Kernel Target Alignment Research Articles

Toward effective SVM sample reduction based on fuzzy membership functions

Support vector machine (SVM) is known for its good generalization performance and wide application in various fields. Despite its success, the learning efficiency of SVM decreases significantly originating from the assumption that the number of training samples increases rapidly. Consequently, the traditional SVM with standard optimization methods faces challenges such as excessive memory requirements and slow training speed, especially for large-scale training sets. To address this issue, this paper draws inspiration from the fuzzy support vector machine (FSVM). Considering that each sample has varying contributions to the decision plane, we propose an effective SVM sample reduction method based on the fuzzy membership function (FMF). This method uses FMF to calculate the fuzzy membership of each training sample. Training samples with low fuzzy memberships are then deleted. Specifically, we propose SVM sample reduction algorithms based on class center distance, kernel target alignment, centered kernel alignment, slack factor, entropy, and bilateral weighted FMF, respectively. Comprehensive experiments on UCI and KEEL datasets demonstrate that our proposed algorithms outperform other comparative methods in terms of accuracy, F-measure, and hinge-loss measures.

Hybrid genetic optimization for quantum feature map design

Kernel methods are an import class of techniques in machine learning. To be effective, good feature maps are crucial for mapping non-linearly separable input data into a higher dimensional (feature) space, thus allowing the data to be linearly separable in feature space. Previous work has shown that quantum feature map design can be automated for a given dataset using NSGA-II, a genetic algorithm, while both minimizing circuit size and maximizing classification accuracy. However, the evaluation of the accuracy achieved by a candidate feature map is costly. In this work, we demonstrate the suitability of kernel-target alignment as a substitute for accuracy in genetic algorithm-based quantum feature map design. Kernel-target alignment is faster to evaluate than accuracy and does not require some data points to be reserved for its evaluation. To further accelerate the evaluation of genetic fitness, we provide a method to approximate kernel-target alignment. To improve kernel-target alignment and root mean squared error, the final trainable parameters of the generated circuits are further trained using COBYLA to determine whether a hybrid approach applying conventional circuit parameter training can easily complement the genetic structure optimization approach. A total of eight new approaches are compared to the original across nine varied binary classification problems from the UCI machine learning repository, showing that kernel-target alignment and its approximation produce feature map circuits enabling comparable accuracy to the previous work but with larger margins on training data (in excess of 20% larger) that improve further with circuit parameter training.

Gaussian RBF Centered Kernel Alignment (CKA) in the Large-Bandwidth Limit.

Centered kernel alignment (CKA), also known as centered kernel-target alignment, is useful as a similarity measure between kernels and as a kernel-based similarity measure between feature representations. We prove that CKA based on a Gaussian RBF kernel converges to linear CKA in the large-bandwidth limit. The result relies on mean-centering of the feature maps and on a Hilbert-Schmidt Independence Criterion (HSIC) identity. We show that convergence onset is sensitive to the geometry of the feature representations, and that a notion of representation eccentricity, ρ, constrains the bandwidth range for which Gaussian CKA can differ noticeably from linear CKA. Our experimental results suggest that Gaussian bandwidths less than ρ should be selected in order to enable nonlinear modeling.

Training quantum embedding kernels on near-term quantum computers

Kernel methods are a cornerstone of classical machine learning. The idea of using quantum computers to compute kernels has recently attracted attention. Quantum embedding kernels (QEKs) constructed by embedding data into the Hilbert space of a quantum computer are a particular quantum kernel technique that allows to gather insights into learning problems and that are particularly suitable for noisy intermediate-scale quantum devices. In this work, we first provide an accessible introduction to quantum embedding kernels and then analyze the practical issues arising when realizing them on a noisy near-term quantum computer. We focus on quantum embedding kernels with variational parameters. These variational parameters are optimized for a given dataset by increasing the kernel-target alignment, a heuristic connected to the achievable classification accuracy. We further show under which conditions noise from device imperfections influences the predicted kernel and provide a strategy to mitigate these detrimental effects which is tailored to quantum embedding kernels. We also address the influence of finite sampling and derive bounds that put guarantees on the quality of the kernel matrix. We illustrate our findings by numerical experiments and tests on actual hardware.

GA-MKB:A Multi-kernel Boosting Learning Method based on Normalized Kernel Target Alignment and Kernel Difference

Concentrates on the problem that the traditional kernel target alignment(KTA) is not invariance under data translation in the feature space, a cosine matrix alignment method is proposed for kernel selection, which is called normalized kernel target alignment(NKTA). On the basis of normalized kernel target alignment and kernel difference, we propose a new multi-kernel boosting. Firstly, the value of NKTA is taken as the election rarget of the kernel function in each iteration of algorithm, which leads to a selective kernel fusion. Secondly, the kernel difference measure is used to construct the combination coefficient to increase the diversity of weak classifiers, and then improve the generalization performance of integrated strong classifiers. Finally, among the 6 data sets, the GA-MKB performed better than MKBoost-D1 under the accuracy of classification, and can improve the generalization performance of the integrated classifier compared with MKBoost-D2.

Kernel-Target Alignment Based Fuzzy Lagrangian Twin Bounded Support Vector Machine

To improve the generalization performance, we develop a new technique for handling the impacts of outliers using Lagrangian twin bounded SVM (TBSVM) with kernel fuzzy membership values, which is termed kernel-target alignment-based fuzzy Lagrangian twin bounded support vector machine (KTA-FLTBSVM). Here, the objective functions are having L2-norm vectors of the slack variable that leads to the optimization problem more convex and yields a unique global solution. Also, the fuzzy membership values are employing the importance of data samples assigned to each sample to minimize the impacts of outlier and noise. Further, we have suggested a linearly convergent iterative approach to obtain the solution of the problem unlike in place to solve the quadratic programming problem in Twin SVM (TSVM) and TBSVM. To investigate the effectiveness of the proposed KTA-FLTBSVM, the comprehensive experiments demonstrate with other reported models on artificial datasets along with benchmark real-life publicly available datasets. Our KTA-FLTBSVM outperforms to other models in terms of better classification accuracy.

Cross-Gram matrices and their use in transfer learning: Application to automatic REM detection using heart rate

Background and objectives: while traditional sleep staging is achieved through the visual - expert-based - annotation of a polysomnography, it has the disadvantages of being unpractical and expensive. Alternatives have been developed over the years to relieve sleep staging from its heavy requirements, through the collection of more easily assessable signals and its automation using machine learning. However, these alternatives have their limitations, some due to variabilities among and between subjects, other inherent to their use of sub-discriminative signals.Many new solutions rely on the evaluation of the Autonomic Nervous System (ANS) activation through the assessment of the heart-rate (HR); the latter is modulated by the aforementioned variabilities, which may result in data and concept shifts between what was learned and what we want to classify. Such adversary effects are usually tackled by Transfer Learning, dealing with problems where there are differences between what is known (source) and what we want to classify (target). In this paper, we propose two new kernel-based methods of transfer learning and assess their performances in Rapid-Eye-Movement (REM) sleep stage detection, using solely the heart rate.Methods: our first contribution is the introduction of Kernel-Cross Alignment (KCA), a measure of similarity between a source and a target, which is a direct extension of Kernel-Target Alignment (KTA). To our knowledge, KCA has currently never been studied in the literature.Our second contribution is two alignment-based methods of transfer learning: Kernel-Target Alignment Transfer Learning (KTATL) and Kernel-Cross Alignment Transfer Learning (KCATL). Both methods differ from KTA, whose traditional use is kernel-tuning: in our methods, the kernel has been fixed beforehand, and our objective is the improvement of the estimation of unknown target labels by taking into account how observations relate to each other, which, as it will be explained, allows to transfer knowledge (transfer learning).Results: we compare performances with transfer learning (KCATL, KTATL) to performances without transfer using a fixed classifier (a Support Vector Classifier - SVC). In most cases, both transfer learning methods result in an improvement of performances (higher detection rates for a fixed false-alarm rate). Our methods do not require iterative computations.Conclusion: we observe improved performances using our transfer methods, which are computationally efficient, as they only require the computation of a kernel matrix and are non-iterative. However, some optimisation aspects are still under investigation.

An Accurate Tool for Uncovering Cancer Subtypes by Fast Kernel Learning Method to Integrate Multiple Profile Data.

In recent years, cancer has become a severe threat to human health. If we can accurately identify the subtypes of cancer, it will be of great significance to the research of anti-cancer drugs, the development of personalized treatment methods, and finally conquer cancer. In this paper, we obtain three feature representation datasets (gene expression profile, isoform expression and DNA methylation data) on lung cancer and renal cancer from the Broad GDAC, which collects the standardized data extracted from The Cancer Genome Atlas (TCGA). Since the feature dimension is too large, Principal Component Analysis (PCA) is used to reduce the feature vector, thus eliminating the redundant features and speeding up the operation speed of the classification model. By multiple kernel learning (MKL), we use Kernel target alignment (KTA), fast kernel learning (FKL), Hilbert-Schmidt Independence Criterion (HSIC), Mean to calculate the weight of kernel fusion. Finally, we put the combined kernel function into the support vector machine (SVM) and get excellent results. Among them, in the classification of renal cell carcinoma subtypes, the maximum accuracy can reach 0.978 by using the method of MKL (HSIC calculation weight), while in the classification of lung cancer subtypes, the accuracy can even reach 0.990 with the same method (FKL calculation weight).

An Optimal Channel Selection for EEG-Based Depression Detection via Kernel-Target Alignment.

Depression is a mental disorder with emotional and cognitive dysfunction. The main clinical characteristic of depression is significant and persistent low mood. As reported, depression is a leading cause of disability worldwide. Moreover, the rate of recognition and treatment for depression is low. Therefore, the detection and treatment of depression are urgent. Multichannel electroencephalogram (EEG) signals, which reflect the working status of the human brain, can be used to develop an objective and promising tool for augmenting the clinical effects in the diagnosis and detection of depression. However, when a large number of EEG channels are acquired, the information redundancy and computational complexity of the EEG signals increase; thus, effective channel selection algorithms are required not only for machine learning feasibility, but also for practicality in clinical depression detection. Consequently, we propose an optimal channel selection method for EEG-based depression detection via kernel-target alignment (KTA) to effectively resolve the abovementioned issues. In this method, we consider a modified version KTA that can measure the similarity between the kernel matrix for channel selection and the target matrix as an objective function and optimize the objective function by a proposed optimal channel selection strategy. Experimental results on two EEG datasets show that channel selection can effectively increase the classification performance and that even if we rely only on a small subset of channels, the results are still acceptable. The selected channels are in line with the expected latent cortical activity patterns in depression detection. Moreover, the experimental results demonstrate that our method outperforms the state-of-the-art channel selection approaches.

Kernel-target alignment based non-linear metric learning

Distance metric learning aims to learn a measure of the pairwise distance between data instances, which is essential for various machine learning algorithms and applications. However, existing linear metric learning methods based on linear transformations fail to capture nonlinear relationships between instances, while most existing kernel-based metric learning algorithms may ignore the correlation between the kernel function and the target learning task during model selection, which results in a suboptimal selection of kernels. To address the aforementioned issues, we propose a method named Kernel Alignment based Metric Learning with Random Fourier Approximation (KAMLRFA). Specifically, on one hand, we utilize the random Fourier features to approximate the shift-invariant kernel function for distance metric learning. On the other hand, we attempt to maximize the degree of agreement between the kernel function and the target learning task. In this way, distance metric learning is performed in a discriminative feature space. Compared with those kernel metric learning algorithms whose kernel functions fall into a suboptimal, KAMLRFA achieves the optimal kernel function for the target task and improves classification accuracy. We develop an efficient solution to solve the proposed optimization problem. Extensive experiments are conducted on several benchmark datasets including NUS-WIDE-LITE, USPS, MNIST, LETTER and 10 UCI datasets to verify the effectiveness of KAMLRFA compared with state-of-the-art methods.

A Novel Triple Matrix Factorization Method for Detecting Drug-Side Effect Association Based on Kernel Target Alignment.

All drugs usually have side effects, which endanger the health of patients. To identify potential side effects of drugs, biological and pharmacological experiments are done but are expensive and time-consuming. So, computation-based methods have been developed to accurately and quickly predict side effects. To predict potential associations between drugs and side effects, we propose a novel method called the Triple Matrix Factorization- (TMF-) based model. TMF is built by the biprojection matrix and latent feature of kernels, which is based on Low Rank Approximation (LRA). LRA could construct a lower rank matrix to approximate the original matrix, which not only retains the characteristics of the original matrix but also reduces the storage space and computational complexity of the data. To fuse multivariate information, multiple kernel matrices are constructed and integrated via Kernel Target Alignment-based Multiple Kernel Learning (KTA-MKL) in drug and side effect space, respectively. Compared with other methods, our model achieves better performance on three benchmark datasets. The values of the Area Under the Precision-Recall curve (AUPR) are 0.677, 0.685, and 0.680 on three datasets, respectively.

Centered kernel alignment inspired fuzzy support vector machine

Support vector machine (SVM) is a theoretically well motivated algorithm developed from statistical learning theory which has shown impressive performance in many fields. In spite of its success, it still suffers from the noise sensitivity problem originating from the assumption that each training point has equal importance or weight in the training process. To relax this problem, the SVM was extended to the fuzzy SVM (FSVM) by applying a fuzzy membership to each training point such that different training points can make different contributions to the learning of the decision surface. Although well-determined fuzzy memberships can improve classification performance, there are no general guidelines for their construction. In this paper, inspired by the centered kernel alignment (CKA), which measures the degree of similarity between two kernels (or kernel matrices), we propose a new fuzzy membership function calculation method in which a heuristic function derived from the CKA is used to calculate the dependence between a data point and its associated label. Although the CKA induced FSVM is similar to the kernel target alignment (KTA) induced FSVM, there is actually a critical difference. Without that centering, the definition of alignment does not correlate well with the performance of learning machines. Extensive experiments are performed on real-world data sets from the UCI benchmark repository and the application domain of computational biology which validate the superiority of the proposed FSVM model in terms of several classification performance measures.

LPI-KTASLP: Prediction of LncRNA-Protein Interaction by Semi-Supervised Link Learning With Multivariate Information

Long non-coding RNA, also known as lncRNA, is a series of single-stranded polynucleotides (no less than 200 nucleotides each), consisting of non-protein coding transcripts. LncRNA plays a crucial role in regulating gene expression, during the transcriptional, post-transcriptional, and epigenetic processes. This is achieved by lncRNA interacts with the corresponding RNA-binding proteins. It has been drawn to a lot of attention that the reduction of the excessive laboratory cost and the increase in speed and accuracy gains benefits from the employment of computational intelligence in lncRNA-protein interaction (LPI) identification. Although numerous pertinent in silico studies of LPI prediction have been proposed, there is still room for enhancing the accuracy of the existing LPI prediction methods. In this paper, we have proposed a novel method for identifying LPI with kernel target alignment based on semi-supervised link prediction (LPI-KTASLP), which adopts multivariate information to predict lncRNAs-proteins interactions. To integrate the heterogeneous kernels, kernel target alignment has been applied to deal with kernel fusion. We have calculated the low-rank approximation matrices of lncRNA and protein, where eigendecomposition is used to reduce computing pressure. The prediction model has been obtained by producing the ultimate LPI prediction matrix. Experimental results show that the prediction ability of the LPI-KTASLP algorithm has surpassed many other LPI prediction schemes. Our method of lncRNA-protein interaction prediction has been evaluated on a standard benchmark dataset of LPIs. We have observed that the highest AUPR of 0.6148 is obtained by our proposed model (LPI-KTASLP). This is superior to the integrated LPLNP (AUPR: 0.4584), the RWR (AUPR: 0.2827), the CF (AUPR: 0.2357), the LPIHN (AUPR: 0.2299), and the LPBNI (AUPR: 0.3302). It is very encouraging that most of the LPI predictions have been confirmed to be close to relevant concentrations.

Independence test and canonical correlation analysis based on the alignment between kernel matrices for multivariate functional data

In the case of vector data, Gretton et al. (Algorithmic learning theory. Springer, Berlin, pp 63–77, 2005) defined Hilbert–Schmidt independence criterion, and next Cortes et al. (J Mach Learn Res 13:795–828, 2012) introduced concept of the centered kernel target alignment (KTA). In this paper we generalize these measures of dependence to the case of multivariate functional data. In addition, based on these measures between two kernel matrices (we use the Gaussian kernel), we constructed independence test and nonlinear canonical variables for multivariate functional data. We show that it is enough to work only on the coefficients of a series expansion of the underlying processes. In order to provide a comprehensive comparison, we conducted a set of experiments, testing effectiveness on two real examples and artificial data. Our experiments show that using functional variants of the proposed measures, we obtain much better results in recognizing nonlinear dependence.

Learning Match Kernels on Grassmann Manifolds for Action Recognition.

Action recognition has been extensively researched in computer vision due to its potential applications in a broad range of areas. The key to action recognition lies in modeling actions and measuring their similarity, which however poses great challenges. In this paper, we propose learning match kernels between actions on Grassmann manifold for action recognition. Specifically, we propose modeling actions as a linear subspace on the Grassmann manifold; the subspace is a set of convolutional neural network (CNN) feature vectors pooled temporally over frames in semantic video clips, which simultaneously captures local discriminant patterns and temporal dynamics of motion. To measure the similarity between actions, we propose Grassmann match kernels (GMK) based on canonical correlations of linear subspaces to directly match videos for action recognition; GMK is learned in a supervised way via kernel target alignment, which is endowed with a great discriminative ability to distinguish actions from different classes. The proposed approach leverages the strengths of CNNs for feature extraction and kernels for measuring similarity, which accomplishes a general learning framework of match kernels for action recognition. We have conducted extensive experiments on five challenging realistic data sets including Youtube, UCF50, UCF101, Penn action, and HMDB51. The proposed approach achieves high performance and substantially surpasses the state-of-the-art algorithms by large margins, which demonstrates the great effectiveness of proposed approach for action recognition.

An algorithm of improving speech emotional perception for hearing aid

In this paper, a speech emotion recognition (SER) algorithm was proposed to improve the emotional perception of hearing-impaired people. The algorithm utilizes multiple kernel technology to overcome the drawback of SVM: slow training speed. Firstly, in order to improve the adaptive performance of Gaussian Radial Basis Function (RBF), the parameter determining the nonlinear mapping was optimized on the basis of Kernel target alignment. Then, the obtained Kernel Function was used as the basis kernel of Multiple Kernel Learning (MKL) with slack variable that could solve the over-fitting problem. However, the slack variable also brings the error into the result. Therefore, a soft-margin MKL was proposed to balance the margin against the error. Moreover, the relatively iterative algorithm was used to solve the combination coefficients and hyper-plane equations. Experimental results show that the proposed algorithm can acquire an accuracy of 90% for five kinds of emotions including happiness, sadness, anger, fear and neutral. Compared with KPCA[Formula: see text]+[Formula: see text]CCA and PIM-FSVM, the proposed algorithm has the highest accuracy.

PD46-01 CAN SURGICAL MENTOR'S TRUST HOLD THE KEY TO TRAINEE PERFORMANCE?

You have accessJournal of UrologySurgical Technology & Simulation: Training & Skills Assessment III1 Apr 2017PD46-01 CAN SURGICAL MENTOR'S TRUST HOLD THE KEY TO TRAINEE PERFORMANCE? Somayeh Shafiei, Ahmed Hussein, Justen Kozlowski, Youssef Ahmed, Sarah Muldoon, and Khurshid Guru Somayeh ShafieiSomayeh Shafiei More articles by this author , Ahmed HusseinAhmed Hussein More articles by this author , Justen KozlowskiJusten Kozlowski More articles by this author , Youssef AhmedYoussef Ahmed More articles by this author , Sarah MuldoonSarah Muldoon More articles by this author , and Khurshid GuruKhurshid Guru More articles by this author View All Author Informationhttps://doi.org/10.1016/j.juro.2017.02.2373AboutPDF ToolsAdd to favoritesDownload CitationsTrack CitationsPermissionsReprints ShareFacebookTwitterLinked InEmail INTRODUCTION AND OBJECTIVES Although robot-assisted surgery (RAS) provides many advantages in terms of ergonomics, visualization, and perioperative outcomes, it adds complexity to the surgical environment, due to its remoteness. Cognition-based trust is related to performance-relevant understandings such as competence, responsibility, reliability, integrity, and dependability. We sought to develop an objective method for evaluation of cognition-based trust during RAS. METHODS We examined EEG data from a mentor who observed 116 Urethro-vesical anastomoses (UVAs) and 98 pelvic lymph node dissections (PLNDs) performed by 3 trainees. The mentor assessed trainee performance using NASA-TLX questionnaires at the end of each step. Procedures were classified as Trustworthy (mentor satisfied) or Concerning (mentor not satisfied) based on the performance score given by the mentor. We tested 68 features extracted from EEG data and applied Kernel Target Alignment (KTA) method to find the most discriminative features. The outcome of the classification was evaluated using the accuracy, sensitivity and specificity of these objective features in their ability to distinguish between trustworthy and concerning procedures. RESULTS Of all features tested, we found that the five most predictive features were Stress, mental workload (MW), Frustration, Surprise and Modularity. These features were significantly different between Trustworthy and Concerning performances, showing higher frustration, stress, MW, surprise and lower modularity while mentoring concerning as opposed to trustworthy performances. CONCLUSIONS Cognition-based Trust can be objectively evaluated using EEG features. This is the first reported study to objectively evaluate trust during RAS by featuring cognitive and brain functioning features. © 2017FiguresReferencesRelatedDetails Volume 197Issue 4SApril 2017Page: e889-e890 Advertisement Copyright & Permissions© 2017MetricsAuthor Information Somayeh Shafiei More articles by this author Ahmed Hussein More articles by this author Justen Kozlowski More articles by this author Youssef Ahmed More articles by this author Sarah Muldoon More articles by this author Khurshid Guru More articles by this author Expand All Advertisement Advertisement PDF downloadLoading ...

A Relevance Vector Machine Prediction Method Based on the Biased Wavelet Kernel Function

Relevance Vector Machine (RVM) is an important learning method in the field of machine learning for its sparsity, global optimality and the ability to solve nonlinear problems by using kernel functions. In this paper, a family of biased wavelets was used to construct the kernel functions of RVM. Biased wavelet have adjustable nonzero mean which makes the kernel of RVM more flexible. With the kernel method of the Centered Kernel Target Alignment (CKTA), the biased parameter was selected to improve the prediction performance of RVM model. The algorithm based on the biased wavelet kernel showed an increased prediction accuracy compared to using wavelet kernel and Cauchy kernel. In short, Relevance Vector Machine with the biased wavelet kernel is a flexible prediction algorithm with high prediction accuracy.

Kernel Target Alignment Parameter: A New Modelability Measure for Regression Tasks.

In this paper, we demonstrate that the kernel target alignment (KTA) parameter can efficiently be used to estimate the relevance of molecular descriptors for QSAR modeling on a given data set, i.e., as a modelability measure. The efficiency of KTA to assess modelability was demonstrated in two series of QSAR modeling studies, either varying different descriptor spaces for one same data set, or comparing various data sets within one same descriptor space. Considered data sets included 25 series of various GPCR binders with ChEMBL-reported pKi values, and a toxicity data set. Employed descriptor spaces covered more than 100 different ISIDA fragment descriptor types, and ChemAxon BCUT terms. Model performances (RMSE) were seen to anticorrelate consistently with the KTA parameter. Two other modelability measures were employed for benchmarking purposes: the Jaccard distance average over the data set (Div), and a measure related to the normalized mean absolute error (MAE) obtained in 1-nearest neighbors calculations on the training set (Sim = 1 - MAE). It has been demonstrated that both Div and Sim perform similarly to KTA. However, a consensus index combining KTA, Div and Sim provides a more robust correlation with RMSE than any of the individual modelability measures.

A Study on Multi-Scale Kernel Optimisation via Centered Kernel-Target Alignment

Kernel mapping is one of the most widespread approaches to intrinsically deriving nonlinear classifiers. With the aim of better suiting a given dataset, different kernels have been proposed and different bounds and methodologies have been studied to optimise them. We focus on the optimisation of a multi-scale kernel, where a different width is chosen for each feature. This idea has been barely studied in the literature, although it has been shown to achieve better performance in the presence of heterogeneous attributes. The large number of parameters in multi-scale kernels makes it computationally unaffordable to optimise them by applying traditional cross-validation. Instead, an analytical measure known as centered kernel-target alignment (CKTA) can be used to align the kernel to the so-called ideal kernel matrix. This paper analyses and compares this and other alternatives, providing a review of the literature in kernel optimisation and some insights into the usefulness of multi-scale kernel optimisation via CKTA. When applied to the binary support vector machine paradigm (SVM), the results using 24 datasets show that CKTA with a multi-scale kernel leads to the construction of a well-defined feature space and simpler SVM models, provides an implicit filtering of non-informative features and achieves robust and comparable performance to other methods even when using random initialisations. Finally, we derive some considerations about when a multi-scale approach could be, in general, useful and propose a distance-based initialisation technique for the gradient-ascent method, which shows promising results.