Motivation Non-negative matrix factorization (NMF) is a common tool for obtaining low-rank approximations of non-negative data matrices and has been widely used in machine learning, e.g. for supporting feature extraction in high-dimensional classification tasks. In its classical form, NMF is an unsupervised method, i.e. the class labels of the training data are not used when computing the NMF. However, incorporating the classification labels into the NMF algorithms allows to specifically guide them toward the extraction of data patterns relevant for discriminating the respective classes. This approach is particularly suited for the analysis of mass spectrometry imaging (MSI) data in clinical applications, such as tumor typing and classification, which are among the most challenging tasks in pathology. Thus, we investigate algorithms for extracting tumor-specific spectral patterns from MSI data by NMF methods. Results In this article, we incorporate a priori class labels into the NMF cost functional by adding appropriate supervised penalty terms. Numerical experiments on a MALDI imaging dataset confirm that the novel supervised NMF methods lead to significantly better classification accuracy and stability as compared with other standard approaches. Availability and implementaton https://gitlab.informatik.uni-bremen.de/digipath/Supervised_NMF_Methods_for_MALDI.git Supplementary information Supplementary data are available at Bioinformatics online.

A primary interest in dynamic inverse problems is to identify the underlying temporal behaviour of the system from outside measurements. In this work, we consider the case, where the target can be represented by a decomposition of spatial and temporal basis functions and hence can be efficiently represented by a low-rank decomposition. We then propose a joint reconstruction and low-rank decomposition method based on the Nonnegative Matrix Factorisation to obtain the unknown from highly undersampled dynamic measurement data. The proposed framework allows for flexible incorporation of separate regularisers for spatial and temporal features. For the special case of a stationary operator, we can effectively use the decomposition to reduce the computational complexity and obtain a substantial speed-up. The proposed methods are evaluated for three simulated phantoms and we compare the obtained results to a separate low-rank reconstruction and subsequent decomposition approach based on the widely used principal component analysis.

Classical approaches in cluster analysis are typically based on a feature space analysis. However, many applications lead to datasets with additional spatial information and a ground truth with spatially coherent classes, which will not necessarily be reconstructed well by standard clustering methods. Motivated by applications in hyperspectral imaging, we introduce in this work clustering models based on Orthogonal Nonnegative Matrix Factorization (ONMF), which include an additional Total Variation (TV) regularization procedure on the cluster membership matrix to enforce the needed spatial coherence in the clusters. We propose several approaches with different optimization techniques, where the TV regularization is either performed as a subsequent post-processing step or included into the clustering algorithm. Finally, we provide a numerical evaluation of 12 different TV regularized ONMF methods on a hyperspectral dataset obtained from a matrix-assisted laser desorption/ionization imaging measurement, which leads to significantly better clustering results compared to classical clustering models.

Motivated by applications in hyperspectral imaging, we investigate methods for approximating a high-dimensional non-negative matrix Y by a product of two lower-dimensional, non-negative matrices K and X. This so-called non-negative matrix factorization is based on defining suitable Tikhonov functionals, which combine a discrepancy measure for Y ≈KX with penalty terms for enforcing additional properties of K and X. The minimization is based on alternating minimization with respect to K and X, where in each iteration step one replaces the original Tikhonov functional by a locally defined surrogate functional. The choice of surrogate functionals is crucial: It should allow a comparatively simple minimization and simultaneously its first-order optimality condition should lead to multiplicative update rules, which automatically preserve non-negativity of the iterates. We review the most standard construction principles for surrogate functionals for Frobenius-norm and Kullback–Leibler discrepancy measures. We extend the known surrogate constructions by a general framework, which allows to add a large variety of penalty terms. The paper finishes by deriving the corresponding alternating minimization schemes explicitly and by applying these methods to MALDI imaging data.

Motivated by the rapidly growing field of mathematics for operator approximation with neural networks, we present a novel universal operator approximation theorem for a broad class of encoder-decoder architectures. In this study, we focus on approximating continuous operators in $\mathcal{C}(\mathcal{X}, \mathcal{Y})$, where $\mathcal{X}$ and $\mathcal{Y}$ are infinite-dimensional normed or metric spaces, and we consider uniform convergence on compact subsets of $\mathcal{X}$. Unlike standard results in the operator learning literature, we investigate the case where the approximating operator sequence can be chosen independently of the compact sets. Taking a topological perspective, we analyze different types of operator approximation and show that compact-set-independent approximation is a strictly stronger property in most relevant operator learning frameworks. To establish our results, we introduce a new approximation property tailored to encoder-decoder architectures, which enables us to prove a universal operator approximation theorem ensuring uniform convergence on every compact subset. This result unifies and extends existing universal operator approximation theorems for various encoder-decoder architectures, including classical DeepONets, BasisONets, special cases of MIONets, architectures based on frames and other related approaches.

A primary interest in dynamic inverse problems is to identify the underlying temporal behaviour of the system from outside measurements. In this work we consider the case, where the target can be represented by a decomposition of spatial and temporal basis functions and hence can be efficiently represented by a low-rank decomposition. We then propose a joint reconstruction and low-rank decomposition method based on the Nonnegative Matrix Factorisation to obtain the unknown from highly undersampled dynamic measurement data. The proposed framework allows for flexible incorporation of separate regularisers for spatial and temporal features. For the special case of a stationary operator, we can effectively use the decomposition to reduce the computational complexity and obtain a substantial speed-up. The proposed methods are evaluated for two simulated phantoms and we compare the obtained results to a separate low-rank reconstruction and subsequent decomposition approach based on the widely used principal component analysis.