Diffusion Kernel Research Articles

Accelerated front propagation for monostable equations with nonlocal diffusion: multidimensional case

We describe acceleration of the front propagation for solutions to a class of monostable nonlinear equations with a nonlocal diffusion in $${{\mathbb {R}}^d}$$, $$d\ge 1$$. We show that the acceleration takes place if either the diffusion kernel or the initial condition has ‘regular’ heavy tails in $${{\mathbb {R}}^d}$$ (in particular, decays slower than exponentially). Under general assumptions which can be verified for particular models, we present sharp estimates for the time-space zone which separates the region of convergence to the unstable zero solution with the region of convergence to the stable positive constant solution. We show the variety of different possible rates of the propagation starting from a little bit faster than a linear one up to the exponential rate. The paper generalizes to the case $$d>1$$ our results for the case $$d=1$$ obtained early in Finkelshtein and Tkachov (Appl Anal 98(4):756–780, 2019).

TOPDRIVER: the novel identifier of cancer driver genes in Gastric cancer and Melanoma

Nowadays, research has found a strong relationship between genomic status and occurrence of disease. Cancer is one of the most common diseases that leads to a high annual mortality rate worldwide, and the disease’s genetic content remains challenging. Detecting driver genes of different cancers could help in early diagnosis and treatment. In this paper, we proposed TOPDRIVER, a network-based algorithm, to detect cancer driver genes in cancers. An initial network was constructed by integrating four different omic datasets: HPRD, NCBI, KEGG, and GTEx. This integration created a gene similarity profile that provided a comprehensive perspective of gene interaction in each subtype of cancer and allocated weights to the edges of the network. The vertex scores were calculated using a gene-disease association dataset (DisGeNet) and a molecular functional disease similarity. In this step, the genes network was jagged and faced with a zero-one gap problem. A diffusion kernel was implemented to smooth the vertex scores to overcome this problem. Finally, potential driver genes were extracted according to the topology of the network, genes overall biological functions, and their involvement in cancer pathways. TOPDRIVER has been applied to two subtypes of gastric cancer and one subtype of melanoma. The method could nominate a considerable number of well-known driver genes of these cancers and also introduce novel driver genes. NKX3-1, KIDINS220, and RIPK4 have introduced for gastrointestinal cancer, UBA3, UBE2M, and RRAGA for hereditary gastric cancer and CIT for invasive melanoma. Biological evidences represents TOPDRIVER’s efficiency in a subtype-specific manner.

Diffusion-based kernel matrix model for face liveness detection

Face recognition and verification systems are vulnerable to video spoofing attacks. In this paper, we present a diffusion-based kernel matrix model for face liveness detection. We use the anisotropic diffusion to enhance the edges of each frame in a video, and the kernel matrix model to extract the video features which we call the diffusion kernel (DK) features. The DK features reflect the inner correlation of the face images in the video. We employ a generalized multiple kernel learning method to fuse the DK features and the deep features extracted from convolution neural networks to achieve better performance. Our experimental evaluation on two publicly available datasets shows that the proposed method outperforms the state-of-art face liveness detection methods.

Network Infusion to Infer Information Sources in Networks

Several significant models have been developed that enable the study of diffusion of signals across biological, social and engineered networks. Within these established frameworks, the inverse problem of identifying the source of the propagated signal is challenging, owing to the numerous alternative possibilities for signal progression through the network. In real world networks, the challenge of determining sources is compounded as the true propagation dynamics are typically unknown, and when they have been directly measured, they rarely conform to the assumptions of any of the well-studied models. In this paper we introduce a method called Network Infusion (NI) that has been designed to circumvent these issues, making source inference practical for large, complex real world networks. The key idea is that to infer the source node in the network, full characterization of diffusion dynamics, in many cases, may not be necessary. This objective is achieved by creating a diffusion kernel that well-approximates standard diffusion models such as the susceptible-infected diffusion model, but lends itself to inversion, by design, via likelihood maximization or error minimization. We apply NI for both single-source and multi-source diffusion, for both single-snapshot and multi-snapshot observations, and for both homogeneous and heterogeneous diffusion setups. We prove the mean-field optimality of NI for different scenarios, and demonstrate its effectiveness over several synthetic networks. Moreover, we apply NI to a real-data application, identifying news sources in the Digg social network, and demonstrate the effectiveness of NI compared to existing methods. Finally, we propose an integrative source inference framework that combines NI with a distance centrality-based method, which leads to a robust performance in cases where the underlying dynamics are unknown.

Watermarking in dot-diffusion halftones using adaptive class-matrix and error diffusion

Digital halftoning deals with transforming a gray or color image into its binary version which is useful in printing applications. Dot diffusion is one of the prominent halftone methods which can yield superior image quality with parallel processing capabilities. In this paper, a rapid watermarking algorithm is proposed for dot-diffusion halftone images using adaptive class-matrix selection and modified error diffusion kernels. To process the image using an adaptive class matrix, the processing order of the class matrix is reversed and transposed, and for error diffusion the coefficients are replaced with different weights. For decoding, an effective strategy is proposed based on a correlation analysis and halftone statistics. The proposed strategy can successfully embed and decode the binary watermark from a single dot-diffused halftone image. From the experimental results, the proposed method is found to be effective in terms of good decoding accuracy, imperceptibility and robustness against various printed distortions.

Transport dynamics of complex fluids

Thermal motion in complex fluids is a complicated stochastic process but ubiquitously exhibits initial ballistic, intermediate subdiffusive, and long-time diffusive motion, unless interrupted. Despite its relevance to numerous dynamical processes of interest in modern science, a unified, quantitative understanding of thermal motion in complex fluids remains a challenging problem. Here, we present a transport equation and its solutions, which yield a unified quantitative explanation of the mean-square displacement (MSD), the non-Gaussian parameter (NGP), and the displacement distribution of complex fluids. In our approach, the environment-coupled diffusion kernel and its time correlation function (TCF) are the essential quantities that determine transport dynamics and characterize mobility fluctuation of complex fluids; their time profiles are directly extractable from a model-free analysis of the MSD and NGP or, with greater computational expense, from the two-point and four-point velocity autocorrelation functions. We construct a general, explicit model of the diffusion kernel, comprising one unbound-mode and multiple bound-mode components, which provides an excellent approximate description of transport dynamics of various complex fluidic systems such as supercooled water, colloidal beads diffusing on lipid tubes, and dense hard disk fluid. We also introduce the concepts of intrinsic disorder and extrinsic disorder that have distinct effects on transport dynamics and different dependencies on temperature and density. This work presents an unexplored direction for quantitative understanding of transport and transport-coupled processes in complex disordered media.

Graph Learning From Filtered Signals: Graph System and Diffusion Kernel Identification

This paper introduces a novel graph signal processing framework for building graph-based models from classes of filtered signals. In our framework, graph-based modeling is formulated as a graph system identification problem, where the goal is to learn a weighted graph (a graph Laplacian matrix) and a graph-based filter (a function of graph Laplacian matrices). In order to solve the proposed problem, an algorithm is developed to jointly identify a graph and a graph-based filter (GBF) from multiple signal/data observations. Our algorithm is valid under the assumption that GBFs are one-to-one functions. The proposed approach can be applied to learn diffusion (heat) kernels, which are popular in various fields for modeling diffusion processes. In addition, for specific choices of graph-based filters, the proposed problem reduces to a graph Laplacian estimation problem. Our experimental results demonstrate that the proposed algorithm outperforms the current state-of-the-art methods. We also implement our framework on a real climate dataset for modeling of temperature signals.

A Novel Method for Identifying the Potential Cancer Driver Genes Based on Molecular Data Integration.

The identification of the cancer driver genes is essential for personalized therapy. The mutation frequency of most driver genes is in the middle (2-20%) or even lower range, which makes it difficult to find the driver genes with low-frequency mutations. Other forms of genomic aberrations, such as copy number variations (CNVs) and epigenetic changes, may also reflect cancer progression. In this work, a method for identifying the potential cancer driver genes (iPDG) based on molecular data integration is proposed. DNA copy number variation, somatic mutation, and gene expression data of matched cancer samples are integrated. In combination with the method of iKEEG, the "key genes" of cancer are identified, and the change in their expression levels is used for auxiliary evaluation of whether the mutated genes are potential drivers. For a mutated gene, the concept of mutational effect is defined, which takes into account the effects of copy number variation, mutation gene itself, and its neighbor genes. The method mainly includes two steps: the first step is data preprocessing. First, DNA copy number variation and somatic mutation data are integrated. Then, the integrated data are mapped to a given interaction network, and the diffusion kernel is used to form the mutation effect matrix. The second step is to obtain the key genes by using the iKGGE method, and construct the connection matrix by means of the gene expression data of the key genes and mutation impact matrix of the mutated genes. Experiments on TCGA breast cancer and Glioblastoma multiforme datasets demonstrate that iPDG is effective not only to identify the known cancer driver genes but also to discover the rare potential driver genes. When measured by functional enrichment analysis, we find that these genes are clearly associated with these two types of cancers.

Fourier Transform on the Homogeneous Space of 3D Positions and Orientations for Exact Solutions to Linear PDEs.

Fokker–Planck PDEs (including diffusions) for stable Lévy processes (including Wiener processes) on the joint space of positions and orientations play a major role in mechanics, robotics, image analysis, directional statistics and probability theory. Exact analytic designs and solutions are known in the 2D case, where they have been obtained using Fourier transform on . Here, we extend these approaches to 3D using Fourier transform on the Lie group of rigid body motions. More precisely, we define the homogeneous space of 3D positions and orientations as the quotient in . In our construction, two group elements are equivalent if they are equal up to a rotation around the reference axis. On this quotient, we design a specific Fourier transform. We apply this Fourier transform to derive new exact solutions to Fokker–Planck PDEs of -stable Lévy processes on . This reduces classical analysis computations and provides an explicit algebraic spectral decomposition of the solutions. We compare the exact probability kernel for (the diffusion kernel) to the kernel for (the Poisson kernel). We set up stochastic differential equations (SDEs) for the Lévy processes on the quotient and derive corresponding Monte-Carlo methods. We verified that the exact probability kernels arise as the limit of the Monte-Carlo approximations.

A longitudinal model for tau aggregation in Alzheimer's disease based on structural connectivity.

Tau tangles are a pathological hallmark of Alzheimer?s disease (AD) with strong correlations existing between tau aggregation and cognitive decline. Studies in mouse models have shown that the characteristic patterns of tau spatial spread associated with AD progression are determined by neural connectivity rather than physical proximity between different brain regions. We present here a network diffusion model for tau aggregation based on longitudinal tau measures from positron emission tomography (PET) and structural connectivity graphs from diffusion tensor imaging (DTI). White matter fiber bundles reconstructed via tractography from the DTI data were used to compute normalized graph Laplacians which served as graph diffusion kernels for tau spread. By linearizing this model and using sparse source localization, we were able to identify distinct patterns of propagative and generative buildup of tau at a population level. A gradient descent approach was used to solve the sparsity-constrained optimization problem. Model fitting was performed on subjects from the Harvard Aging Brain Study cohort. The fitted model parameters include a scalar factor controlling the network-based tau spread and a network-independent seed vector representing seeding in different regions-of-interest. This parametric model was validated on an independent group of subjects from the same cohort. We were able to predict with reasonably high accuracy the tau buildup at a future time-point. The network diffusion model, therefore, successfully identifies two distinct mechanisms for tau buildup in the aging brain and offers a macroscopic perspective on tau spread.

A Cross-Domain Recommender System With Kernel-Induced Knowledge Transfer for Overlapping Entities.

The aim of recommender systems is to automatically identify user preferences within collected data, then use those preferences to make recommendations that help with decisions. However, recommender systems suffer from data sparsity problem, which is particularly prevalent in newly launched systems that have not yet had enough time to amass sufficient data. As a solution, cross-domain recommender systems transfer knowledge from a source domain with relatively rich data to assist recommendations in the target domain. These systems usually assume that the entities either fully overlap or do not overlap at all. In practice, it is more common for the entities in the two domains to partially overlap. Moreover, overlapping entities may have different expressions in each domain. Neglecting these two issues reduces prediction accuracy of cross-domain recommender systems in the target domain. To fully exploit partially overlapping entities and improve the accuracy of predictions, this paper presents a cross-domain recommender system based on kernel-induced knowledge transfer, called KerKT. Domain adaptation is used to adjust the feature spaces of overlapping entities, while diffusion kernel completion is used to correlate the non-overlapping entities between the two domains. With this approach, knowledge is effectively transferred through the overlapping entities, thus alleviating data sparsity issues. Experiments conducted on four data sets, each with three sparsity ratios, show that KerKT has 1.13%-20% better prediction accuracy compared with six benchmarks. In addition, the results indicate that transferring knowledge from the source domain to the target domain is both possible and beneficial with even small overlaps.

Practical shape analysis and segmentation methods for point cloud models

Current point cloud processing algorithms do not have the capability to automatically extract semantic information from the observed scenes, except in very specialized cases. Furthermore, existing mesh analysis paradigms cannot be directly employed to automatically perform typical shape analysis tasks directly on point cloud models.We present a potent framework for shape analysis, similarity and segmentation of noisy point cloud models for real objects of engineering interest, models that may be incomplete. The proposed framework relies on spectral methods and the heat diffusion kernel to construct compact shape signatures, and we show that the framework supports a variety of clustering techniques that have traditionally been applied only on mesh models. We developed and implemented one practical and convergent estimate of the Laplace–Beltrami operator for point clouds as well as a number of clustering techniques adapted to work directly on point clouds to produce geometric features of engineering interest. The key advantage of this framework is that it supports practical shape analysis capabilities that operate directly on point cloud models of objects without requiring surface reconstruction or global meshing. We show that the proposed technique is robust against typical noise present in possibly incomplete point clouds, and segment point clouds scanned by depth cameras (e.g. Kinect) into semantically-meaningful sub-shapes.

Resting state dynamics meets anatomical structure: Temporal multiple kernel learning (tMKL) model

Over the last decade there has been growing interest in understanding the brain activity, in the absence of any task or stimulus, captured by the resting-state functional magnetic resonance imaging (rsfMRI). The resting state patterns have been observed to be exhibiting complex spatio-temporal dynamics and substantial effort has been made to characterize the dynamic functional connectivity (dFC) configurations. However, the dynamics governing the state transitions that the brain undergoes and their relationship to stationary functional connectivity still remains an open problem.One class of approaches attempts to characterize the dynamics in terms of finite number of latent brain states, however, such attempts are yet to amalgamate the underlying anatomical structural connectivity (SC) with the dynamics. Another class of methods links individual dynamic FCs with the underlying SC but does not characterize the temporal evolution of FC. Further, the latent states discovered by previous approaches could not be directly linked to the SC, thereby motivating us to discover the underlying lower-dimensional manifold that represents the temporal structure. In the proposed approach, the discovered manifold is further parameterized as a set of local density distributions, or latent transient states. We propose an innovative method that learns parameters specific to the latent states using a graph-theoretic model (temporal Multiple Kernel Learning, tMKL) that inherently links dynamics to the structure and finally predicts the grand average FC of the test subjects by leveraging a state transition Markov model.The proposed solution does not make strong assumptions about the underlying data and is generally applicable to resting or task data for learning subject-specific state transitions and for successfully characterizing SC-dFC-FC relationship through a unifying framework. Training and testing were done using the rs-fMRI data of 46 healthy participants. tMKL model performs significantly better than the existing models for predicting resting state functional connectivity based on whole-brain dynamic mean-field model (DMF), single diffusion kernel (SDK) model and multiple kernel learning (MKL) model. Further, the learned model was tested on an independent cohort of 100 young, healthy participants from the Human Connectome Project (HCP) and the results establish the generalizability of the proposed solution. More importantly, the model retains sensitivity toward subject-specific anatomy, a unique contribution towards a holistic approach for SC-FC characterization.

A generalized thermoelastic diffusion problem with memory-dependent derivative

Based on the generalized thermoelastic diffusion theory with memory-dependent derivative in both the generalized heat conduction law and the generalized diffusion law, the transient response is investigated of a half-space subjected to a thermal shock and a chemical potential shock on its bounding surface. The coupled governing equations containing time-delay factors and kernel functions, which can be chosen freely according to specific problems, are solved by the Laplace transform together with its numerical inversion. The non-dimensional temperature, chemical potential, displacement, stress, as well as concentration at different values of time, time-delay factors, and kernel functions are obtained and illustrated graphically. The results show that: all the considered variables have obvious changes with the passage of time; the thermal time-delay and kernel functions hardly influence the distributions of the non-dimensional chemical potential, modestly influence the distributions of the non-dimensional concentration, while appreciably influence the distributions of the non-dimensional temperature, displacement, and stress; the diffusion time-delay and kernel functions barely affect the distributions of the non-dimensional temperature, slightly affect the distributions of the non-dimensional concentration, while remarkably affect the distributions of the non-dimensional chemical potential, displacement and stress.

Diffusion representations

Diffusion Maps framework is a kernel based method for manifold learning and data analysis that defines diffusion similarities by imposing a Markovian process on the given dataset. Analysis by this process uncovers the intrinsic geometric structures in the data. Recently, it was suggested to replace the standard kernel by a measure-based kernel that incorporates information about the density of the data. Thus, the manifold assumption is replaced by a more general measure-based assumption. The measure-based diffusion kernel incorporates two separate independent representations. The first determines a measure that correlates with a density that represents normal behaviors and patterns in the data. The second consists of the analyzed multidimensional data points. In this paper, we present a representation framework for data analysis of datasets that is based on a closed-form decomposition of the measure-based kernel. The proposed representation preserves pairwise diffusion distances that does not depend on the data size while being invariant to scale. For a stationary data, no out-of-sample extension is needed for embedding newly arrived data points in the representation space. Several aspects of the presented methodology are demonstrated on analytically generated data.

Koopman analysis of the long-term evolution in a turbulent convection cell

We analyse the long-time evolution of the three-dimensional flow in a closed cubic turbulent Rayleigh–Bénard convection cell via a Koopman eigenfunction analysis. A data-driven basis derived from diffusion kernels known in machine learning is employed here to represent a regularized generator of the unitary Koopman group in the sense of a Galerkin approximation. The resulting Koopman eigenfunctions can be grouped into subsets in accordance with the discrete symmetries in a cubic box. In particular, a projection of the velocity field onto the first group of eigenfunctions reveals the four stable large-scale circulation (LSC) states in the convection cell. We recapture the preferential circulation rolls in diagonal corners and the short-term switching through roll states parallel to the side faces which have also been seen in other simulations and experiments. The diagonal macroscopic flow states can last as long as 1000 convective free-fall time units. In addition, we find that specific pairs of Koopman eigenfunctions in the secondary subset obey enhanced oscillatory fluctuations for particular stable diagonal states of the LSC. The corresponding velocity-field structures, such as corner vortices and swirls in the midplane, are also discussed via spatiotemporal reconstructions.

Symmetric and antisymmetric properties of solutions to kernel-based machine learning problems

Abstract A particularly interesting instance of supervised learning with kernels is when each training example is associated with two objects, as in pairwise classification (Brunner et al., 2012), and in supervised learning of preference relations (Herbrich et al., 1998). In these cases, one may want to embed additional prior knowledge into the optimization problem associated with the training of the learning machine, modeled, respectively, by the symmetry of its optimal solution with respect to an exchange of order between the two objects, and by its antisymmetry. Extending the approach proposed in Brunner et al. (2012) (where the only symmetric case was considered), we show, focusing on support vector binary classification, how such embedding is possible through the choice of a suitable pairwise kernel, which takes as inputs the individual feature vectors and also the group feature vectors associated with the two objects. We also prove that the symmetry/antisymmetry constraints still hold when considering the sequence of suboptimal solutions generated by one version of the Sequential Minimal Optimization (SMO) algorithm, and we present numerical results supporting the theoretical findings. We conclude discussing extensions of the main results to support vector regression, to transductive support vector machines, and to several kinds of graph kernels, including diffusion kernels.

Multiple Kernel Learning Model for Relating Structural and Functional Connectivity in the Brain

A challenging problem in cognitive neuroscience is to relate the structural connectivity (SC) to the functional connectivity (FC) to better understand how large-scale network dynamics underlying human cognition emerges from the relatively fixed SC architecture. Recent modeling attempts point to the possibility of a single diffusion kernel giving a good estimate of the FC. We highlight the shortcomings of the single-diffusion-kernel model (SDK) and propose a multi-scale diffusion scheme. Our multi-scale model is formulated as a reaction-diffusion system giving rise to spatio-temporal patterns on a fixed topology. We hypothesize the presence of inter-regional co-activations (latent parameters) that combine diffusion kernels at multiple scales to characterize how FC could arise from SC. We formulated a multiple kernel learning (MKL) scheme to estimate the latent parameters from training data. Our model is analytically tractable and complex enough to capture the details of the underlying biological phenomena. The parameters learned by the MKL model lead to highly accurate predictions of subject-specific FCs from test datasets at a rate of 71%, surpassing the performance of the existing linear and non-linear models. We provide an example of how these latent parameters could be used to characterize age-specific reorganization in the brain structure and function.

Global stability and pattern formation in a nonlocal diffusive Lotka–Volterra competition model

A diffusive Lotka–Volterra competition model with nonlocal intraspecific and interspecific competition between species is formulated and analyzed. The nonlocal competition strength is assumed to be determined by a diffusion kernel function to model the movement pattern of the biological species. It is shown that when there is no nonlocal intraspecific competition, the dynamics properties of nonlocal diffusive competition problem are similar to those of classical diffusive Lotka–Volterra competition model regardless of the strength of nonlocal interspecific competition. Global stability of nonnegative constant equilibria are proved using Lyapunov or upper–lower solution methods. On the other hand, strong nonlocal intraspecific competition increases the system spatiotemporal dynamic complexity. For the weak competition case, the nonlocal diffusive competition model may possess nonconstant positive equilibria for some suitably large nonlocal intraspecific competition coefficients.

Study on monostable and bistable reaction–diffusion equations by iteration of travelling wave maps

In this paper, based on the iterative properties of travelling wave maps, we develop a new method to obtain spreading speeds and asymptotic propagation for monostable and bistable reaction–diffusion equations. Precisely, for Dirichlet problems of monostable reaction–diffusion equations on the half line, by making links between travelling wave maps and integral operators associated with the Dirichlet diffusion kernel (the latter is NOT invariant under translation), we obtain some iteration properties of the Dirichlet diffusion and some a priori estimates on nontrivial solutions of Dirichlet problems under travelling wave transformation. We then provide the asymptotic behavior of nontrivial solutions in the space–time region for Dirichlet problems. These enable us to develop a unified method to obtain results on heterogeneous steady states, travelling waves, spreading speeds, and asymptotic spreading behavior for Dirichlet problem of monostable reaction–diffusion equations on R+ as well as of monostable/bistable reaction–diffusion equations on R.