Related papers: Large Sets Avoiding Rough Patterns
One of the distinguishing characteristics of modern deep learning systems is that they typically employ neural network architectures that utilize enormous numbers of parameters, often in the millions and sometimes even in the billions.…
A deep neural network model is a powerful framework for learning representations. Usually, it is used to learn the relation $x \to y$ by exploiting the regularities in the input $x$. In structured output prediction problems, $y$ is…
This paper is one of a series whose goal is to enumerate the avoiders, in the sense of classical pattern avoidance, for each triple of 4-letter patterns. There are 317 symmetry classes of triples of 4-letter patterns, avoiders of 267 of…
Predictive rate-distortion analysis suffers from the curse of dimensionality: clustering arbitrarily long pasts to retain information about arbitrarily long futures requires resources that typically grow exponentially with length. The…
In [BabStein] Babson and Steingr\'{\i}msson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. In [Kit1] Kitaev considered simultaneous…
We study relationships between permutation statistics and pattern-functions, counting the number of times particular patterns occur in a permutation. This allows us to write several familiar statistics as linear combinations of pattern…
Many inverse problems include nuisance parameters which, while not of direct interest, are required to recover primary parameters. Structure present in these problems allows efficient optimization strategies - a well known example is…
Finding the optimal size of deep learning models is very actual and of broad impact, especially in energy-saving schemes. Very recently, an unexpected phenomenon, the ``double descent'', has caught the attention of the deep learning…
In this work we approach attractor neural networks from a machine learning perspective: we look for optimal network parameters by applying a gradient descent over a regularized loss function. Within this framework, the optimal…
In this paper, we compute the distributions of the statistic number of crossings over permutations avoiding one of the pairs $\{321,231\}$, $\{123,132\}$ and $\{123,213\}$. The obtained results are new combinatorial interpretations of two…
High-dimensional big data appears in many research fields such as image recognition, biology and collaborative filtering. Often, the exploration of such data by classic algorithms is encountered with difficulties due to `curse of…
Missing data are an unavoidable complication in many machine learning tasks. When data are `missing at random' there exist a range of tools and techniques to deal with the issue. However, as machine learning studies become more ambitious,…
In this paper we study the enumeration and the construction of particular binary words avoiding the pattern $1^{j+1}0^j$. By means of the theory of Riordan arrays, we solve the enumeration problem and we give a particular succession rule,…
Regularization, whether explicit in terms of a penalty in the loss or implicit in the choice of algorithm, is a cornerstone of modern machine learning. Indeed, controlling the complexity of the model class is particularly important when…
Fuzzy rough feature selection (FRFS) is an effective means of addressing the curse of dimensionality in high-dimensional data. By removing redundant and irrelevant features, FRFS helps mitigate classifier overfitting, enhance generalization…
In a previous paper [Adcock & Huybrechs, 2019] we described the numerical approximation of functions using redundant sets and frames. Redundancy in the function representation offers enormous flexibility compared to using a basis, but…
Two pressing topics in the theory of deep learning are the interpretation of feature learning (FL) mechanisms and the determination of implicit bias of networks in the rich regime. Current theories of rich FL often appear in the form of…
Enumeration of various types of lattice polygons and in particular polyominoes is of primary importance in many machine learning, pattern recognition, and geometric analysis problems. In this work, we develop a large deviation principle for…
We study a model of machine teaching where the teacher mapping is constructed from a size function on both concepts and examples. The main question in machine teaching is the minimum number of examples needed for any concept, the so-called…
This paper is devoted to the construction of order reduced method of fourth order problems. A framework is presented such that a problem on a high-regularity space can be deduced in a constructive way to an equivalent problem on three…