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Numerical evaluations of the memory capacity (MC) of recurrent neural networks reported in the literature often contradict well-established theoretical bounds. In this paper, we study the case of linear echo state networks, for which the total memory capacity has been proven to be equal to the rank of the corresponding Kalman controllability matrix. We shed light on various reasons for the inaccurate numerical estimations of the memory, and we show that these issues, often overlooked in the recent literature, are of an exclusively numerical nature. More explicitly, we prove that when the Krylov structure of the linear MC is ignored, a gap between the theoretical MC and its empirical counterpart is introduced. As a solution, we develop robust numerical approaches by exploiting a result of MC neutrality with respect to the input mask matrix. Simulations show that the memory curves that are recovered using the proposed methods fully agree with the theory.

This paper extends the gradient-based reconstruction approach of Chamarthi \cite{chamarthi2023gradient} to genuine high-order accuracy for inviscid test cases involving smooth flows. A seventh-order accurate scheme is derived using the same stencil as of the explicit fourth-order scheme proposed in Ref. \cite{chamarthi2023gradient}, which also has low dissipation properties. The proposed method is seventh-order accurate under the assumption that the variables at the \textit{cell centres are point values}. A problem-independent discontinuity detector is used to obtain high-order accuracy. Accordingly, primitive or conservative variable reconstruction is performed around regions of discontinuities, whereas smooth solution regions apply flux reconstruction. The proposed approach can still share the derivatives between the inviscid and viscous fluxes, which is the main idea behind the gradient-based reconstruction. Several standard benchmark test cases are presented. The proposed method is more efficient than the seventh-order weighted compact nonlinear scheme (WCNS) for the test cases considered in this paper.

The Immersed Boundary (IB) method of Peskin (J. Comput. Phys., 1977) is useful for problems involving fluid-structure interactions or complex geometries. By making use of a regular Cartesian grid that is independent of the geometry, the IB framework yields a robust numerical scheme that can efficiently handle immersed deformable structures. Additionally, the IB method has been adapted to problems with prescribed motion and other PDEs with given boundary data. IB methods for these problems traditionally involve penalty forces which only approximately satisfy boundary conditions, or they are formulated as constraint problems. In the latter approach, one must find the unknown forces by solving an equation that corresponds to a poorly conditioned first-kind integral equation. This operation can require a large number of iterations of a Krylov method, and since a time-dependent problem requires this solve at each time step, this method can be prohibitively inefficient without preconditioning. In this work, we introduce a new, well-conditioned IB formulation for boundary value problems, which we call the Immersed Boundary Double Layer (IBDL) method. We present the method as it applies to Poisson and Helmholtz problems to demonstrate its efficiency over the original constraint method. In this double layer formulation, the equation for the unknown boundary distribution corresponds to a well-conditioned second-kind integral equation that can be solved efficiently with a small number of iterations of a Krylov method. Furthermore, the iteration count is independent of both the mesh size and immersed boundary point spacing. The method converges away from the boundary, and when combined with a local interpolation, it converges in the entire PDE domain. Additionally, while the original constraint method applies only to Dirichlet problems, the IBDL formulation can also be used for Neumann conditions.

We define new graph parameters, called flip-width, that generalize treewidth, degeneracy, and generalized coloring numbers for sparse graphs, and clique-width and twin-width for dense graphs. The flip-width parameters are defined using variants of the Cops and Robber game, in which the robber has speed bounded by a fixed constant $r\in\mathbb N\cup\{\infty\}$, and the cops perform flips (or perturbations) of the considered graph. We then propose a new notion of tameness of a graph class, called bounded flip-width, which is a dense counterpart of classes of bounded expansion of Ne\v{s}etril and Ossona de Mendez, and includes classes of bounded twin-width of Bonnet, Kim, Thomass{\'e}, and Watrigant. This unifies Sparsity Theory and Twin-width Theory, providing a common language for studying the central notions of the two theories, such as weak coloring numbers and twin-width -- corresponding to winning strategies of one player -- or dense shallow minors, rich divisions, or well-linked sets, corresponding to winning strategies of the other player. We prove that boundedness of flip-width is preserved by first-order interpretations, or transductions, generalizing previous results concerning classes of bounded expansion and bounded twin-width. We provide an algorithm approximating the flip-width of a given graph, which runs in slicewise polynomial time (XP) in the size of the graph. Finally, we propose a more general notion of tameness, called almost bounded flip-width, which is a dense counterpart of nowhere dense classes. We conjecture, and provide evidence, that classes with almost bounded flip-width coincide with monadically dependent (or monadically NIP) classes, introduced by Shelah in model theory. We also provide evidence that classes of almost bounded flip-width characterise the hereditary graph classes for which the model-checking problem is fixed-parameter tractable.

There has recently been widespread discussion of whether large language models might be sentient or conscious. Should we take this idea seriously? I will break down the strongest reasons for and against. Given mainstream assumptions in the science of consciousness, there are significant obstacles to consciousness in current models: for example, their lack of recurrent processing, a global workspace, and unified agency. At the same time, it is quite possible that these obstacles will be overcome in the next decade or so. I conclude that while it is somewhat unlikely that current large language models are conscious, we should take seriously the possibility that successors to large language models may be conscious in the not-too-distant future.

We characterize the uniqueness condition in the hardcore model for bipartite graphs with degree bounds only on one side, and provide a nearly linear time sampling algorithm that works up to the uniqueness threshold. We show that the uniqueness threshold for bipartite graph has almost the same form of the tree uniqueness threshold for general graphs, except with degree bounds only on one side of the bipartition. The hardcore model from statistical physics can be seen as a weighted enumeration of independent sets. Its bipartite version (#BIS) is a central open problem in approximate counting. Compared to the same problem in a general graph, surprising tractable regime have been identified that are believed to be hard in general. This is made possible by two lines of algorithmic approach: the high-temperature algorithms starting from Liu and Lu (STOC 2015), and the low-temperature algorithms starting from Helmuth, Perkins, and Regts (STOC 2019). In this work, we study the limit of these algorithms in the high-temperature case. Our characterization of the uniqueness condition is obtained by proving decay of correlations for arguably the best possible regime, which involves locating fixpoints of multivariate iterative rational maps and showing their contraction. We also give a nearly linear time sampling algorithm based on simulating field dynamics only on one side of the bipartite graph that works up to the uniqueness threshold. Our algorithm is very different from the original high-temperature algorithm of Liu and Lu, and it makes use of a connection between correlation decay and spectral independence of Markov chains. Last but not the least, we are able to show that the standard Glauber dynamics on both side of the bipartite graph mixes in polynomial time up to the uniqueness.

In this paper, we find a sample complexity bound for learning a simplex from noisy samples. Assume a dataset of size $n$ is given which includes i.i.d. samples drawn from a uniform distribution over an unknown simplex in $\mathbb{R}^K$, where samples are assumed to be corrupted by a multi-variate additive Gaussian noise of an arbitrary magnitude. We prove the existence of an algorithm that with high probability outputs a simplex having a $\ell_2$ distance of at most $\varepsilon$ from the true simplex (for any $\varepsilon>0$). Also, we theoretically show that in order to achieve this bound, it is sufficient to have $n\ge\left(K^2/\varepsilon^2\right)e^{\Omega\left(K/\mathrm{SNR}^2\right)}$ samples, where $\mathrm{SNR}$ stands for the signal-to-noise ratio. This result solves an important open problem and shows as long as $\mathrm{SNR}\ge\Omega\left(K^{1/2}\right)$, the sample complexity of the noisy regime has the same order to that of the noiseless case. Our proofs are a combination of the so-called sample compression technique in \citep{ashtiani2018nearly}, mathematical tools from high-dimensional geometry, and Fourier analysis. In particular, we have proposed a general Fourier-based technique for recovery of a more general class of distribution families from additive Gaussian noise, which can be further used in a variety of other related problems.

We present algorithms based on satisfiability problem (SAT) solving, as well as answer set programming (ASP), for solving the problem of determining inconsistency degrees in propositional knowledge bases. We consider six different inconsistency measures whose respective decision problems lie on the first level of the polynomial hierarchy. Namely, these are the contension inconsistency measure, the forgetting-based inconsistency measure, the hitting set inconsistency measure, the max-distance inconsistency measure, the sum-distance inconsistency measure, and the hit-distance inconsistency measure. In an extensive experimental analysis, we compare the SAT-based and ASP-based approaches with each other, as well as with a set of naive baseline algorithms. Our results demonstrate that overall, both the SAT-based and the ASP-based approaches clearly outperform the naive baseline methods in terms of runtime. The results further show that the proposed ASP-based approaches perform superior to the SAT-based ones with regard to all six inconsistency measures considered in this work. Moreover, we conduct additional experiments to explain the aforementioned results in greater detail.

In order to overcome the expressive limitations of graph neural networks (GNNs), we propose the first method that exploits vector flows over graphs to develop globally consistent directional and asymmetric aggregation functions. We show that our directional graph networks (DGNs) generalize convolutional neural networks (CNNs) when applied on a grid. Whereas recent theoretical works focus on understanding local neighbourhoods, local structures and local isomorphism with no global information flow, our novel theoretical framework allows directional convolutional kernels in any graph. First, by defining a vector field in the graph, we develop a method of applying directional derivatives and smoothing by projecting node-specific messages into the field. Then we propose the use of the Laplacian eigenvectors as such vector field, and we show that the method generalizes CNNs on an n-dimensional grid, and is provably more discriminative than standard GNNs regarding the Weisfeiler-Lehman 1-WL test. Finally, we bring the power of CNN data augmentation to graphs by providing a means of doing reflection, rotation and distortion on the underlying directional field. We evaluate our method on different standard benchmarks and see a relative error reduction of 8\% on the CIFAR10 graph dataset and 11% to 32% on the molecular ZINC dataset. An important outcome of this work is that it enables to translate any physical or biological problems with intrinsic directional axes into a graph network formalism with an embedded directional field.

Graph convolution networks (GCN) are increasingly popular in many applications, yet remain notoriously hard to train over large graph datasets. They need to compute node representations recursively from their neighbors. Current GCN training algorithms suffer from either high computational costs that grow exponentially with the number of layers, or high memory usage for loading the entire graph and node embeddings. In this paper, we propose a novel efficient layer-wise training framework for GCN (L-GCN), that disentangles feature aggregation and feature transformation during training, hence greatly reducing time and memory complexities. We present theoretical analysis for L-GCN under the graph isomorphism framework, that L-GCN leads to as powerful GCNs as the more costly conventional training algorithm does, under mild conditions. We further propose L^2-GCN, which learns a controller for each layer that can automatically adjust the training epochs per layer in L-GCN. Experiments show that L-GCN is faster than state-of-the-arts by at least an order of magnitude, with a consistent of memory usage not dependent on dataset size, while maintaining comparable prediction performance. With the learned controller, L^2-GCN can further cut the training time in half. Our codes are available at //github.com/Shen-Lab/L2-GCN.