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Morten Mørup

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17 papers

2 author rows

TMLR Journal 2026 Journal Article

E$^2$M: Double Bounded $\alpha$-Divergence Optimization for Tensor-based Discrete Density Estimation

Kazu Ghalamkari
Jesper Løve Hinrich
Morten Mørup

Tensor-based discrete density estimation requires flexible modeling and proper divergence criteria to enable effective learning; however, traditional approaches using α-divergence face analytical challenges due to the α-power terms in the objective function, which hinder the derivation of closed-form update rules. We present a generalization of the expectation-maximization~(EM) algorithm, called the E2M algorithm. It circumvents this issue by first relaxing the optimization into the minimization of a surrogate objective based on the Kullback–Leibler (KL) divergence, which is tractable via the standard EM algorithm, and subsequently applying a tensor many-body approximation in the M-step to enable simultaneous closed-form updates of all parameters. Our approach offers flexible modeling for not only a variety of low-rank structures, including the CP, Tucker, and Tensor Train formats, but also their mixtures, thus allowing us to leverage the strengths of different low-rank structures. We evaluate the effectiveness of our approach on synthetic and real datasets, highlighting its comparable convergence to gradient-based procedures, robustness to outliers, and favorable density estimation performance compared to prominent existing tensor-based methods.

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TMLR Journal 2025 Journal Article

Estimating the Event-Related Potential from Few EEG Trials

Anders Vestergaard Nørskov
Kasper Jørgensen
Alexander Neergaard Zahid
Morten Mørup

Event-related potentials (ERP) are measurements of brain activity with wide applications in basic and clinical neuroscience, that are typically estimated using the average of many trials of electroencephalography signals (EEG) to sufficiently reduce noise and signal variability. We introduce EEG2ERP, a novel uncertainty-aware autoencoder approach that maps an arbitrary number of EEG trials to their associated ERP. To account for the ERP uncertainty we use bootstrapped training targets and introduce a separate variance decoder to model the uncertainty of the estimated ERP. We evaluate our approach in the challenging zero-shot scenario of generalizing to new subjects considering three different publicly available data sources; i) the comprehensive ERP CORE dataset that includes over 50,000 EEG trials across six ERP paradigms from 40 subjects, ii) the large P300 Speller BCI dataset, and iii) a neuroimaging dataset on face perception consisting of both EEG and magnetoencephalography (MEG) data. We consistently find that our method in the few trial regime provides substantially better ERP estimates than commonly used conventional and robust averaging procedures. EEG2ERP is the first deep learning approach to map EEG signals to their associated ERP, moving toward reducing the number of trials necessary for ERP research. Code is available at https://github.com/andersxa/EEG2ERP.

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ICLR Conference 2025 Conference Paper

How Low Can You Go? Searching for the Intrinsic Dimensionality of Complex Networks using Metric Node Embeddings

Nikolaos Nakis
Niels Raunkjær Holm
Andreas Lyhne Fiehn
Morten Mørup

Low-dimensional embeddings are essential for machine learning tasks involving graphs, such as node classification, link prediction, community detection, network visualization, and network compression. Although recent studies have identified exact low-dimensional embeddings, the limits of the required embedding dimensions remain unclear. We presently prove that lower dimensional embeddings are possible when using Euclidean metric embeddings as opposed to vector-based Logistic PCA (LPCA) embeddings. In particular, we provide an efficient logarithmic search procedure for identifying the exact embedding dimension and demonstrate how metric embeddings enable inference of the exact embedding dimensions of large-scale networks by exploiting that the metric properties can be used to provide linearithmic scaling. Empirically, we show that our approach extracts substantially lower dimensional representations of networks than previously reported for small-sized networks. For the first time, we demonstrate that even large-scale networks can be effectively embedded in very low-dimensional spaces, and provide examples of scalable, exact reconstruction for graphs with up to a million nodes. Our approach highlights that the intrinsic dimensionality of networks is substantially lower than previously reported and provides a computationally efficient assessment of the exact embedding dimension also of large-scale networks. The surprisingly low dimensional representations achieved demonstrate that networks in general can be losslessly represented using very low dimensional feature spaces, which can be used to guide existing network analysis tasks from community detection and node classification to structure revealing exact network visualizations.

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AAAI Conference 2024 Conference Paper

Continuous-Time Graph Representation with Sequential Survival Process

Abdulkadir Çelikkanat
Nikolaos Nakis
Morten Mørup

Over the past two decades, there has been a tremendous increase in the growth of representation learning methods for graphs, with numerous applications across various fields, including bioinformatics, chemistry, and the social sciences. However, current dynamic network approaches focus on discrete-time networks or treat links in continuous-time networks as instantaneous events. Therefore, these approaches have limitations in capturing the persistence or absence of links that continuously emerge and disappear over time for particular durations. To address this, we propose a novel stochastic process relying on survival functions to model the durations of links and their absences over time. This forms a generic new likelihood specification explicitly accounting for intermittent edge-persistent networks, namely GraSSP: Graph Representation with Sequential Survival Process. We apply the developed framework to a recent continuous time dynamic latent distance model characterizing network dynamics in terms of a sequence of piecewise linear movements of nodes in latent space. We quantitatively assess the developed framework in various downstream tasks, such as link prediction and network completion, demonstrating that the developed modeling framework accounting for link persistence and absence well tracks the intrinsic trajectories of nodes in a latent space and captures the underlying characteristics of evolving network structure.

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NeurIPS Conference 2023 Conference Paper

CSLP-AE: A Contrastive Split-Latent Permutation Autoencoder Framework for Zero-Shot Electroencephalography Signal Conversion

Anders Nørskov
Alexander Neergaard Zahid
Morten Mørup

Electroencephalography (EEG) is a prominent non-invasive neuroimaging technique providing insights into brain function. Unfortunately, EEG data exhibit a high degree of noise and variability across subjects hampering generalizable signal extraction. Therefore, a key aim in EEG analysis is to extract the underlying neural activation (content) as well as to account for the individual subject variability (style). We hypothesize that the ability to convert EEG signals between tasks and subjects requires the extraction of latent representations accounting for content and style. Inspired by recent advancements in voice conversion technologies, we propose a novel contrastive split-latent permutation autoencoder (CSLP-AE) framework that directly optimizes for EEG conversion. Importantly, the latent representations are guided using contrastive learning to promote the latent splits to explicitly represent subject (style) and task (content). We contrast CSLP-AE to conventional supervised, unsupervised (AE), and self-supervised (contrastive learning) training and find that the proposed approach provides favorable generalizable characterizations of subject and task. Importantly, the procedure also enables zero-shot conversion between unseen subjects. While the present work only considers conversion of EEG, the proposed CSLP-AE provides a general framework for signal conversion and extraction of content (task activation) and style (subject variability) components of general interest for the modeling and analysis of biological signals.

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YNIMG Journal 2022 Journal Article

Psilocybin modulation of time-varying functional connectivity is associated with plasma psilocin and subjective effects

Anders S. Olsen
Anders Lykkebo-Valløe
Brice Ozenne
Martin K. Madsen
Dea S. Stenbæk
Sophia Armand
Morten Mørup
Melanie Ganz

BACKGROUND: Psilocin, the neuroactive metabolite of psilocybin, is a serotonergic psychedelic that induces an acute altered state of consciousness, evokes lasting changes in mood and personality in healthy individuals, and has potential as an antidepressant treatment. Examining the acute effects of psilocin on resting-state time-varying functional connectivity implicates network-level connectivity motifs that may underlie acute and lasting behavioral and clinical effects. AIM: Evaluate the association between resting-state time-varying functional connectivity (tvFC) characteristics and plasma psilocin level (PPL) and subjective drug intensity (SDI) before and right after intake of a psychedelic dose of psilocybin in healthy humans. METHODS: Fifteen healthy individuals completed the study. Before and at multiple time points after psilocybin intake, we acquired 10-minute resting-state blood-oxygen-level-dependent functional magnetic resonance imaging scans. Leading Eigenvector Dynamics Analysis (LEiDA) and diametrical clustering were applied to estimate discrete, sequentially active brain states. We evaluated associations between the fractional occurrence of brain states during a scan session and PPL and SDI using linear mixed-effects models. We examined associations between brain state dwell time and PPL and SDI using frailty Cox proportional hazards survival analysis. RESULTS: Fractional occurrences for two brain states characterized by lateral frontoparietal and medial fronto-parietal-cingulate coherence were statistically significantly negatively associated with PPL and SDI. Dwell time for these brain states was negatively associated with SDI and, to a lesser extent, PPL. Conversely, fractional occurrence and dwell time of a fully connected brain state partly associated with motion was positively associated with PPL and SDI. CONCLUSION: Our findings suggest that the acute perceptual psychedelic effects induced by psilocybin may stem from drug-level associated decreases in the occurrence and duration of lateral and medial frontoparietal connectivity motifs. We apply and argue for a modified approach to modeling eigenvectors produced by LEiDA that more fully acknowledges their underlying structure. Together these findings contribute to a more comprehensive neurobiological framework underlying acute effects of serotonergic psychedelics.

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YNIMG Journal 2021 Journal Article

Using connectomics for predictive assessment of brain parcellations

Kristoffer J. Albers
Karen S. Ambrosen
Matthew G. Liptrot
Tim B. Dyrby
Mikkel N. Schmidt
Morten Mørup

The organization of the human brain remains elusive, yet is of great importance to the mechanisms of integrative brain function. At the macroscale, its structural and functional interpretation is conventionally assessed at the level of cortical units. However, the definition and validation of such cortical parcellations are problematic due to the absence of a true gold standard. We propose a framework for quantitative evaluation of brain parcellations via statistical prediction of connectomics data. Specifically, we evaluate the extent in which the network representation at the level of cortical units (defined as parcels) accounts for high-resolution brain connectivity. Herein, we assess the pertinence and comparative ranking of ten existing parcellation atlases to account for functional (FC) and structural connectivity (SC) data based on data from the Human Connectome Project (HCP), and compare them to data-driven as well as spatially-homogeneous geometric parcellations including geodesic parcellations with similar size distributions as the atlases. We find substantial discrepancy in parcellation structures that well characterize FC and SC and differences in what well represents an individual's functional connectome when compared against the FC structure that is preserved across individuals. Surprisingly, simple spatial homogenous parcellations generally provide good representations of both FC and SC, but are inferior when their within-parcellation distribution of individual parcel sizes is matched to that of a valid atlas. This suggests that the choice of fine grained and coarse representations used by existing atlases are important. However, we find that resolution is more critical than the exact border location of parcels.

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YNIMG Journal 2020 Journal Article

Validation of structural brain connectivity networks: The impact of scanning parameters

Karen S. Ambrosen
Simon F. Eskildsen
Max Hinne
Kristine Krug
Henrik Lundell
Mikkel N. Schmidt
Marcel A.J. van Gerven
Morten Mørup

Evaluation of the structural connectivity (SC) of the brain based on tractography has mainly focused on the choice of diffusion model, tractography algorithm, and their respective parameter settings. Here, we systematically validate SC derived from a post mortem monkey brain, while varying key acquisition parameters such as the b-value, gradient angular resolution and image resolution. As gold standard we use the connectivity matrix obtained invasively with histological tracers by Markov et al. (2014). As performance metric, we use cross entropy as a measure that enables comparison of the relative tracer labeled neuron counts to the streamline counts from tractography. We find that high angular resolution and high signal-to-noise ratio are important to estimate SC, and that SC derived from low image resolution (1. 03 mm3) are in better agreement with the tracer network, than those derived from high image resolution (0. 53 mm3) or at an even lower image resolution (2. 03 mm3). In contradiction, sensitivity and specificity analyses suggest that if the angular resolution is sufficient, the balanced compromise in which sensitivity and specificity are identical remains 60–64% regardless of the other scanning parameters. Interestingly, the tracer graph is assumed to be the gold standard but by thresholding, the balanced compromise increases to 70–75%. Hence, by using performance metrics based on binarized tracer graphs, one risks losing important information, changing the performance of SC graphs derived by tractography and their dependence of different scanning parameters.

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YNICL Journal 2019 Journal Article

Individuals with 22q11.2 deletion syndrome show intact prediction but reduced adaptation in responses to repeated sounds: Evidence from Bayesian mapping

Kit Melissa Larsen
Morten Mørup
Michelle Rosgaard Birknow
Elvira Fischer
Line Olsen
Michael Didriksen
William Frans Christiaan Baaré
Thomas Mears Werge

One of the most common copy number variants, the 22q11.2 microdeletion, confers an increased risk for schizophrenia. Since schizophrenia has been associated with an aberrant neural response to repeated stimuli through both reduced adaptation and prediction, we here hypothesized that this may also be the case in nonpsychotic individuals with a 22q11.2 deletion. We recorded high-density EEG from 19 individuals with 22q11.2 deletion syndrome (12-25 years), as well as 27 healthy volunteers with comparable age and sex distribution, while they listened to a sequence of sounds arranged in a roving oddball paradigm. Using posterior probability maps and dynamic causal modelling we tested three different models accounting for repetition dependent changes in cortical responses as well as in effective connectivity; namely an adaptation model, a prediction model, and a model including both adaptation and prediction. Repetition-dependent changes were parametrically modulated by a combination of adaptation and prediction and were apparent in both cortical responses and in the underlying effective connectivity. This effect was reduced in individuals with a 22q11.2 deletion and was negatively correlated with negative symptom severity. Follow-up analysis showed that the reduced effect of the combined adaptation and prediction model seen in individuals with 22q11.2 deletion was driven by reduced adaptation rather than prediction failure. Our findings suggest that adaptation is reduced in individuals with a 22q11.2 deletion, which can be interpreted in light of the framework of predictive coding as a failure to suppress prediction errors.

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YNIMG Journal 2018 Journal Article

Predictive assessment of models for dynamic functional connectivity

Søren F.V. Nielsen
Mikkel N. Schmidt
Kristoffer H. Madsen
Morten Mørup

In neuroimaging, it has become evident that models of dynamic functional connectivity (dFC), which characterize how intrinsic brain organization changes over time, can provide a more detailed representation of brain function than traditional static analyses. Many dFC models in the literature represent functional brain networks as a meta-stable process with a discrete number of states; however, there is a lack of consensus on how to perform model selection and learn the number of states, as well as a lack of understanding of how different modeling assumptions influence the estimated state dynamics. To address these issues, we consider a predictive likelihood approach to model assessment, where models are evaluated based on their predictive performance on held-out test data. Examining several prominent models of dFC (in their probabilistic formulations) we demonstrate our framework on synthetic data, and apply it on two real-world examples: a face recognition EEG experiment and resting-state fMRI. Our results evidence that both EEG and fMRI are better characterized using dynamic modeling approaches than by their static counterparts, but we also demonstrate that one must be cautious when interpreting dFC because parameter settings and modeling assumptions, such as window lengths and emission models, can have a large impact on the estimated states and consequently on the interpretation of the brain dynamics.

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NeurIPS Conference 2016 Conference Paper

Completely random measures for modelling block-structured sparse networks

Tue Herlau
Mikkel Schmidt
Morten Mørup

Statistical methods for network data often parameterize the edge-probability by attributing latent traits such as block structure to the vertices and assume exchangeability in the sense of the Aldous-Hoover representation theorem. These assumptions are however incompatible with traits found in real-world networks such as a power-law degree-distribution. Recently, Caron & Fox (2014) proposed the use of a different notion of exchangeability after Kallenberg (2005) and obtained a network model which permits edge-inhomogeneity, such as a power-law degree-distribution whilst retaining desirable statistical properties. However, this model does not capture latent vertex traits such as block-structure. In this work we re-introduce the use of block-structure for network models obeying Kallenberg’s notion of exchangeability and thereby obtain a collapsed model which both admits the inference of block-structure and edge inhomogeneity. We derive a simple expression for the likelihood and an efficient sampling method. The obtained model is not significantly more difficult to implement than existing approaches to block-modelling and performs well on real network datasets.

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YNIMG Journal 2014 Journal Article

Non-parametric Bayesian graph models reveal community structure in resting state fMRI

Kasper Winther Andersen
Kristoffer H. Madsen
Hartwig Roman Siebner
Mikkel N. Schmidt
Morten Mørup
Lars Kai Hansen

Modeling of resting state functional magnetic resonance imaging (rs-fMRI) data using network models is of increasing interest. It is often desirable to group nodes into clusters to interpret the communication patterns between nodes. In this study we consider three different nonparametric Bayesian models for node clustering in complex networks. In particular, we test their ability to predict unseen data and their ability to reproduce clustering across datasets. The three generative models considered are the Infinite Relational Model (IRM), Bayesian Community Detection (BCD), and the Infinite Diagonal Model (IDM). The models define probabilities of generating links within and between clusters and the difference between the models lies in the restrictions they impose upon the between-cluster link probabilities. IRM is the most flexible model with no restrictions on the probabilities of links between clusters. BCD restricts the between-cluster link probabilities to be strictly lower than within-cluster link probabilities to conform to the community structure typically seen in social networks. IDM only models a single between-cluster link probability, which can be interpreted as a background noise probability. These probabilistic models are compared against three other approaches for node clustering, namely Infomap, Louvain modularity, and hierarchical clustering. Using 3 different datasets comprising healthy volunteers' rs-fMRI we found that the BCD model was in general the most predictive and reproducible model. This suggests that rs-fMRI data exhibits community structure and furthermore points to the significance of modeling heterogeneous between-cluster link probabilities.

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AILAW Journal 2013 Journal Article

Cross-categorization of legal concepts across boundaries of legal systems: in consideration of inferential links

Fumiko Kano Glückstad
Tue Herlau
Mikkel N. Schmidt
Morten Mørup

Abstract This work contrasts Giovanni Sartor’s view of inferential semantics of legal concepts (Sartor in Artif Intell Law 17: 217–251, 2009 ) with a probabilistic model of theory formation (Kemp et al. in Cognition 114: 165–196, 2010 ). The work further explores possibilities of implementing Kemp’s probabilistic model of theory formation in the context of mapping legal concepts between two individual legal systems. For implementing the legal concept mapping, we propose a cross-categorization approach that combines three mathematical models: the Bayesian Model of Generalization (BMG; Tenenbaum and Griffiths in Behav Brain Sci 4: 629–640, 2001 ), the probabilistic model of theory formation, i. e. , the Infinite Relational Model (IRM) first introduced by Kemp et al. (The twenty-first national conference on artificial intelligence, 2006, Cognition 114: 165–196, 2010 ) and its extended model, i. e. , the normal-IRM (n-IRM) proposed by Herlau et al. (IEEE International Workshop on Machine Learning for Signal Processing, 2012 ). We apply our cross-categorization approach to datasets where legal concepts related to educational systems are respectively defined by the Japanese- and the Danish authorities according to the International Standard Classification of Education. The main contribution of this work is the proposal of a conceptual framework of the cross-categorization approach that, inspired by Sartor (Artif Intell Law 17: 217–251, 2009 ), attempts to explain reasoner’s inferential mechanisms.

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ICML Conference 2013 Conference Paper

Modeling Temporal Evolution and Multiscale Structure in Networks

Tue Herlau
Morten Mørup
Mikkel N. Schmidt

Many real-world networks exhibit both temporal evolution and multiscale structure. We propose a model for temporally correlated multifurcating hierarchies in complex networks which jointly capture both effects. We use the Gibbs fragmentation tree as prior over multifurcating trees and a change-point model to account for the temporal evolution of each vertex. We demonstrate that our model is able to infer time-varying multiscale structure in synthetic as well as three real world time-evolving complex networks. Our modeling of the temporal evolution of hierarchies brings new insights into the changing roles and position of entities and possibilities for better understanding these dynamic complex systems.

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NeurIPS Conference 2010 Conference Paper

Infinite Relational Modeling of Functional Connectivity in Resting State fMRI

Morten Mørup
Kristoffer Madsen
Anne-Marie Dogonowski
Hartwig Siebner
Lars Hansen

Functional magnetic resonance imaging (fMRI) can be applied to study the functional connectivity of the neural elements which form complex network at a whole brain level. Most analyses of functional resting state networks (RSN) have been based on the analysis of correlation between the temporal dynamics of various regions of the brain. While these models can identify coherently behaving groups in terms of correlation they give little insight into how these groups interact. In this paper we take a different view on the analysis of functional resting state networks. Starting from the definition of resting state as functional coherent groups we search for functional units of the brain that communicate with other parts of the brain in a coherent manner as measured by mutual information. We use the infinite relational model (IRM) to quantify functional coherent groups of resting state networks and demonstrate how the extracted component interactions can be used to discriminate between functional resting state activity in multiple sclerosis and normal subjects.

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YNIMG Journal 2008 Journal Article

Shift-invariant multilinear decomposition of neuroimaging data

Morten Mørup
Lars Kai Hansen
Sidse Marie Arnfred
Lek-Heng Lim
Kristoffer Hougaard Madsen

We present an algorithm for multilinear decomposition that allows for arbitrary shifts along one modality. The method is applied to neural activity arranged in the three modalities space, time, and trial. Thus, the algorithm models neural activity as a linear superposition of components with a fixed time course that may vary across either trials or space in its overall intensity and latency. Its utility is demonstrated on simulated data as well as actual EEG, and fMRI data. We show how shift-invariant multilinear decompositions of multiway data can successfully cope with variable latencies in data derived from neural activity — a problem that has caused degenerate solutions especially in modeling neuroimaging data with instantaneous multilinear decompositions. Our algorithm is available for download at www. erpwavelab. org.

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YNIMG Journal 2006 Journal Article

Parallel Factor Analysis as an exploratory tool for wavelet transformed event-related EEG

Morten Mørup
Lars Kai Hansen
Christoph S. Herrmann
Josef Parnas
Sidse M. Arnfred

In the decomposition of multi-channel EEG signals, principal component analysis (PCA) and independent component analysis (ICA) have widely been used. However, as both methods are based on handling two-way data, i. e. two-dimensional matrices, multi-way methods might improve the interpretation of frequency transformed multi-channel EEG of channel × frequency × time data. The multi-way decomposition method Parallel Factor (PARAFAC), also named Canonical Decomposition (CANDECOMP), was recently used to decompose the wavelet transformed ongoing EEG of channel × frequency × time (Miwakeichi, F. , Martinez-Montes, E. , Valdes-Sosa, P. A. , Nishiyama, N. , Mizuhara, H. , Yamaguchi, Y. , 2004. Decomposing EEG data into space–time–frequency components using parallel factor analysis. Neuroimage 22, 1035–1045). In this article, PARAFAC is used for the first time to decompose wavelet transformed event-related EEG given by the inter-trial phase coherence (ITPC) encompassing ANOVA analysis of differences between conditions and 5-way analysis of channel × frequency × time × subject × condition. A flow chart is presented on how to perform data exploration using the PARAFAC decomposition on multi-way arrays. This includes (A) channel × frequency × time 3-way arrays of F test values from a repeated measures analysis of variance (ANOVA) between two stimulus conditions; (B) subject-specific 3-way analyses; and (C) an overall 5-way analysis of channel × frequency × time × subject × condition. The PARAFAC decompositions were able to extract the expected features of a previously reported ERP paradigm: namely, a quantitative difference of coherent occipital gamma activity between conditions of a visual paradigm. Furthermore, the method revealed a qualitative difference which has not previously been reported. The PARAFAC decomposition of the 3-way array of ANOVA F test values clearly showed the difference of regions of interest across modalities, while the 5-way analysis enabled visualization of both quantitative and qualitative differences. Consequently, PARAFAC is a promising data exploratory tool in the analysis of the wavelets transformed event-related EEG.

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