Maxime Peralta

Pierre Jannin

John S H Baxter
email: jbaxter@univ-rennes1.fr

Machine learning in deep brain stimulation: a systematic review 

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Deep Brain Stimulation (DBS) is a neurosurgical procedure, introduced in 1987 by Pr.

Benabid [START_REF] Benabid | Combined (thalamotomy and stimulation) stereotactic surgery of the vim thalamic nucleus for bilateral parkinson disease[END_REF], in which electrodes are implanted into deep regions of the brain to correct for abnormal neural behavior. Continuous stimulation of these regions typically greatly enhances the quality of life of the patient by reducing the severity their symptoms. Parkinson's Disease (PD), Essential Tremor (ET), dystonia, Tourette's syndrome or Obsessive Compulsive Disorders (OCD) are all among the pathologies now commonly treated using DBS, and the most commonly targeted structures are the bilateral Ventral Intermediate nucleus of the thalamus (VIM), Globus Pallidus internus (GPi) and Subthalamic Nucleus (STN).

DBS interventions have a complex clinical workflow, involving several steps (presented in

Figure 1) that involve many challenges, both clinical challenges and challenges for the use of computer assistance. A large amount of expertise and domain knowledge is crucial for the success of this procedure, and computer-assisted tools, referred to as Computational Decision Support System (CDSS), have been designed to support clinicians in neurosurgery since the 1980s [START_REF] Reed T Sutton | An overview of clinical decision support systems: benefits, risks, and strategies for success[END_REF]. CDSS's are commonly divided in two categories: knowledge-based and non-knowledge-based. In knowledge-based CDSS's, the tool's intelligence is explicitly integrated into the system by a human, often the system programmer. Decision rules are explicitly programmed according to medical domain knowledge such as guidelines and definitions [START_REF] Sim | Clinical decision support systems for the practice of evidence-based medicine[END_REF], and the purpose of the CDSS is solely to retrieve the data, to evaluate the rule and to display the result with an User Interface (UI).

In non-knowledge-based CDSS's, machine learning replaces expert medical knowledge in order to address new challenges and to reach new levels of performance: the intelligence is implicitly generated by learning from a database using Machine Learning (ML) tools designed by data scientists, without requiring external domain knowledge. The term 'machine learning' appeared for the first time in 1959 in the works of Samuel et al. [START_REF] Arthur | Some studies in machine learning using the game of checkers[END_REF]. It is a branch of Artificial Intelligence (AI) which consists in constructing an algorithm that learns how to perform a task using a database of experiences without requiring any explicit programming of the user or knowledge of the task. On its most common form, called supervised learning, it consists in predicting an output from a set of inputs, called features. To model the link between the inputs and the output, the algorithm processes a dataset of multiple known pairs of inputs and output. If the training has been successful, the trained model should predict the unknown output of a new sample from its known inputs. The other common form of machine learning is called unsupervised learning, where there is no specific output to predict. In this case, the model performs a task, such as clustering, based solely on the set of inputs without an explicit function determining the applicability of the learned representations to the larger problem. Thus, these unsupervised methods require some additional programming to apply the results of the ML model to a particular clinical problem One common use of ML models in CDSS's is to automate a process that would otherwise be performed by a human operator, such as determining the location of a DBS electrode for example. Another use is to predict the outcome of the stimulation itself, allowing the clinical team to explore a broader array of alternatives in a non-invasive manner.

The interest of ML to assist clinicians in healthcare has been underlined for a long time.

Celtikci et al. [START_REF] Celtikci | A systematic review on machine learning in neurosurgery: the future of decision-making in patient care[END_REF] and Bulchlak et al. [START_REF] Quinlan D Buchlak | Machine learning applications to clinical decision support in neurosurgery: an artificial intelligence augmented systematic review[END_REF] conducted systematic reviews of the utilization of ML to assist the decision-making in neurosurgery. They highlighted how ML can outperform some traditional statistical methods for the analysis of retrospective data, for example by leveraging non-linearities in high-dimensional and large databases. Senders et al. [START_REF] Joeky T Senders | An introduction and overview of machine learning in neurosurgical care[END_REF] also conducted a review of the utilization of ML in neurosurgery. They also underlined the applicability of ML in neurosurgical care, notably by saying that "In the last few decades, the volume and complexity of bio-medical data have grown beyond the physician's ability to extract all meaningful data patterns using conventional statistical methods alone. [...] The complex diagnostic and therapeutic modalities used in neurosurgery provide a vast amount of data that is ideally suited for ML models." [START_REF] Joeky T Senders | An introduction and overview of machine learning in neurosurgical care[END_REF] In another review, they compared the performance of ML to human experts for diagnosis, surgical planning, and outcome prediction in neurosurgery [START_REF] Joeky | Natural and artificial intelligence in neurosurgery: a systematic review[END_REF]. They concluded that machine intelligence had overall superior results, but pointed out a publication bias which tends to overestimate the performance of ML, as negative results are less likely to be published. These reviews have elucidated the current and potential use cases of ML in neurosurgery, highlighting interesting ML applications, but none of them focused specifically on DBS. The use of ML in DBS is a broad area of research as the methods, data modalities and clinical problems addressed are numerous. In order to draw a landscape of this research field, extracting major trends, better identifying recurring methodological limits, we conducted a systematic review focused on ML in DBS, answering the broad question of how ML is currently used to address clinical problems in DBS.

The next sections will present the methodology used to compile a corpus of paper to analyse, the data acquired from each paper, the results of said analysis, and a discussion about these results.

Material and Methods

We performed this systematic review by following the Preferred Reporting Items for Systematic review and Meta-Analysis Protocols (PRISMA-P) [START_REF] Moher | Preferred reporting items for systematic review and meta-analysis protocols (prisma-p) 2015 statement[END_REF] relevant and applicable recommendations. Figure 2 presents the workflow used to select papers for further analysis.

Initial literature selection strategy

We researched relevant papers in the literature through three queries on two search engines, on September 18th 2021: • Google Scholar, with the following query: ("machine learning" OR "deep learning" OR "neural networks" OR "data-driven" OR "learning-based" OR "artificial intelligence")("deep brain stimulation")

• Google Scholar, with the following query: ("prediction")("deep brain stimulation")

We chose to use PubMed with MeSH method as it is a proven way of browsing papers in medical research, providing that they are tagged with the appropriate MeSH terms. We did not add the MeSH term 'Deep Learning' to the query as it is already included in the 'Machine Learning' hierarchy. We turned off automatic explosion of the MeSH term 'Artificial Intelligence' in order not to include other unrelated topics such as 'Robotics'. We made two different queries on Google Scholar as it is more comprehensive than PubMed. The first one was composed of targeted keywords, and the second one consisted of a broader term ('prediction') in order to include additional papers that could have been missed by the first two queries. Due to the high number of papers returned by queries on Google Scholar, we analyzed the results page by page and stopped when we retained no new papers on two consecutive pages after title and abstract screening (the results being sorted by relevance). We decided not to merge both Google Scholar queries into a single as the term 'prediction' is broad and returned a lot of irrelevant papers. Therefore, including this term in the first Google Scholar query could have made relevant items sparser.

In order to keep a fully systematic methodology, to avoid flaws in the results and to make our screening method reproducible, we chose not to manually include additional papers in the corpus.

Selection process

The first author screened each paper by reading the title and abstract with the following criteria:

• The paper must be methodological, ı.e. validate at least one method.

• ML must be at the heart of the methodology employed.

• The paper must be validated on patients. Papers validated with synthetic data, or using a non-human cohort were discarded.

• It must address a clearly identified clinical problem.

• The paper must be peer-reviewed. If we couldn't obtain the published version, the pre-print version was used. Thesis manuscripts and reviews were discarded.

• It must be written in English.

The number of papers returned by each query and the number of papers kept after title and abstract screening is indicated in Figure 2. We merged the results provided by the three queries, removed duplicates, and obtained a corpus of 117 papers. The first author then re-screened each paper by reading the full text according to the same selection criteria. 44 papers were discarded (notably six papers for being a preliminary version of another retained paper).

Data obtained from each paper

Each of the 73 papers in the final corpus was described according to four classes, as presented on Figure 3. The 'data' class describes the cohort used in the experiment, as well as the nature of the inputs of the ML model. We evaluated the following items:

• The input data modality type, such as imaging or Micro-electrode Recordings (MER).

• The number of patients in the cohort.

• The pathology of the patients in the cohort, such as PD or ET.

The 'application' class corresponds to the clinical problem addressed. We evaluated the following items:

• The stage of the DBS workflow this problem is encountered, using the instances 'screening', 'planning', 'surgery' and 'post-op', as showed in Figure 1.

• The nature of the task that has to be addressed by the method, such as 'classification', 'segmentation', 'regression' or 'clustering'.

The 'method' class describes the specific ML algorithm or framework employed in the paper and is composed of the following items:

• The method used to handle input data, decomposed into three sub-items: the data compression method used (if any), whether or not an automatic feature selection method was employed, and if the method is feature-based in which the input data was transformed into a synthetic set of features requiring a significant amount of feature engineering. For example, we did not consider common domain transformations (such as the Fourier or wavelet domains) as feature-based unless additional operations were performed on said domain which would require additional domain knowledge (such as the selection of particular frequency bands).

• The ML model used to perform the task (ie. the classification, the regression, etc.). If several models were benchmarked, we only reported the one(s) giving the better results

or the one(s) highlighted in the paper's abstract, discussion and conclusion sections.

The 'validation' class describes how the methods were evaluated, according to the following items:

• The method employed to split the data between training and testing sets, such as 'hold-out' or 'LOOCV'.

• Whether or not the validation method is performed in a patient-wise manner, implying that data collected from a single patient cannot be simultaneously in the training set and the testing set.

• The primary metric used to evaluate the performance of the method(s). If several metrics were used, we reported the one highlighted in the paper's abstract, discussion and conclusion sections, or the one the most extensively used in the experiments.

Results

Data obtained from each study was recorded on an Excel spreadsheet and is reproduced in Tables 1 and2. Figure 4b shows the distribution of cohort sizes. The average cohort size was 43 patients, with a median at 17. We counted 10 papers with cohorts larger than 100 patients, with a maximum at 501 and six papers that used data from a single patient.

Paper

Most cohorts (56 occurrences) are solely composed of PD patients. Cohorts of ET patients come second, with a total of five occurrences. Seven papers used a more heterogeneous cohort by mixing patients suffering from different pathologies, by mixing PD and ET patients, and/or by also studying patients suffering from dystonia or Tourette's. One paper used a cohort of patients suffering from dystonia, and one from OCD. Finally, three papers did not explicitly communicate the patients' condition.

Application

Figure 5a presents the distribution of the clinical phase studied in the corpus. We can observe that the post-operative phase was the most extensively studied with 31 occurrences. The surgery phase and the planning phase came next with 21 and 13 occurrences respectively. Finally, the screening phase was the less studied with only nine occurrences. second, with 26 occurrences in total. This strategy consists in partitioning the datapoints into k different disjoint sets. An independent training procedure is then performed k times, each using a different set as testing data and the remaining 1 k  as training data. When performed correctly (i.e. ensuring that the training procedures are completely independent and isolating from each other) this has the benefit of evaluating the algorithm on all the data possible although at the expense of re-training the algorithm several times. Leave-one-out CV (LOOCV) is a specific case k-CV for very small datasets, in which k is equal to the number of datapoints (or the number of patients, if the validation was done patient-wise) in the database. LOOCV is beneficial in that it also ensures that a maximum amount of data is present in the training set at any given time.

LOOCV has been used 13 times. Two papers do not validate their method on a separated testing set, and one did not specify the validation method employed. Finally, bootstrapping refers to evaluation procedures in which the algorithm is trained once, but the effect of any individual datapoint can be isolated and removed from the model, thus allowing for said datapoint to be used in evaluating the reduced model. For methods that are inherently based on averaging an ensemble of simpler models (such as Random Forests), bootstrapping can be easily performed by removing the simpler models that had access to a particular datapoint in training time, and then evaluating the performance of the remaining simple models on said datapoint. This process is repeat and averaged over all datapoints to estimate the perform of the model as a whole. Due to being highly model-specific, bootstrapping was only used once.

Figure 7b presents the evaluation metrics, as well as the corresponding task, for the proposed methods. For classification tasks, accuracy is the most consistently used metric (42 occurrences), followed by sensitivity and specificity ( R -ratio and  -ratio were used only once each. The metrics mostly used for regression were Mean Absolute Error (MAE) and Mean Squared Error (MSE), with six occurrences and three occurrence respectively. Pearson's r was used twice, and correlation coefficient (R) and coefficient of determination ( 2 R ) were used once each. For segmentation, the Dice coefficient was the most consistently used metric with five occurrences, followed by Mean Squared Distance (MSD) and MAE (two occurrences each). Finally, accuracy, Intersection-over-Union (IoU), failure rate and Contour Mean Distance (CMD) were used once each. For clustering, sensitivity, specificity and MAE were each used once.

Discussion

Clinical problems

Post-operative problems

The most consistent post-operative problem addressed with ML methods is the real-time analysis of LFP signals recorded by macro-electrode, in the perspective of delivering adaptive (rather than continuous) stimulation [START_REF] Wu | Prediction of parkinson's disease tremor onset using a radial basis function neural network based on particle swarm optimization[END_REF][START_REF] Jiang | Time-frequency analysis of brain electrical signals for behvior recognition in patients with parkinson's disease[END_REF][START_REF] Niketeghad | Single trial behavioral task classification using subthalamic nucleus local field potential signals[END_REF][START_REF] Mohammed | Toward on-demand deep brain stimulation using online parkinsonâĂŹs disease prediction driven by dynamic detection[END_REF][START_REF] Hosein M Golshan | A hierarchical structure for human behavior classification using stn local field potentials[END_REF][START_REF] Syed | Parkinsonian tremor detection from subthalamic nucleus local field potentials for closed-loop deep brain stimulation[END_REF][START_REF] Yao | Resting tremor detection in parkinson's disease with machine learning and kalman filtering[END_REF][START_REF] Hosein M Golshan | Studying the effects of deep brain stimulation and medication on the dynamics of stn-lfp signals for human behavior analysis[END_REF][START_REF] Wang | Towards adaptive deep brain stimulation in parkinson's disease: Lfp-based feature analysis and classification[END_REF][START_REF] Chen | Automatic sleep stage classification based on subthalamic local field potentials[END_REF][START_REF] Tan | Decoding voluntary movements and postural tremor based on thalamic lfps as a basis for closed-loop stimulation for essential tremor[END_REF][START_REF] Camara | Non-linear dynamical analysis of resting tremor for demand-driven deep brain stimulation[END_REF][START_REF] Mohammed | A framework for adapting deep brain stimulation using parkinsonian state estimates[END_REF][START_REF] Hosein M Golshan | LFP-Net: A deep learning framework to recognize human behavioral activities using brain STN-LFP signals[END_REF][START_REF] Yao | Improved detection of Parkinsonian resting tremor with feature engineering and Kalman filtering[END_REF].

Indeed, the ability to process and analyze LFP signals in real time could allow for the design of closed-loop stimulation systems that deliver the therapy only when needed, thus limiting undesirable side effects and extending battery life. A complementary approach was investigated by Loukas et al. [START_REF] Loukas | A pc-based system for predicting movement from deep brain signals in parkinson's disease[END_REF] who proposed a complete system to record, process, and display LFP signals.

Houston et al. [START_REF] Brady C Houston | Classifier-based closed-loop deep brain stimulation for essential tremor[END_REF][START_REF] Houston | A machine-learning approach to volitional control of a closed-loop deep brain stimulation system[END_REF] designed a close-loop system using cortical activity analysis, and Shukla et al. [START_REF] Shukla | A neural network-based design of an on-off adaptive control for deep brain stimulation in movement disorders[END_REF] and Khobragade et al. [START_REF] Khobragade | Towards fully automated closed-loop deep brain stimulation in parkinson's disease patients: a lamstar-based tremor predictor[END_REF][START_REF] Khobragade | On the need for adaptive learning in on-demand deep brain stimulation for movement disorders[END_REF] The second most common post-operative clinical problem is the analysis and quantification of the motor symptomatology of patients with external sensors (wearable sensors [START_REF] Shivanthan Ac Yohanandan | Evaluating machine learning algorithms estimating tremor severity ratings on the bain-findley scale[END_REF][START_REF] Angeles | Automated assessment of symptom severity changes during deep brain stimulation (dbs) therapy for parkinson's disease[END_REF][START_REF] Lemoyne | Implementation of a smartphone as a wearable and wireless accelerometer and gyroscope platform for ascertaining deep brain stimulation treatment efficacy of parkinsonâĂŹs disease through machine learning classification[END_REF][START_REF] Henrique | On the use of t-distributed stochastic neighbor embedding for data visualization and classification of individuals with parkinsonâĂŹs disease[END_REF][START_REF] Lemoyne | Evaluation of machine learning algorithms for classifying deep brain stimulation respective of âĂŸonâĂŹand âĂŸoffâĂŹstatus[END_REF] or a force platform [START_REF] Muniz | Assessment of the effects of subthalamic stimulation in parkinson disease patients by artificial neural network[END_REF]). Such automatic systems can be clinically valuable by providing objective, automatic and quick feedback for therapies in order to, for example, compare several therapy parameters and combinations. Indeed, the degree of treatment parameter tuning, including stimulation parameters and drug dosage, is large and can't be assessed exhaustively, which may result in sub-optimal configurations. Other works have been done toward automatizing this post-operative phase: Connolly et al. [START_REF] Allison T Connolly | Guiding deep brain stimulation contact selection using local field potentials sensed by a chronically implanted device in parkinson's disease patients[END_REF] proposed a system predicting which contact is optimal from LFP recordings. Boutet et al. [START_REF] Boutet | Predicting optimal deep brain stimulation parameters for ParkinsonâĂŹs disease using functional MRI and machine learning[END_REF] proposed a method to predict whether the contact and stimulation voltage are optimal or not by analysing post-operative functional Magnetic

Resonance Imaging (MRI). Shamir et al. [START_REF] Reuben R Shamir | Machine learning approach to optimizing combined stimulation and medication therapies for parkinson's disease[END_REF] proposed a predictive system using inputs from several modalities (clinical, therapy (medication and stimulation) and demographic data) in order to narrow the research space both for medication dosage and stimulation parameters. Lastly, Stuart et al. [START_REF] Stuart | Machine learning for deep brain stimulation efficacy using dense array eeg[END_REF] used EEG to predict effective stimulation in real time.

Surgery problems

The second most commonly investigated phase is the surgery itself, with the clinical problem frequently being the inter-operative identification of the DBS target. Every paper addressing this problem used MER analysis [START_REF] Orozco | Identification of spike sources using proximity analysis through hidden markov models[END_REF][START_REF] Guillén | Characterization of subcortical structures during deep brain stimulation utilizing support vector machines[END_REF][START_REF] Vikram Rajpurohit | Optimizing computational feature sets for subthalamic nucleus localization in dbs surgery with feature selection[END_REF][START_REF] Guillén-Rondon | Deep brain stimulation signal classification using deep belief networks[END_REF][START_REF] Valsky | Stop! border ahead: A utomatic detection of subthalamic exit during deep brain stimulation surgery[END_REF][START_REF] Khosravi | Electrophysiological signal processing for intraoperative localization of subthalamic nucleus during deep brain stimulation surgery[END_REF][START_REF] Darío | Multi-task learning for subthalamic nucleus identification in deep brain stimulation[END_REF][START_REF] Ciecierski | Unsupervised machine learning in classification of neurobiological data[END_REF][START_REF] Valsky | Real-time machine learning classification of pallidal borders during deep brain stimulation surgery[END_REF][START_REF] Khosravi | Intraoperative localization of STN during DBS surgery using a data-driven model[END_REF][START_REF] Peralta | SepaConvNet for Localizing the Subthalamic Nucleus using One Second Micro-Electrode Recordings[END_REF][START_REF] Pa Karthick | Automated detection of subthalamic nucleus in deep brain stimulation surgery for ParkinsonâĂŹs disease using microelectrode recordings and wavelet packet features[END_REF][START_REF] Hosny | Deep convolutional neural network for the automated detection of Subthalamic nucleus using MER signals[END_REF][START_REF] Thibault Martin | Extending convolutional neural networks for localizing the subthalamic nucleus from micro-electrode recordings in ParkinsonâĂŹs disease[END_REF][START_REF] Thibault Martin | Adapting the listening time for micro-electrode recordings in deep brain stimulation interventions[END_REF]. Instead of helping clinicians to aim for an anatomical structure, Lu et al. [START_REF] Charles W Lu | High density microelectrode recording predicts span of therapeutic tissue activation volumes in subthalamic deep brain stimulation for Parkinson disease[END_REF] proposed a method to predict, by analyzing MER, whether or not the electrode lead is inside a clinically predefined therapeutic site of activation. Complementarily, Wong et al. [START_REF] Wong | Functional localization and visualization of the subthalamic nucleus from microelectrode recordings acquired during dbs surgery with unsupervised machine learning[END_REF] proposed a method to project MER in a 2D plan in order for clinicians an alternative way to visualize and interpret it.

Park et al. [START_REF] Kwang | Clinical outcome prediction from analysis of microelectrode recordings using deep learning in subthalamic deep brain stimulation for Parkinsons disease[END_REF] proposed a method to predict motor outcomes six months after surgery by analysing MERs, which was used to find the optimal lead location during surgery. From the perspective of curating these signals for downstream analysis, Klempivr et al. [START_REF] Ond R Ej Klempíř | Identification of microrecording artifacts with wavelet analysis and convolutional neural network: An image recognition approach[END_REF] and Hosny et al. [START_REF] Hosny | A novel deep lstm network for artifacts detection in microelectrode recordings[END_REF] proposed systems for artifact detection and correction. Kostoglou et al. [START_REF] Kostoglou | Classification and prediction of clinical improvement in deep brain stimulation from intraoperative microelectrode recordings[END_REF] used MER features, a few clinical scores, demographics and contact location to predict the clinical improvement that could result from the stimulation of various locations, in a perspective of placing the electrode based on functional criteria rather than anatomical ones.

As MERs are electrophysiological signals, these papers are similar in methodology to those using LFPs.

Planning problems

The planning phase comes third, the clinical problem being how to select the stimulation targets and determine the electrode trajectories prior to the operation. A first strategy is to assist the surgeon by automatically segmenting, or localising the subcortical structures of interest from pre-operative images ( [START_REF] Kim | Robust prediction of clinical deep brain stimulation target structures via the estimation of influential high-field mr atlases[END_REF][START_REF] Liu | Multi-modal learning-based pre-operative targeting in deep brain stimulation procedures[END_REF][START_REF] Milletari | Hough-cnn: deep learning for segmentation of deep brain regions in mri and ultrasound[END_REF][START_REF] Kim | Automatic localization of the subthalamic nucleus on patient-specific clinical mri by incorporating 7 t mri and machine learning: Application in deep brain stimulation[END_REF][START_REF] Park | Deep learning-based deep brain stimulation targeting and clinical applications[END_REF][START_REF] Baxter | Localisation of the subthalamic nucleus in mri via convolutional neural networks for deep brain stimulation planning[END_REF][START_REF] Baxter | Segmentation of the subthalamic nucleus in MRI via Convolutional Neural Networks for deep brain stimulation planning[END_REF][START_REF] Solomon | Deep-learning based fully automatic segmentation of the globus pallidus interna and externa using ultra-high 7 Tesla MRI[END_REF]). Such works are important as the surgical targets are often small with low contrast in clinical images. An alternative strategy is to propose to the surgeon functional criteria instead of anatomical ones for the choice of a stimulation site.

Baumgarten et al. [START_REF] Baumgarten | Image-guided preoperative prediction of pyramidal tract side effect in deep brain stimulation: proof of concept and application to the pyramidal tract side effect induced by pallidal stimulation[END_REF][START_REF] Baumgarten | Improvement of pyramidal tract side effect prediction using a data-driven method in subthalamic stimulation[END_REF][START_REF] Baumgarten | Data-driven prediction of the therapeutic window during subthalamic deep brain stimulation surgery[END_REF] and Bermudez et al. [START_REF] Bermudez | Towards machine learning prediction of deep brain stimulation (dbs) intra-operative efficacy maps[END_REF] proposed clinical efficacy probability maps to visualize the expected clinical effects of stimulation of several locations around the structure of interest. Singer et al. [START_REF] Singer | Post-operative electrode placement prediction in deep brain stimulation using support vector regression[END_REF] went even further in this idea by directly predicting the optimal electrode location without the use of intermediate representations such as anatomical segmentations or clinical-effect probability maps.

Screening problems

Finally, the least commonly addressed phase using ML is the screening phase. Oliveira et al. [START_REF] Henrique | On the use of t-distributed stochastic neighbor embedding for data visualization and classification of individuals with parkinsonâĂŹs disease[END_REF] proposed a method to visualize on a two-dimension space the motor symptomatology of the patient from electromyography sensors in order to facilitate the clinical interpretation of patient motor scores. Habets et al. [START_REF] Jeroen | Machine learning prediction of motor response after deep brain stimulation in parkinson's disease[END_REF] proposed a predictive system to identify weak motor responders from pre-operative clinical data and demographics for patient selection purposes. In the same screening assisting tool objectives, Koch et al. [START_REF] Koch | Automated machine learning for eeg-based classification of parkinsonâĂŹs disease patients[END_REF] proposed a system to classify patients regarding their cognition from EEG. Using pre-operative EEG too, Geraedts et al. [START_REF] Vj Geraedts | Machine learning for automated EEG-based biomarkers of cognitive impairment during Deep Brain Stimulation screening in patients with ParkinsonâĂŹs Disease[END_REF] proposed a method to predict post-operative cognitive functions. Farrokhi et al. [START_REF] Farrokhi | Investigating risk factors and predicting complications in deep brain stimulation surgery with machine learning algorithms[END_REF] attempted to find factors of surgery adverse outcomes, such as infections or hemorrhages. Shah et al. [START_REF] Syed | Application of machine learning using decision trees for prognosis of deep brain stimulation of globus pallidus internus for children with dystonia[END_REF] proposed a method to predict DBS outcomes from pre-operative demographics, clinical tests, expert interpretation of anatomical abnormalities using MRI, and TMS. Shang et al. [START_REF] Shang | Connectome-Based Model Predicts Deep Brain Stimulation Outcome in Parkinson's Disease[END_REF] proposed a method to predict post-operative motor outcomes with functional connectivity. Peralta et al. [START_REF] Peralta | PassFlow: a multimodal workflow for predicting Deep Brain Stimulation Outcomes[END_REF] proposed a method to predict 21 different clinical scores (including motor, cognitiive, and quality-of-life scores) three months, six months, one year, and three years after surgery by using pre-operative clinical tests, demographic information, and the shape of particular anatomical structures using T1-MRI. Finally, Liebrand et al. [START_REF] Luka C Liebrand | Deep brain stimulation response in obsessive-compulsive disorder is associated with preoperative nucleus accumbens volume[END_REF] attempted to predict DBS good responders from pre-operative MRI but did not obtain satisfactory results.

Wide variety of models

When it comes to the choice of ML model, the most widely used remains the SVM which is not surprising as SVMs are known to perform well and are simple to train for both classification and regression problems, even for small databases. Among the papers comparing several models, SVMs were amongst the top-performing [START_REF] Lemoyne | Evaluation of machine learning algorithms for classifying deep brain stimulation respective of âĂŸonâĂŹand âĂŸoffâĂŹstatus[END_REF][START_REF] Stuart | Machine learning for deep brain stimulation efficacy using dense array eeg[END_REF]. Shallow feed-forward ANNs come second, likely due to the recent advances in deep learning and its growing popularity in the research community.

More specialized ANN structures, such as CNNs and RNNs were also used. CNNs were used 13 times in total, notably six times for image analysis: three times for subcortical structures segmentation or localisation with the VGG model [START_REF] Park | Deep learning-based deep brain stimulation targeting and clinical applications[END_REF], a modified ResNet structure [START_REF] Bermudez | Towards machine learning prediction of deep brain stimulation (dbs) intra-operative efficacy maps[END_REF], a custom structure called Hough-CNN [START_REF] Milletari | Hough-cnn: deep learning for segmentation of deep brain regions in mri and ultrasound[END_REF] based on Hough voting, and U-net-based structures [START_REF] Baxter | Localisation of the subthalamic nucleus in mri via convolutional neural networks for deep brain stimulation planning[END_REF][START_REF] Baxter | Segmentation of the subthalamic nucleus in MRI via Convolutional Neural Networks for deep brain stimulation planning[END_REF][START_REF] Solomon | Deep-learning based fully automatic segmentation of the globus pallidus interna and externa using ultra-high 7 Tesla MRI[END_REF].

CNNs are also extensively used for MER spectrogram analysis: three times with a structure based on 1D separable convolutions [START_REF] Peralta | SepaConvNet for Localizing the Subthalamic Nucleus using One Second Micro-Electrode Recordings[END_REF][START_REF] Thibault Martin | Extending convolutional neural networks for localizing the subthalamic nucleus from micro-electrode recordings in ParkinsonâĂŹs disease[END_REF][START_REF] Thibault Martin | Adapting the listening time for micro-electrode recordings in deep brain stimulation interventions[END_REF], once with the AlexNet model [START_REF] Ond R Ej Klempíř | Identification of microrecording artifacts with wavelet analysis and convolutional neural network: An image recognition approach[END_REF], once with a CNN based on VGG16 and trained with multi-task learning [START_REF] Kwang | Clinical outcome prediction from analysis of microelectrode recordings using deep learning in subthalamic deep brain stimulation for Parkinsons disease[END_REF], and once with a custom structure based on 1D-convolution [START_REF] Hosny | Deep convolutional neural network for the automated detection of Subthalamic nucleus using MER signals[END_REF]. RNNs were used once with LSTM for MER artifact detection [START_REF] Hosny | A novel deep lstm network for artifacts detection in microelectrode recordings[END_REF].

We can also mention the usage of an interesting technique called EL, which consists in using several models in combination to make a more accurate prediction. Bagging was used eight times with RFs, and once by Kim et al. [START_REF] Kim | Robust prediction of clinical deep brain stimulation target structures via the estimation of influential high-field mr atlases[END_REF]. Gradient boosting was used four times, twice with the XGB model [START_REF] Yao | Resting tremor detection in parkinson's disease with machine learning and kalman filtering[END_REF][START_REF] Farrokhi | Investigating risk factors and predicting complications in deep brain stimulation surgery with machine learning algorithms[END_REF] and twice with GBRT [START_REF] Shang | Connectome-Based Model Predicts Deep Brain Stimulation Outcome in Parkinson's Disease[END_REF] or GBDT [START_REF] Yao | Improved detection of Parkinsonian resting tremor with feature engineering and Kalman filtering[END_REF], and stacking was used by Golshan et al. [START_REF] Hosein M Golshan | A hierarchical structure for human behavior classification using stn local field potentials[END_REF] and Shamir et al. [START_REF] Reuben R Shamir | Machine learning approach to optimizing combined stimulation and medication therapies for parkinson's disease[END_REF].

The methods used in the literature seem relatively independent of the input data used as shown in Figure 6b performed to validate the model's performance. To this extent, Graphics Processing Units (GPU), computational clusters, or external computational clouds can be necessary to train some methods.

The prominent role of pre-processing and feature engineering for handling high-dimension input data

An important consideration for ML studies is the curse of dimensionality: the greater the input dimensionality is, the exponentially greater the number of training samples are required to guarantee that data points are not too sparse in the input space. In DBS, the number of training samples are usually limited. Therefore, limiting the dimensionality of the input space to ease the training process can be an interesting strategy, even if it comes at the cost of reducing the amount of information available to the ML model.

Two common strategies can be to unsupervisedly compress the data and/or to automatically select the features, but such approaches are not in the majority. The most common strategy is to transform the original, raw input space into a set of features thanks to expert knowledge of the domain. For example, Kostoglou et al. [START_REF] Kostoglou | Classification and prediction of clinical improvement in deep brain stimulation from intraoperative microelectrode recordings[END_REF] synthesized the MER signals by computing domain-specific features such as the mean inter-spike interval, or the power band ratio of the signal in different pre-determined frequency bands.

A minority of papers chose a fully data-driven method by straightforwardly feeding the model with raw data, without reducing it in the form of features, compressing it or automatically selecting some of them. Naturally, the 13 papers using CNNs fall under this umbrella. Indeed, the purpose of a CNN is to take advantage of the spatial inter-correlations in the input images by applying a series of learnable convolution kernels to it. Therefore, CNNs learn an optimal dimensionality reduction strategy in a supervised manner. Among the other occurrences are the three papers from Baumgarten et al. [START_REF] Baumgarten | Image-guided preoperative prediction of pyramidal tract side effect in deep brain stimulation: proof of concept and application to the pyramidal tract side effect induced by pallidal stimulation[END_REF][START_REF] Baumgarten | Improvement of pyramidal tract side effect prediction using a data-driven method in subthalamic stimulation[END_REF][START_REF] Baumgarten | Data-driven prediction of the therapeutic window during subthalamic deep brain stimulation surgery[END_REF] and the paper from Habets et al. [START_REF] Jeroen | Machine learning prediction of motor response after deep brain stimulation in parkinson's disease[END_REF], because the number of inputs is low (respectively four stimulation parameters and 15 clinical scores and demographics).

We can interestingly mention five papers which compared a feature-based method with a fully data-driven method. First, for STN localisation using MER's, Khosravi et al. [START_REF] Khosravi | Electrophysiological signal processing for intraoperative localization of subthalamic nucleus during deep brain stimulation surgery[END_REF][START_REF] Khosravi | Intraoperative localization of STN during DBS surgery using a data-driven model[END_REF] compared the utilization of state-of-the-art features versus the raw signal's Fourier coefficients, and Hosny et al. [START_REF] Hosny | Deep convolutional neural network for the automated detection of Subthalamic nucleus using MER signals[END_REF] compared the utilization of state-of-the-art features versus the raw signal directly. Second, Baumgarten et al. Feature engineering could be thought of as a particular, very involved type of pre-processing, that is, the modification of the input data before it is presented to the machine learning algorithm. From this perspective, pre-processing can be seen as a matter of degree. For example, for machine learning methods that process MERs as input, some use a spectral representation of the entire signal [START_REF] Khosravi | Electrophysiological signal processing for intraoperative localization of subthalamic nucleus during deep brain stimulation surgery[END_REF][START_REF] Valsky | Real-time machine learning classification of pallidal borders during deep brain stimulation surgery[END_REF], whereas others use a spectrogram [START_REF] Hosny | A novel deep lstm network for artifacts detection in microelectrode recordings[END_REF] or Haar wavelets [START_REF] Orozco | Identification of spike sources using proximity analysis through hidden markov models[END_REF], i.e. a mixed temporal/frequential representation, and others process the temporal signal directly without explicitly representing the frequency components [START_REF] Guillén | Characterization of subcortical structures during deep brain stimulation utilizing support vector machines[END_REF]. These can be interpreted as a gradation between methods that require a large amount of pre-processing in order to make the informative components more accessible to the machine learning algorithm and others that rely more heavily on the algorithm itself. That being said, some amount of pre-processing appears to always be necessary, even for complex machine learning approaches, to ensure that the input data is meaningful to the particular algorithm or adheres to some pre-defined scope, however that scope has appeared to get broader as newer methods are developed, which is evidenced by half of the feature-selection-free methods being published in the last two years.

Inter-patient variability: the elephant in the room

Small cohorts are problematic in ML because they limit the performance of predictive systems (several papers [START_REF] Hosein M Golshan | A hierarchical structure for human behavior classification using stn local field potentials[END_REF][START_REF] Chen | Automatic sleep stage classification based on subthalamic local field potentials[END_REF][START_REF] Stuart | Machine learning for deep brain stimulation efficacy using dense array eeg[END_REF][START_REF] Mohammed | A framework for adapting deep brain stimulation using parkinsonian state estimates[END_REF][START_REF] Farrokhi | Investigating risk factors and predicting complications in deep brain stimulation surgery with machine learning algorithms[END_REF] stated lack of data as a limitation). Furthermore, the pathologies treated by DBS are heterogeneous causing a high inter-patient variability, on top of intra-patient variability (as the clinical state of the patients can fluctuate or because the recording conditions may vary). Therefore, a system trained on one patient or on one recording configuration is not likely to have good performance on another one, or later in time, which limits its prospective usability.

The contribution of Khobragade et al. [START_REF] Khobragade | On the need for adaptive learning in on-demand deep brain stimulation for movement disorders[END_REF] illustrates this phenomenon well. They gathered surface electromyography and accelerometer data from two patients through several trials spread on different sessions, with at least a week between consecutive sessions. They did two experiments: the first one by training one model per patient and per session, therefore testing the model on trials of the same sessions. For the second one, they trained one model per patient, but trained the model on a set of sessions and tested it on other sessions. They obtained a perfect accuracy for the first experiment, but the median performance dropped to 46.15% for the second one. On the same extent, Rajpurohit et al. [START_REF] Vikram Rajpurohit | Optimizing computational feature sets for subthalamic nucleus localization in dbs surgery with feature selection[END_REF] reported results with a patient-specific feature normalization scheme (therefore not applicable prospectively), and with a patient-independent normalization scheme. Not surprisingly, the classification error rates of the patient-specific scheme are much lower (in the range of 0.0711 to 0.1353) than the patient-independent scheme error rates (in the range of 0.102 to 0.1979).

The majority of the papers did not employ a patient-wise validation method [START_REF] Orozco | Identification of spike sources using proximity analysis through hidden markov models[END_REF][START_REF] Wu | Prediction of parkinson's disease tremor onset using a radial basis function neural network based on particle swarm optimization[END_REF][START_REF] Guillén | Characterization of subcortical structures during deep brain stimulation utilizing support vector machines[END_REF][START_REF] Shukla | A neural network-based design of an on-off adaptive control for deep brain stimulation in movement disorders[END_REF][START_REF] Loukas | A pc-based system for predicting movement from deep brain signals in parkinson's disease[END_REF][START_REF] Jiang | Time-frequency analysis of brain electrical signals for behvior recognition in patients with parkinson's disease[END_REF][START_REF] Niketeghad | Single trial behavioral task classification using subthalamic nucleus local field potential signals[END_REF][START_REF] Allison T Connolly | Guiding deep brain stimulation contact selection using local field potentials sensed by a chronically implanted device in parkinson's disease patients[END_REF][START_REF] Reuben R Shamir | Machine learning approach to optimizing combined stimulation and medication therapies for parkinson's disease[END_REF][START_REF] Khobragade | Towards fully automated closed-loop deep brain stimulation in parkinson's disease patients: a lamstar-based tremor predictor[END_REF][START_REF] Shivanthan Ac Yohanandan | Evaluating machine learning algorithms estimating tremor severity ratings on the bain-findley scale[END_REF][START_REF] Baumgarten | Image-guided preoperative prediction of pyramidal tract side effect in deep brain stimulation: proof of concept and application to the pyramidal tract side effect induced by pallidal stimulation[END_REF][START_REF] Angeles | Automated assessment of symptom severity changes during deep brain stimulation (dbs) therapy for parkinson's disease[END_REF][START_REF] Brady C Houston | Classifier-based closed-loop deep brain stimulation for essential tremor[END_REF][START_REF] Guillén-Rondon | Deep brain stimulation signal classification using deep belief networks[END_REF][START_REF] Mohammed | Toward on-demand deep brain stimulation using online parkinsonâĂŹs disease prediction driven by dynamic detection[END_REF][START_REF] Hosein M Golshan | A hierarchical structure for human behavior classification using stn local field potentials[END_REF][START_REF] Khosravi | Electrophysiological signal processing for intraoperative localization of subthalamic nucleus during deep brain stimulation surgery[END_REF][START_REF] Lemoyne | Implementation of a smartphone as a wearable and wireless accelerometer and gyroscope platform for ascertaining deep brain stimulation treatment efficacy of parkinsonâĂŹs disease through machine learning classification[END_REF][START_REF] Darío | Multi-task learning for subthalamic nucleus identification in deep brain stimulation[END_REF][START_REF] Khobragade | On the need for adaptive learning in on-demand deep brain stimulation for movement disorders[END_REF][START_REF] Henrique | On the use of t-distributed stochastic neighbor embedding for data visualization and classification of individuals with parkinsonâĂŹs disease[END_REF][START_REF] Syed | Parkinsonian tremor detection from subthalamic nucleus local field potentials for closed-loop deep brain stimulation[END_REF][START_REF] Yao | Resting tremor detection in parkinson's disease with machine learning and kalman filtering[END_REF][START_REF] Hosein M Golshan | Studying the effects of deep brain stimulation and medication on the dynamics of stn-lfp signals for human behavior analysis[END_REF][START_REF] Wang | Towards adaptive deep brain stimulation in parkinson's disease: Lfp-based feature analysis and classification[END_REF][START_REF] Houston | A machine-learning approach to volitional control of a closed-loop deep brain stimulation system[END_REF][START_REF] Chen | Automatic sleep stage classification based on subthalamic local field potentials[END_REF] [START_REF] Reuben R Shamir | Machine learning approach to optimizing combined stimulation and medication therapies for parkinson's disease[END_REF][START_REF] Koch | Automated machine learning for eeg-based classification of parkinsonâĂŹs disease patients[END_REF], that further validation has to be done on other recording conditions or other centers [START_REF] Valsky | Stop! border ahead: A utomatic detection of subthalamic exit during deep brain stimulation surgery[END_REF][START_REF] Syed | Application of machine learning using decision trees for prognosis of deep brain stimulation of globus pallidus internus for children with dystonia[END_REF][START_REF] Peralta | SepaConvNet for Localizing the Subthalamic Nucleus using One Second Micro-Electrode Recordings[END_REF][START_REF] Thibault Martin | Extending convolutional neural networks for localizing the subthalamic nucleus from micro-electrode recordings in ParkinsonâĂŹs disease[END_REF][START_REF] Solomon | Deep-learning based fully automatic segmentation of the globus pallidus interna and externa using ultra-high 7 Tesla MRI[END_REF] or that inter-patient variability was a limiting problem [START_REF] Houston | A machine-learning approach to volitional control of a closed-loop deep brain stimulation system[END_REF][START_REF] Khosravi | Intraoperative localization of STN during DBS surgery using a data-driven model[END_REF].

A reproducibility and comparison problem

Another recurring problem is the disparity in the validation methods which keeps from comparing the results of different contributions together. As an example, all of the following contributions report the results of the location of the STN with MER but differ in the validation process:

• Guillén-Rondon et al. [START_REF] Guillén | Characterization of subcortical structures during deep brain stimulation utilizing support vector machines[END_REF][START_REF] Guillén-Rondon | Deep brain stimulation signal classification using deep belief networks[END_REF] mixed patients and different portions of the same signals in training and test sets, which is an identified source of data leakage, leading to likely overestimated performance.

• Khosravi et al. [START_REF] Khosravi | Electrophysiological signal processing for intraoperative localization of subthalamic nucleus during deep brain stimulation surgery[END_REF] didn't specify if the validation was done on a separated set of patients.

• Rajpurohit et al. [START_REF] Vikram Rajpurohit | Optimizing computational feature sets for subthalamic nucleus localization in dbs surgery with feature selection[END_REF] used a leave-one-patient-one validation strategy.

While the contributions on the same topic are proliferating, it is often difficult or impossible to directly compare the quantitative results presented as the methods are validated in such heterogeneous ways with varying levels of bias and data leakage.

On the same extent, methods are not easily reproducible for benchmarking purposes, only three [START_REF] Henrique | On the use of t-distributed stochastic neighbor embedding for data visualization and classification of individuals with parkinsonâĂŹs disease[END_REF][START_REF] Shang | Connectome-Based Model Predicts Deep Brain Stimulation Outcome in Parkinson's Disease[END_REF][START_REF] Boutet | Predicting optimal deep brain stimulation parameters for ParkinsonâĂŹs disease using functional MRI and machine learning[END_REF] mentioned having publicly shared, or being willing to share on request their code, their data or the features they computed on their data.

One factor that exacerbates this issue is a lack of openly available annotated datasets that multiple groups could use to ensure that a higher proportion of the patient variability is captured in training. To the best of our knowledge, there are no openly accessible databases of MER or LFP signals for DBS. For images and clinical questionnaires, there is an open dataset specific to Parkinson's disease, the Parkinson's Progression Markers Initiative (PPMI) [START_REF] Marek | The parkinson progression marker initiative (ppmi)[END_REF]. However, because it is not specific to DBS, it lacks annotations for DBS-specific problems, and many of the individuals contained in the dataset may not have received DBS at all. The creation of these open datasets would be an obvious area for the community to develop, especially given how much of the literature involves the collection of their own datasets which, if combined, would represent a sizeable if heterogeneous database for most ML problems associated with DBS.

One potential impediment to such a dataset is the relatively closed nature of medical data in general. Centres may be wary of releasing denser types of patient data such as MR images (and even less dense data such as clinical questionnaires) due to concerns about the future potential for de-anonymisation which would be ethically problematic [START_REF] Emam | Anonymising and sharing individual patient data[END_REF].

Conclusion

We conducted, to the best of our knowledge, the first systematic review on ML for DBS, and identified several common methodological threads and limitations from the analysis of a corpus of 73 papers.

First, we have seen that only a few studies concern the screening phase. To us, there is a real opportunity for ML here because, as there exist several clinical challenges and a lot of complex and high-dimensional data arising from several pre-operative modalities (such as clinical testing, patient questionnaires, imaging data or demographics). These factors make this phase challenging to address but also ripe for data-driven methods providing that adequate methods are employed and large enough databases become available.

Second, the majority of studies use simple models with either a few features as input or aggressive dimensionality reduction methods (with automatic feature selection or feature-engineering). While it ensures the input data is readily usable for simple models, it also limits drastically the information that can be leveraged by the model, and therefore limits the performance and the practicable complexity of the prediction task. We think more ambitious problems, or higher levels of performance could be achieved by employing more bottom-up, data-driven paradigms. Four papers comparing a feature-oriented method to a data-driven one support this hypothesis, showing better results for the latter approach [START_REF] Baumgarten | Improvement of pyramidal tract side effect prediction using a data-driven method in subthalamic stimulation[END_REF][START_REF] Khosravi | Electrophysiological signal processing for intraoperative localization of subthalamic nucleus during deep brain stimulation surgery[END_REF][START_REF] Khosravi | Intraoperative localization of STN during DBS surgery using a data-driven model[END_REF][START_REF] Hosny | Deep convolutional neural network for the automated detection of Subthalamic nucleus using MER signals[END_REF], where only one reported the opposite [START_REF] Yao | Improved detection of Parkinsonian resting tremor with feature engineering and Kalman filtering[END_REF].

Third, small cohorts are often used: a lot of studies collect data that is not usually collected in the clinical routine, thus limiting the number of patients. Small cohorts, first, are problematic because they limit the generalization performance of the model and impose limits regarding the dimensionality of the input, because of the curse of dimensionality. Some papers overcame this problem by employing data augmentation when possible (for example, by splitting a ten seconds LFP signal into several two seconds ones). Unfortunately, this strategy does not address the second problem caused by small cohorts, which is that they cannot cover the large heterogeneity of the studied populations. Consequently, the performances reported are likely not representative of a prospective use of the system. Additionally, a number of contributions validated their method in a non-patient-wise manner which could lead to potential data leakage and thus inflated accuracy measures. Although the degree of this bias is not fully known, one study has found it to be extremely highly significant.

Lastly, to us, more effort has to be dedicated to the validation method employed and, if possible, to share the code and the data to facilitate reproducibility. We found that numerous papers are sharing similar, if not identical objectives but have not found any that benchmark different approaches by comparing the performance of their method to those proposed in the literature. This may be a result of the aforementioned problem of reproducibiliy, as it would imply reproducing and evaluating them with the same database and validation method. We think this area of research would benefit from more uniform validation methods and explicit validation guidelines, and further work would be required in this direction.

Overall, our survey indicates that ML is growing in this field, and we expect many of these issues to be resolved in large part with this particular research field maturing.

Figure 1 :

 1 Figure 1: Surgical workflow of a typical DBS, composed of four majors steps.

Figure 2 :

 2 Figure 2: Workflow to select the corpus of 73 papers to classify.

Figure 3 :

 3 Figure 3: Data was acquired concerning four classes: data used, clinical application, method, and

  (a) Input data modality type. (b) Cumulative plot showing the number of patients in each paper's cohort. The blue dashed line shows the median value (17 patients).

Figure 4 :

 4 Figure 4: Charts presenting the results for the 'data' class.

Figure 5b presents the

  Figure 5b presents the distribution of the tasks addressed. Classification is, by far, the most

Figure 5 :

 5 Figure 5: Chart presenting the results for the 'application' class.

Figure 6 :

 6 Figure 6: Charts presenting the results for the 'method' class.

Figure

  Figure6bshows the model used, or the best performing model(s) if several were

Figure 7 :

 7 Figure 7: Charts presenting the results for the 'validation' class.

  using surface electromyography (sEMG) and accelerometer signals. These works towards designing closed-loop stimulation systems represent 19 of the 28 papers focused on the post-operative phase of DBS interventions.

  , which outlines a great heterogeneity of model used for each input data type, with the notable exception of CNNs, which were used 12 out of 13 times to analyze MER spectrograms and imaging data, and once to analyze LFPs. This is not a surprising result as CNNs are tailored to find spatial patterns automatically in high-dimensional data. Nevertheless, even though teh use of Deep Learning (DL) models is becoming more and more common, other ML models (along with handcrafted features) still represent the majority of the methods employed. This is outlined by the analysis of site-specific electrophysiological signals such as MER and LFP where both options are possible, but where non-DL based methods are still used more than 85% of the time. One difficulty in model training and optimization that particularly affects DL-based methods is the extensive amount of computational time and resources required. Indeed, on top of being usually heavyweight, CNNs require the tuning of several architectural and training hyper-parameters, necessitating the training of possibly hundreds of neural networks for a typical research paper. This is exacerbated when k-fold CV is employed, as k training iterations are

[ 29 ]

 29 compared two methods for predicting the occurrence of Pyramidal Tract Side Effect (PTSE) during the stimulation: a Volume of Tissue Activated (VTA)-based method versus a method using the raw information straightforwardly, which is the three-dimensional location of the contact and the stimulation voltage. All of the four papers showed a superiority of the fully data-driven paradigm which outperformed the feature-engineering approaches. Lastly, Yao et al.[START_REF] Yao | Improved detection of Parkinsonian resting tremor with feature engineering and Kalman filtering[END_REF] compared the utilisation of handcrafted features versus a CNN for LFP signals classification, and got better results with handcrafted features, likely due to overfitting.

Table 1

 1 Data obtained from each of the 55 papers in the corpus for the 'data' and 'application' classes.

			Data		Application
		Input Modality C. size	Pathology	Phase	Task
	Orozco et al.	MER		PD	surgery	class.
	(2006) [10]
	Muniz et al.	external sensors 45	PD	post-op.	class.
	(2009) [11]
	Wong et al.	MER	27	PD	surgery	clust.
	(2009) [12]
	Wu et al. (2010) LFP	1	PD	post-op.	class.

Table 2

 2 Data obtained from each of the 55 papers in the corpus for the 'method' and 'validation' classes.

'FB' stands for 'feature-based'. 'FS' stands for 'feature selection'.

  , 50, 52, 53, 56, 59, 60, 67, 68, 71]. A large number of those papers reported performance on a single patient, training a different model for each patient and validating it on that same patient. This is often because a singular patient has multiple signals and the model uses some for training and others for evaluation, treating the learning of a model as a form of calibration. However, there were some papers that simply did not separate training and testing patients, could lead to biased measurements of their performance. The results reported by these contributions are interesting as preliminary work, but cannot safely be considered as representative of a real, prospective usage.Others measured their performance by employing a patient-wise validation method[START_REF] Muniz | Assessment of the effects of subthalamic stimulation in parkinson disease patients by artificial neural network[END_REF][START_REF] Wong | Functional localization and visualization of the subthalamic nucleus from microelectrode recordings acquired during dbs surgery with unsupervised machine learning[END_REF][START_REF] Vikram Rajpurohit | Optimizing computational feature sets for subthalamic nucleus localization in dbs surgery with feature selection[END_REF][START_REF] Kim | Robust prediction of clinical deep brain stimulation target structures via the estimation of influential high-field mr atlases[END_REF][START_REF] Liu | Multi-modal learning-based pre-operative targeting in deep brain stimulation procedures[END_REF][START_REF] Kostoglou | Classification and prediction of clinical improvement in deep brain stimulation from intraoperative microelectrode recordings[END_REF][START_REF] Milletari | Hough-cnn: deep learning for segmentation of deep brain regions in mri and ultrasound[END_REF][START_REF] Baumgarten | Improvement of pyramidal tract side effect prediction using a data-driven method in subthalamic stimulation[END_REF][START_REF] Valsky | Stop! border ahead: A utomatic detection of subthalamic exit during deep brain stimulation surgery[END_REF][START_REF] Baumgarten | Data-driven prediction of the therapeutic window during subthalamic deep brain stimulation surgery[END_REF][START_REF] Koch | Automated machine learning for eeg-based classification of parkinsonâĂŹs disease patients[END_REF][START_REF] Kim | Automatic localization of the subthalamic nucleus on patient-specific clinical mri by incorporating 7 t mri and machine learning: Application in deep brain stimulation[END_REF][START_REF] Park | Deep learning-based deep brain stimulation targeting and clinical applications[END_REF][START_REF] Stuart | Machine learning for deep brain stimulation efficacy using dense array eeg[END_REF][START_REF] Jeroen | Machine learning prediction of motor response after deep brain stimulation in parkinson's disease[END_REF][START_REF] Singer | Post-operative electrode placement prediction in deep brain stimulation using support vector regression[END_REF][START_REF] Bermudez | Towards machine learning prediction of deep brain stimulation (dbs) intra-operative efficacy maps[END_REF][START_REF] Hosny | A novel deep lstm network for artifacts detection in microelectrode recordings[END_REF][START_REF] Farrokhi | Investigating risk factors and predicting complications in deep brain stimulation surgery with machine learning algorithms[END_REF][START_REF] Valsky | Real-time machine learning classification of pallidal borders during deep brain stimulation surgery[END_REF][START_REF] Baxter | Localisation of the subthalamic nucleus in mri via convolutional neural networks for deep brain stimulation planning[END_REF][START_REF] Syed | Application of machine learning using decision trees for prognosis of deep brain stimulation of globus pallidus internus for children with dystonia[END_REF][START_REF] Shang | Connectome-Based Model Predicts Deep Brain Stimulation Outcome in Parkinson's Disease[END_REF][START_REF] Charles W Lu | High density microelectrode recording predicts span of therapeutic tissue activation volumes in subthalamic deep brain stimulation for Parkinson disease[END_REF][START_REF] Peralta | SepaConvNet for Localizing the Subthalamic Nucleus using One Second Micro-Electrode Recordings[END_REF][START_REF] Pa Karthick | Automated detection of subthalamic nucleus in deep brain stimulation surgery for ParkinsonâĂŹs disease using microelectrode recordings and wavelet packet features[END_REF][START_REF] Kwang | Clinical outcome prediction from analysis of microelectrode recordings using deep learning in subthalamic deep brain stimulation for Parkinsons disease[END_REF][START_REF] Peralta | PassFlow: a multimodal workflow for predicting Deep Brain Stimulation Outcomes[END_REF][START_REF] Boutet | Predicting optimal deep brain stimulation parameters for ParkinsonâĂŹs disease using functional MRI and machine learning[END_REF][START_REF] Hosny | Deep convolutional neural network for the automated detection of Subthalamic nucleus using MER signals[END_REF][START_REF] Thibault Martin | Extending convolutional neural networks for localizing the subthalamic nucleus from micro-electrode recordings in ParkinsonâĂŹs disease[END_REF][START_REF] Thibault Martin | Adapting the listening time for micro-electrode recordings in deep brain stimulation interventions[END_REF][START_REF] Baxter | Segmentation of the subthalamic nucleus in MRI via Convolutional Neural Networks for deep brain stimulation planning[END_REF][START_REF] Vj Geraedts | Machine learning for automated EEG-based biomarkers of cognitive impairment during Deep Brain Stimulation screening in patients with ParkinsonâĂŹs Disease[END_REF][START_REF] Luka C Liebrand | Deep brain stimulation response in obsessive-compulsive disorder is associated with preoperative nucleus accumbens volume[END_REF][START_REF] Solomon | Deep-learning based fully automatic segmentation of the globus pallidus interna and externa using ultra-high 7 Tesla MRI[END_REF]. While it does not ensure the complete absence of data leakage (that could occur, for example, by mixing the validation and testing sets, or by selecting or normalize features with the whole database), these results can be considered as more reliable, and more representative of prospective-usage performance. Inter-patient variability remains an open problem for most of the contributions and is likely only solvable thanks to extensive data collection. Several papers stated that the lack of variability in the cohort limits the generalizability of the results

Acknowledgments

Maxime Peralta's PhD is funded by the Fondation pour la Recherche Médicale (FRM). John Baxter is supported by a Post-Doctoral Fellowship from the Natural Sciences and Research Council of Canada (NSERC) and by the Institut des Neurosciences Cliniques de Rennes (INCR).

These institutions were not involved in protocol development or in the data analysis.

Declarations

Conflicts of interest: The authors have no relevant financial or non-financial interests to disclose.

Highlights


Matthieu Kowalski
email: kowalski@cmi.univ-mrs.fr

Emmanuel Vincent
email: emmanuel.vincent@irisa.fr

Rémi Gribonval
email: remi.gribonval@irisa.fr

UNDER-DETERMINED SOURCE SEPARATION VIA MIXED-NORM REGULARIZED MINIMIZATION

We consider the problem of extracting the source signals from an under-determined convolutive mixture assuming that the mixing filters are known. We wish to exploit the sparsity and approximate disjointness of the time-frequency representations of the sources. However, classical time-frequency masking techniques cannot be directly applied due to the convolutive nature of the mixture. To address this problem, we first formulate it as the minimization of a functional combining a classical ℓ 2 discrepancy term between the observed mixture and the mixture reconstructed from the estimated sources and a sparse regularization term defined in terms of mixed ℓ 2 /ℓ 1 norms of source coefficients in a time-frequency domain. The minimum of the functional is then obtained by a thresholded Landweber iteration algorithm. Preliminary results are discussed for two synthetic audio mixtures.

UNDER-DETERMINED SOURCE SEPARATION

We consider the source separation problem for convolutive mixtures of the form

x = A ⋆ s , (1)
with x = (x 1 , . . . , x M ) T , x m ∈ R T x ∀m, the channels of the observed mixture, s = (s 1 , . . . , s N ) T , s n ∈ R T s ∀n, the unknown source signals, and A the mixing filter system. In the case of an under-determined mixture (M < N), this problem is generally split into two distinct steps, namely the estimation of the mixing system A and the extraction of the source signals s given A and x. Both steps are equally challenging. In the following, we focus on the estimation of s, assuming that A is known. Since the linear system (1) is under-determined, additional hypotheses are needed in order to "invert it" and estimate the original sources. A classical assumption is to suppose that only few sources are simultaneously active. This assumption is often roughly satisfied after a properly chosen transform. In the case of audio sources, this is generally the case after switching to the time-frequency domain using the short-time Fourier transform (STFT). Taking the STFT of both hand sides of (1) yields the narrowband approximation

x( f ,t) ≈ Â( f )ŝ( f ,t) , (2)
where x (resp. ŝ) is the STFT transform of x (resp. s) and  is the Fourier transform of A. The sparsity assumption means that for each time-frequency point ( f ,t), few sources are simultaneously active. A classical approach [START_REF] Zibulevsky | Blind source separation by sparse decomposition in a signal dictionary[END_REF][START_REF] Bofill | Underdetermined blind source separation using sparse representations[END_REF] to identify their contributions is to solve for each ( f ,t) the optimization problem

min ŝ ŝ( f ,t) p p = min ŝ N ∑ n=1 | ŝn (t, f )| p (3) subject to x( f ,t) = Â( f )ŝ( f ,t) ,
where the ℓ p norm is a measure of sparsity for p ≤ 1. Algorithms such as FOCUSS [START_REF] Rao | An affine scaling methodology for best basis selection[END_REF] or the study made in [START_REF] Vincent | Complex nonconvex l p norm minimization for underdetermined source separation[END_REF] can be used to solve [START_REF] Rao | An affine scaling methodology for best basis selection[END_REF]. While this approach results in M active sources per time-frequency point for small p [4], a single active source can be selected instead using a time-frequency masking algorithm such as DUET [START_REF] Yilmaz | Blind separation of speech mixtures via time-frequency masking[END_REF]. A variable number of active sources between 0 and M can also be determined via ℓ 1 -regularized minimization [START_REF] Togami | Sound source separation of overcomplete convolutive mixture using generalized sparseness[END_REF] or via a probabilistic approach [START_REF] Vielva | Underdetermined blind source separation using a probabilistic source sparsity model[END_REF].

The above source separation techniques cannot easily be applied to convolutive mixtures, or even anechoic mixtures with rather long delays, because the approximation (2) becomes to crude. Indeed, even if the sources happen to have sparse and almost disjoint time-frequency representations, their contributions a n ⋆ s n to the mixture (where a n is the n-th column of the filter matrix A) may have significantly more overlapping time-frequency representations. In this paper, we give a general approach to estimate s from (1), avoiding the approximation made in (2). Our approach is based on the minimization of a suitable functional, discussed in section 2. We then derive a thresholded Landweber iteration algorithm that minimizes this functional in section 3. Section 4 provides some results for audio data.

SOURCE SEPARATION BY MINIMIZATION OF A SUITABLE FUNCTIONAL

The goal of this section is to provide, under the form of an optimization problem, a simple formulation to estimate the original sources. The problem of source separation (1) can be viewed as a linear inverse problem. A classical estimate for s is then given by minimizing the discrepancy

x -A ⋆ s 2 2 , (4)
where the ℓ 2 norm is the Froebenius norm (the sum of the energy across all channels). In a Bayesian point of view, it corresponds to a Gaussian prior on the residual.

However, minimizing the discrepancy term (4) is often insufficient and does not provide a good separation of the sources. In particular, when the mixture is under-determined, there are infinitely many solutions and one must introduce additional knowledge about the sources, which can be done by choosing a suitable regularization term.

Usually, the source signals are modeled by their expansion s = ∑ k s k φ T k = sΦ T in a dictionary Φ associated with a time-frequency transform, such as the modified discrete cosine transform (MDCT) [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF], and the regularization term is defined in terms of the transformed coefficients s. This leads to the following functional min

s x -A ⋆ sΦ T 2 2 + λ E(s) , ( 5
)
where E is the regularization term, and λ ∈ R + a free penalty factor. The choice of the regularization term E depends on the assumptions made about the sources.

ℓ 1 regularization

One possible assumption is that the signals are sparse in the time-frequency domain [START_REF] Zibulevsky | Blind source separation by sparse decomposition in a signal dictionary[END_REF]. This assumption has been used in particular by the Multichannel Morphological Component Analysis (MMCA) [START_REF] Bobin | Sparsity and morphological diversity in blind source separation[END_REF] algorithm, which relies on the following functional for instantaneous over-determined source separation min

s x -A sΦ T 2 2 + λ s 1 . (6)
where Φ is chosen as a union of frames (or bases) adapted to different components of the sources. However, additional experiments showed us that this functional was not adapted to the under-determined case. In this case, the functional tends to threshold out the sources at high frequencies, and keep them all at low frequencies since they have higher energy. In other words, the sources are distorted at high frequencies and remain mixed together at low frequencies.

ℓ 1,2 regularization

In order to achieve better separation, we follow the usual assumption made in the under-determined case: only few sources are active for each time-frequency index. However we would like to avoid "over-sparsifying", that is thresholding out time-frequency regions which have less energy. The mixed norm defined hereafter is well adapted to this aim. Definition 1 Let p ≥ 1 and q ≥ 1. Let x ∈ R L be labeled by a double index (n, k) such that L = N × K. We call mixed norm of x, the norm ℓ p,q defined by

x p,q =   K ∑ k=1 N ∑ n=1 |x n,k | p q/p   1/q . ( 7
)
The cases p = +∞ and q = +∞ can be obtained by replacing the corresponding norm by the supremum.

This type of norm was studied in the context of functional spaces (see e.g [START_REF] Samarah | A Schur test for weighted mixed-norm spaces[END_REF] and references therein). Such norms allow the "structuring" of sparsity on the coefficients: contrary to the simple ℓ p norms, the coefficients are not considered to be independently distributed. The ℓ p,1 norm was used in the context of multichannel signal processing [START_REF]Piecewise linear source separation[END_REF][START_REF] Gribonval | Average case analysis of multichannel thresholding[END_REF][START_REF] Cotter | Sparse solutions to linear inverse problems with multiple measurement vectors[END_REF] and under the name of joint-sparsity [START_REF] Teschke | An iterative algorithm for nonlinear inverse problems with joint sparsity constraints in vector valued regimes and an application to color image inpainting[END_REF][START_REF] Fornasier | Recovery algorithm for vector-valued data with joint sparsity constraints[END_REF][START_REF] Fornasier | Iterative thresholding algorithms[END_REF]. The ℓ 2,1 norm is used in the statistical community for the group-lasso [START_REF] Yuan | Model selection and estimation in regression with grouped variables[END_REF] estimate.

In the context of source separation, the ℓ 1,2 norm can be used to model the sparsity through the channels. Indeed, let k be a time-frequency index and n be the channel index, and consider the following ℓ 1,2 norms on the sources:

s 2 1,2 = ∑ k ∑ n |s n,k | 2 .
Minimizing such a quantity will enforce sparsity across channels for each time-frequency index k, but not necessarily sparsity across time-frequency indices: for a given timefrequency index, we hope to keep only the coefficients corresponding to the most significant channels, but we expect some coefficients to be kept for most time-frequency indices. This behavior will be enlightened at the end of section 3.

Using this mixed norm as the regularization term in the functional (5), we obtain the following optimization problem

min s Ψ(s) := x -A ⋆ sΦ T 2 2 + λ s 2 1,2 . (8)
Such a functional remains convex and can be minimized by a thresholded Landweber iteration following [START_REF] Daubechies | An iterative thresholding algorithm for linear inverse problems with a sparsity constraint[END_REF]. This is detailed in the next section.

ITERATIVE MINIMIZATION ALGORITHM

For the sake of simplicity, we introduce the following linear operator, which reconstructs a mixture from a representation of the sources by applying first the source reconstruction operator •Φ T , then the filter matrix A:

F : R N×T s → R M×T x s → F (s) = A ⋆ sΦ T . (9)
Then, one can apply the thresholded Landweber iteration algorithm described in [START_REF] Kowalski | Sparse regression using mixed norms[END_REF], to minimize [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF]. We recall the iteration hereafter in the context of source separation.

If we differentiate the functional Ψ of ( 8) with respect to s n,k , the resulting system of variational equations is not readily solvable. Indeed these equations are coupled due to the presence of convolution products. In order to decouple the variational equations, we introduce the surrogate functional:

Ψ sur (s, z) = x -F (s)) 2 2 +C s -z 2 2 -F (s) -F (z) 2 2 + λ s 2 1,2 , ( 10
)
where C is chosen greater than the square of the operator norm of F , i.e. such that F (s) 2 2 < C s 2 2 for all s. Let us introduce the following notations: x := (A * ⋆ x)Φ and z := {A * ⋆ (A ⋆ (zΦ T ))}Φ, where we denoted by A * the adjoint operator of A, that is obtained by transposition of the channel indices and the source indices, and time reversal of A. Then, if we denote by

y n,k = xn,k +Cs n,k -zn,k C ,
the variational equations associated with Ψ sur are the following

∀(n, k) s n,k = y n,k -sgn(s n,k ) λ C s k 1 , (11)
where s k := (s 1,k , . . . , s N,k ) . Applying the theorem 3 of [START_REF] Kowalski | Sparse regression using mixed norms[END_REF], the solution of this equation and then the argmin ŝ of Ψ sur (s, z) with respect to s, is given coordinatewise by 1 ŝn,k = sgn y n,k y n,k -λ /C

1 + L k λ C |y k | + ∀(n, k) , (12)
with

|y k | = L k ∑ l=1
yk,l , where yk,l denote the coefficients |y k,l | ordered by descending order. The quantity L k is the number such that yk,L k +1 ≤ λ

L k +1 ∑ ℓ=1 yk,ℓ -yk,L k +1 and yk,L k > λ L k ∑ ℓ=1 yk,ℓ -yk,L k .
(13) Knowing this new "generalized thresholding operator", one can apply the thresholded Landweber iteration which is, starting from any s (0) ,

s (m+1) = argmin s Ψ sur s, s (m) .
These iterations lead then to the following practical algorithm for source separation, with the mixing filter system A known.

Algorithm 1 Let x = ( x1 , . . . , xN ) := (A * ⋆ x)Φ, s (0) = 0 m = 0 do s(m) = s(m) 1 , . . . , s (m)
n,k = xn,k +Cs (m) n,k - s(m) n,k C s (m+1) n,k = sgn y (m) n,k y (m) n,k - λ /C 1 + L (m) k λ C y (m) k + with L (m)
k and y (m) k as defined in (13) and (12). endfor until convergence It has been shown in [START_REF] Kowalski | Sparse regression using mixed norms[END_REF] that this algorithm converges to a global minimum of the original functional Ψ of (8).

We shall stress that in the extreme case λ = +∞, for a given index k, at least one coefficient will be kept among (y (m) 1,k , . . . , y (m) N,k ) at each iterations. Furthermore, equation [START_REF]Piecewise linear source separation[END_REF] assures that y (m) k = 0 if and only if s (m+1) k = 0. This remark guaranties that once the algorithm converged, at least one coefficient is non zero for each time-frequency index if xk = 0.

1 The operator + is defined as follow: x + = x if x ≥ 0, and 0 if x < 0 .

SOURCE SEPARATION RESULTS

We assessed the source separation performance of our algorithm on an anechoic audio mixture and a convolutive audio mixture obtained with simulated room impulse responses. We compared the performance of our algorithm with an implementation of DUET based on the narrowband approximation (2), using a STFT with a Gaussian window and defining Â( f ) as the Fourier transform of the known mixing filters. The window size for DUET has an influence on the performance: the window must be long enough in order for (2) to hold, but a too long window results in greater overlap of the sources. 1024 samples seems a good compromise for both examples. The resulting audio signals can be downloaded from [20] for listening comparisons.

We would like to stress again that the mixing filter system A is supposed to be known.

Anechoic mixture

The anechoic mixture consisted of four musical sources. Each source was sampled at 44100 Hz and had a duration of 2 17 samples (about 3 s ). The mixing system was obtained by combining the following instantaneous matrix (available on [21]) with delays between 0 and 512 samples 0.3420 0.6428 0.7934 0.9239 0.9397 0.7660 0.6088 0.3827 .

The chosen dictionary Φ was a MDCT basis with a sine window of 2048 samples (about 46 ms). Note that, similarly as above, a different window length could affect the results but that the chosen length seems a good compromise. We display in Figure 1 the SDR (Signal to Distortion Ratio), SIR (Signal to Interference Ratio) and SAR (Signal to Artifact Ratio) [START_REF] Vincent | Performance measurement in blind audio source separation[END_REF], averaged over the four sources, as a function of the penalty factor λ . A larger value of SDR/SIR/SAR means a better quality of the separation. As one could expect, the SIR of DUET is always better than the SIR of our algorithm: since DUET uses only one source for each time-frequency index, it avoids as much as possible interference between sources. This comes to the price of a larger mismatch between the mixture and its reconstruction from the estimated sources. On the opposite, the SAR obtained by our algorithm is systematically better, because of the discrepancy term.

The SDR curve gives an idea of the global performance, taking into account both artifacts and interferences. One can see that, for well chosen λ , our iterative algorithm performs better than DUET. Having to choose the penalty factor λ is certainly one of the main drawbacks of the functional approach [START_REF] Yilmaz | Blind separation of speech mixtures via time-frequency masking[END_REF]. On Figure 2 we plot the percentage of coefficients which are set to zero, and observe that the algorithm performs best when about 50 % coefficients are zeroed. Here, this means that on the average, at each time-frequency point two sources are considered active and two are set to zero. This seems a good compromise for a stereo mixture, and we believe that such a heuristic could be used as an adaptive strategy to tune the penalty factor in other settings.

Convolutive mixture

The convolutive mixture was generated by simulation of a recording in a meeting room with 250 ms reverberation time using a pair of omnidirectional microphones spaced by 1 m. The mixing filters were simulated by the image method and had a length of 4000 samples. The sources were four female speech recordings, sampled at 16000 Hz, and 2 17 samples long (about 8 s). The matrix Φ corresponded to a MDCT basis, with a window of 512 samples (about 32 ms). All the data (sources and mixing system) are available on [21].

As for the anechoic mixture, we display in Figure 3 the the SDR, SIR and SAR obtained by our algorithm, averaged over four sources, as a function of the penalty factor. These are compared to the ones obtained by DUET. Figure 4 shows the percentage of coefficients which are set to zero.

In this case too, one can remark that the iterative algorithm performs best when about 50 % of coefficients are set to zero, even if the point λ = 10 4 seems to perform better than the others in terms of SAR (and correspond to 75 % of coefficients set to zero). Indeed, these curves are made in average, and, by chance, the first and the second sources are well estimated for this value of λ . These two sources are the best estimated ones by DUET, and that can explain why the iterative algorithm performs well. Further experiments showed that, in general, the iterative algorithm performs best when 50 % of coefficients are set to zero 2 . We also compared briefly the results obtained by our minimization to the results given by an approach such (3), with p = 1 and p = 10 -3 . It appears that the Landweber iteration gives better results in term of SDR/SIR/SAR in average on the sources, but can provide lower SDR/SIR on particular sources (typically, on the third source of the used mixture).

Computational efficiency

The computation time of the proposed algorithm depends mainly on the efficiency of the time-frequency transform and that of the convolution by the mixing system and its adjoint. The convergence of the thresholded Landwebver iterative algorithm itself is known to be quite slow, but potential speedups are proposed in the literature [START_REF] Daubechies | Accelerated projected gradient methods for linear inverse problems with sparsity constraints[END_REF][START_REF] Fornasier | Domain decomposition methods for linear inverse problems with sparsity constraints[END_REF]. In the above experiments, the overall computation time was about 5 min in 2 For a stereo mixture of four sources.

the anechoic case and between 1 h for small values of λ and 5 min for very large values of λ in the convolutive case.

CONCLUSION

In this paper we developed an iterative optimization approach to separate convolutive mixtures using sparse source models in a time-frequency dictionary, when the mixing filter system is supposed to be known. Instead of the classical ℓ 1 penalty, we showed that the ℓ 1,2 mixed norm appears to be well adapted to respect the sparse hypothesis over the channels: for each time-frequency index, only few sources are active. The functional we proposed can be easily minimized by a thresholded Landweber iteration and leads to a simple algorithm that one can quickly implement.

We showed that this approach gives interesting results compared to DUET: despite an expected lower SIR, the SAR is always better, and the SDR can be improved too. Moreover, this algorithm works for any number of mixture and any number of sources to estimate, i.e. for any M, N ∈ N with M < N.

One of the major benefits of the proposed approach, is that one can make additional assumptions on the sources, and take them into account in the regularization term. Indeed, the next step should be to try to incorporate some structures on each source, as, for example, harmonicity for music sources.

N:=

  {A * ⋆ (A ⋆ (s (m) Φ T ))}Φ, for each time-frequency index k and each source index n y (m)

Figure 1 :

 1 Figure 1: SDR, SIR, SAR of the proposed iterative algorithm and DUET as a function of the penalty factor λ (log scale).

Figure 2 :

 2 Figure 2: Percentage of coefficients set to zero as a function of the penalty factor λ (log scale).

Figure 3 :

 3 Figure 3: SDR, SIR, SAR of the proposed iterative algorithm and DUET as function of the penalty factor λ (log scale).

Figure 4 :

 4 Figure 4: Percentage of coefficients set to zero as a function of the penalty factor λ (log scale).

Acknowledgments

We wish to thank B. Torrésani for interesting discussions and his interest to this work.