Paul Bendich

Abraham D. Smith, Michael J. Catanzaro, Gabrielle Angeloro et al.

For safety and robustness of AI systems, we introduce topological parallax as a theoretical and computational tool that compares a trained model to a reference dataset to determine whether they have similar multiscale geometric structure. Our proofs and examples show that this geometric similarity between dataset and model is essential to trustworthy interpolation and perturbation, and we conjecture that this new concept will add value to the current debate regarding the unclear relationship between overfitting and generalization in applications of deep-learning. In typical DNN applications, an explicit geometric description of the model is impossible, but parallax can estimate topological features (components, cycles, voids, etc.) in the model by examining the effect on the Rips complex of geodesic distortions using the reference dataset. Thus, parallax indicates whether the model shares similar multiscale geometric features with the dataset. Parallax presents theoretically via topological data analysis [TDA] as a bi-filtered persistence module, and the key properties of this module are stable under perturbation of the reference dataset.

Elchanan Solomon, Paul Bendich

In this paper, we consider topological featurizations of data defined over simplicial complexes, like images and labeled graphs, obtained by convolving this data with various filters before computing persistence. Viewing a convolution filter as a local motif, the persistence diagram of the resulting convolution describes the way the motif is distributed across the simplicial complex. This pipeline, which we call convolutional persistence, extends the capacity of topology to observe patterns in such data. Moreover, we prove that (generically speaking) for any two labeled complexes one can find some filter for which they produce different persistence diagrams, so that the collection of all possible convolutional persistence diagrams is an injective invariant. This is proven by showing convolutional persistence to be a special case of another topological invariant, the Persistent Homology Transform. Other advantages of convolutional persistence are improved stability, greater flexibility for data-dependent vectorizations, and reduced computational complexity for certain data types. Additionally, we have a suite of experiments showing that convolutions greatly improve the predictive power of persistence on a host of classification tasks, even if one uses random filters and vectorizes the resulting diagrams by recording only their total persistences.

Gary Koplik, Nathan Borggren, Sam Voisin et al.

As Internet of Things (IoT) devices become both cheaper and more powerful, researchers are increasingly finding solutions to their scientific curiosities both financially and computationally feasible. When operating with restricted power or communications budgets, however, devices can only send highly-compressed data. Such circumstances are common for devices placed away from electric grids that can only communicate via satellite, a situation particularly plausible for environmental sensor networks. These restrictions can be further complicated by potential variability in the communications budget, for example a solar-powered device needing to expend less energy when transmitting data on a cloudy day. We propose a novel, topology-based, lossy compression method well-equipped for these restrictive yet variable circumstances. This technique, Topological Signal Compression, allows sending compressed signals that utilize the entirety of a variable communications budget. To demonstrate our algorithm's capabilities, we perform entropy calculations as well as a classification exercise on increasingly topologically simplified signals from the Free-Spoken Digit Dataset and explore the stability of the resulting performance against common baselines.

Alexander Wagner, Elchanan Solomon, Paul Bendich

We propose a novel approach to dimensionality reduction combining techniques of metric geometry and distributed persistent homology, in the form of a gradient-descent based method called DIPOLE. DIPOLE is a dimensionality-reduction post-processing step that corrects an initial embedding by minimizing a loss functional with both a local, metric term and a global, topological term. By fixing an initial embedding method (we use Isomap), DIPOLE can also be viewed as a full dimensionality-reduction pipeline. This framework is based on the strong theoretical and computational properties of distributed persistent homology and comes with the guarantee of almost sure convergence. We observe that DIPOLE outperforms popular methods like UMAP, t-SNE, and Isomap on a number of popular datasets, both visually and in terms of precise quantitative metrics.

Elchanan Solomon, Alexander Wagner, Paul Bendich

What is the "right" topological invariant of a large point cloud X? Prior research has focused on estimating the full persistence diagram of X, a quantity that is very expensive to compute, unstable to outliers, and far from a sufficient statistic. We therefore propose that the correct invariant is not the persistence diagram of X, but rather the collection of persistence diagrams of many small subsets. This invariant, which we call "distributed persistence," is perfectly parallelizable, more stable to outliers, and has a rich inverse theory. The map from the space of point clouds (with the quasi-isometry metric) to the space of distributed persistence invariants (with the Hausdorff-Bottleneck distance) is a global quasi-isometry. This is a much stronger property than simply being injective, as it implies that the inverse of a small neighborhood is a small neighborhood, and is to our knowledge the only result of its kind in the TDA literature. Moreover, the quasi-isometry bounds depend on the size of the subsets taken, so that as the size of these subsets goes from small to large, the invariant interpolates between a purely geometric one and a topological one. Lastly, we note that our inverse results do not actually require considering all subsets of a fixed size (an enormous collection), but a relatively small collection satisfying certain covering properties that arise with high probability when randomly sampling subsets. These theoretical results are complemented by two synthetic experiments demonstrating the use of distributed persistence in practice.

Elchanan Solomon, Alexander Wagner, Paul Bendich

Topological statistics, in the form of persistence diagrams, are a class of shape descriptors that capture global structural information in data. The mapping from data structures to persistence diagrams is almost everywhere differentiable, allowing for topological gradients to be backpropagated to ordinary gradients. However, as a method for optimizing a topological functional, this backpropagation method is expensive, unstable, and produces very fragile optima. Our contribution is to introduce a novel backpropagation scheme that is significantly faster, more stable, and produces more robust optima. Moreover, this scheme can also be used to produce a stable visualization of dots in a persistence diagram as a distribution over critical, and near-critical, simplices in the data structure.

Elchanan Solomon, Paul Bendich

We introduce geometric and topological methods to develop a new framework for fusing multi-sensor time series. This framework consists of two steps: (1) a joint delay embedding, which reconstructs a high-dimensional state space in which our sensors correspond to observation functions, and (2) a simple orthogonalization scheme, which accounts for tangencies between such observation functions, and produces a more diversified geometry on the embedding space. We conclude with some synthetic and real-world experiments demonstrating that our framework outperforms traditional metric fusion methods.

Lihan Yao, Paul Bendich

We propose a graph spectral representation of time series data that 1) is parsimoniously encoded to user-demanded resolution; 2) is unsupervised and performant in data-constrained scenarios; 3) captures event and event-transition structure within the time series; and 4) has near-linear computational complexity in both signal length and ambient dimension. This representation, which we call Laplacian Events Signal Segmentation (LESS), can be computed on time series of arbitrary dimension and originating from sensors of arbitrary type. Hence, time series originating from sensors of heterogeneous type can be compressed to levels demanded by constrained-communication environments, before being fused at a common center. Temporal dynamics of the data is summarized without explicit partitioning or probabilistic modeling. As a proof-of-principle, we apply this technique on high dimensional wavelet coefficients computed from the Free Spoken Digit Dataset to generate a memory efficient representation that is interpretable. Due to its unsupervised and non-parametric nature, LESS representations remain performant in the digit classification task despite the absence of labels and limited data.

Christopher J. Tralie, Paul Bendich, John Harer

In this work, we address fusion of heterogeneous sensor data using wavelet-based summaries of fused self-similarity information from each sensor. The technique we develop is quite general, does not require domain specific knowledge or physical models, and requires no training. Nonetheless, it can perform surprisingly well at the general task of differentiating classes of time-ordered behavior sequences which are sensed by more than one modality. As a demonstration of our capabilities in the audio to video context, we focus on the differentiation of speech sequences. Data from two or more modalities first are represented using self-similarity matrices(SSMs) corresponding to time-ordered point clouds in feature spaces of each of these data sources; we note that these feature spaces can be of entirely different scale and dimensionality. A fused similarity template is then derived from the modality-specific SSMs using a technique called similarity network fusion (SNF). We investigate pipelines using SNF as both an upstream (feature-level) and a downstream (ranking-level) fusion technique. Multiscale geometric features of this template are then extracted using a recently-developed technique called the scattering transform, and these features are then used to differentiate speech sequences. This method outperforms unsupervised techniques which operate directly on the raw data, and it also outperforms stovepiped methods which operate on SSMs separately derived from the distinct modalities. The benefits of this method become even more apparent as the simulated peak signal to noise ratio decreases.

Christopher J. Tralie, Abraham Smith, Nathan Borggren et al.

In this work, we address the problem of cross-modal comparison of aerial data streams. A variety of simulated automobile trajectories are sensed using two different modalities: full-motion video, and radio-frequency (RF) signals received by detectors at various locations. The information represented by the two modalities is compared using self-similarity matrices (SSMs) corresponding to time-ordered point clouds in feature spaces of each of these data sources; we note that these feature spaces can be of entirely different scale and dimensionality. Several metrics for comparing SSMs are explored, including a cutting-edge time-warping technique that can simultaneously handle local time warping and partial matches, while also controlling for the change in geometry between feature spaces of the two modalities. We note that this technique is quite general, and does not depend on the choice of modalities. In this particular setting, we demonstrate that the cross-modal distance between SSMs corresponding to the same trajectory type is smaller than the cross-modal distance between SSMs corresponding to distinct trajectory types, and we formalize this observation via precision-recall metrics in experiments. Finally, we comment on promising implications of these ideas for future integration into multiple-hypothesis tracking systems.

Abraham Smith, Paul Bendich, John Harer et al.

We introduce a new algorithm, called CDER, for supervised machine learning that merges the multi-scale geometric properties of Cover Trees with the information-theoretic properties of entropy. CDER applies to a training set of labeled pointclouds embedded in a common Euclidean space. If typical pointclouds corresponding to distinct labels tend to differ at any scale in any sub-region, CDER can identify these differences in (typically) linear time, creating a set of distributional coordinates which act as a feature extraction mechanism for supervised learning. We describe theoretical properties and implementation details of CDER, and illustrate its benefits on several synthetic examples.

Christopher J. Tralie, Paul Bendich

We introduce a novel low level feature for identifying cover songs which quantifies the relative changes in the smoothed frequency spectrum of a song. Our key insight is that a sliding window representation of a chunk of audio can be viewed as a time-ordered point cloud in high dimensions. For corresponding chunks of audio between different versions of the same song, these point clouds are approximately rotated, translated, and scaled copies of each other. If we treat MFCC embeddings as point clouds and cast the problem as a relative shape sequence, we are able to correctly identify 42/80 cover songs in the "Covers 80" dataset. By contrast, all other work to date on cover songs exclusively relies on matching note sequences from Chroma derived features.

Paul Bendich, Ellen Gasparovic, John Harer et al.

We introduce a method called multi-scale local shape analysis, or MLSA, for extracting features that describe the local structure of points within a dataset. The method uses both geometric and topological features at multiple levels of granularity to capture diverse types of local information for subsequent machine learning algorithms operating on the dataset. Using synthetic and real dataset examples, we demonstrate significant performance improvement of classification algorithms constructed for these datasets with correspondingly augmented features.

Paul Bendich, Sang Chin, Jesse Clarke et al.

We introduce the first unified theory for target tracking using Multiple Hypothesis Tracking, Topological Data Analysis, and machine learning. Our string of innovations are 1) robust topological features are used to encode behavioral information, 2) statistical models are fitted to distributions over these topological features, and 3) the target type classification methods of Wigren and Bar Shalom et al. are employed to exploit the resulting likelihoods for topological features inside of the tracking procedure. To demonstrate the efficacy of our approach, we test our procedure on synthetic vehicular data generated by the Simulation of Urban Mobility package.