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Incorporating inductive biases is a promising approach for tackling challenging robot learning domains with sample-efficient solutions. This paper identifies partially observable domains where symmetries can be a useful inductive bias for efficient learning. Specifically, by encoding the equivariance regarding specific group symmetries into the neural networks, our actor-critic reinforcement learning agents can reuse solutions in the past for related scenarios. Consequently, our equivariant agents outperform non-equivariant approaches significantly in terms of sample efficiency and final performance, demonstrated through experiments on a range of robotic tasks in simulation and real hardware., Comment: Conference on Robot Learning, 2023

Simulating strongly coupled gauge theories at finite temperature and density is a longstanding challenge in nuclear and high-energy physics that also has fundamental implications for condensed matter physics. In this work, we investigate the utility of minimally entangled typical thermal state (METTS) approaches to facilitate both classical and quantum computational studies of such systems. METTS techniques combine classical random sampling with imaginary time evolution, which can be performed on either a classical or a quantum computer, to estimate thermal averages of observables. We study the simplest model of a confining gauge theory, namely $\mathbb{Z}_2$ gauge theory coupled to spinless fermionic matter in 1+1 dimensions, which can be directly mapped to a local quantum spin chain with two- and three-body interactions. We benchmark both a classical matrix-product-state implementation of METTS and a recently proposed adaptive variational approach to METTS that is a promising candidate for implementation on near-term quantum devices, focusing on the equation of state as well as on various measures of fermion confinement. Of particular importance is the choice of basis for obtaining new METTS samples, which impacts both the classical sampling complexity (a key factor in both classical and quantum simulation applications) and complexity of circuits used in the quantum computing approach. Our work sets the stage for future studies of strongly coupled gauge theories with both classical and quantum hardware., Comment: 14 pages, 8 figures

This paper introduces the Bosch street dataset (BSD), a novel multi-modal large-scale dataset aimed at promoting highly automated driving (HAD) and advanced driver-assistance systems (ADAS) research. Unlike existing datasets, BSD offers a unique integration of high-resolution imaging radar, lidar, and camera sensors, providing unprecedented 360-degree coverage to bridge the current gap in high-resolution radar data availability. Spanning urban, rural, and highway environments, BSD enables detailed exploration into radar-based object detection and sensor fusion techniques. The dataset is aimed at facilitating academic and research collaborations between Bosch and current and future partners. This aims to foster joint efforts in developing cutting-edge HAD and ADAS technologies. The paper describes the dataset's key attributes, including its scalability, radar resolution, and labeling methodology. Key offerings also include initial benchmarks for sensor modalities and a development kit tailored for extensive data analysis and performance evaluation, underscoring our commitment to contributing valuable resources to the HAD and ADAS research community.

Sample efficiency is critical when applying learning-based methods to robotic manipulation due to the high cost of collecting expert demonstrations and the challenges of on-robot policy learning through online Reinforcement Learning (RL). Offline RL addresses this issue by enabling policy learning from an offline dataset collected using any behavioral policy, regardless of its quality. However, recent advancements in offline RL have predominantly focused on learning from large datasets. Given that many robotic manipulation tasks can be formulated as rotation-symmetric problems, we investigate the use of $SO(2)$-equivariant neural networks for offline RL with a limited number of demonstrations. Our experimental results show that equivariant versions of Conservative Q-Learning (CQL) and Implicit Q-Learning (IQL) outperform their non-equivariant counterparts. We provide empirical evidence demonstrating how equivariance improves offline learning algorithms in the low-data regime.

Non-alcoholic fatty liver disease (NAFLD) is a prevalent chronic liver disorder characterized by the excessive accumulation of fat in the liver in individuals who do not consume significant amounts of alcohol, including risk factors like obesity, insulin resistance, type 2 diabetes, etc. We aim to identify subgroups of NAFLD patients based on demographic, clinical, and genetic characteristics for precision medicine. The genomic and phenotypic data (3,408 cases and 4,739 controls) for this study were gathered from participants in Mayo Clinic Tapestry Study (IRB#19-000001) and their electric health records, including their demographic, clinical, and comorbidity data, and the genotype information through whole exome sequencing performed at Helix using the Exome+$^\circledR$ Assay according to standard procedure (www$.$helix$.$com). Factors highly relevant to NAFLD were determined by the chi-square test and stepwise backward-forward regression model. Latent class analysis (LCA) was performed on NAFLD cases using significant indicator variables to identify subgroups. The optimal clustering revealed 5 latent subgroups from 2,013 NAFLD patients (mean age 60.6 years and 62.1% women), while a polygenic risk score based on 6 single-nucleotide polymorphism (SNP) variants and disease outcomes were used to analyze the subgroups. The groups are characterized by metabolic syndrome, obesity, different comorbidities, psychoneurological factors, and genetic factors. Odds ratios were utilized to compare the risk of complex diseases, such as fibrosis, cirrhosis, and hepatocellular carcinoma (HCC), as well as liver failure between the clusters. Cluster 2 has a significantly higher complex disease outcome compared to other clusters. Keywords: Fatty liver disease; Polygenic risk score; Precision medicine; Deep phenotyping; NAFLD comorbidities; Latent class analysis., Comment: Extended Abstract presented at Machine Learning for Health (ML4H) symposium 2023, December 10th, 2023, New Orleans, United States, 11 pages

Hurwitz algebras are unital composition algebras widely known in algebra and mathematical physics for their useful applications. In this paper, inspired by works of Lesenby and Hitzer, we show how to embed all seven Hurwitz algebras (division and split) in 3D geometric algebras, i.e. $\mathcal{G}\left(p,q\right)$ with $p+q=3$. This is achieved studying the even subalgebra, which is always of quaternionic or split-quaternionic type, and the algebra generated by the pseudoscalar which is always of complex or split-complex type. Reversion, inversion and Clifford conjugation correspond to biquaternionic, complex and quaternionic conjugation respectively. Octonionic algebras, i.e., octonions and split-octonions, are obtained in two ways introducing two different products, replicating a variation of the Cayley-Dickson process. Octonionic conjugation is achieved in the geometric algebra formalism through the involution defined by the full grade inversion.

The transport of conserved quantities like spin and charge is fundamental to characterizing the behavior of quantum many-body systems. Numerically simulating such dynamics is generically challenging, which motivates the consideration of quantum computing strategies. However, the relatively high gate errors and limited coherence times of today's quantum computers pose their own challenge, highlighting the need to be frugal with quantum resources. In this work we report simulations on quantum hardware of infinite-temperature energy transport in the mixed-field Ising chain, a paradigmatic many-body system that can exhibit a range of transport behaviors at intermediate times. We consider a chain with $L=12$ sites and find results broadly consistent with those from ideal circuit simulators over 90 Trotter steps, containing up to 990 entangling gates. To obtain these results, we use two key problem-tailored insights. First, we identify a convenient basis$\unicode{x2013}$the Pauli $Y$ basis$\unicode{x2013}$in which to sample the infinite-temperature trace and provide theoretical and numerical justifications for its efficiency relative to, e.g., the computational basis. Second, in addition to a variety of problem-agnostic error mitigation strategies, we employ a renormalization strategy that compensates for global nonconservation of energy due to device noise. We discuss the applicability of the proposed sampling approach beyond the mixed-field Ising chain and formulate a variational method to search for a sampling basis with small sample-to-sample fluctuations for an arbitrary Hamiltonian. This opens the door to applying these techniques in more general models., Comment: 10 + 11 pages, 4 + 8 figures. Change to creative commons licenses

Learning about the three-dimensional world from two-dimensional images is a fundamental problem in computer vision. An ideal neural network architecture for such tasks would leverage the fact that objects can be rotated and translated in three dimensions to make predictions about novel images. However, imposing SO(3)-equivariance on two-dimensional inputs is difficult because the group of three-dimensional rotations does not have a natural action on the two-dimensional plane. Specifically, it is possible that an element of SO(3) will rotate an image out of plane. We show that an algorithm that learns a three-dimensional representation of the world from two dimensional images must satisfy certain geometric consistency properties which we formulate as SO(2)-equivariance constraints. We use the induced and restricted representations of SO(2) on SO(3) to construct and classify architectures which satisfy these geometric consistency constraints. We prove that any architecture which respects said consistency constraints can be realized as an instance of our construction. We show that three previously proposed neural architectures for 3D pose prediction are special cases of our construction. We propose a new algorithm that is a learnable generalization of previously considered methods. We test our architecture on three pose predictions task and achieve SOTA results on both the PASCAL3D+ and SYMSOL pose estimation tasks.

In this work, we define quasicrystalline spin networks as a subspace within the standard Hilbert space of loop quantum gravity, effectively constraining the states to coherent states that align with quasicrystal geometry structures. We introduce quasicrystalline spin foam amplitudes, a variation of the EPRL spin foam model, in which the internal spin labels are constrained to correspond to the boundary data of quasicrystalline spin networks. Within this framework, the quasicrystalline spin foam amplitudes encode the dynamics of quantum geometries that exhibit aperiodic structures. Additionally, we investigate the coupling of fermions within the quasicrystalline spin foam amplitudes. We present calculations for three-dimensional examples and then explore the 600-cell construction, which is a fundamental component of the four-dimensional Elser-Sloane quasicrystal derived from the E8 root lattice.

We generalize Koopman-von Neumann classical mechanics to poly-symplectic fields and recover De Donder-Weyl theory. Comparing with Dirac's Hamiltonian density inspires a new Hamiltonian formulation with a canonical momentum field that is Lorentz covariant with symplectic geometry. We provide commutation relations for the classical and quantum fields that generalize the Koopman-von Neumann and Heisenberg algebras. The classical algebra requires four fields that generalize space-time, energy-momentum, frequency-wavenumber, and the Fourier conjugate of energy-momentum. We clarify how 1st and 2nd quantization can be found by simply mapping between operators in classical and quantum commutator algebras., Comment: 27 pages including appendices and references

In this paper we present a general setting for aperiodic Jordan algebras arising from icosahedral quasicrystals that are obtainable as model sets of a cut-and-project scheme with a convex acceptance window. In these hypothesis, we show the existence of an aperiodic Jordan algebra structure whose generators are in one-to-one correspondence with elements of the quasicrystal. Moreover, if the acceptance window enjoys a non-crystallographic symmetry arising from H2, H3 or H4 then the resulting Jordan algebra enjoys the same H2, H3 or H4 symmetry. Finally, we present as special cases some examples of Jordan algebras over a Fibonacci-chain quasicrystal, a Penrose tiling, and the Elser-Sloane quasicrystal.

We present three new coset manifolds named Dixon-Rosenfeld lines that are similar to Rosenfeld projective lines except over the Dixon algebra $\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$. Three different Lie groups are found as isometry groups of these coset manifolds using Tits' formula. We demonstrate how Standard Model interactions with the Dixon algebra in recent work from Furey and Hughes can be uplifted to tensor products of division algebras and Jordan algebras for a single generation of fermions. The Freudenthal-Tits construction clarifies how the three Dixon-Rosenfeld projective lines are contained within $\mathbb{C}\otimes\mathbb{H}\otimes J_{2}(\mathbb{O})$, $\mathbb{O}\otimes J_{2}(\mathbb{C}\otimes\mathbb{H})$, and $\mathbb{C}\otimes\mathbb{O}\otimes J_{2}(\mathbb{H})$., Comment: 46 pages with references, 1 figure, 3 tables

The microscopic structure of two amorphous materials with extreme nonlinear optical properties has been studied. One of these materials exhibits second harmonic generation, while another material of similar molecular structure emits brilliant white light if being irradiated with a simple IR laser diode. Structural differences were investigated using X-ray scattering and EXAFS combined with molecular RMC. Transmission electron microscopy and scanning precession electron diffraction were used to understand specific structural differences on all length scales, from mesoscopic down to mutual molecular arrangements. Characteristic differences were found at all scales. Close core-core spacing between {SnS} clusters as well as characteristic cluster distortions appear to be characteristic features of the white light emitting material. In the other material, cores are undistorted and core distances are larger. There, the formation of nanocrystalline structures in the amorphous matrix could also be identified as reason for the WLG suppression., Comment: 12 pages, 8 figures

Predicting the pose of objects from a single image is an important but difficult computer vision problem. Methods that predict a single point estimate do not predict the pose of objects with symmetries well and cannot represent uncertainty. Alternatively, some works predict a distribution over orientations in $\mathrm{SO}(3)$. However, training such models can be computation- and sample-inefficient. Instead, we propose a novel mapping of features from the image domain to the 3D rotation manifold. Our method then leverages $\mathrm{SO}(3)$ equivariant layers, which are more sample efficient, and outputs a distribution over rotations that can be sampled at arbitrary resolution. We demonstrate the effectiveness of our method at object orientation prediction, and achieve state-of-the-art performance on the popular PASCAL3D+ dataset. Moreover, we show that our method can model complex object symmetries, without any modifications to the parameters or loss function. Code is available at https://dmklee.github.io/image2sphere.

Aperiodic algebras are infinite dimensional algebras with generators corresponding to an element of the aperiodic set. These algebras proved to be an useful tool in studying elementary excitations that can propagate in multilayered structures and in the construction of some integrable models in quantum mechanics. Starting from the works of Patera and Twarock we present three aperiodic algebras based on Fibonacci-chain quasicrystals: a quasicrystal Lie algebra, an aperiodic Witt algebra and, finally, an aperiodic Jordan algebra. While a quasicrystal Lie algebra was already constructed from a modification of the Fibonacci chain, we here present an aperiodic algebra that matches exactly the original quasicrystal. Moreover, this is the first time to our knowledge, that an aperiodic Jordan algebra is presented leaving room for both theoretical and applicative developments., Comment: 13 pages 1 figure

We propose a variational quantum algorithm for estimating microcanonical expectation values in models obeying the eigenstate thermalization hypothesis. Using a relaxed criterion for convergence of the variational optimization loop, the algorithm generates weakly entangled superpositions of eigenstates at a given target energy density. An ensemble of these variational states is then used to estimate microcanonical averages of local operators, with an error whose dominant contribution decreases initially as a power law in the size of the ensemble and is ultimately limited by a small bias. We apply the algorithm to the one-dimensional mixed-field Ising model, where it converges for ansatz circuits of depth roughly linear in system size. The most accurate thermal estimates are produced for intermediate energy densities. In our error analysis, we find connections with recent works investigating the underpinnings of the eigenstate thermalization hypothesis. In particular, the failure of energy-basis matrix elements of local operators to behave as \textit{independent} random variables is a potential source of error that the algorithm can overcome by averaging over an ensemble of variational states., Comment: 27 pages, 20 figures, latest version contains a revised error analysis

In his study on the geometry of Lie groups, Rosenfeld postulated a strict relation between all real forms of exceptional Lie groups and the isometries of projective and hyperbolic spaces over the (rank-2) tensor product of Hurwitz algebras taken with appropriate conjugations. Unfortunately, the procedure carried out by Rosenfeld was not rigorous, since many of the theorems he had been using do not actually hold true in the case of algebras that are not alternative nor power-associative. A more rigorous approach to the definition of all the planes presented more than thirty years ago by Rosenfeld in terms of their isometry group, can be considered within the theory of coset manifolds, which we exploit in this work, by making use of all real forms of Magic Squares of order three and two over Hurwitz normed division algebras and their split versions. Within our analysis, we find 7 pseudo-Riemannian symmetric coset manifolds which seemingly cannot have any interpretation within Rosenfeld's framework. We carry out a similar analysis for Rosenfeld lines, obtaining that there are a number of pseudo-Riemannian symmetric cosets which do not have any interpretation \`a la Rosenfeld., Comment: 28 pages, 3 figures