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Higher-order unification has been shown to be undecidable. Miller discovered the pattern fragment and subsequently showed that higher-order pattern unification is decidable and has most general unifiers. We extend the algorithm to…
A matrix framework is presented for the solution of ODEs, including initial-, boundary and inner-value problems. The framework enables the solution of the ODEs for arbitrary nodes. There are four key issues involved in the formulation of…
Given a unitary representation of a finite group on a finite-dimensional Hilbert space, we show how to find a state whose translates under the group are distinguishable with the highest probability. We apply this to several quantum oracle…
A complete classification of the computational complexity of the fixed-point existence problem for boolean dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}, is presented. For function classes F and graph…
The combination of higher-order theories and fuzzy logic can be useful in decision-making tasks that involve reasoning across abstract functions and predicates, where exact matches are often rare or unnecessary. Developing efficient…
Clustering is an unsupervised learning problem that aims to partition unlabelled data points into groups with similar features. Traditional clustering algorithms provide limited insight into the groups they find as their main focus is…
Tensor decomposition plays a key role in identifying common features across a collection of matrices in many areas of science. A fundamental need in big data research is to process data tabulated as large-scale matrices using eigenvectors.…
A system of nested dichotomies is a method of decomposing a multi-class problem into a collection of binary problems. Such a system recursively applies binary splits to divide the set of classes into two subsets, and trains a binary…
Deep Neural Networks miss a principled model of their operation. A novel framework for supervised learning based on Topological Quantum Field Theory that looks particularly well suited for implementation on quantum processors has been…
Despite extensive research spanning several decades, class imbalance is still considered a profound difficulty for both machine learning and deep learning models. While data oversampling is the foremost technique to address this issue,…
Neural networks predictions are unreliable when the input sample is out of the training distribution or corrupted by noise. Being able to detect such failures automatically is fundamental to integrate deep learning algorithms into robotics.…
Deep ReLU neural networks admit nontrivial functional symmetries: vastly different architectures and parameters (weights and biases) can realize the same function. We address the complete identification problem -- given a function f,…
In this paper, we address the scenario where nodes with sensor data are connected in a tree network, and every node wants to compute a given symmetric Boolean function of the sensor data. We first consider the problem of computing a…
Function approximation from input and output data pairs constitutes a fundamental problem in supervised learning. Deep neural networks are currently the most popular method for learning to mimic the input-output relationship of a general…
Inferring network topology from smooth signals is a significant problem in data science and engineering. A common challenge in real-world scenarios is the availability of only partially observed nodes. While some studies have considered…
This paper argues that the ideas underlying the renormalization group technique used to characterize phase transitions in condensed matter systems could be useful for distinguishing computational complexity classes. The paper presents a…
Solving partial differential equations (PDEs) has been a fundamental problem in computational science and of wide applications for both scientific and engineering research. Due to its universal approximation property, neural network is…
Generalized moment problems optimize functional expectation over a class of distributions with generalized moment constraints, i.e., the function in the moment can be any measurable function. These problems have recently attracted growing…
The theory of complex trees is introduced as a new approach to study a broad class of self-similar sets. Systems of equations encoded by complex trees tip-to-tip equivalence relations are used to obtain one-parameter families of connected…
We propose a universal approach to a range of enumeration problems in graphs. The key point is in contracting suitably chosen symmetric tensors placed at the vertices of a graph along the edges. In particular, this leads to an algorithm…