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We have developed a reinforcement learning agent that often finds a minimal sequence of unknotting crossing changes for a knot diagram with up to 200 crossings, hence giving an upper bound on the unknotting number. We have used this to determine the unknotting number of 57k knots. We took diagrams of connected sums of such knots with oppositely signed signatures, where the summands were overlaid. The agent has found examples where several of the crossing changes in an unknotting collection of crossings result in hyperbolic knots. Based on this, we have shown that, given knots $K$ and $K'$ that satisfy some mild assumptions, there is a diagram of their connected sum and $u(K) + u(K')$ unknotting crossings such that changing any one of them results in a prime knot. As a by-product, we have obtained a dataset of 2.6 million distinct hard unknot diagrams; most of them under 35 crossings. Assuming the additivity of the unknotting number, we have determined the unknotting number of 43 at most 12-crossing knots for which the unknotting number is unknown., Comment: 29 pages, 17 figures

In this paper, we develop a class of interacting particle Langevin algorithms to solve inverse problems for partial differential equations (PDEs). In particular, we leverage the statistical finite elements (statFEM) formulation to obtain a finite-dimensional latent variable statistical model where the parameter is that of the (discretised) forward map and the latent variable is the statFEM solution of the PDE which is assumed to be partially observed. We then adapt a recently proposed expectation-maximisation like scheme, interacting particle Langevin algorithm (IPLA), for this problem and obtain a joint estimation procedure for the parameters and the latent variables. We consider three main examples: (i) estimating the forcing for linear Poisson PDE, (ii) estimating the forcing for nonlinear Poisson PDE, and (iii) estimating diffusivity for linear Poisson PDE. We provide computational complexity estimates for forcing estimation in the linear case. We also provide comprehensive numerical experiments and preconditioning strategies that significantly improve the performance, showing that the proposed class of methods can be the choice for parameter inference in PDE models., Comment: 23 pages, 6 figures

Variational inference (VI) is a computationally efficient and scalable methodology for approximate Bayesian inference. It strikes a balance between accuracy of uncertainty quantification and practical tractability. It excels at generative modelling and inversion tasks due to its built-in Bayesian regularisation and flexibility, essential qualities for physics related problems. Deriving the central learning objective for VI must often be tailored to new learning tasks where the nature of the problems dictates the conditional dependence between variables of interest, such as arising in physics problems. In this paper, we provide an accessible and thorough technical introduction to VI for forward and inverse problems, guiding the reader through standard derivations of the VI framework and how it can best be realized through deep learning. We then review and unify recent literature exemplifying the creative flexibility allowed by VI. This paper is designed for a general scientific audience looking to solve physics-based problems with an emphasis on uncertainty quantification.

Quantum error correction provides a path to reach practical quantum computing by combining multiple physical qubits into a logical qubit, where the logical error rate is suppressed exponentially as more qubits are added. However, this exponential suppression only occurs if the physical error rate is below a critical threshold. In this work, we present two surface code memories operating below this threshold: a distance-7 code and a distance-5 code integrated with a real-time decoder. The logical error rate of our larger quantum memory is suppressed by a factor of $\Lambda$ = 2.14 $\pm$ 0.02 when increasing the code distance by two, culminating in a 101-qubit distance-7 code with 0.143% $\pm$ 0.003% error per cycle of error correction. This logical memory is also beyond break-even, exceeding its best physical qubit's lifetime by a factor of 2.4 $\pm$ 0.3. We maintain below-threshold performance when decoding in real time, achieving an average decoder latency of 63 $\mu$s at distance-5 up to a million cycles, with a cycle time of 1.1 $\mu$s. To probe the limits of our error-correction performance, we run repetition codes up to distance-29 and find that logical performance is limited by rare correlated error events occurring approximately once every hour, or 3 $\times$ 10$^9$ cycles. Our results present device performance that, if scaled, could realize the operational requirements of large scale fault-tolerant quantum algorithms., Comment: 10 pages, 4 figures, Supplementary Information

The principal benefit of unsupervised representation learning is that a pre-trained model can be fine-tuned where data or labels are scarce. Existing approaches for graph representation learning are domain specific, maintaining consistent node and edge features across the pre-training and target datasets. This has precluded transfer to multiple domains. In this work we present Topology Only Pre-Training, a graph pre-training method based on node and edge feature exclusion. Separating graph learning into two stages, topology and features, we use contrastive learning to pre-train models over multiple domains. These models show positive transfer on evaluation datasets from multiple domains, including domains not present in pre-training data. On 75% of experiments, ToP models perform significantly ($P \leq 0.01$) better than a supervised baseline. These results include when node and edge features are used in evaluation, where performance is significantly better on 85.7% of tasks compared to single-domain or non-pre-trained models. We further show that out-of-domain topologies can produce more useful pre-training than in-domain. We show better transfer from non-molecule pre-training, compared to molecule pre-training, on 79% of molecular benchmarks., Comment: 23 pages, 5 figures, 11 tables. For in-development code see https://github.com/neutralpronoun/general-gcl

Quantum error-correction is a prerequisite for reliable quantum computation. Towards this goal, we present a recurrent, transformer-based neural network which learns to decode the surface code, the leading quantum error-correction code. Our decoder outperforms state-of-the-art algorithmic decoders on real-world data from Google's Sycamore quantum processor for distance 3 and 5 surface codes. On distances up to 11, the decoder maintains its advantage on simulated data with realistic noise including cross-talk, leakage, and analog readout signals, and sustains its accuracy far beyond the 25 cycles it was trained on. Our work illustrates the ability of machine learning to go beyond human-designed algorithms by learning from data directly, highlighting machine learning as a strong contender for decoding in quantum computers.

Large graphs are present in a variety of domains, including social networks, civil infrastructure, and the physical sciences to name a few. Graph generation is similarly widespread, with applications in drug discovery, network analysis and synthetic datasets among others. While GNN (Graph Neural Network) models have been applied in these domains their high in-memory costs restrict them to small graphs. Conversely less costly rule-based methods struggle to reproduce complex structures. We propose HIGGS (Hierarchical Generation of Graphs) as a model-agnostic framework of producing large graphs with realistic local structures. HIGGS uses GNN models with conditional generation capabilities to sample graphs in hierarchies of resolution. As a result HIGGS has the capacity to extend the scale of generated graphs from a given GNN model by quadratic order. As a demonstration we implement HIGGS using DiGress, a recent graph-diffusion model, including a novel edge-predictive-diffusion variant edge-DiGress. We use this implementation to generate categorically attributed graphs with tens of thousands of nodes. These HIGGS generated graphs are far larger than any previously produced using GNNs. Despite this jump in scale we demonstrate that the graphs produced by HIGGS are, on the local scale, more realistic than those from the rule-based model BTER., Comment: Presented at the NeurIPS 2023 Synthetic Data Generation with Generative AI workshop

Social network analysis faces profound difficulties in sharing data between researchers due to privacy and security concerns. A potential remedy to this issue are synthetic networks, that closely resemble their real counterparts, but can be freely distributed. generating synthetic networks requires the creation of network topologies that, in application, function as realistically as possible. Widely applied models are currently rule-based and can struggle to reproduce structural dynamics. Lead by recent developments in Graph Neural Network (GNN) models for network generation we evaluate the potential of GNNs for synthetic social networks. Our GNN use is specifically within a reasonable use-case and includes empirical evaluation using Maximum Mean Discrepancy (MMD). We include social network specific measurements which allow evaluation of how realistically synthetic networks behave in typical social network analysis applications. We find that the Gated Recurrent Attention Network (GRAN) extends well to social networks, and in comparison to a benchmark popular rule-based generation Recursive-MATrix (R-MAT) method, is better able to replicate realistic structural dynamics. We find that GRAN is more computationally costly than R-MAT, but is not excessively costly to employ, so would be effective for researchers seeking to create datasets of synthetic social networks., Comment: 12 pages, 4 figures

Incorporating unstructured data into physical models is a challenging problem that is emerging in data assimilation. Traditional approaches focus on well-defined observation operators whose functional forms are typically assumed to be known. This prevents these methods from achieving a consistent model-data synthesis in configurations where the mapping from data-space to model-space is unknown. To address these shortcomings, in this paper we develop a physics-informed dynamical variational autoencoder ($\Phi$-DVAE) to embed diverse data streams into time-evolving physical systems described by differential equations. Our approach combines a standard, possibly nonlinear, filter for the latent state-space model and a VAE, to assimilate the unstructured data into the latent dynamical system. Unstructured data, in our example systems, comes in the form of video data and velocity field measurements, however the methodology is suitably generic to allow for arbitrary unknown observation operators. A variational Bayesian framework is used for the joint estimation of the encoding, latent states, and unknown system parameters. To demonstrate the method, we provide case studies with the Lorenz-63 ordinary differential equation, and the advection and Korteweg-de Vries partial differential equations. Our results, with synthetic data, show that $\Phi$-DVAE provides a data efficient dynamics encoding methodology which is competitive with standard approaches. Unknown parameters are recovered with uncertainty quantification, and unseen data are accurately predicted., Comment: 29 pages, 9 figures, updated version

We introduce a new real-valued invariant called the natural slope of a hyperbolic knot in the 3-sphere, which is defined in terms of its cusp geometry. We show that twice the knot signature and the natural slope differ by at most a constant times the hyperbolic volume divided by the cube of the injectivity radius. This inequality was discovered using machine learning to detect relationships between various knot invariants. It has applications to Dehn surgery and to 4-ball genus. We also show a refined version of the inequality where the upper bound is a linear function of the volume, and the slope is corrected by terms corresponding to short geodesics that link the knot an odd number of times., Comment: 28 pages, 13 figures. v3: revised final version. Accepted by Geometry & Topology

Kazhdan-Lusztig polynomials are important and mysterious objects in representation theory. Here we present a new formula for their computation for symmetric groups based on the Bruhat graph. Our approach suggests a solution to the combinatorial invariance conjecture for symmetric groups, a well-known conjecture formulated by Lusztig and Dyer in the 1980s., Comment: 47 pages, comments welcome

We present Random Partition Kernels, a new class of kernels derived by demonstrating a natural connection between random partitions of objects and kernels between those objects. We show how the construction can be used to create kernels from methods that would not normally be viewed as random partitions, such as Random Forest. To demonstrate the potential of this method, we propose two new kernels, the Random Forest Kernel and the Fast Cluster Kernel, and show that these kernels consistently outperform standard kernels on problems involving real-world datasets. Finally, we show how the form of these kernels lend themselves to a natural approximation that is appropriate for certain big data problems, allowing $O(N)$ inference in methods such as Gaussian Processes, Support Vector Machines and Kernel PCA.

The Internet has enabled the creation of a growing number of large-scale knowledge bases in a variety of domains containing complementary information. Tools for automatically aligning these knowledge bases would make it possible to unify many sources of structured knowledge and answer complex queries. However, the efficient alignment of large-scale knowledge bases still poses a considerable challenge. Here, we present Simple Greedy Matching (SiGMa), a simple algorithm for aligning knowledge bases with millions of entities and facts. SiGMa is an iterative propagation algorithm which leverages both the structural information from the relationship graph as well as flexible similarity measures between entity properties in a greedy local search, thus making it scalable. Despite its greedy nature, our experiments indicate that SiGMa can efficiently match some of the world's largest knowledge bases with high precision. We provide additional experiments on benchmark datasets which demonstrate that SiGMa can outperform state-of-the-art approaches both in accuracy and efficiency., Comment: 10 pages + 2 pages appendix; 5 figures -- initial preprint