Dn) 1 as n, which means the most posterior mass falls in the neighbourhood of true parameter. Remarkonthenotation: ν() is similar toν() defined in Section 2.1 of the main text. Thenotationsweusedinthis proof are the same as in the proof of Theorem 2.1. Theorem 2.2 implies that a faithful prediction interval can be constructed for the sparse neural network learned by the proposed algorithms. In practice, for a normal regression problem with noise N(0,σ2), to construct the prediction interval for a test pointx0, the terms σ2 and Σ = γ µ(β,x0)TH 1 γ µ(β,x0) in Theorem 2.2 need to be estimated from data.

We set the output of each rule as the majority class (resp.

Optical Character Recognition (OCR) is crucial to the National Library of Norway's (NLN) digitisation process as it converts scanned documents into machine-readable text. However, for the S\'ami documents in NLN's collection, the OCR accuracy is insufficient. Given that OCR quality affects downstream processes, evaluating and improving OCR for text written in S\'ami languages is necessary to make these resources accessible. To address this need, this work fine-tunes and evaluates three established OCR approaches, Transkribus, Tesseract and TrOCR, for transcribing S\'ami texts from NLN's collection. Our results show that Transkribus and TrOCR outperform Tesseract on this task, while Tesseract achieves superior performance on an out-of-domain dataset. Furthermore, we show that fine-tuning pre-trained models and supplementing manual annotations with machine annotations and synthetic text images can yield accurate OCR for S\'ami languages, even with a moderate amount of manually annotated data.

Digital tools for text analysis have long been essential for the searchability and accessibility of digitised library collections. Recent computer vision advances have introduced similar capabilities for visual materials, with deep learning-based embeddings showing promise for analysing visual heritage. Given that many books feature visuals in addition to text, taking advantage of these breakthroughs is critical to making library collections open and accessible. In this work, we present a proof-of-concept image search application for exploring images in the National Library of Norway's pre-1900 books, comparing Vision Transformer (ViT), Contrastive Language-Image Pre-training (CLIP), and Sigmoid loss for Language-Image Pre-training (SigLIP) embeddings for image retrieval and classification. Our results show that the application performs well for exact image retrieval, with SigLIP embeddings slightly outperforming CLIP and ViT in both retrieval and classification tasks. Additionally, SigLIP-based image classification can aid in cleaning image datasets from a digitisation pipeline.

Learning how to automatically solve optimization problems has the potential to provide the next big leap in optimization technology. The performance of automatically learned heuristics on routing problems has been steadily improving in recent years, but approaches based purely on machine learning are still outperformed by state-of-the-art optimization methods. To close this performance gap, we propose a novel large neighborhood search (LNS) framework for vehicle routing that integrates learned heuristics for generating new solutions. The learning mechanism is based on a deep neural network with an attention mechanism and has been especially designed to be integrated into an LNS search setting. We evaluate our approach on the capacitated vehicle routing problem (CVRP) and the split delivery vehicle routing problem (SDVRP). On CVRP instances with up to 297 customers our approach significantly outperforms an LNS that uses only handcrafted heuristics and a well-known heuristic from the literature. Furthermore, we show for the CVRP and the SDVRP that our approach surpasses the performance of existing machine learning approaches and comes close to the performance of state-of-the-art optimization approaches.

Recent years have witnessed the great success of deep neural networks in many research areas. The fundamental idea behind the design of most neural networks is to learn similarity patterns from data for prediction and inference, which lacks the ability of logical reasoning. However, the concrete ability of logical reasoning is critical to many theoretical and practical problems. In this paper, we propose Neural Logic Network (NLN), which is a dynamic neural architecture that builds the computational graph according to input logical expressions. It learns basic logical operations as neural modules, and conducts propositional logical reasoning through the network for inference. Experiments on simulated data show that NLN achieves significant performance on solving logical equations. Further experiments on real-world data show that NLN significantly outperforms state-of-the-art models on collaborative filtering and personalized recommendation tasks.

We propose a novel learning paradigm for Deep Neural Networks (DNN) by using Boolean logic algebra. We first present the basic differentiable operators of a Boolean system such as conjunction, disjunction and exclusive-OR and show how these elementary operators can be combined in a simple and meaningful way to form Neural Logic Networks (NLNs). We examine the effectiveness of the proposed NLN framework in learning Boolean functions and discrete-algorithmic tasks. We demonstrate that, in contrast to the implicit learning in MLP approach, the proposed neural logic networks can learn the logical functions explicitly that can be verified and interpreted by human. In particular, we propose a new framework for learning the inductive logic programming (ILP) problems by exploiting the explicit representational power of NLN. We show the proposed neural ILP solver is capable of feats such as predicate invention and recursion and can outperform the current state of the art neural ILP solvers using a variety of benchmark tasks such as decimal addition and multiplication, and sorting on ordered list.

Coinduction has recently been introduced into logic programming by Simon et al. The resulting paradigm, termed coinductive logic programming (co-LP), allows one to model and reason about infinite processes and objects. Co-LP extended with negation has many interesting applications: for instance in developing top-down, goaldirected evaluation strategies for Answer Set Programming. In this paper we show yet another application of co-LP, namely, elegantly realizing Boolean SAT solvers