Laplacian Estimation Research Articles

Prototype Bayesian Meta-Learning for Few-Shot Image Classification.

Meta-learning aims to leverage prior knowledge from related tasks to enable a base learner to quickly adapt to new tasks with limited labeled samples. However, traditional meta-learning methods have limitations as they provide an optimal initialization for all new tasks, disregarding the inherent uncertainty induced by few-shot tasks and impeding task-specific self-adaptation initialization. In response to this challenge, this article proposes a novel probabilistic meta-learning approach called prototype Bayesian meta-learning (PBML). PBML focuses on meta-learning variational posteriors within a Bayesian framework, guided by prototype-conditioned prior information. Specifically, to capture model uncertainty, PBML treats both meta-and task-specific parameters as random variables and integrates their posterior estimates into hierarchical Bayesian modeling through variational inference (VI). During model inference, PBML employs Laplacian estimation to approximate the integral term over the likelihood loss, deriving a rigorous upper-bound for generalization errors. To enhance the model's expressiveness and enable task-specific adaptive initialization, PBML proposes a data-driven approach to model the task-specific variational posteriors. This is achieved by designing a generative model structure that incorporates prototype-conditioned task-dependent priors into the random generation of task-specific variational posteriors. Additionally, by performing latent embedding optimization, PBML decouples the gradient-based meta-learning from the high-dimensional variational parameter space. Experimental results on benchmark datasets for few-shot image classification illustrate that PBML attains state-of-the-art or competitive performance when compared to other related works. Versatility studies demonstrate the adaptability and applicability of PBML in addressing diverse and challenging few-shot tasks. Furthermore, ablation studies validate the performance gains attributed to the inference and model components.

Looking for optimal concentric ring electrodes: influence of design aspects on their performance

Concentric ring electrodes (CREs) allow improved spatial resolution, reduced crosstalk and interference, and increased bandwidth in the sensing of bioelectrical activity. A wide variety of designs have been used, but their selection is rarely well-founded. The aim of this work is to assess the implications of aspects of CRE design such as the distance between poles, their width and their maximum diameter on aspects such as the signal amplitude (and, therefore, quality), Laplacian estimation error and spatial selectivity (SS). For this purpose, a finite dimensional model of the CRE was used, and its response to the activity of an electric dipole of variable depth was simulated via finite element method modeling. Our results show that increasing the electrode size increases the error to a greater extent than the signal amplitude increases. Pole widths should be as small as possible. The middle ring of the tripolar CRE should be as far away as possible from the central disc. Tripolar CREs typically outperform bipolar CREs of the same outer diameter, significantly reducing the Laplacian estimation error and improving the SS at the cost of a small decrease in signal amplitude. Our results also show that the design of current commercial versions of CREs can be optimized. Furthermore, we propose a methodology that facilitates the selection of an adequate CRE configuration based on the specifications for CRE performance and practical aspects, such as the depth of activity sources to be recorded from and/or the maximum size of electrodes to be used. The monitoring and analysis of bioelectrical signals in a wide range of applications can benefit from the enhanced electrode design and methodology proposed in this work.

Model-based Estimators of QT Series Time Delay in Following Heart-Rate Changes.

Sudden cardiac death is the leading cause of death among cardiovascular diseases. Markers for patient risk stratification focusing on QT-interval dynamics in response to heart-rate (HR) changes can be characterized in terms of parametric QT to RR dependence and QT/RR hysteresis. The QT/RR hysteresis can be quantified by the time delay the QT interval takes to accommodate for the HR changes. The exercise stress test has been proposed as a proper test, with large HR dynamics, to evaluate the QT/RR hysteresis. The present study aims at evaluating several time-delay estimators based on noise statistic (Gaussian or Laplacian) and HR changes profile at stress test (gradual transition change). The estimator's performance was assessed on a simulated QT transition contaminated by noise and in a clinical study including patients affected by coronary arteries disease (CAD). As expected, the Laplacian and Gaussian estimators yield the best results when noise follows the respective distribution. Further, the Laplacian estimator showed greater discriminative power in classifying different levels of cardiac risk in CAD patients, suggesting that real data fit better the Laplacian distribution than the Gaussian one. The Laplacian estimator appears to be the choice for time-delay estimation of QT/RR hysteresis lag in response to HR changes in stress test.Clinical Relevance-The proposed time-delay estimator of QT/RR hysteresis lag improves its significance as biomarkers for coronary artery diseases risk stratification.

Two-dimensional phase unwrapping based on Fourier transforms and the Yukawa potential spectrum.

The two-dimensional phase unwrapping problem (PHUP) has been solved with discrete Fourier transforms (FTs) and many other techniques traditionally. Nevertheless, a formal way of solving the continuous Poisson equation for the PHUP, with the use of continuous FT and based on distribution theory, has not been reported yet, to our knowledge. The well-known specific solution of this equation is given in general by a convolution of a continuous Laplacian estimate with a particular Green function, whose FT does not exist mathematically. However, an alternative Green function called the Yukawa potential, with a guaranteed Fourier spectrum, can be considered for solving an approximated Poisson equation, inducing a standard procedure of a FT-based unwrapping algorithm. Thus, the general steps for this approach are described in this work by considering some reconstructions with synthetic and real data.

Graph learning from band-limited data by graph Fourier transform analysis

A graph provides an effective means to represent the statistical dependence or similarity among signals observed at different vertices. A critical challenge is to excavate graphs underlying observed signals, because of non-convex problem structure and associated high computational requirements. This paper presents a new graph learning technique that is able to efficiently infer the graph structure underlying observed graph signals. The key idea is that we reveal the intrinsic relation between the frequency-domain representation of general band-limited graph signals, and the graph Fourier transform (GFT) basis. Accordingly, we derive a new closed-form analytic expression for the GFT basis, which depends deterministically on the observed signals (as opposed to being solved numerically and approximately in the literature). Given the GFT basis, the estimation of the graph Laplacian, more explicitly, its eigenvalues, is convex and efficiently solved using the alternating direction method of multipliers (ADMM). Simulations based on synthetic data and experiments based on public weather and brain signal datasets show that the new technique outperforms the state of the art in accuracy and efficiency.

Distributed Graph Estimation under Laplacian Constraints

In recent years, distributed estimation of graph Laplacian matrices for smooth graph signals has received much attention. Traditional methods for estimating the graph Laplacian matrices usually estimate the global parameters in a centralized manner, which is computationally intensive and hard to apply in large-scale networks. In this paper, in order to reduce the computational complexity and meanwhile maintain the estimation accuracy, we propose a distributed graph Laplacian matrix estimation method called the distributed combinatorial graph Laplacian estimation (DCGL). In our method, a local parameter estimation problem is first formulated for each vertex by maximizing the marginal likelihood of the data collected from the neighborhood of the vertex. Then, by discussing the connectivity and Laplacian property of the marginal precision matrix, the Laplacian and structural constraints are added to each local estimation problem to resolve the non-convexity between the local and global estimates. Finally, through a simple and single message-passing rule, the global graph Laplacian matrix is obtained by extracting, combining, and symmetrizing the locally estimated parameters. Experiments on synthetic and real datasets demonstrate that the proposed distributed estimator is asymptotically consistent in the classical regime while having advantages in the high-dimensional regime.

Comprehensive Optimization of the Tripolar Concentric Ring Electrode Based on Its Finite Dimensions Model and Confirmed by Finite Element Method Modeling

The optimization performed in this study is based on the finite dimensions model of the concentric ring electrode as opposed to the negligible dimensions model used in the past. This makes the optimization problem comprehensive, as all of the electrode parameters including, for the first time, the radius of the central disc and individual widths of concentric rings, are optimized simultaneously. The optimization criterion used is maximizing the accuracy of the surface Laplacian estimation, as the ability to estimate the Laplacian at each electrode constitutes primary biomedical significance of concentric ring electrodes. For tripolar concentric ring electrodes, the optimal configuration was compared to previously proposed linearly increasing inter-ring distances and constant inter-ring distances configurations of the same size and based on the same finite dimensions model. The obtained analytic results suggest that previously proposed configurations correspond to almost two-fold and more than three-fold increases in the Laplacian estimation error compared with the optimal configuration proposed in this study, respectively. These analytic results are confirmed using finite element method modeling, which was adapted to the finite dimensions model of the concentric ring electrode for the first time. Moreover, the finite element method modeling results suggest that optimal electrode configuration may also offer improved sensitivity and spatial resolution.

Fast two-dimensional nuclear magnetic resonance resolution enhancement by use of a Laplacian estimator

A simple and fast data processing algorithm for the resolution enhancement of two dimensional (2 D) nuclear magnetic resonance spectra is proposed. It is based on a modified 2 D Laplacian operator and can be used as a standalone algorithm or embedded within peak picking routines. It can also be combined with other processing techniques such as linear prediction filling or covariance.

Validating the Comparison Framework for the Finite Dimensions Model of Concentric Ring Electrodes Using Human Electrocardiogram Data

While progress has been made in design optimization of concentric ring electrodes maximizing the accuracy of the surface Laplacian estimation, it was based exclusively on the negligible dimensions model of the electrode. Recent proof of concept of the new finite dimensions model that adds the radius of the central disc and the widths of concentric rings to the previously included number of rings and inter-ring distances provides an opportunity for more comprehensive design optimization. In this study, the aforementioned proof of concept was developed into a framework allowing direct comparison of any two concentric ring electrodes of the same size and with the same number of rings. The proposed framework is illustrated on constant and linearly increasing inter-ring distances tripolar concentric ring electrode configurations and validated on electrocardiograms from 20 human volunteers. In particular, ratios of truncation term coefficients between the two electrode configurations were used to demonstrate the similarity between the negligible and the finite dimension models analytically (p = 0.077). Laplacian estimates based on the two models were calculated on electrocardiogram data for emulation of linearly increasing inter-ring distances tripolar concentric ring electrode. The difference between the estimates was not statistically significant (p >> 0.05) which is consistent with the analytic result.

Evaluation of Bipolar, Tripolar, and Quadripolar Laplacian Estimates of Electrocardiogram via Concentric Ring Electrodes.

Surface Laplacian estimates via concentric ring electrodes (CREs) have proven to enhance spatial resolution compared to conventional disc electrodes, which is of great importance for P-wave analysis. In this study, Laplacian estimates for traditional bipolar configuration (BC), two tripolar configurations with linearly decreasing and increasing inter-ring distances (TCLDIRD and TCLIIRD, respectively), and quadripolar configuration (QC) were obtained from cardiac recordings with pentapolar CREs placed at CMV1 and CMV2 positions. Normalized P-wave amplitude (NAP) was computed to assess the contrast to study atrial activity. Signals were of good quality (20–30 dB). Atrial activity was more emphasized at CMV1 (NAP ≃ 0.19–0.24) compared to CMV2 (NAP ≃ 0.08–0.10). Enhanced spatial resolution of TCLIIRD and QC resulted in higher NAP values than BC and TCLDIRD. Comparison with simultaneous standard 12-lead ECG proved that Laplacian estimates at CMV1 outperformed all the limb and chest standard leads in the contrast to study P-waves. Clinical recordings with CRE at this position could allow more detailed observation of atrial activity and facilitate the diagnosis of associated pathologies. Furthermore, such recordings would not require additional electrodes on limbs and could be performed wirelessly, so it should also be suitable for ambulatory monitoring, for example, using cardiac Holter monitors.

Optimization Algorithms for Graph Laplacian Estimation via ADMM and MM

In this paper, we study the graph Laplacian estimation problem under a given connectivity topology. We aim at enriching the unified graph learning framework proposed by Egilmez et al. and improve the optimality performance of the combinatorial graph Laplacian (CGL) case. We apply the well-known alternating direction method of multipliers (ADMM) and majorization–minimization (MM) algorithmic frameworks and propose two algorithms, namely, GLE-ADMM and GLE-MM, for graph Laplacian estimation. Both algorithms can achieve an optimality gap as low as $10^{-4}$ , around three orders of magnitude more accurate than the benchmark. In addition, we find that GLE-ADMM is more computationally efficient in a dense topology (e.g., an almost complete graph), while GLE-MM is more suitable for sparse graphs (e.g., trees). Furthermore, we consider exploiting the leading eigenvectors of the sample covariance matrix as a nominal eigensubspace and propose a third algorithm, named GLENE, which is also based on ADMM. Numerical experiments show that the inclusion of a nominal eigensubspace significantly improves the estimation of the graph Laplacian, which is more evident when the sample size is smaller than or comparable to the problem dimension.

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.

Solving the general inter-ring distances optimization problem for concentric ring electrodes to improve Laplacian estimation

BackgroundSuperiority of noninvasive tripolar concentric ring electrodes over conventional disc electrodes in accuracy of surface Laplacian estimation has been demonstrated in a range of electrophysiological measurement applications. Recently, a general approach to Laplacian estimation for an (n + 1)-polar electrode with n rings using the (4n + 1)-point method has been proposed and used to introduce novel multipolar and variable inter-ring distances electrode configurations. While only linearly increasing and linearly decreasing inter-ring distances have been considered previously, this paper defines and solves the general inter-ring distances optimization problem for the (4n + 1)-point method.ResultsGeneral inter-ring distances optimization problem is solved for tripolar (n = 2) and quadripolar (n = 3) concentric ring electrode configurations through minimizing the truncation error of Laplacian estimation. For tripolar configuration with middle ring radius αr and outer ring radius r the optimal range of values for α was determined to be 0 < α ≤ 0.22 while for quadripolar configuration with an additional middle ring with radius βr the optimal range of values for α and β was determined by inequalities 0 < α < β < 1 and αβ ≤ 0.21. Finite element method modeling and full factorial analysis of variance were used to confirm statistical significance of Laplacian estimation accuracy improvement due to optimization of inter-ring distances (p < 0.0001).ConclusionsObtained results suggest the potential of using optimization of inter-ring distances to improve the accuracy of surface Laplacian estimation via concentric ring electrodes. Identical approach can be applied to solving corresponding inter-ring distances optimization problems for electrode configurations with higher numbers of concentric rings. Solutions of the proposed inter-ring distances optimization problem define the class of the optimized inter-ring distances electrode designs. These designs may result in improved noninvasive sensors for measurement systems that use concentric ring electrodes to acquire electrical signals such as from the brain, intestines, heart or uterus for diagnostic purposes.

&lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \mathcal{W}$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds

In this paper, we prove Perelman type $\mathcal{W}$-entropy formulae and global differential Harnack estimates for positive solutions to porous medium equation on the closed Riemannian manifolds with Ricci curvature bounded below. As applications, we derive Harnack inequalities and Laplacian estimates.

Learning Graphs With Monotone Topology Properties and Multiple Connected Components

Recent papers have formulated the problem of learning graphs from data as an inverse covariance estimation with graph Laplacian constraints. While such problems are convex, existing methods cannot guarantee that solutions will have specific graph topology properties (e.g., being $k$-partite), which are desirable for some applications. In fact, the problem of learning a graph with given topology properties, e.g., finding the $k$-partite graph that best matches the data, is in general non-convex. In this paper, we develop novel theoretical results that provide performance guarantees for an approach to solve these problems. Our solution decomposes this problem into two sub-problems, for which efficient solutions are known. Specifically, a graph topology inference (GTI) step is employed to select a feasible graph topology, i.e., one having the desired property. Then, a graph weight estimation (GWE) step is performed by solving a generalized graph Laplacian estimation problem, where edges are constrained by the topology found in the GTI step. Our main result is a bound on the error of the GWE step as a function of the error in the GTI step. This error bound indicates that the GTI step should be solved using an algorithm that approximates the similarity matrix by another matrix whose entries have been thresholded to zero to have the desired type of graph topology. The GTI stage can leverage existing methods (e.g., state of the art approaches for graph coloring) which are typically based on minimizing the total weight of removed edges. Since the GWE stage is formulated as an inverse covariance estimation problem with linear constraints, it can be solved using existing convex optimization methods. We demonstrate that our two step approach can achieve good results for both synthetic and texture image data.

Proof of concept Laplacian estimate derived for noninvasive tripolar concentric ring electrode with incorporated radius of the central disc and the widths of the concentric rings.

Tripolar concentric ring electrodes are showing great promise in a range of applications including braincomputer interface and seizure onset detection due to their superiority to conventional disc electrodes, in particular, in accuracy of surface Laplacian estimation. Recently, we proposed a general approach to estimation of the Laplacian for an (n + 1)-polar electrode with n rings using the (4n + 1)-point method for n ≥ 2 that allows cancellation of all the truncation terms up to the order of 2n. This approach has been used to introduce novel multipolar and variable inter-ring distances concentric ring electrode configurations verified using finite element method. The obtained results suggest their potential to improve Laplacian estimation compared to currently used constant interring distances tripolar concentric ring electrodes. One of the main limitations of the proposed (4n + 1)-point method is that the radius of the central disc and the widths of the concentric rings are not included and therefore cannot be optimized. This study incorporates these two parameters by representing the central disc and both concentric rings as clusters of points with specific radius and widths respectively as opposed to the currently used single point and concentric circles. A proof of concept Laplacian estimate is derived for a tripolar concentric ring electrode with non-negligible radius of the central disc and non-negligible widths of the concentric rings clearly demonstrating how both of these parameters can be incorporated into the (4n + 1)-point method.

Analysis of variance to assess statistical significance of Laplacian estimation accuracy improvement due to novel variable inter-ring distances concentric ring electrodes.

Concentric ring electrodes have shown promise in non-invasive electrophysiological measurement demonstrating their superiority to conventional disc electrodes, in particular, in accuracy of Laplacian estimation. Recently, we have proposed novel variable inter-ring distances concentric ring electrodes. Analytic and finite element method modeling results for linearly increasing distances electrode configurations suggested they may decrease the truncation error resulting in more accurate Laplacian estimates compared to currently used constant inter-ring distances configurations. This study assesses statistical significance of Laplacian estimation accuracy improvement due to novel variable inter-ring distances concentric ring electrodes. Full factorial design of analysis of variance was used with one categorical and two numerical factors: the inter-ring distances, the electrode diameter, and the number of concentric rings in the electrode. The response variables were the Relative Error and the Maximum Error of Laplacian estimation computed using a finite element method model for each of the combinations of levels of three factors. Effects of the main factors and their interactions on Relative Error and Maximum Error were assessed and the obtained results suggest that all three factors have statistically significant effects in the model confirming the potential of using inter-ring distances as a means of improving accuracy of Laplacian estimation.

Finite element method modeling to assess Laplacian estimates via novel variable inter-ring distances concentric ring electrodes.

Noninvasive concentric ring electrodes are a promising alternative to conventional disc electrodes. Currently, superiority of tripolar concentric ring electrodes over disc electrodes, in particular, in accuracy of Laplacian estimation has been demonstrated in a range of applications. In our recent work we have shown that accuracy of Laplacian estimation can be improved with multipolar concentric ring electrodes using a general approach to estimation of the Laplacian for an (n + 1)-polar electrode with n rings using the (4n + 1)-point method for n ≥ 2. This paper takes the next step toward further improving the Laplacian estimate by proposing novel variable inter-ring distances concentric ring electrodes. Derived using a modified (4n + 1)-point method, linearly increasing and decreasing inter-ring distances tripolar (n = 2) and quadripolar (n = 3) electrode configurations are compared to their constant inter-ring distances counterparts using finite element method modeling. Obtained results suggest that increasing inter-ring distances electrode configurations may decrease the estimation error resulting in more accurate Laplacian estimates compared to respective constant inter-ring distances configurations. For currently used tripolar electrode configuration the estimation error may be decreased more than two-fold while for the quadripolar configuration more than six-fold decrease is expected.

Analytic assessment of Laplacian estimates via novel variable interring distances concentric ring electrodes.

Noninvasive concentric ring electrodes are a promising alternative to conventional disc electrodes. Currently, superiority of tripolar concentric ring electrodes over disc electrodes, in particular, in accuracy of Laplacian estimation has been demonstrated in a range of applications. In our recent work we have shown that accuracy of Laplacian estimation can be improved with multipolar concentric ring electrodes using a general approach to estimation of the Laplacian for an (n + 1)-polar electrode with n rings using the (4n + 1)-point method for n ≥ 2. This paper takes the next step toward further improving the Laplacian estimate by proposing novel variable inter-ring distances concentric ring electrodes. Derived using a modified (4n + 1)-point method, linearly increasing inter-ring distances tripolar (n = 2) and quadripolar (n = 3) electrode configurations are analytically compared to their constant inter-ring distances counterparts using coefficients of the Taylor series truncation terms. Obtained results suggest that increasing inter-ring distances electrode configurations may decrease the truncation error of the Laplacian estimation resulting in more accurate Laplacian estimates compared to respective constant inter-ring distances configurations. For currently used tripolar electrode configuration the truncation error may be decreased more than two-fold while for the quadripolar more than seven-fold decrease is expected.

Improving the Accuracy of Laplacian Estimation with Novel Variable Inter-Ring Distances Concentric Ring Electrodes.

Noninvasive concentric ring electrodes are a promising alternative to conventional disc electrodes. Currently, the superiority of tripolar concentric ring electrodes over disc electrodes, in particular, in accuracy of Laplacian estimation, has been demonstrated in a range of applications. In our recent work, we have shown that accuracy of Laplacian estimation can be improved with multipolar concentric ring electrodes using a general approach to estimation of the Laplacian for an (n + 1)-polar electrode with n rings using the (4n + 1)-point method for n ≥ 2. This paper takes the next step toward further improving the Laplacian estimate by proposing novel variable inter-ring distances concentric ring electrodes. Derived using a modified (4n + 1)-point method, linearly increasing and decreasing inter-ring distances tripolar (n = 2) and quadripolar (n = 3) electrode configurations are compared to their constant inter-ring distances counterparts. Finite element method modeling and analytic results are consistent and suggest that increasing inter-ring distances electrode configurations may decrease the truncation error resulting in more accurate Laplacian estimates compared to respective constant inter-ring distances configurations. For currently used tripolar electrode configuration, the truncation error may be decreased more than two-fold, while for the quadripolar configuration more than a six-fold decrease is expected.