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Encoding geospatial data is crucial for enabling machine learning (ML) models to perform tasks that require spatial reasoning, such as identifying the topological relationships between two different geospatial objects. However, existing encoding methods are limited as they are typically customized to handle only specific types of spatial data, which impedes their applicability across different downstream tasks where multiple data types coexist. To address this, we introduce Poly2Vec, an encoding framework that unifies the modeling of different geospatial objects, including 2D points, polylines, and polygons, irrespective of the downstream task. We leverage the power of the 2D Fourier transform to encode useful spatial properties, such as shape and location, from geospatial objects into fixed-length vectors. These vectors are then inputted into neural network models for spatial reasoning tasks.This unified approach eliminates the need to develop and train separate models for each distinct spatial type. We evaluate Poly2Vec on both synthetic and real datasets of mixed geometry types and verify its consistent performance across several downstream spatial reasoning tasks.

Simulating human mobility data is essential for various application domains, including transportation, urban planning, and epidemic control, since real data are often inaccessible to researchers due to expensive costs and privacy issues. Several existing deep generative solutions propose learning from real trajectories to generate synthetic ones. Despite the progress, most of them suffer from training stability issues and scale poorly with growing data size. More importantly, they generally lack control mechanisms to steer the generated trajectories based on spatiotemporal constraints such as fixing specific visits. To address such limitations, we formally define the controlled trajectory generation problem with spatiotemporal constraints and propose Geo-Llama. This novel LLM-inspired framework enforces explicit visit constraints in a contextually coherent way. It fine-tunes pre-trained LLMs on trajectories with a visit-wise permutation strategy where each visit corresponds to a time and location. This enables the model to capture the spatiotemporal patterns regardless of visit orders and allows flexible and in-context constraint integration through prompts during generation. Extensive experiments on real-world and synthetic datasets validate the effectiveness of Geo-Llama, demonstrating its versatility and robustness in handling a broad range of constraints to generate more realistic trajectories compared to existing methods.

Cardiac Cine Magnetic Resonance Imaging (MRI) provides an accurate assessment of heart morphology and function in clinical practice. However, MRI requires long acquisition times, with recent deep learning-based methods showing great promise to accelerate imaging and enhance reconstruction quality. Existing networks exhibit some common limitations that constrain further acceleration possibilities, including single-domain learning, reliance on a single regularization term, and equal feature contribution. To address these limitations, we propose to embed information from multiple domains, including low-rank, image, and k-space, in a novel deep learning network for MRI reconstruction, which we denote as A-LIKNet. A-LIKNet adopts a parallel-branch structure, enabling independent learning in the k-space and image domain. Coupled information sharing layers realize the information exchange between domains. Furthermore, we introduce attention mechanisms into the network to assign greater weights to more critical coils or important temporal frames. Training and testing were conducted on an in-house dataset, including 91 cardiovascular patients and 38 healthy subjects scanned with 2D cardiac Cine using retrospective undersampling. Additionally, we evaluated A-LIKNet on the real-time 8x prospectively undersampled data from the OCMR dataset. The results demonstrate that our proposed A-LIKNet outperforms existing methods and provides high-quality reconstructions. The network can effectively reconstruct highly retrospectively undersampled dynamic MR images up to 24x accelerations, indicating its potential for single breath-hold imaging.

Accurate motion estimation at high acceleration factors enables rapid motion-compensated reconstruction in Magnetic Resonance Imaging (MRI) without compromising the diagnostic image quality. In this work, we introduce an attention-aware deep learning-based framework that can perform non-rigid pairwise registration for fully sampled and accelerated MRI. We extract local visual representations to build similarity maps between the registered image pairs at multiple resolution levels and additionally leverage long-range contextual information using a transformer-based module to alleviate ambiguities in the presence of artifacts caused by undersampling. We combine local and global dependencies to perform simultaneous coarse and fine motion estimation. The proposed method was evaluated on in-house acquired fully sampled and accelerated data of 101 patients and 62 healthy subjects undergoing cardiac and thoracic MRI. The impact of motion estimation accuracy on the downstream task of motion-compensated reconstruction was analyzed. We demonstrate that our model derives reliable and consistent motion fields across different sampling trajectories (Cartesian and radial) and acceleration factors of up to 16x for cardiac motion and 30x for respiratory motion and achieves superior image quality in motion-compensated reconstruction qualitatively and quantitatively compared to conventional and recent deep learning-based approaches. The code is publicly available at https://github.com/lab-midas/GMARAFT., Comment: 14 pages, 7 figures

Motivated by a uniform boundedness conjecture of Morton and Silverman, we study the graphs of pre-periodic points for maps in three families of dynamical systems, namely the collections of rational functions of degree two having a periodic critical point of period $n$, where $n\in\{2,3,4\}$. In particular, we provide a conjecturally complete list of possible graphs of rational pre-periodic points in the case $n=4$, analogous to well-known work of Poonen for $n=1$, and we strengthen earlier results of Canci and Vishkautsan for $n\in\{2,3\}$. In addition, we address the problem of determining the representability of a given graph in our list by infinitely many distinct linear conjugacy classes of maps.

With the impressive progress of deep learning, applications relying on machine learning are increasingly being integrated into daily life. However, most deep learning models have an opaque, oracle-like nature making it difficult to interpret and understand their decisions. This problem led to the development of the field known as eXplainable Artificial Intelligence (XAI). One method in this field known as Projective Simulation (PS) models a chain-of-thought as a random walk of a particle on a graph with vertices that have concepts attached to them. While this description has various benefits, including the possibility of quantization, it cannot be naturally used to model thoughts that combine several concepts simultaneously. To overcome this limitation, we introduce Multi-Excitation Projective Simulation (mePS), a generalization that considers a chain-of-thought to be a random walk of several particles on a hypergraph. A definition for a dynamic hypergraph is put forward to describe the agent's training history along with applications to AI and hypergraph visualization. An inductive bias inspired by the remarkably successful few-body interaction models used in quantum many-body physics is formalized for our classical mePS framework and employed to tackle the exponential complexity associated with naive implementations of hypergraphs. We prove that our inductive bias reduces the complexity from exponential to polynomial, with the exponent representing the cutoff on how many particles can interact. We numerically apply our method to two toy environments and a more complex scenario modelling the diagnosis of a broken computer. These environments demonstrate the resource savings provided by an appropriate choice of inductive bias, as well as showcasing aspects of interpretability. A quantum model for mePS is also briefly outlined and some future directions for it are discussed., Comment: 24 pages, 8 figures; Code repository at https://github.com/MariusKrumm/ManyBodyMEPS. Added figures and shortened computer maintenance section text for better readability

Measurement-based quantum computation (MBQC) offers a fundamentally unique paradigm to design quantum algorithms. Indeed, due to the inherent randomness of quantum measurements, the natural operations in MBQC are not deterministic and unitary, but are rather augmented with probabilistic byproducts. Yet, the main algorithmic use of MBQC so far has been to completely counteract this probabilistic nature in order to simulate unitary computations expressed in the circuit model. In this work, we propose designing MBQC algorithms that embrace this inherent randomness and treat the random byproducts in MBQC as a resource for computation. As a natural application where randomness can be beneficial, we consider generative modeling, a task in machine learning centered around generating complex probability distributions. To address this task, we propose a variational MBQC algorithm equipped with control parameters that allow to directly adjust the degree of randomness to be admitted in the computation. Our numerical findings indicate that this additional randomness can lead to significant gains in learning performance in certain generative modeling tasks. These results highlight the potential advantages in exploiting the inherent randomness of MBQC and motivate further research into MBQC-based algorithms., Comment: 12 pages, 7 figures

Mobility data captures the locations of moving objects such as humans, animals, and cars. With the availability of GPS-equipped mobile devices and other inexpensive location-tracking technologies, mobility data is collected ubiquitously. In recent years, the use of mobility data has demonstrated significant impact in various domains including traffic management, urban planning, and health sciences. In this paper, we present the emerging domain of mobility data science. Towards a unified approach to mobility data science, we envision a pipeline having the following components: mobility data collection, cleaning, analysis, management, and privacy. For each of these components, we explain how mobility data science differs from general data science, we survey the current state of the art and describe open challenges for the research community in the coming years., Comment: Updated to reflect the major revision for ACM Transactions on Spatial Algorithms and Systems (TSAS). This version reflects the final version accepted by ACM TSAS

We prove the existence of transcendental entire functions $f$ having a property studied by Mahler, namely that $f(\overline{\mathbb{Q}})\subseteq \overline{\mathbb{Q}}$ and $f^{-1}(\overline{\mathbb{Q}})\subseteq \overline{\mathbb{Q}}$, and in addition having a prescribed number of $k$-periodic algebraic orbits, for all $k\geq 1$. Under a suitable topology, such functions are shown to be dense in the set of all entire transcendental functions.

For every nonconstant rational function $\phi\in\mathbb{Q}(x)$, the Galois groups of the dynatomic polynomials of $\phi$ encode various properties of $\phi$ that are of interest in the subject of arithmetic dynamics. We study here the structure of these Galois groups as $\phi$ varies in a particular one-parameter family of maps, namely the quadratic rational maps having a critical point of period 2. In particular, we provide explicit descriptions of the third and fourth dynatomic Galois groups for maps in this family.

Variational quantum algorithms represent a promising approach to quantum machine learning where classical neural networks are replaced by parametrized quantum circuits. However, both approaches suffer from a clear limitation, that is a lack of interpretability. Here, we present a variational method to quantize projective simulation (PS), a reinforcement learning model aimed at interpretable artificial intelligence. Decision making in PS is modeled as a random walk on a graph describing the agent's memory. To implement the quantized model, we consider quantum walks of single photons in a lattice of tunable Mach-Zehnder interferometers trained via variational algorithms. Using an example from transfer learning, we show that the quantized PS model can exploit quantum interference to acquire capabilities beyond those of its classical counterpart. Finally, we discuss the role of quantum interference for training and tracing the decision making process, paving the way for realizations of interpretable quantum learning agents., Comment: 11+8 pages, 6+9 figures, 2 tables. F. Flamini and M. Krumm contributed equally to this work

Among all the dynamical modular curves associated to quadratic polynomial maps, we determine which curves have infinitely many quadratic points. This yields a classification statement on preperiodic points for quadratic polynomials over quadratic fields, extending previous work of Poonen, Faber, and the authors., Comment: Improvements to the exposition; stronger version of what is now Proposition 4.5

Learning a hidden property of a quantum system typically requires a series of interactions. In this work, we formalise such multi-round learning processes using a generalisation of classical-quantum states, called classical-quantum combs. Here, "classical" refers to a random variable encoding the hidden property to be learnt, and "quantum" refers to the quantum comb describing the behaviour of the system. The optimal strategy for learning the hidden property can be quantified by applying the comb min-entropy (Chiribella and Ebler, NJP, 2016) to classical-quantum combs. To demonstrate the power of this approach, we focus attention on an array of problems derived from measurement-based quantum computation (MBQC) and related applications. Specifically, we describe a known blind quantum computation (BQC) protocol using the combs formalism and thereby leverage the min-entropy to provide a proof of single-shot security for multiple rounds of the protocol, extending the existing result in the literature. Furthermore, we consider a range of operationally motivated examples related to the verification of a partially unknown MBQC device. These examples involve learning the features of the device necessary for its correct use, including learning its internal reference frame for measurement calibration. We also introduce a novel connection between MBQC and quantum causal models that arises in this context., Comment: 24 + 17 pages, 11 figures; Modifications to the presentation; some additional results added. Accepted in Quantum 2023-12-01