Graph Neural Network (GNN) with its ability to integrate graph information has been widely used for data analyses. However, the expressive power of GNN has only been studied for graph-level tasks but not for node-level tasks, such as node classification, where one tries to interpolate missing nodal labels from the observed ones. In this paper, we study the expressive power of GNN for the said classification task, which is in essence a function interpolation problem. Explicitly, we derive the number of weights and layers needed for a GNN to interpolate a band-limited function in $\mathbb{R}^d$. Our result shows that, the number of weights needed to $\epsilon$-approximate a bandlimited function using the GNN architecture is much fewer than the best known one using a fully connected neural network (NN) - in particular, one only needs $O((\log \epsilon^{-1})^{d})$ weights using a GNN trained by $O((\log \epsilon^{-1})^{d})$ samples to $\epsilon$-approximate a discretized bandlimited signal in $\mathbb{R}^d$. The result is obtained by drawing a connection between the GNN structure and the classical sampling theorems, making our work the first attempt in this direction.
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In literature, NAND and NOR are two logic gates that display functional completeness, hence regarded as Universal gates. So, the present effort is focused on exploring a library of universal gates in binary that are still unexplored in literature along with a broad and systematic approach to classify the logic connectives. The study shows that the number of Universal Gates in any logic system grows exponentially with the number of input variables $N$. It is revealed that there are $56$ Universal gates in binary for $N=3$. It is shown that the ratio of the count of Universal gates to the total number of Logic gates is $\approx $ $\frac{1}{4}$ or 0.25. Adding constants $0,1$ allow for the creation of $4$ additional (for $N=2$) and $169$ additional Universal Gates (for $N=3$). In this article, the mathematical and logical underpinnings of the concept of universal logic gates are presented, along with a search strategy $ULG_{SS}$ exploring multiple paths leading to their identification. A fast-track approach has been introduced that uses the hexadecimal representation of a logic gate to quickly ascertain its attribute.
In this study, Synthetic Aperture Radar (SAR) and optical data are both considered for Earth surface classification. Specifically, the integration of Sentinel-1 (S-1) and Sentinel-2 (S-2) data is carried out through supervised Machine Learning (ML) algorithms implemented on the Google Earth Engine (GEE) platform for the classification of a particular region of interest. Achieved results demonstrate how in this case radar and optical remote detection provide complementary information, benefiting surface cover classification and generally leading to increased mapping accuracy. In addition, this paper works in the direction of proving the emerging role of GEE as an effective cloud-based tool for handling large amounts of satellite data.
In this paper, we introduce the flexible interpretable gamma (FIG) distribution which has been derived by Weibullisation of the body-tail generalised normal distribution. The parameters of the FIG have been verified graphically and mathematically as having interpretable roles in controlling the left-tail, body, and right-tail shape. The generalised gamma (GG) distribution has become a staple model for positive data in statistics due to its interpretable parameters and tractable equations. Although there are many generalised forms of the GG which can provide better fit to data, none of them extend the GG so that the parameters are interpretable. Additionally, we present some mathematical characteristics and prove the identifiability of the FIG parameters. Finally, we apply the FIG model to hand grip strength and insurance loss data to assess its flexibility relative to existing models.
To minimize the average of a set of log-convex functions, the stochastic Newton method iteratively updates its estimate using subsampled versions of the full objective's gradient and Hessian. We contextualize this optimization problem as sequential Bayesian inference on a latent state-space model with a discriminatively-specified observation process. Applying Bayesian filtering then yields a novel optimization algorithm that considers the entire history of gradients and Hessians when forming an update. We establish matrix-based conditions under which the effect of older observations diminishes over time, in a manner analogous to Polyak's heavy ball momentum. We illustrate various aspects of our approach with an example and review other relevant innovations for the stochastic Newton method.
Conway's Game of Life is a two-dimensional cellular automaton. As a dynamical system, it is well-known to be computationally universal, i.e.\ capable of simulating an arbitrary Turing machine. We show that in a sense taking a single backwards step of Game of Life is a computationally universal process, by constructing patterns whose preimage computation encodes an arbitrary circuit-satisfaction problem, or (equivalently) any tiling problem. As a corollary, we obtain for example that the set of orphans is coNP-complete, exhibit a $6210 \times 37800$-periodic configuration that admits a preimage but no periodic one, and prove that the existence of a preimage for a periodic point is undecidable. Our constructions were obtained by a combination of computer searches and manual design.
Quantization summarizes continuous distributions by calculating a discrete approximation. Among the widely adopted methods for data quantization is Lloyd's algorithm, which partitions the space into Vorono\"i cells, that can be seen as clusters, and constructs a discrete distribution based on their centroids and probabilistic masses. Lloyd's algorithm estimates the optimal centroids in a minimal expected distance sense, but this approach poses significant challenges in scenarios where data evaluation is costly, and relates to rare events. Then, the single cluster associated to no event takes the majority of the probability mass. In this context, a metamodel is required and adapted sampling methods are necessary to increase the precision of the computations on the rare clusters.
Deep neural operators (DNOs) have been utilized to approximate nonlinear mappings between function spaces. However, DNOs face the challenge of increased dimensionality and computational cost associated with unaligned observation data. In this study, we propose a hybrid Decoder-DeepONet operator regression framework to handle unaligned data effectively. Additionally, we introduce a Multi-Decoder-DeepONet, which utilizes an average field of training data as input augmentation. The consistencies of the frameworks with the operator approximation theory are provided, on the basis of the universal approximation theorem. Two numerical experiments, Darcy problem and flow-field around an airfoil, are conducted to validate the efficiency and accuracy of the proposed methods. Results illustrate the advantages of Decoder-DeepONet and Multi-Decoder-DeepONet in handling unaligned observation data and showcase their potentials in improving prediction accuracy.
A central problem in computational statistics is to convert a procedure for sampling combinatorial from an objects into a procedure for counting those objects, and vice versa. Weconsider sampling problems coming from *Gibbs distributions*, which are probability distributions of the form $\mu^\Omega_\beta(\omega) \propto e^{\beta H(\omega)}$ for $\beta$ in an interval $[\beta_\min, \beta_\max]$ and $H( \omega ) \in \{0 \} \cup [1, n]$. The *partition function* is the normalization factor $Z(\beta)=\sum_{\omega \in\Omega}e^{\beta H(\omega)}$. Two important parameters are the log partition ratio $q = \log \tfrac{Z(\beta_\max)}{Z(\beta_\min)}$ and the vector of counts $c_x = |H^{-1}(x)|$. Our first result is an algorithm to estimate the counts $c_x$ using roughly $\tilde O( \frac{q}{\epsilon^2})$ samples for general Gibbs distributions and $\tilde O( \frac{n^2}{\epsilon^2} )$ samples for integer-valued distributions (ignoring some second-order terms and parameters). We show this is optimal up to logarithmic factors. We illustrate with improved algorithms for counting connected subgraphs and perfect matchings in a graph. We develop a key subroutine for global estimation of the partition function. Specifically, we produce a data structure to estimate $Z(\beta)$ for \emph{all} values $\beta$, without further samples. Constructing the data structure requires $O(\frac{q \log n}{\epsilon^2})$ samples for general Gibbs distributions and $O(\frac{n^2 \log n}{\epsilon^2} + n \log q)$ samples for integer-valued distributions. This improves over a prior algorithm of Kolmogorov (2018) which computes the single point estimate $Z(\beta_\max)$ using $\tilde O(\frac{q}{\epsilon^2})$ samples. We also show that this complexity is optimal as a function of $n$ and $q$ up to logarithmic terms.
The goal of explainable Artificial Intelligence (XAI) is to generate human-interpretable explanations, but there are no computationally precise theories of how humans interpret AI generated explanations. The lack of theory means that validation of XAI must be done empirically, on a case-by-case basis, which prevents systematic theory-building in XAI. We propose a psychological theory of how humans draw conclusions from saliency maps, the most common form of XAI explanation, which for the first time allows for precise prediction of explainee inference conditioned on explanation. Our theory posits that absent explanation humans expect the AI to make similar decisions to themselves, and that they interpret an explanation by comparison to the explanations they themselves would give. Comparison is formalized via Shepard's universal law of generalization in a similarity space, a classic theory from cognitive science. A pre-registered user study on AI image classifications with saliency map explanations demonstrate that our theory quantitatively matches participants' predictions of the AI.
Most algorithms for representation learning and link prediction in relational data have been designed for static data. However, the data they are applied to usually evolves with time, such as friend graphs in social networks or user interactions with items in recommender systems. This is also the case for knowledge bases, which contain facts such as (US, has president, B. Obama, [2009-2017]) that are valid only at certain points in time. For the problem of link prediction under temporal constraints, i.e., answering queries such as (US, has president, ?, 2012), we propose a solution inspired by the canonical decomposition of tensors of order 4. We introduce new regularization schemes and present an extension of ComplEx (Trouillon et al., 2016) that achieves state-of-the-art performance. Additionally, we propose a new dataset for knowledge base completion constructed from Wikidata, larger than previous benchmarks by an order of magnitude, as a new reference for evaluating temporal and non-temporal link prediction methods.
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