Diego Marcondes

Diego Marcondes, Adilson Simonis, Junior Barrera

Science consists on conceiving hypotheses, confronting them with empirical evidence, and keeping only hypotheses which have not yet been falsified. Under deductive reasoning they are conceived in view of a theory and confronted with empirical evidence in an attempt to falsify it, and under inductive reasoning they are conceived based on observation, confronted with empirical evidence and a theory is established based on the not falsified hypotheses. When the hypotheses testing can be performed with quantitative data, the confrontation can be achieved with Machine Learning methods, whose quality is highly dependent on the hypotheses' complexity, hence on the proper insertion of prior information into the set of hypotheses seeking to decrease its complexity without loosing good hypotheses. However, Machine Learning tools have been applied under the pragmatic view of instrumentalism, which is concerned only with the performance of the methods and not with the understanding of their behavior, leading to methods which are not fully understood. In this context, we discuss how prior information and computational power can be employed to solve a learning problem, but while prior information and a careful design of the hypotheses space has as advantage the interpretability of the results, employing high computational power has the advantage of a higher performance. We discuss why learning methods which combine both should work better from an understanding and performance perspective, arguing in favor of basic theoretical research on Machine Learning, in special about how properties of classifiers may be identified in parameters of modern learning models.

Diego Marcondes, Cláudia Peixoto

Cross-validation techniques for risk estimation and model selection are widely used in statistics and machine learning. However, the understanding of the theoretical properties of learning via model selection with cross-validation risk estimation is quite low in face of its widespread use. In this context, this paper presents learning via model selection with cross-validation risk estimation as a general systematic learning framework within classical statistical learning theory and establishes distribution-free deviation bounds in terms of VC dimension, giving detailed proofs of the results and considering both bounded and unbounded loss functions. In particular, we investigate how the generalization of learning via model selection may be increased by modeling the collection of candidate models. We define the Learning Spaces as a class of candidate models in which the partial order by inclusion reflects the models complexities, and we formalize a manner of defining them based on domain knowledge. We illustrate this modeling in a worst-case scenario of learning a classifier with finite domain and a typical scenario of linear regression. Through theoretical insights and concrete examples, we aim to provide guidance on selecting the family of candidate models based on domain knowledge to increase generalization.

Diego Marcondes, Junior Barrera

The machine learning of lattice operators has three possible bottlenecks. From a statistical standpoint, it is necessary to design a constrained class of operators based on prior information with low bias, and low complexity relative to the sample size. From a computational perspective, there should be an efficient algorithm to minimize an empirical error over the class. From an understanding point of view, the properties of the learned operator need to be derived, so its behavior can be theoretically understood. The statistical bottleneck can be overcome due to the rich literature about the representation of lattice operators, but there is no general learning algorithm for them. In this paper, we discuss a learning paradigm in which, by overparametrizing a class via elements in a lattice, an algorithm for minimizing functions in a lattice is applied to learn. We present the stochastic lattice descent algorithm as a general algorithm to learn on constrained classes of operators as long as a lattice overparametrization of it is fixed, and we discuss previous works which are proves of concept. Moreover, if there are algorithms to compute the basis of an operator from its overparametrization, then its properties can be deduced and the understanding bottleneck is also overcome. This learning paradigm has three properties that modern methods based on neural networks lack: control, transparency and interpretability. Nowadays, there is an increasing demand for methods with these characteristics, and we believe that mathematical morphology is in a unique position to supply them. The lattice overparametrization paradigm could be a missing piece for it to achieve its full potential within modern machine learning.

Diego Marcondes, Mariana Feldman, Junior Barrera

There have been attempts to insert mathematical morphology (MM) operators into convolutional neural networks (CNN), and the most successful endeavor to date has been the morphological neural networks (MNN). Although MNN have performed better than CNN in solving some problems, they inherit their black-box nature. Furthermore, in the case of binary images, they are approximations that loose the Boolean lattice structure of MM operators and, thus, it is not possible to represent a specific class of W-operators with desired properties. In a recent work, we proposed the Discrete Morphological Neural Networks (DMNN) for binary image transformation to represent specific classes of W-operators and estimate them via machine learning. We also proposed a stochastic lattice descent algorithm (SLDA) to learn the parameters of Canonical Discrete Morphological Neural Networks (CDMNN), whose architecture is composed only of operators that can be decomposed as the supremum, infimum, and complement of erosions and dilations. In this paper, we propose an algorithm to learn unrestricted sequential DMNN, whose architecture is given by the composition of general W-operators. We illustrate the algorithm in a practical example.

Diego Marcondes

The dual norm characterisation of weak solutions of second-order linear elliptic partial differential equations is mathematically natural but computationally intractable: evaluating the $H^{-1}$ norm of a residual requires a supremum over an infinite-dimensional function space. We prove that the $H^{-1}$ norm of any functional is equivalent to its expected squared evaluation against a random test function whose distribution depends only on the domain. Crucially, realisations of this random test function have negative Sobolev regularity for $d \geq 2$, yet this roughness is not an obstacle: averaging over the distribution exactly recovers the correct weak topology, independently of the differential operator. This equivalence introduces the notion of stochastically weak solutions, which coincide with classical weak solutions, and motivates stochastic variational physics-informed neural networks (SV-PINNs): neural networks trained by minimising an empirical approximation of the stochastic norm of the PDE residual. Although instantiated here with neural networks as trial spaces, the underlying principle is independent of the approximation architecture and suggests a broader paradigm for numerical methods based on stochastic rather than deterministic test spaces. The framework extends naturally to higher-order elliptic, parabolic and hyperbolic equations and to abstract operator equations on Hilbert spaces. As a proof of concept, we present numerical experiments on eight challenging second-order linear elliptic problems spanning high-frequency and multi-scale solutions, indefinite operators, variable coefficients, and non-standard domains, in which SV-PINNs consistently and significantly outperform standard PINNs, recovering solutions to within one percent relative error in hundreds of L-BFGS steps.

Diego Marcondes

Physics-informed statistical learning (PISL) integrates empirical data with physical knowledge to enhance the statistical performance of estimators. While PISL methods are widely used in practice, a comprehensive theoretical understanding of how informed regularization affects statistical properties is still missing. Specifically, two fundamental questions have yet to be fully addressed: (1) what is the trade-off between considering soft penalties versus hard constraints, and (2) what is the statistical gain of incorporating physical knowledge compared to purely data-driven empirical error minimisation. In this paper, we address these questions for PISL in convex classes of functions under physical knowledge expressed as linear equations by developing appropriate complexity dependent error rates based on the small-ball method. We show that, under suitable assumptions, (1) the error rates of physics-informed estimators are comparable to those of hard constrained empirical error minimisers, differing only by constant terms, and that (2) informed penalization can effectively reduce model complexity, akin to dimensionality reduction, thereby improving learning performance. This work establishes a theoretical framework for evaluating the statistical properties of physics-informed estimators in convex classes of functions, contributing to closing the gap between statistical theory and practical PISL, with potential applications to cases not yet explored in the literature.

Diego Marcondes

This paper introduces the class of grey-scale image stack operators as those that (a) map binary-images into binary-images and (b) commute on average with cross-sectioning. Equivalently, stack operators are 1-Lipchitz extensions of set operators which can be represented by applying a characteristic set operator to the cross-sections of the image and adding. In particular, they are a generalisation of stack filters, for which the characteristic set operators are increasing. Our main result is that stack operators inherit lattice properties of the characteristic set operators. We focus on the case of translation-invariant and locally defined stack operators and show the main result by deducing the characteristic function, kernel, and basis representation of stack operators. The results of this paper have implications on the design of image operators, since imply that to solve some grey-scale image processing problems it is enough to design an operator for performing the desired transformation on binary images, and then considering its extension given by a stack operator. We leave many topics for future research regarding the machine learning of stack operators and the characterisation of the image processing problems that can be solved by them.

Diego Marcondes, Ulisses Braga-Neto

We propose generalized resubstitution error estimators for regression, a broad family of estimators, each corresponding to a choice of empirical probability measures and loss function. The usual sum of squares criterion is a special case corresponding to the standard empirical probability measure and the quadratic loss. Other choices of empirical probability measure lead to more general estimators with superior bias and variance properties. We prove that these error estimators are consistent under broad assumptions. In addition, procedures for choosing the empirical measure based on the method of moments and maximum pseudo-likelihood are proposed and investigated. Detailed experimental results using polynomial regression demonstrate empirically the superior finite-sample bias and variance properties of the proposed estimators. The R code for the experiments is provided.

Diego Marcondes, Junior Barrera

A classical approach to designing binary image operators is Mathematical Morphology (MM). We propose the Discrete Morphological Neural Networks (DMNN) for binary image analysis to represent W-operators and estimate them via machine learning. A DMNN architecture, which is represented by a Morphological Computational Graph, is designed as in the classical heuristic design of morphological operators, in which the designer should combine a set of MM operators and Boolean operations based on prior information and theoretical knowledge. Then, once the architecture is fixed, instead of adjusting its parameters (i.e., structural elements or maximal intervals) by hand, we propose a lattice descent algorithm (LDA) to train these parameters based on a sample of input and output images under the usual machine learning approach. We also propose a stochastic version of the LDA that is more efficient, is scalable and can obtain small error in practical problems. The class represented by a DMNN can be quite general or specialized according to expected properties of the target operator, i.e., prior information, and the semantic expressed by algebraic properties of classes of operators is a differential relative to other methods. The main contribution of this paper is the merger of the two main paradigms for designing morphological operators: classical heuristic design and automatic design via machine learning. As a proof-of-concept, we apply the DMNN to recognize the boundary of digits with noise, and we discuss many topics for future research.

Diego Marcondes, Adilson Simonis, Junior Barrera

This paper proposes a data-driven systematic, consistent and non-exhaustive approach to Model Selection, that is an extension of the classical agnostic PAC learning model. In this approach, learning problems are modeled not only by a hypothesis space $\mathcal{H}$, but also by a Learning Space $\mathbb{L}(\mathcal{H})$, a poset of subspaces of $\mathcal{H}$, which covers $\mathcal{H}$ and satisfies a property regarding the VC dimension of related subspaces, that is a suitable algebraic search space for Model Selection algorithms. Our main contributions are a data-driven general learning algorithm to perform implicitly regularized Model Selection on $\mathbb{L}(\mathcal{H})$ and a framework under which one can, theoretically, better estimate a target hypothesis with a given sample size by properly modeling $\mathbb{L}(\mathcal{H})$ and employing high computational power. A remarkable consequence of this approach are conditions under which a non-exhaustive search of $\mathbb{L}(\mathcal{H})$ can return an optimal solution. The results of this paper lead to a practical property of Machine Learning, that the lack of experimental data may be mitigated by a high computational capacity. In a context of continuous popularization of computational power, this property may help understand why Machine Learning has become so important, even where data is expensive and hard to get.

Diego Marcondes

We propose a robust parameter estimation method for dynamical systems based on Statistical Learning techniques which aims to estimate a set of parameters that well fit the dynamics in order to obtain robust evidences about the qualitative behaviour of its trajectory. The method is quite general and flexible, since it does not rely on any specific property of the dynamical system, and represents a reinterpretation of Approximate Bayesian Computation methods through the lens of Statistical Learning. The method is specially useful for estimating parameters in epidemiological compartmental models in order to obtain qualitative properties of a disease evolution. We apply it to simulated and real data about COVID-19 spread in the US in order to evaluate qualitatively its evolution over time, showing how one may assess the effectiveness of measures implemented to slow the spread and some qualitative features of the disease current and future evolution.

Diego Marcondes, Adilson Simonis, Junior Barrera

In part \textit{I} we proposed a structure for a general Hypotheses Space $\mathcal{H}$, the Learning Space $\mathbb{L}(\mathcal{H})$, which can be employed to avoid \textit{overfitting} when estimating in a complex space with relative shortage of examples. Also, we presented the U-curve property, which can be taken advantage of in order to select a Hypotheses Space without exhaustively searching $\mathbb{L}(\mathcal{H})$. In this paper, we carry further our agenda, by showing the consistency of a model selection framework based on Learning Spaces, in which one selects from data the Hypotheses Space on which to learn. The method developed in this paper adds to the state-of-the-art in model selection, by extending Vapnik-Chervonenkis Theory to \textit{random} Hypotheses Spaces, i.e., Hypotheses Spaces learned from data. In this framework, one estimates a random subspace $\hat{\mathcal{M}} \in \mathbb{L}(\mathcal{H})$ which converges with probability one to a target Hypotheses Space $\mathcal{M}^{\star} \in \mathbb{L}(\mathcal{H})$ with desired properties. As the convergence implies asymptotic unbiased estimators, we have a consistent framework for model selection, showing that it is feasible to learn the Hypotheses Space from data. Furthermore, we show that the generalization errors of learning on $\hat{\mathcal{M}}$ are lesser than those we commit when learning on $\mathcal{H}$, so it is more efficient to learn on a subspace learned from data.

Diego Marcondes, Adilson Simonis, Junior Barrera

This paper presents an extension of the classical agnostic PAC learning model in which learning problems are modelled not only by a Hypothesis Space $\mathcal{H}$, but also by a Learning Space $\mathbb{L}(\mathcal{H})$, which is a cover of $\mathcal{H}$, constrained by a VC-dimension property, that is a suitable domain for Model Selection algorithms. Our main contribution is a data driven general learning algorithm to perform regularized Model Selection on $\mathbb{L}(\mathcal{H})$. A remarkable, formally proved, consequence of this approach are conditions on $\mathbb{L}(\mathcal{H})$ and on the loss function that lead to estimated out-of-sample error surfaces which are true U-curves on $\mathbb{L}(\mathcal{H})$ chains, enabling a more efficient search on $\mathbb{L}(\mathcal{H})$. To our knowledge, this is the first rigorous result asserting that a non exhaustive search of a family of candidate models can return an optimal solution. In this new framework, an U-curve optimization algorithm becomes a natural component of Model Selection, hence of learning algorithms. The abstract general framework proposed here may have important implications on modern learning models and on areas such as Neural Architecture Search.

Diego Marcondes, Adilson Simonis, Junior Barrera

This paper uses a classical approach to feature selection: minimization of a cost function applied on estimated joint distributions. However, the search space in which such minimization is performed is extended. In the original formulation, the search space is the Boolean lattice of features sets (BLFS), while, in the present formulation, it is a collection of Boolean lattices of ordered pairs (features, associated value) (CBLOP), indexed by the elements of the BLFS. In this approach, we may not only select the features that are most related to a variable Y, but also select the values of the features that most influence the variable or that are most prone to have a specific value of Y. A local formulation of Shanon's mutual information is applied on a CBLOP to select features, namely, the Local Lift Dependence Scale, an scale for measuring variable dependence in multiple resolutions. The main contribution of this paper is to define and apply this local measure, which permits to analyse local properties of joint distributions that are neglected by the classical Shanon's global measure. The proposed approach is applied to a dataset consisting of student performances on a university entrance exam, as well as on undergraduate courses. The approach is also applied to two datasets of the UCI Machine Learning Repository.