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Motivated by results on generic-case complexity in group theory, we apply the ideas of effective Baire category and effective measure theory to study complexity classes of functions which are "fractionally computable" by a partial…

We introduce partial differential encodings of Boolean functions as a way of measuring the complexity of Boolean functions. These encodings enable us to derive from group actions non-trivial bounds on the Chow-Rank of polynomials used to…

Why depth yields a genuine computational advantage over shallow methods remains a central open question in learning theory. We study this question in a controlled high-dimensional Gaussian setting, focusing on compositional target…

We propose an approach to determine the continual progression of algorithmic efficiency, as an alternative to standard calculations of time complexity, likely, but not exclusively, when dealing with data structures with unknown maximum…

One of the most ubiquitous problems in optimization is that of finding all the elements of a finite set at which a function $f$ attains its minimum (or maximum). When the codomain of $f$ is equipped with a total order, it is easy to…

We adapt previous research on category theory and topological unsupervised learning to develop a functorial perspective on manifold learning, also known as nonlinear dimensionality reduction. We first characterize manifold learning…

Since many real-world data can be described from multiple views, multi-view learning has attracted considerable attention. Various methods have been proposed and successfully applied to multi-view learning, typically based on matrix…

We present a class of algorithms capable of directly training deep neural networks with respect to large families of task-specific performance measures such as the F-measure and the Kullback-Leibler divergence that are structured and…

We propose a categorical semantics of gradient-based machine learning algorithms in terms of lenses, parametrised maps, and reverse derivative categories. This foundation provides a powerful explanatory and unifying framework: it…

Derivative-free optimization (DFO) consists in finding the best value of an objective function without relying on derivatives. To tackle such problems, one may build approximate derivatives, using for instance finite-difference estimates.…

While the universal approximation property holds both for hierarchical and shallow networks, we prove that deep (hierarchical) networks can approximate the class of compositional functions with the same accuracy as shallow networks but with…

Recently, it has been shown that many functions on sets can be represented by sum decompositions. These decompositons easily lend themselves to neural approximations, extending the applicability of neural nets to set-valued inputs---Deep…

Fine-grained domain generalization (FGDG) is a more challenging task than traditional DG tasks due to its small inter-class variations and relatively large intra-class disparities. When domain distribution changes, the vulnerability of…

The concept of data depth in non-parametric multivariate descriptive statistics is the generalization of the univariate rank method to multivariate data. Halfspace depth is a measure of data depth. Given a set S of points and a point p, the…

Despite their widespread success, the application of deep neural networks to functional data remains scarce today. The infinite dimensionality of functional data means standard learning algorithms can be applied only after appropriate…

Systems that rely on Machine Learning (ML systems) have differing demands on system quality compared to traditional systems. Such quality demands, known as non-functional requirements (NFRs), may differ in their definition, scope, and…

We consider sequential maximization of performance metrics that are general functions of a confusion matrix of a classifier (such as precision, F-measure, or G-mean). Such metrics are, in general, non-decomposable over individual instances,…

We consider integration of functions with values in a partially ordered vector space, and two notions of extension of the space of integrable functions. Applying both extensions to the space of real valued simple functions on a measure…

Robust estimation of location is a fundamental problem in statistics, particularly in scenarios where data contamination by outliers or model misspecification is a concern. In univariate settings, methods such as the sample median and…

Machine learning typically relies on the assumption that training and testing distributions are identical and that data is centrally stored for training and testing. However, in real-world scenarios, distributions may differ significantly…