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Reduced order modelling relies on representing complex dynamical systems using simplified modes, which can be achieved through Koopman operator analysis. However, computing Koopman eigen pairs for high-dimensional observable data can be inefficient. This paper proposes using deep autoencoders, a type of deep learning technique, to perform non-linear geometric transformations on raw data before computing Koopman eigen vectors. The encoded data produced by the deep autoencoder is diffeomorphic to a manifold of the dynamical system, and has a significantly lower dimension than the raw data. To handle high-dimensional time series data, Takens's time delay embedding is presented as a pre-processing technique. The paper concludes by presenting examples of these techniques in action.

The proliferation of automated data collection schemes and the advances in sensorics are increasing the amount of data we are able to monitor in real-time. However, given the high annotation costs and the time required by quality inspections, data is often available in an unlabeled form. This is fostering the use of active learning for the development of soft sensors and predictive models. In production, instead of performing random inspections to obtain product information, labels are collected by evaluating the information content of the unlabeled data. Several query strategy frameworks for regression have been proposed in the literature but most of the focus has been dedicated to the static pool-based scenario. In this work, we propose a new strategy for the stream-based scenario, where instances are sequentially offered to the learner, which must instantaneously decide whether to perform the quality check to obtain the label or discard the instance. The approach is inspired by the optimal experimental design theory and the iterative aspect of the decision-making process is tackled by setting a threshold on the informativeness of the unlabeled data points. The proposed approach is evaluated using numerical simulations and the Tennessee Eastman Process simulator. The results confirm that selecting the examples suggested by the proposed algorithm allows for a faster reduction in the prediction error.

Following a fast initial breakthrough in graph based learning, Graph Neural Networks (GNNs) have reached a widespread application in many science and engineering fields, prompting the need for methods to understand their decision process. GNN explainers have started to emerge in recent years, with a multitude of methods both novel or adapted from other domains. To sort out this plethora of alternative approaches, several studies have benchmarked the performance of different explainers in terms of various explainability metrics. However, these earlier works make no attempts at providing insights into why different GNN architectures are more or less explainable, or which explainer should be preferred in a given setting. In this survey, we fill these gaps by devising a systematic experimental study, which tests ten explainers on eight representative architectures trained on six carefully designed graph and node classification datasets. With our results we provide key insights on the choice and applicability of GNN explainers, we isolate key components that make them usable and successful and provide recommendations on how to avoid common interpretation pitfalls. We conclude by highlighting open questions and directions of possible future research.

Recently there has been great interest in operator learning, where networks learn operators between function spaces from an essentially infinite-dimensional perspective. In this work we present results for when the operators learned by these networks are injective and surjective. As a warmup, we combine prior work in both the finite-dimensional ReLU and operator learning setting by giving sharp conditions under which ReLU layers with linear neural operators are injective. We then consider the case the case when the activation function is pointwise bijective and obtain sufficient conditions for the layer to be injective. We remark that this question, while trivial in the finite-rank case, is subtler in the infinite-rank case and is proved using tools from Fredholm theory. Next, we prove that our supplied injective neural operators are universal approximators and that their implementation, with finite-rank neural networks, are still injective. This ensures that injectivity is not `lost' in the transcription from analytical operators to their finite-rank implementation with networks. Finally, we conclude with an increase in abstraction and consider general conditions when subnetworks, which may be many layers deep, are injective and surjective and provide an exact inversion from a `linearization.' This section uses general arguments from Fredholm theory and Leray-Schauder degree theory for non-linear integral equations to analyze the mapping properties of neural operators in function spaces. These results apply to subnetworks formed from the layers considered in this work, under natural conditions. We believe that our work has applications in Bayesian UQ where injectivity enables likelihood estimation and in inverse problems where surjectivity and injectivity corresponds to existence and uniqueness, respectively.

Neuroradiologists and neurosurgeons increasingly opt to use functional magnetic resonance imaging (fMRI) to map functionally relevant brain regions for noninvasive presurgical planning and intraoperative neuronavigation. This application requires a high degree of spatial accuracy, but the fMRI signal-to-noise ratio (SNR) decreases as spatial resolution increases. In practice, fMRI scans can be collected at multiple spatial resolutions, and it is of interest to make more accurate inference on brain activity by combining data with different resolutions. To this end, we develop a new Bayesian model to leverage both better anatomical precision in high resolution fMRI and higher SNR in standard resolution fMRI. We assign a Gaussian process prior to the mean intensity function and develop an efficient, scalable posterior computation algorithm to integrate both sources of data. We draw posterior samples using an algorithm analogous to Riemann manifold Hamiltonian Monte Carlo in an expanded parameter space. We illustrate our method in analysis of presurgical fMRI data, and show in simulation that it infers the mean intensity more accurately than alternatives that use either the high or standard resolution fMRI data alone.

A key feature of out-of-distribution (OOD) detection is to exploit a trained neural network by extracting statistical patterns and relationships through the multi-layer classifier to detect shifts in the expected input data distribution. Despite achieving solid results, several state-of-the-art methods rely on the penultimate or last layer outputs only, leaving behind valuable information for OOD detection. Methods that explore the multiple layers either require a special architecture or a supervised objective to do so. This work adopts an original approach based on a functional view of the network that exploits the sample's trajectories through the various layers and their statistical dependencies. It goes beyond multivariate features aggregation and introduces a baseline rooted in functional anomaly detection. In this new framework, OOD detection translates into detecting samples whose trajectories differ from the typical behavior characterized by the training set. We validate our method and empirically demonstrate its effectiveness in OOD detection compared to strong state-of-the-art baselines on computer vision benchmarks.

This paper introduces an approach, named DFormer, for universal image segmentation. The proposed DFormer views universal image segmentation task as a denoising process using a diffusion model. DFormer first adds various levels of Gaussian noise to ground-truth masks, and then learns a model to predict denoising masks from corrupted masks. Specifically, we take deep pixel-level features along with the noisy masks as inputs to generate mask features and attention masks, employing diffusion-based decoder to perform mask prediction gradually. At inference, our DFormer directly predicts the masks and corresponding categories from a set of randomly-generated masks. Extensive experiments reveal the merits of our proposed contributions on different image segmentation tasks: panoptic segmentation, instance segmentation, and semantic segmentation. Our DFormer outperforms the recent diffusion-based panoptic segmentation method Pix2Seq-D with a gain of 3.6% on MS COCO val2017 set. Further, DFormer achieves promising semantic segmentation performance outperforming the recent diffusion-based method by 2.2% on ADE20K val set. Our source code and models will be publicly on //github.com/cp3wan/DFormer

Despite significant advances in the field of deep learning in ap-plications to various areas, an explanation of the learning pro-cess of neural network models remains an important open ques-tion. The purpose of this paper is a comprehensive comparison and description of neural network architectures in terms of ge-ometry and topology. We focus on the internal representation of neural networks and on the dynamics of changes in the topology and geometry of a data manifold on different layers. In this paper, we use the concepts of topological data analysis (TDA) and persistent homological fractal dimension. We present a wide range of experiments with various datasets and configurations of convolutional neural network (CNNs) architectures and Transformers in CV and NLP tasks. Our work is a contribution to the development of the important field of explainable and interpretable AI within the framework of geometrical deep learning.

Generative modeling has been the dominant approach for large-scale pretraining and zero-shot generalization. In this work, we challenge this convention by showing that discriminative approaches perform substantially better than generative ones on a large number of NLP tasks. Technically, we train a single discriminator to predict whether a text sample comes from the true data distribution, similar to GANs. Since many NLP tasks can be formulated as selecting from a few options, we use this discriminator to predict the concatenation of input and which option has the highest probability of coming from the true data distribution. This simple formulation achieves state-of-the-art zero-shot results on the T0 benchmark, outperforming T0 by 16.0\%, 7.8\%, and 11.5\% respectively on different scales. In the finetuning setting, our approach also achieves new state-of-the-art results on a wide range of NLP tasks, with only 1/4 parameters of previous methods. Meanwhile, our approach requires minimal prompting efforts, which largely improves robustness and is essential for real-world applications. Furthermore, we also jointly train a generalized UD in combination with generative tasks, which maintains its advantage on discriminative tasks and simultaneously works on generative tasks.

The goal of few-shot learning is to learn a classifier that generalizes well even when trained with a limited number of training instances per class. The recently introduced meta-learning approaches tackle this problem by learning a generic classifier across a large number of multiclass classification tasks and generalizing the model to a new task. Yet, even with such meta-learning, the low-data problem in the novel classification task still remains. In this paper, we propose Transductive Propagation Network (TPN), a novel meta-learning framework for transductive inference that classifies the entire test set at once to alleviate the low-data problem. Specifically, we propose to learn to propagate labels from labeled instances to unlabeled test instances, by learning a graph construction module that exploits the manifold structure in the data. TPN jointly learns both the parameters of feature embedding and the graph construction in an end-to-end manner. We validate TPN on multiple benchmark datasets, on which it largely outperforms existing few-shot learning approaches and achieves the state-of-the-art results.