test the corresponding predictions, finding very good agreement, where appropriate, with the corresponding partial differential equation results. Where the two findings differ, we explain the source of disparities. Finally, we offer a first glimpse of the interplay of harmonic and biharmonic effects on the results of kink–antikink collisions and the corresponding single- and multi-bounce windows.

Improved thermal multiple-relaxation-time lattice Boltzmann model for liquid-vapor phase change

In this paper, an improved thermal multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is proposed for simulating liquid-vapor phase change. A temperature equation is first derived for liquid-vapor phase change, where the latent heat of vaporization is decoupled with the equation of state. Therefore, the latent heat of vaporization can be arbitrarily specified in practice, which significantly improves the flexibility of the present LB model for liquid-vapor phase change. The Laplacian term of temperature is avoided in the proposed temperature equation and the gradient term of temperature is calculated through a local scheme. To solve the temperature equation accurately and efficiently, an improved MRT LB equation with nondiagonal relaxation matrix is developed. The implicit calculation of the temperature, caused by the source term and encountered in previous works, is avoided by approximating the source term with its value at the previous time step. As demonstrated by numerical tests, the results by the present LB model agree well with analytical results, experimental results, or the results by the finite difference method where the fourth-order Runge-Kutta method is employed to implement the discretization of time.

Drug-Target Interaction Prediction via Dual Laplacian Graph Regularized Logistic Matrix Factorization.

Drug-target interactions provide useful information for biomedical drug discovery as well as drug development. However, it is costly and time consuming to find drug-target interactions by experimental methods. As a result, developing computational approaches for this task is necessary and has practical significance. In this study, we establish a novel dual Laplacian graph regularized logistic matrix factorization model for drug-target interaction prediction, referred to as DLGrLMF briefly. Specifically, DLGrLMF regards the task of drug-target interaction prediction as a weighted logistic matrix factorization problem, in which the experimentally validated interactions are allocated with larger weights. Meanwhile, by considering that drugs with similar chemical structure should have interactions with similar targets and targets with similar genomic sequence similarity should in turn have interactions with similar drugs, the drug pairwise chemical structure similarities as well as the target pairwise genomic sequence similarities are fully exploited to serve the matrix factorization problem by using a dual Laplacian graph regularization term. In addition, we design a gradient descent algorithm to solve the resultant optimization problem. Finally, the efficacy of DLGrLMF is validated on various benchmark datasets and the experimental results demonstrate that DLGrLMF performs better than other state-of-the-art methods. Case studies are also conducted to validate that DLGrLMF can successfully predict most of the experimental validated drug-target interactions.

An Anti-Clustering Model for Stability Enhancement of a 3D Moving Particle Semi-Implicit Method and Two-Phase Coupling between MPS and Euler Grids

As a Lagrangian gridless particle method, the MPS (Moving Particle Semi-implicit) method has a wide engineering application. However, for complex 3D flows, unphysical pressure oscillations often occur and result in the failure of simulations. This paper compares the stability enhancement methods proposed by different researchers to develop a 3D, stable MPS method. The results indicate that the proposed methods are incapable of eliminating the particle clustering that leads to instability as the main source in coarser particle spacing cases. An anti-clustering model, referring to the SPH (Smoothed Particle Hydrodynamics) artificial viscosity model, is proposed to further reduce instability. Combining various proposed methods and models, several typical examples are simulated comparatively. The results are compared with those of the VOF (Volume of Fluid) model using commercial software to validate the accuracy and stability of the combination of the proposed methods and models. It is concluded that (1) 3D cases that adopt a high-order Laplacian model and high-order source terms in PPE are more accurate than those adopting the low-order operators; (2) the proposed anti-clustering model can produce a tuned interparticle force to prevent particle clustering and introduce no additional viscosity effects in the flow of the normal state, which plays a very positive role for further stability enhancement of MPS; (3) particle resolution significantly maintains simulation accuracy given the stable algorithms by the combination of stability enhancement methods. The 3D MPS method is coupled with the Euler grid (FLUENT V17 software, ANSYS, Pittsburgh, PA, USA) in two phases. In particular, the 3D MPS algorithm is used to calculate the liquid-phase change from the continuous to the dispersed, and the finite volume method based on the Euler grid is adopted to measure the corresponding gas-phase motion. The atomization of the liquid jet under static air flow is calculated and compared with the results of the VOF method, which can capture the continuous interface.

De Novo Prediction of Drug-Target Interactions Using Laplacian Regularized Schatten p-Norm Minimization.

In pharmaceutical sciences, a crucial step of the drug discovery is the identification of drug-target interactions (DTIs). However, only a small portion of the DTIs have been experimentally validated. Moreover, it is an extremely laborious, expensive, and time-consuming procedure to capture new interactions between drugs and targets through traditional biochemical experiments. Therefore, designing computational methods for predicting potential interactions to guide the experimental verification is of practical significance, especially for de novo situation. In this article, we propose a new algorithm, namely Laplacian regularized Schatten p-norm minimization (LRSpNM), to predict potential target proteins for novel drugs and potential drugs for new targets where there are no known interactions. Specifically, we first take advantage of the drug and target similarity information to dynamically prefill the partial unknown interactions. Then based on the assumption that the interaction matrix is low-rank, we use Schatten p-norm minimization model combined with Laplacian regularization terms to improve prediction performance in the new drug/target cases. Finally, we numerically solve the LRSpNM model by an efficient alternating direction method of multipliers algorithm. We evaluate LRSpNM on five data sets and an extensive set of numerical experiments show that LRSpNM achieves better and more robust performance than five state-of-the-art DTIs prediction algorithms. In addition, we conduct two case studies for new drug and new target prediction, which illustrates that LRSpNM can successfully predict most of the experimental validated DTIs.

JCS: An Explainable Surface Defects Detection Method for Steel Sheet by Joint Classification and Segmentation

For surface defect images that captured from a practical steel production line, different shape, size, location and texture of defect object may cause inter-class similarity and intra-class difference of defect images. Despite attractive results have been achieved in some surface methods for defect classification and segmentation, it is still far from meeting the needs of real-world applications due to lack of adaptiveness of these methods. Considering the surface defect image can be decomposed into defect foreground image and defect-free background image, the paper develops a novel joint classification and segmentation (JCS) approach to perform surface defects detection for steel sheet. It comprises of the classification method based on a class-specific and shared discriminative dictionary learning (CASDDL) and the segmentation method based on a double low-rank based matrix decomposition (DLMD), respectively. For the proposed CASDDL method, we learn a shared sub-dictionary as well as several class-specific sub-dictionaries to explicitly capture common information shared by all classes and class-specific information belonging to corresponding class. We adopt a mutual incoherence constrain for each sub-dictionary, a Fisher-like discriminative criterion and low-rank constrain on coding vector to improve the discriminative ability of learned dictionary. For the proposed DLMD method, we formulate the segmentation task as a double low-rank based matrix factorization problem, and the Laplacian and sparse regularization terms are introduced into the matrix decomposition framework. Experimental results demonstrate that our proposed JCS method achieve a comparable or better performance than the state-of- the-art methods in classifying and segmenting surface defects of steel sheet.

An optimal chemoprophylaxis and treatment control for a spatiotemporal tuberculosis model

In this work, we propose a Susceptible-Exposed-Infected-Recovered (SEIR) spatiotemporal model that characterizes the dynamics of tuberculosis disease by taking into consideration the spatial heterogeneity; in order to provide a realistic description of this disease. Based on an existing model, we add the Laplacian term in each class to describe the spatial mobility of its individuals, which led us to a SEIR reaction-diffusion system. Then, controls with treatment and chemoprophylaxis are incorporated to reduce the latently infected (exposed) and actively infected individual populations to fight against the spread of the disease. Theoretically the existence, positivity and boundness of state systems have been proved, also the existence of controls has been shown, and a characterization of the controls in terms of the state functions and the adjoint functions has been provided. To illustrate the effectiveness of our theoretical results, we give numerical simulations for several scenarios. Our results indicate that the control effect is effective if both control strategies controls on treatment and chemoprophylaxis strategies are used simultaneously.

Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms

<p style='text-indent:20px;'>In this paper, we prove two new improved Sobolev inequalities involving weighted Morrey norms in <inline-formula><tex-math id="M1">\begin{document}$ {\dot{H}}^s( \mathbb{R}^{n})\times {\dot{H}}^s( \mathbb{R}^{n}) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ {D}^{1, p}( \mathbb{R}^{n})\times{D}^{1, p}( \mathbb{R}^{n}) $\end{document}</tex-math></inline-formula>. For instance, the corresponding inequality in <inline-formula><tex-math id="M3">\begin{document}$ {\dot{H}}^s( \mathbb{R}^{n})\!\times\! {\dot{H}}^s( \mathbb{R}^{n}) $\end{document}</tex-math></inline-formula> states that: there exists <inline-formula><tex-math id="M4">\begin{document}$ C\! = \!C(n, s, \alpha, \eta_1, \eta_2)\!&gt;\!0 $\end{document}</tex-math></inline-formula> such that for each <inline-formula><tex-math id="M5">\begin{document}$ (u, v) \!\in\! {\dot{H}}^s( \mathbb{R}^{n})\!\times\! {\dot{H}}^s( \mathbb{R}^{n}) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ p\!\in\![2, 2^*_{s}(\alpha)) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ \theta \!\in\! (\bar{\theta}, \frac{2\eta_1}{2^*_{s}(\alpha)}) $\end{document}</tex-math></inline-formula>, it holds that <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \Big( \int_{ \mathbb{R}^{n} } \frac{ |u|^{\eta_1} |v|^{\eta_2} } { |y|^{\alpha} } dy \Big)^{ \frac{1}{ 2^*_{s} (\alpha) }} \nonumber \\ \!\leq\! C ||u||_{{\dot{H}}^s(\mathbb{R}^{n})}^{\frac{\theta}{2}} ||v||_{{\dot{H}}^s(\mathbb{R}^{n})}^{\frac{\theta}{2}+\frac{\eta_2-\eta_1}{2^*_{s} (\alpha)}} ||(uv)||^{\frac{\eta_1}{2^*_{s} (\alpha)}-\frac{\theta}{2}}_{ L^{\frac{p}{2}, \frac{p}{2}(n-2s+r)}(\mathbb{R}^{n}, |y|^{-\frac{p}{2}r}) }, ~~~~(0)$ \end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M8">\begin{document}$ s \!\in\! (0, 1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ 0\!&lt;\!\alpha\!&lt;\!2s\!&lt;\!n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ \eta_1\!+\!\eta_2\! = \!2^*_{s}(\alpha)\!: = \!\frac{2(n-\alpha)}{n-2s} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M11">\begin{document}$ 1\!&lt;\!\eta_1\!\leq\!\eta_2\!&lt;\!\eta_1\!+\!\frac{\alpha}{s} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ \bar{\theta}\! = \!\max \Big\{ \frac{2}{2^*_{s}(\alpha)}, \frac{2\eta_1}{2^*_{s}(\alpha)} -\frac{2t(\frac{\alpha}{2s}-\frac{\alpha}{n})}{2^*_{s}(\alpha) -\frac{2\alpha}{n}}\Big\} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M13">\begin{document}$ t\! = \!1\!-\!\frac{(\eta_2-\eta_1)s}{\alpha} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M14">\begin{document}$ r\! = \!\frac{2\alpha}{ 2^*_{s}(\alpha) } $\end{document}</tex-math></inline-formula>. This inequality, together with its counterpart in <inline-formula><tex-math id="M15">\begin{document}$ {D}^{1, p}( \mathbb{R}^{n})\!\times\!{D}^{1, p}( \mathbb{R}^{n}) $\end{document}</tex-math></inline-formula> extend similar Sobolev inequality in <inline-formula><tex-math id="M16">\begin{document}$ {\dot{H}}^s( \mathbb{R}^{n}) $\end{document}</tex-math></inline-formula> as well as in <inline-formula><tex-math id="M17">\begin{document}$ {D}^{1, p}( \mathbb{R}^{n}) $\end{document}</tex-math></inline-formula> obtained by G. Palatucci and A. Pisante [Calc. Var., <b>50</b> (2014)] to the product spaces <inline-formula><tex-math id="M18">\begin{document}$ {\dot{H}}^s( \mathbb{R}^{n})\times {\dot{H}}^s( \mathbb{R}^{n}) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M19">\begin{document}$ {D}^{1, p}( \mathbb{R}^{n})\times{D}^{1, p}( \mathbb{R}^{n}) $\end{document}</tex-math></inline-formula>, respectively. <p style='text-indent:20px;'>With the help of the inequality (1), we succeed in obtaining some new existence results for doubly critical elliptic systems involving fractional Laplacian and Hardy terms.

Laplacian regularized low-rank sparse representation transfer learning

In unsupervised transfer learning, it is extremely valuable to effectively extract knowledge from the vast amount of untagged data that exists by utilizing tagged data from other similar databases. In general, the data in the real world often resides in the low-dimensional manifold embedded in the high-dimensional environment space. However, the current subspace transfer learning methods do not consider the nonlinear geometry structure inside the data, so the local similarity information between the data may be lost in the learning process. In order to improve this respect, we propose a new subspace transfer learning algorithm, namely Laplacian Regularized Low-Rank Sparse Representation Transfer Learning (LRLRSR-TL). After introducing the low-rank representation and sparse constraints, the method incorporates Laplacian regularization term to represent the global low-dimensional structure and capture the inherent nonlinear geometry information of the data. Experimental investigation conducted based on five different cross-domain visual image datasets shows that the proposed method has outstanding performance compared with several state-of-the-art transfer learning methods.

Learning representation from multiple media domains for enhanced event discovery

In this paper, we focus on event discovery by utilizing data distributed in multiple media domains, such as news media and social media. To this end, we propose an in-domain and cross-domain Laplacian regularization (ICLR) model, which can learn effective data representation for both textual news reports contributed by journalists in news media domain, and image posts shared by amateur users in social media domain. The achieved data representation can be used by classification and clustering strategies for existing and new event discovery, respectively. More specifically, ICLR constructs respective Laplacian regularization terms considering the property of inter-domain and intra-domain label consistency, which can be optimized by employing an alternating optimization strategy with theoretical guarantee for convergence. In particular, we collect and release a multi-domain and multimodal dataset for evaluations and public use.

Monitoring Lévy-process crossovers

The crossover among two or more types of diffusive processes represents a vibrant theme in nonequilibrium statistical physics. In this work we propose two models to generate crossovers among different Lévy processes: in the first model we change gradually the order of the derivative in the Laplacian term of the diffusion equation, whereas in the second one we consider a combination of fractional-derivative diffusive terms characterized by coefficients that change in time. The proposals are illustrated by considering semi-analytical (i.e., analytical together with numerical) procedures to follow the time-dependent solutions. We find changes between two different regimes and it is shown that, far from the crossover regime, both models yield qualitatively similar results, although these changes may occur in different forms for the two models. The models introduced herein are expected to be useful for describing crossovers among distinct diffusive regimes that occur frequently in complex systems.

A new look at the fractional Poisson problem via the logarithmic Laplacian

We analyze the s-dependence of solutions us to the family of fractional Poisson problems(−Δ)su=fin Ω,u≡0on RN∖Ω in an open bounded set Ω⊂RN, s∈(0,1). In the case where Ω is of class C2 and f∈Cα(Ω‾) for some α>0, we show that the map (0,1)→L∞(Ω), s↦us is of class C1, and we characterize the derivative ∂sus in terms of the logarithmic Laplacian of f. As a corollary, we derive pointwise monotonicity properties of the solution map s↦us under suitable assumptions on f and Ω. Moreover, we derive explicit bounds for the corresponding Green operator on arbitrary bounded domains which are new even for the case s=1, i.e., for the local Dirichlet problem −Δu=f in Ω, u≡0 on ∂Ω.

Diffusion NMR in periodic media: efficient computation and spectral properties

The Bloch–Torrey equation governs the evolution of the transverse magnetization in diffusion magnetic resonance imaging, where two mechanisms are at play: diffusion of spins (Laplacian term) and their precession in a magnetic field gradient (imaginary potential term). In this paper, we study this equation in a periodic medium: a unit cell repeated over the nodes of a lattice. Although the gradient term of the equation is not invariant by lattice translations, the equation can be analyzed within a single unit cell by replacing a continuous-time gradient profile by narrow pulses. In this approximation, the effects of precession and diffusion are separated and the problem is reduced to the study of a sequence of diffusion equations with pseudo-periodic boundary conditions. This representation allows for efficient numerical computations as well as new theoretical insights into the formation of the signal in periodic media. In particular, we study the eigenmodes and eigenvalues of the Bloch–Torrey operator. We show how the localization of eigenmodes is related to branching points in the spectrum and we discuss low- and high-gradient asymptotic behaviors. The range of validity of the approximation is discussed; interestingly the method turns out to be more accurate and efficient at high gradient, being thus an important complementary tool to conventional numerical methods that are most accurate at low gradients.

Numerical Modeling of Momentum Dispersion in Porous Media Based on the Pore Scale Prevalence Hypothesis

A macroscopic model that accounts for the effect of momentum dispersion on flows in porous media is proposed. The model is based on the pore scale prevalence hypothesis (PSPH). The effects of macroscopic velocity gradient on momentum transport are approximated using a Laplacian term. A local Reynolds number Red, which characterizes the strength of momentum dispersion, is introduced to calculate the effective viscosity. The characteristic length used in defining Red is the pore size, while the characteristic velocity is the mixing velocity. A Taylor expansion is made for the effective viscosity with respect to Red. The two leading-order terms of the Taylor series are adopted in the present PSPH momentum-dispersion model. The model constants are determined from the direct numerical simulation results of a flow in the same porous medium bounded by two walls. The effective viscosity approaches the molecular viscosity when the porosity is increased to 1. It approaches infinity when the porosity approaches 0. The benchmark studies show that the effects of the macroscopic velocity gradient can be approximated by the Laplacian term. The proposed PSPH momentum-dispersion model is highly accurate in a wide range of Reynolds and Darcy numbers as well as porosities.

Graph transform learning

Transform learning is a new representation learning framework where we learn an operator/transform that analyses the data to generate the coefficient/representation. We propose a variant of it called the graph transform learning; in this we explicitly account for the correlation in the dataset in terms of graph Laplacian. We will give two variants; in the first one the graph is computed from the data and fixed during the operation. In the second, the graph is learnt iteratively from the data during operation. The first technique will be applied for clustering, and the second one for solving inverse problems.