Many (if not all) real-world applications require efficient handling of uncertainty and a compact representation of a wide variety of knowledge. Indeed, complex concepts and relationships that typically comprise expert knowledge may be difficult to express in graphical models but can be represented compactly using classical logic.

In this paper, we provide a theoretical analysis of the inductive biases in convolutional neural networks (CNNs). We start by examining the universality of CNNs, i.e., the ability to approximate any continuous functions. We prove that a depth of $\mathcal{O}(\log d)$ suffices for deep CNNs to achieve this universality, where $d$ in the input dimension. Additionally, we establish that learning sparse functions with CNNs requires only $\widetilde{\mathcal{O}}(\log^2d)$ samples, indicating that deep CNNs can efficiently capture {\em long-range} sparse correlations. These results are made possible through a novel combination of the multichanneling and downsampling when increasing the network depth.

Logical Credal Networks or LCNs were recently introduced as a powerful probabilistic logic framework for representing and reasoning with imprecise knowledge. Unlike many existing formalisms, LCNs have the ability to represent cycles and allow specifying marginal and conditional probability bounds on logic formulae which may be important in many realistic scenarios. Previous work on LCNs has focused exclusively on marginal inference, namely computing posterior lower and upper probability bounds on a query formula. In this paper, we explore abductive reasoning tasks such as solving MAP and Marginal MAP queries in LCNs given some evidence. We first formally define the MAP and Marginal MAP tasks for LCNs and subsequently show how to solve these tasks exactly using search-based approaches. We then propose several approximate schemes that allow us to scale MAP and Marginal MAP inference to larger problem instances. An extensive empirical evaluation demonstrates the effectiveness of our algorithms on both random LCN instances as well as LCNs derived from more realistic use-cases.

Logical Credal Networks or LCNs were recently introduced as a powerful probabilistic logic framework for representing and reasoning with imprecise knowledge.

Logical Credal Networks or LCNs were recently introduced as a powerful probabilistic logic framework for representing and reasoning with imprecise knowledge.

Logical Credal Networks or LCNs were recently introduced as a powerful probabilistic logic framework for representing and reasoning with imprecise knowledge. Unlike many existing formalisms, LCNs have the ability to represent cycles and allow specifying marginal and conditional probability bounds on logic formulae which may be important in many realistic scenarios. Previous work on LCNs has focused exclusively on marginal inference, namely computing posterior lower and upper probability bounds on a query formula. In this paper, we explore abductive reasoning tasks such as solving MAP and Marginal MAP queries in LCNs given some evidence. We first formally define the MAP and Marginal MAP tasks for LCNs and subsequently show how to solve these tasks exactly using search-based approaches.

The widespread use of Multi-layer perceptrons (MLPs) often relies on a fixed activation function (e.g., ReLU, Sigmoid, Tanh) for all nodes within the hidden layers. While effective in many scenarios, this uniformity may limit the networks ability to capture complex data patterns. We argue that employing the same activation function at every node is suboptimal and propose leveraging different activation functions at each node to increase flexibility and adaptability. To achieve this, we introduce Local Control Networks (LCNs), which leverage B-spline functions to enable distinct activation curves at each node. Our mathematical analysis demonstrates the properties and benefits of LCNs over conventional MLPs. In addition, we demonstrate that more complex architectures, such as Kolmogorov-Arnold Networks (KANs), are unnecessary in certain scenarios, and LCNs can be a more efficient alternative. Empirical experiments on various benchmarks and datasets validate our theoretical findings. In computer vision tasks, LCNs achieve marginal improvements over MLPs and outperform KANs by approximately 5\%, while also being more computationally efficient than KANs. In basic machine learning tasks, LCNs show a 1\% improvement over MLPs and a 0.6\% improvement over KANs. For symbolic formula representation tasks, LCNs perform on par with KANs, with both architectures outperforming MLPs. Our findings suggest that diverse activations at the node level can lead to improved performance and efficiency.