Topological deep learning (TDL) is a rapidly evolving field that uses topological features to understand and design deep learning models. This paper posits that TDL is the new frontier for relational learning. TDL may complement graph representation learning and geometric deep learning by incorporating topological concepts, and can thus provide a natural choice for various machine learning settings. To this end, this paper discusses open problems in TDL, ranging from practical benefits to theoretical foundations. For each problem, it outlines potential solutions and future research opportunities. At the same time, this paper serves as an invitation to the scientific community to actively participate in TDL research to unlock the potential of this emerging field.

This work introduces the Topological CNN (TCNN), which encompasses several topologically defined convolutional methods. Manifolds with important relationships to the natural image space are used to parameterize image filters which are used as convolutional weights in a TCNN. These manifolds also parameterize slices in layers of a TCNN across which the weights are localized. We show evidence that TCNNs learn faster, on less data, with fewer learned parameters, and with greater generalizability and interpretability than conventional CNNs. We introduce and explore TCNN layers for both image and video data. We propose extensions to 3D images and 3D video.

Persistent homology, a powerful mathematical tool for data analysis, summarizes the shape of data through tracking topological features across changes in different scales. Classical algorithms for persistent homol-ogy are often constrained by running times and mem-ory requirements that grow exponentially on the number of data points. To surpass this problem, two quantum algorithms of persistent homology have been developed based on two different approaches. However, both of these quantum algorithms consider a data set in the form of a point cloud, which can be restrictive considering that many data sets come in the form of time series. In this paper, we alleviate this issue by establishing a quantum Takens's delay embedding algorithm, which turns a time series into a point cloud by considering a pertinent embedding into a higher dimensional space. Having this quantum transformation of time series to point clouds, then one may use a quantum persistent homology algorithm to extract the topological features from the point cloud associated with the original times series.

Investigation of human brain states through electroencephalograph (EEG) signals is a crucial step in human-machine communications. However, classifying and analyzing EEG signals are challenging due to their noisy, nonlinear and nonstationary nature. Current methodologies for analyzing these signals often fall short because they have several regularity assumptions baked in. This work provides an effective, flexible and noise-resilient scheme to analyze EEG by extracting pertinent information while abiding by the 3N (noisy, nonlinear and nonstationary) nature of data. We implement a topological tool, namely persistent homology, that tracks the evolution of topological features over time intervals and incorporates individual's expectations as prior knowledge by means of a Bayesian framework to compute posterior distributions. Relying on these posterior distributions, we apply Bayes factor classification to noisy EEG measurements. The performance of this Bayesian classification scheme is then compared with other existing methods for EEG signals.

Topological data analysis refers to a broad set of techniques that are used to make inferences about the shape of data. A popular topological summary is the persistence diagram. Through the language of random sets, we describe a notion of global probability density function for persistence diagrams that fully characterizes their behavior and in part provides a noise likelihood model. Our approach encapsulates the number of topological features and considers the appearance or disappearance of those near the diagonal in a stable fashion. In particular, the structure of our kernel individually tracks long persistence features, while considering those near the diagonal as a collective unit. The choice to describe short persistence features as a group reduces computation time while simultaneously retaining accuracy. Indeed, we prove that the associated kernel density estimate converges to the true distribution as the number of persistence diagrams increases and the bandwidth shrinks accordingly. We also establish the convergence of the mean absolute deviation estimate, defined according to the bottleneck metric. Lastly, examples of kernel density estimation are presented for typical underlying datasets as well as for virtual electroencephalographic data related to cognition.

Microscopy imaging allows detailed observations of intracellular movements and the acquisition of large datasets that can be fully analyzed only by automated algorithms. Here, we develop a computational method for the automatic identification and reconstruction of trajectories followed by subcellular particles captured in microscopy image data. The method operates on stacks of raw image data and computes the complete set of contained trajectories. The method utilizes topological data analysis and standard image processing techniques and makes no assumptions about the underlying dynamics besides continuity. We test the developed method successfully against artificial and experimental datasets. Application of the method on the experimental data reveals good agreement with manual tracking and benchmarking yields performance scores competitive to the existing state-of-the-art tracking methods.

Sensor mobility and sparse sensor data correlations are exploited in this work for the problem of tracking multiple targets. Sparse matrix decomposition is integrated with the design of proper kinematic rules to identify informative sensors, associate them with the targets and enable them to follow closely the moving targets. The modified barrier method is employed to minimize proper error covariance matrices obtained by extended Kalman filtering recursions. Distributed updating recursive rules are obtained enabling only the informative sensors to update their location, and follow closely the corresponding targets while staying connected. This is to be contrasted with existing alternatives where all sensors need to move constantly. Numerical tests corroborate that the proposed scheme outperforms existing alternatives while accurately tracks multiple targets by imposing mobility only on target-informative sensors.

Particle filtering is combined with sparse matrix decomposition techniques to address the problem of tracking multiple targets using nonlinear sensor observations measuring signal strength. The unknown number of targets may be time-varying, while sensors are spatially scattered. Norm-one regularized matrix factorization is employed to decompose the sensing data covariance matrix into sparse factors whose support facilitates the task of associating the targets with sensor measurements. The novel sensors-to-targets association scheme is developed using distributed optimization which is further integrated with particle filtering mechanisms to perform accurate tracking. Numerical tests demonstrate the tracking superiority of the proposed algorithm over alternative approaches.

Estimation of the principal eigenspace of a data covariance matrix is instrumental in applications such as data dimensionality reduction and denoising. In sensor networks the acquired data are spatially scattered which further calls for the development of distributed principal subspace estimation algorithms. Toward this end, the standard principal component analysis framework is reformulated as a separable constrained minimization problem which is solved by utilizing coordinate descent techniques combined with the alternating direction method of multipliers. Computationally simple local updating recursions are obtained that involve only single-hop inter-sensor communications and allow sensors to estimate the principal covariance eigenspace in a distributed fashion. Adaptive implementations are also considered that allow online information processing. Numerical tests demonstrate that the novel algorithm has the potential to achieve a considerably faster convergence rate and better steady-state estimation performance compared to existing alternatives.

The problem of joint detection and state estimation of a Markov signal when a variable number of noisy measurements can be taken is here considered. In particular, the signal-observation sequence {Xi, Zi}iisinNis a hidden Markov process (HMP) while, if the signal is absent, the measurement {Zi}i2isinNis an i.i.d. process. In this framework, two coupled detection and estimation procedures are introduced for the cases of discrete and continuous state space. Bounds on the performance of the proposed procedures in terms of the thresholds are derived, similar to the classical bounds for the sequential probability ratio test (SPRT). Moreover, it is shown that, under a set of rather mild conditions, the procedures end with probability one and the stopping time is almost surely minimized in the class of tests with the same or smaller error probabilities.