Autonomous morphology, such as inflection class systems and paradigmatic distribution patterns, is widespread and diachronically resilient in natural language. Why this should be so has remained unclear given that autonomous morphology imposes learning costs, offers no clear benefit relative to its absence and could easily be removed by the analogical forces which are constantly reshaping it. Here we propose an explanation for the resilience of autonomous morphology, in terms of a diachronic dynamic of attraction and repulsion between morphomic categories, which emerges spontaneously from a simple paradigm cell filling process. Employing computational evolutionary models, our key innovation is to bring to light the role of `dissociative evidence', i.e., evidence for inflectional distinctiveness which a rational reasoner will have access to during analogical inference. Dissociative evidence creates a repulsion dynamic which prevents morphomic classes from collapsing together entirely, i.e., undergoing complete levelling. As we probe alternative models, we reveal the limits of conditional entropy as a measure for predictability in systems that are undergoing change. Finally, we demonstrate that autonomous morphology, far from being `unnatural' (e.g. \citealt{Aronoff1994}), is rather the natural (emergent) consequence of a natural (rational) process of inference applied to inflectional systems.
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Machine learning interatomic potentials (MLIPs) often neglect long-range interactions, such as electrostatic and dispersion forces. In this work, we introduce a straightforward and efficient method to account for long-range interactions by learning a latent variable from local atomic descriptors and applying an Ewald summation to this variable. We demonstrate that in systems including charged and polar molecular dimers, bulk water, and water-vapor interface, standard short-ranged MLIPs can lead to unphysical predictions even when employing message passing. The long-range models effectively eliminate these artifacts, with only about twice the computational cost of short-range MLIPs.
Neuromorphic computing aims to replicate the brain's capabilities for energy efficient and parallel information processing, promising a solution to the increasing demand for faster and more efficient computational systems. Efficient training of neural networks on neuromorphic hardware requires the development of training algorithms that retain the sparsity of spike-based communication during training. Here, we report on the first implementation of event-based backpropagation on the SpiNNaker2 neuromorphic hardware platform. We use EventProp, an algorithm for event-based backpropagation in spiking neural networks (SNNs), to compute exact gradients using sparse communication of error signals between neurons. Our implementation computes multi-layer networks of leaky integrate-and-fire neurons using discretized versions of the differential equations and their adjoints, and uses event packets to transmit spikes and error signals between network layers. We demonstrate a proof-of-concept of batch-parallelized, on-chip training of SNNs using the Yin Yang dataset, and provide an off-chip implementation for efficient prototyping, hyper-parameter search, and hybrid training methods.
We propose a method utilizing physics-informed neural networks (PINNs) to solve Poisson equations that serve as control variates in the computation of transport coefficients via fluctuation formulas, such as the Green--Kubo and generalized Einstein-like formulas. By leveraging approximate solutions to the Poisson equation constructed through neural networks, our approach significantly reduces the variance of the estimator at hand. We provide an extensive numerical analysis of the estimators and detail a methodology for training neural networks to solve these Poisson equations. The approximate solutions are then incorporated into Monte Carlo simulations as effective control variates, demonstrating the suitability of the method for moderately high-dimensional problems where fully deterministic solutions are computationally infeasible.
We present a novel class of projected gradient (PG) methods for minimizing a smooth but not necessarily convex function over a convex compact set. We first provide a novel analysis of the "vanilla" PG method, achieving the best-known iteration complexity for finding an approximate stationary point of the problem. We then develop an "auto-conditioned" projected gradient (AC-PG) variant that achieves the same iteration complexity without requiring the input of the Lipschitz constant of the gradient or any line search procedure. The key idea is to estimate the Lipschitz constant using first-order information gathered from the previous iterations, and to show that the error caused by underestimating the Lipschitz constant can be properly controlled. We then generalize the PG methods to the stochastic setting, by proposing a stochastic projected gradient (SPG) method and a variance-reduced stochastic gradient (VR-SPG) method, achieving new complexity bounds in different oracle settings. We also present auto-conditioned stepsize policies for both stochastic PG methods and establish comparable convergence guarantees.
In order to determine an optimal plan for a complex task, one often deals with dynamic and hierarchical relationships between several entities. Traditionally, such problems are tackled with optimal control, which relies on the optimization of cost functions; instead, a recent biologically-motivated proposal casts planning and control as an inference process. Active inference assumes that action and perception are two complementary aspects of life whereby the role of the former is to fulfill the predictions inferred by the latter. In this study, we present a solution, based on active inference, for complex control tasks. The proposed architecture exploits hybrid (discrete and continuous) processing, and it is based on three features: the representation of potential body configurations related to the objects of interest; the use of hierarchical relationships that enable the agent to flexibly expand its body schema for tool use; the definition of potential trajectories related to the agent's intentions, used to infer and plan with dynamic elements at different temporal scales. We evaluate this deep hybrid model on a habitual task: reaching a moving object after having picked a moving tool. We show that the model can tackle the presented task under different conditions. This study extends past work on planning as inference and advances an alternative direction to optimal control.
A statistical network model with overlapping communities can be generated as a superposition of mutually independent random graphs of varying size. The model is parameterized by the number of nodes, the number of communities, and the joint distribution of the community size and the edge probability. This model admits sparse parameter regimes with power-law limiting degree distributions and non-vanishing clustering coefficients. This article presents large-scale approximations of clique and cycle frequencies for graph samples generated by the model, which are valid for regimes with unbounded numbers of overlapping communities. Our results reveal the growth rates of these subgraph frequencies and show that their theoretical densities can be reliably estimated from data.
Statistical learning under distribution shift is challenging when neither prior knowledge nor fully accessible data from the target distribution is available. Distributionally robust learning (DRL) aims to control the worst-case statistical performance within an uncertainty set of candidate distributions, but how to properly specify the set remains challenging. To enable distributional robustness without being overly conservative, in this paper, we propose a shape-constrained approach to DRL, which incorporates prior information about the way in which the unknown target distribution differs from its estimate. More specifically, we assume the unknown density ratio between the target distribution and its estimate is isotonic with respect to some partial order. At the population level, we provide a solution to the shape-constrained optimization problem that does not involve the isotonic constraint. At the sample level, we provide consistency results for an empirical estimator of the target in a range of different settings. Empirical studies on both synthetic and real data examples demonstrate the improved accuracy of the proposed shape-constrained approach.
We prove, for stably computably enumerable formal systems, direct analogues of the first and second incompleteness theorems of G\"odel. A typical stably computably enumerable set is the set of Diophantine equations with no integer solutions, and in particular such sets are generally not computably enumerable. And so this gives the first extension of the second incompleteness theorem to non classically computable formal systems. Let's motivate this with a somewhat physical application. Let $\mathcal{H} $ be the suitable infinite time limit (stabilization in the sense of the paper) of the mathematical output of humanity, specializing to first order sentences in the language of arithmetic (for simplicity), and understood as a formal system. Suppose that all the relevant physical processes in the formation of $\mathcal{H} $ are Turing computable. Then as defined $\mathcal{H} $ may \emph{not} be computably enumerable, but it is stably computably enumerable. Thus, the classical G\"odel disjunction applied to $\mathcal{H} $ is meaningless, but applying our incompleteness theorems to $\mathcal{H} $ we then get a sharper version of G\"odel's disjunction: assume $\mathcal{H} \vdash PA$ then either $\mathcal{H} $ is not stably computably enumerable or $\mathcal{H} $ is not 1-consistent (in particular is not sound) or $\mathcal{H} $ cannot prove a certain true statement of arithmetic (and cannot disprove it if in addition $\mathcal{H} $ is 2-consistent).
Online optimisation studies the convergence of optimisation methods as the data embedded in the problem changes. Based on this idea, we propose a primal dual online method for nonlinear time-discrete inverse problems. We analyse the method through regret theory and demonstrate its performance in real-time monitoring of moving bodies in a fluid with Electrical Impedance Tomography (EIT). To do so, we also prove the second-order differentiability of the Complete Electrode Model (CEM) solution operator on $L^\infty$.
Many combinatorial optimization problems can be formulated as the search for a subgraph that satisfies certain properties and minimizes the total weight. We assume here that the vertices correspond to points in a metric space and can take any position in given uncertainty sets. Then, the cost function to be minimized is the sum of the distances for the worst positions of the vertices in their uncertainty sets. We propose two types of polynomial-time approximation algorithms. The first one relies on solving a deterministic counterpart of the problem where the uncertain distances are replaced with maximum pairwise distances. We study in details the resulting approximation ratio, which depends on the structure of the feasible subgraphs and whether the metric space is Ptolemaic or not. The second algorithm is a fully-polynomial time approximation scheme for the special case of $s-t$ paths.
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