Optimal transport (OT) is a widely used technique in machine learning, graphics, and vision that aligns two distributions or datasets using their relative geometry. In symmetry-rich settings, however, OT alignments based solely on pairwise geometric distances between raw features can ignore the intrinsic coherence structure of the data. We introduce Bis-pectral Optimal Transport, a symmetry-aware extension of discrete OT that compares elements using their representation using the bispectrum, a group Fourier invariant that preserves all signal structure while removing only the variation due to group actions. Empirically, we demonstrate that the transport plans computed with Bispectral OT achieve greater class preservation accuracy than naive feature OT on benchmark datasets transformed with visual symmetries, improving the quality of meaningful correspondences that capture the underlying semantic label structure in the dataset while removing nuisance variation not affecting class or content.

Driver stress is a major cause of car accidents and death worldwide. Furthermore, persistent stress is a health problem, contributing to hypertension and other diseases of the cardiovascular system. Stress has a measurable impact on heart and breathing rates and stress levels can be inferred from such measurements. Galvanic skin response is a common test to measure the perspiration caused by both physiological and psychological stress, as well as extreme emotions. In this paper, galvanic skin response is used to estimate the ground truth stress levels. A feature selection technique based on the minimal redundancy-maximal relevance method is then applied to multiple heart rate variability and breathing rate metrics to identify a novel and optimal combination for use in detecting stress. The support vector machine algorithm with a radial basis function kernel was used along with these features to reliably predict stress. The proposed method has achieved a high level of accuracy on the target dataset.

We present a neural network architecture, Bispectral Neural Networks (BNNs) for learning representations that are invariant to the actions of compact commutative groups on the space over which a signal is defined. The model incorporates the ansatz of the bispectrum, an analytically defined group invariant that is complete--that is, it preserves all signal structure while removing only the variation due to group actions. Here, we demonstrate that BNNs are able to simultaneously learn groups, their irreducible representations, and corresponding equivariant and complete-invariant maps purely from the symmetries implicit in data. Further, we demonstrate that the completeness property endows these networks with strong invariance-based adversarial robustness. This work establishes Bispectral Neural Networks as a powerful computational primitive for robust invariant representation learning. A fundamental problem of intelligence is to model the transformation structure of the natural world. In the context of vision, translation, rotation, and scaling define symmetries of object categorization--the transformations that leave perceived object identity invariant. In audition, pitch and timbre define symmetries of speech recognition. Biological neural systems have learned these symmetries from the statistics of the natural world--either through evolution or accumulated experience. Here, we tackle the problem of learning symmetries in artificial neural networks. At the heart of the challenge lie two requirements that are frequently in tension: invariance to transformation structure and selectivity to pattern structure. In deep networks, operations such as max or average are commonly employed to achieve invariance to local transformations. Such operations are invariant to many natural transformations; however, they are also invariant to unnatural transformations that destroy image structure, such as pixel permutations.

Producing thousands of simulations of the dark matter distribution in the Universe with increasing precision is a challenging but critical task to facilitate the exploitation of current and forthcoming cosmological surveys. Many inexpensive substitutes to full $N$-body simulations have been proposed, even though they often fail to reproduce the statistics of the smaller, non-linear scales. Among these alternatives, a common approximation is represented by the lognormal distribution, which comes with its own limitations as well, while being extremely fast to compute even for high-resolution density fields. In this work, we train a generative deep learning model, mainly made of convolutional layers, to transform projected lognormal dark matter density fields to more realistic dark matter maps, as obtained from full $N$-body simulations. We detail the procedure that we follow to generate highly correlated pairs of lognormal and simulated maps, which we use as our training data, exploiting the information of the Fourier phases. We demonstrate the performance of our model comparing various statistical tests with different field resolutions, redshifts and cosmological parameters, proving its robustness and explaining its current limitations. When evaluated on 100 test maps, the augmented lognormal random fields reproduce the power spectrum up to wavenumbers of $1 \ h \ \rm{Mpc}^{-1}$, and the bispectrum within 10%, and always within the error bars, of the fiducial target simulations. Finally, we describe how we plan to integrate our proposed model with existing tools to yield more accurate spherical random fields for weak lensing analysis.

This article introduces a method of evaluating subsamples until any prescribed level of classification accuracy is attained, thus obtaining arbitrary accuracy. A logarithmic reduction in error rate is obtained with a linear increase in sample count. The technique is applied to specific emitter identification on a published dataset of physically recorded over-the-air signals from 16 ostensibly identical high-performance radios. The technique uses a multi-channel deep learning convolutional neural network acting on the bispectra of I/Q signal subsamples each consisting of 56 parts per million (ppm) of the original signal duration. High levels of accuracy are obtained with minimal computation time: in this application, each addition of eight samples decreases error by one order of magnitude.

Abstract--Motivated by the task of 2 -D classification in single particle reconstruction by cryo-electron microscopy (cryo-EM), we consider the problem of heterogeneous multireference alignment of images. In this problem, the goal is to estimate a (typically small) set of target images from a (typically large) collection of observations. Each observation is a rotated, noisy version of one of the target images. For each individual observation, neither the rotation nor which target image has been rotated are known. As the noise level in cryo-EM data is high, clustering the observations and estimating individual rotations is challenging. We propose a framework to estimate the target images directly from the observations, completely bypassing the need to cluster or register the images. The framework consists of two steps. First, we estimate rotation-invariant features of the images, such as the bispectrum. These features can be estimated to any desired accuracy, at any noise level, provided sufficiently many observations are collected. Then, we estimate the images from the invariant features. Numerical experiments on synthetic cryo-EM datasets demonstrate the effectiveness of the method. Ultimately, we outline future developments required to apply this method to experimental data. INGLE particle reconstruction using cryo-EM is a high-resolution imaging technique used in structural biology to image 3-D structures of macromolecules [1], [2]. In a cryo-EM experiment, multiple samples of a particle are frozen in a thin layer of vitreous ice. Within the ice, the samples are randomly oriented and positioned.