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Physical neural networks are promising candidates for next generation artificial intelligence hardware. In such architectures, neurons and connections are physically realized and do not leverage digital, i.e. practically infinite signal-to-noise ratio digital concepts. They therefore are prone to noise, and base don analytical derivations we here introduce connectivity topologies, ghost neurons as well as pooling as noise mitigation strategies. Finally, we demonstrate the effectiveness of the combined methods based on a fully trained neural network classifying the MNIST handwritten digits.

Spectral independence is a recently-developed framework for obtaining sharp bounds on the convergence time of the classical Glauber dynamics. This new framework has yielded optimal $O(n \log n)$ sampling algorithms on bounded-degree graphs for a large class of problems throughout the so-called uniqueness regime, including, for example, the problems of sampling independent sets, matchings, and Ising-model configurations. Our main contribution is to relax the bounded-degree assumption that has so far been important in establishing and applying spectral independence. Previous methods for avoiding degree bounds rely on using $L^p$-norms to analyse contraction on graphs with bounded connective constant (Sinclair, Srivastava, Yin; FOCS'13). The non-linearity of $L^p$-norms is an obstacle to applying these results to bound spectral independence. Our solution is to capture the $L^p$-analysis recursively by amortising over the subtrees of the recurrence used to analyse contraction. Our method generalises previous analyses that applied only to bounded-degree graphs. As a main application of our techniques, we consider the random graph $G(n,d/n)$, where the previously known algorithms run in time $n^{O(\log d)}$ or applied only to large $d$. We refine these algorithmic bounds significantly, and develop fast $n^{1+o(1)}$ algorithms based on Glauber dynamics that apply to all $d$, throughout the uniqueness regime.

Graph Convolutional Networks (GCNs) are one of the most popular architectures that are used to solve classification problems accompanied by graphical information. We present a rigorous theoretical understanding of the effects of graph convolutions in multi-layer networks. We study these effects through the node classification problem of a non-linearly separable Gaussian mixture model coupled with a stochastic block model. First, we show that a single graph convolution expands the regime of the distance between the means where multi-layer networks can classify the data by a factor of at least $1/\sqrt[4]{\mathbb{E}{\rm deg}}$, where $\mathbb{E}{\rm deg}$ denotes the expected degree of a node. Second, we show that with a slightly stronger graph density, two graph convolutions improve this factor to at least $1/\sqrt[4]{n}$, where $n$ is the number of nodes in the graph. Finally, we provide both theoretical and empirical insights into the performance of graph convolutions placed in different combinations among the layers of a network, concluding that the performance is mutually similar for all combinations of the placement. We present extensive experiments on both synthetic and real-world data that illustrate our results.

In this paper we present a deep learning method to predict the temporal evolution of dissipative dynamic systems. We propose using both geometric and thermodynamic inductive biases to improve accuracy and generalization of the resulting integration scheme. The first is achieved with Graph Neural Networks, which induces a non-Euclidean geometrical prior with permutation invariant node and edge update functions. The second bias is forced by learning the GENERIC structure of the problem, an extension of the Hamiltonian formalism, to model more general non-conservative dynamics. Several examples are provided in both Eulerian and Lagrangian description in the context of fluid and solid mechanics respectively, achieving relative mean errors of less than 3% in all the tested examples. Two ablation studies are provided based on recent works in both physics-informed and geometric deep learning.

We consider the question of adaptive data analysis within the framework of convex optimization. We ask how many samples are needed in order to compute $\epsilon$-accurate estimates of $O(1/\epsilon^2)$ gradients queried by gradient descent, and we provide two intermediate answers to this question. First, we show that for a general analyst (not necessarily gradient descent) $\Omega(1/\epsilon^3)$ samples are required. This rules out the possibility of a foolproof mechanism. Our construction builds upon a new lower bound (that may be of interest of its own right) for an analyst that may ask several non adaptive questions in a batch of fixed and known $T$ rounds of adaptivity and requires a fraction of true discoveries. We show that for such an analyst $\Omega (\sqrt{T}/\epsilon^2)$ samples are necessary. Second, we show that, under certain assumptions on the oracle, in an interaction with gradient descent $\tilde \Omega(1/\epsilon^{2.5})$ samples are necessary. Our assumptions are that the oracle has only \emph{first order access} and is \emph{post-hoc generalizing}. First order access means that it can only compute the gradients of the sampled function at points queried by the algorithm. Our assumption of \emph{post-hoc generalization} follows from existing lower bounds for statistical queries. More generally then, we provide a generic reduction from the standard setting of statistical queries to the problem of estimating gradients queried by gradient descent. These results are in contrast with classical bounds that show that with $O(1/\epsilon^2)$ samples one can optimize the population risk to accuracy of $O(\epsilon)$ but, as it turns out, with spurious gradients.

Given the prevalence of large-scale graphs in real-world applications, the storage and time for training neural models have raised increasing concerns. To alleviate the concerns, we propose and study the problem of graph condensation for graph neural networks (GNNs). Specifically, we aim to condense the large, original graph into a small, synthetic and highly-informative graph, such that GNNs trained on the small graph and large graph have comparable performance. We approach the condensation problem by imitating the GNN training trajectory on the original graph through the optimization of a gradient matching loss and design a strategy to condense node futures and structural information simultaneously. Extensive experiments have demonstrated the effectiveness of the proposed framework in condensing different graph datasets into informative smaller graphs. In particular, we are able to approximate the original test accuracy by 95.3% on Reddit, 99.8% on Flickr and 99.0% on Citeseer, while reducing their graph size by more than 99.9%, and the condensed graphs can be used to train various GNN architectures.Code is released at //github.com/ChandlerBang/GCond.

We propose a novel concise function representation for graphical models, a central theoretical framework that provides the basis for many reasoning tasks. We then show how we exploit our concise representation based on deterministic finite state automata within Bucket Elimination (BE), a general approach based on the concept of variable elimination that can be used to solve many inference and optimisation tasks, such as most probable explanation and constrained optimisation. We denote our version of BE as FABE. By using our concise representation within FABE, we dramatically improve the performance of BE in terms of runtime and memory requirements. Results achieved by comparing FABE with state of the art approaches for most probable explanation (i.e., recursive best-first and structured message passing) and constrained optimisation (i.e., CPLEX, GUROBI, and toulbar2) following an established methodology confirm the efficacy of our concise function representation, showing runtime improvements of up to 5 orders of magnitude in our tests.

A High-dimensional and sparse (HiDS) matrix is frequently encountered in a big data-related application like an e-commerce system or a social network services system. To perform highly accurate representation learning on it is of great significance owing to the great desire of extracting latent knowledge and patterns from it. Latent factor analysis (LFA), which represents an HiDS matrix by learning the low-rank embeddings based on its observed entries only, is one of the most effective and efficient approaches to this issue. However, most existing LFA-based models perform such embeddings on a HiDS matrix directly without exploiting its hidden graph structures, thereby resulting in accuracy loss. To address this issue, this paper proposes a graph-incorporated latent factor analysis (GLFA) model. It adopts two-fold ideas: 1) a graph is constructed for identifying the hidden high-order interaction (HOI) among nodes described by an HiDS matrix, and 2) a recurrent LFA structure is carefully designed with the incorporation of HOI, thereby improving the representa-tion learning ability of a resultant model. Experimental results on three real-world datasets demonstrate that GLFA outperforms six state-of-the-art models in predicting the missing data of an HiDS matrix, which evidently supports its strong representation learning ability to HiDS data.

We introduce Universal Solution Manifold Network (USM-Net), a novel surrogate model, based on Artificial Neural Networks (ANNs), which applies to differential problems whose solution depends on physical and geometrical parameters. Our method employs a mesh-less architecture, thus overcoming the limitations associated with image segmentation and mesh generation required by traditional discretization methods. Indeed, we encode geometrical variability through scalar landmarks, such as coordinates of points of interest. In biomedical applications, these landmarks can be inexpensively processed from clinical images. Our approach is non-intrusive and modular, as we select a data-driven loss function. The latter can also be modified by considering additional constraints, thus leveraging available physical knowledge. Our approach can also accommodate a universal coordinate system, which supports the USM-Net in learning the correspondence between points belonging to different geometries, boosting prediction accuracy on unobserved geometries. Finally, we present two numerical test cases in computational fluid dynamics involving variable Reynolds numbers as well as computational domains of variable shape. The results show that our method allows for inexpensive but accurate approximations of velocity and pressure, avoiding computationally expensive image segmentation, mesh generation, or re-training for every new instance of physical parameters and shape of the domain.

Causality can be described in terms of a structural causal model (SCM) that carries information on the variables of interest and their mechanistic relations. For most processes of interest the underlying SCM will only be partially observable, thus causal inference tries to leverage any exposed information. Graph neural networks (GNN) as universal approximators on structured input pose a viable candidate for causal learning, suggesting a tighter integration with SCM. To this effect we present a theoretical analysis from first principles that establishes a novel connection between GNN and SCM while providing an extended view on general neural-causal models. We then establish a new model class for GNN-based causal inference that is necessary and sufficient for causal effect identification. Our empirical illustration on simulations and standard benchmarks validate our theoretical proofs.