arXiv subjects

Alexandre L. M. Levada

Publications and source records attributed to Alexandre L. M. Levada.

14 recordsLinked to original sources

arXiv2026

Curvature-Aware Radius Shrinkage for Adaptive Nearest Neighbor Classification

Nearest neighbor classification relies fundamentally on how locality is defined, yet conventional $k$-NN imposes the same neighborhood cardinality throughout the feature space. This assumption can be inadequate for data whose local geometry varies substantially across the underlying manifold. We introduce Curvature-Aware Radius Shrinkage for Adaptive Nearest Neighbor Classification (CARSANN), a geometry-driven framework that adapts the spatial support of each neighborhood according to local geometric complexity. CARSANN first estimates intrinsic dimensionality using TwoNN and constructs an intrinsic representation through principal component analysis. Local mean curvature is then estimated using a shape-operator-based formulation and controls neighborhood scale: highly curved regions receive stronger radius shrinkage, whereas approximately flat regions retain broader spatial support. Unlike methods that modify only the number of neighbors or the local metric, CARSANN explicitly adapts the spatial extent of local evidence. Experiments on more than 70 real-world OpenML datasets show that CARSANN consistently improves upon standard $k$-NN and is competitive with adaptive nearest-neighbor methods. In a controlled comparison using the same base neighborhood size, CARSANN achieves higher balanced accuracy on 40 of 45 datasets, increasing mean balanced accuracy from 0.6506 to 0.7528. The advantage also persists against $k$-NN with fixed $k=5$. Friedman and Nemenyi tests confirm that the improvements are statistically significant. These results indicate that local manifold curvature can serve as an effective geometric control variable for adapting neighborhood support, providing a complementary paradigm to cardinality-based nearest-neighbor adaptation.

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arXiv2026

Shape Operator PCA: Curvature-Aware Projections for Geometric Machine Learning

In this paper, we propose SHOPCA (Shape Operator-based Principal Component Analysis), a novel method for unsupervised metric learning and dimensionality reduction that incorporates differential geometric information into the covariance structure of classical PCA. SHOPCA regularizes the global covariance matrix using the mean shape operator, defined as the average of the absolute local shape operators estimated from the data manifold, steering principal components toward directions of both maximum variance and informative curvature. A single trace-normalized mixing coefficient $α$ controls the regularization, recovering standard PCA at $α= 0$ and a curvature-driven embedding as $α\to \infty$. We further introduce a fully unsupervised criterion for selecting $α$ based on the spectral eigengap of the regularized covariance matrix, maximizing the relative separation between the top-$d$ and remaining eigenvalues without using class labels. We evaluate SHOPCA on more than 50 real-world benchmark datasets, comparing it with PCA, ISOMAP, and UMAP using Adjusted Rand Index (ARI), Normalized Mutual Information (NMI), Fowlkes-Mallows index (FM), and V-measure. Results show that SHOPCA consistently improves clustering quality over PCA across a broad range of datasets and surpasses UMAP on small-sample settings, where iterative neighborhood-based manifold estimation can degrade. SHOPCA is computationally tractable, parameter-efficient, and applicable to domains requiring fully unsupervised, geometry-aware dimensionality reduction.

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arXiv2026

CuBAS: Information Geometric Curvature-Based Adaptive Sampling for Supervised Classification

The informativeness of a training set is as consequential as its size, yet most sampling strategies remain agnostic to the intrinsic geometry of the data distribution. We introduce CuBAS (Curvature-Based Adaptive Sampling), an information-geometric framework for adaptive data selection in supervised classification, grounded in the q-state Potts Markov random field (MRF) model. The central insight is that a labeled dataset can be viewed as a statistical manifold, on which local curvature, estimated via the ratio of second to first-order observed Fisher information, faithfully encodes the geometric complexity of the data distribution. We construct a k-nearest-neighbor graph over the labeled data and derive a closed-form curvature score at each vertex from the Potts sufficient statistics. This curvature signal partitions the graph into two complementary regimes: low-curvature regions, corresponding to smooth, homogeneous clusters, and high-curvature regions, concentrated around decision boundaries that are disproportionately informative for classification. By selecting nodes from both regimes, CuBAS constructs compact yet maximally informative training subsets. Empirical evaluation across more than 60 benchmark datasets demonstrates consistent and statistically significant improvements over random sampling and uncertainty-based baselines, across a wide range of labeling budgets and classifier architectures. CuBAS is computationally efficient (linear in the number of k-NN graph edges), theoretically grounded in the differential geometry of statistical manifolds, and interpretable in terms of the local shape operator of the data manifold.

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arXiv2026

Efficient Mean Curvature Computation on High-Dimensional Data Manifolds

Estimating local mean curvature at each point of a high-dimensional dataset is a key ingredient of geometry-aware machine learning algorithms, such as the Mean Curvature Boundary Points (MCBP) method. The naive implementation of this computation, based on a local shape operator approximated from k-nearest neighbor patches, involves an explicit construction of a matrix $H$ whose trace form yields an $O(m^4)$ cost per point, rendering the approach intractable for datasets with more than a few dozen features. This paper introduces two complementary contributions that together reduce this cost by several orders of magnitude. The first contribution is an exact algebraic identity. This identity, derived from the orthogonality of the eigenvectors of the covariance matrix and the cyclicity of the trace operator, eliminates $H$ entirely and reduces the per-point cost to $O(m^2)$ after the eigendecomposition. The second contribution addresses the remaining $O(m^3)$ bottleneck of the full eigendecomposition. Since the local covariance matrix has rank at most $k-1 \ll m$, we replace it with a truncated SVD of the $k \times m$ centered data matrix, an $O(k^2 m)$ operation, and derive an analytical approximation for the contribution of the null-space eigenvectors based on the expected value of their outer product under the Haar measure. The resulting estimator has total cost $O(k^2 m + k m p^2)$, where $p = k-1$. Experiments on real-world datasets confirm speedups of 50 to 300 times relative to the original implementation, with negligible loss when the fast estimator is used to replace the original version. By providing a scalable and data-driven estimate of local curvature, the proposed method establishes curvature as a practical geometric feature for a broad range of machine learning tasks, from classical to modern deep learning pipelines.

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arXiv2026

A Mean Curvature Approach to Boundary Detection: Geometric Insights for Unsupervised Learning

Accurate boundary detection in high-dimensional data remains a central challenge in unsupervised learning, particularly in the presence of non-linear structures and heterogeneous densities. In this work, we introduce Mean Curvature Boundary Points (MCBP), a novel geometric framework grounded in Geometric Machine Learning that departs from traditional density-based approaches by explicitly modeling the intrinsic curvature of the data manifold. The method relies on a discrete approximation of the shape operator, estimated from local k-nearest neighbor patches, to compute pointwise mean curvature without requiring explicit manifold parametrization. The key insight of MCBP is to use mean curvature as a principled descriptor of boundary structure: high-curvature regions naturally correspond to transitions between clusters, geometric irregularities, and low-density interfaces. This yields a unified geometric interpretation of boundary, outlier, and transition points. We further introduce an adaptive percentile-based thresholding scheme that enables multiscale boundary extraction without relying on ad hoc density parameters. Beyond detection, we propose a curvature-driven data decomposition that separates samples into smooth (low-curvature) and boundary (high-curvature) subsets, effectively acting as a non-linear geometric filtering mechanism. This representation enhances cluster separability and improves the robustness of downstream unsupervised algorithms. Extensive experiments on synthetic and real-world datasets demonstrate that MCBP consistently improves clustering performance, particularly in complex and high-dimensional scenarios. These results position MCBP as a concrete contribution to Geometric Machine Learning, highlighting the potential of curvature-aware analysis as a unifying paradigm bridging differential geometry and data-driven modeling.

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arXiv2026

Curvature-Aware PCA with Geodesic Tangent Space Aggregation for Semi-Supervised Learning

Principal Component Analysis (PCA) is a fundamental tool for representation learning, but its global linear formulation fails to capture the structure of data supported on curved manifolds. In contrast, manifold learning methods model nonlinearity but often sacrifice the spectral structure and stability of PCA. We propose \emph{Geodesic Tangent Space Aggregation PCA (GTSA-PCA)}, a geometric extension of PCA that integrates curvature awareness and geodesic consistency within a unified spectral framework. Our approach replaces the global covariance operator with curvature-weighted local covariance operators defined over a $k$-nearest neighbor graph, yielding local tangent subspaces that adapt to the manifold while suppressing high-curvature distortions. We then introduce a geodesic alignment operator that combines intrinsic graph distances with subspace affinities to globally synchronize these local representations. The resulting operator admits a spectral decomposition whose leading components define a geometry-aware embedding. We further incorporate semi-supervised information to guide the alignment, improving discriminative structure with minimal supervision. Experiments on real datasets show consistent improvements over PCA, Kernel PCA, Supervised PCA and strong graph-based baselines such as UMAP, particularly in small sample size and high-curvature regimes. Our results position GTSA-PCA as a principled bridge between statistical and geometric approaches to dimensionality reduction.

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arXiv2024

An information-geometric approach for network decomposition using the q-state Potts model

Complex networks are critical in many scientific, technological, and societal contexts due to their ability to represent and analyze intricate systems with interdependent components. Often, after labeling the nodes of a network with a community detection algorithm, its modular organization emerges, allowing a better understanding of the underlying structure by uncovering hidden relationships. In this paper, we introduce a novel information-geometric framework for the filtering and decomposition of networks whose nodes have been labeled. Our approach considers the labeled network as the outcome of a Markov random field modeled by a q-state Potts model. According to information geometry, the first and second order Fisher information matrices are related to the metric and curvature tensor of the parametric space of a statistical model. By computing an approximation to the local shape operator, the proposed methodology is able to identify low and high information nodes, allowing the decomposition of the labeled network in two complementary subgraphs. Hence, we call this method as the LO-HI decomposition. Experimental results with several kinds of networks show that the high information subgraph is often related to edges and boundaries, while the low information subgraph is a smoother version of the network, in the sense that the modular structure is improved.

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arXiv2024

The geodesic dispersion phenomenon in random fields dynamics

Random fields are ubiquitous mathematical structures in physics, with applications ranging from thermodynamics and statistical physics to quantum field theory and cosmology. Recent works on information geometry of Gaussian random fields proposed mathematical expressions for the components of the metric tensor of the underlying parametric space, allowing the computation of the curvature in each point of the manifold. In this study, our hypothesis is that time irreversibility in Gaussian random fields dynamics is a direct consequence of intrinsic geometric properties (curvature) of their parametric space. In order to validate this hypothesis, we compute the components of the metric tensor and derive the twenty seven Christoffel symbols of the metric to define the Euler-Lagrange equations, a system of partial differential equations that are used to build geodesic curves in Riemannian manifolds. After that, by the application of the fourth-order Runge-Kutta method and Markov Chain Monte Carlo simulation, we numerically build geodesic curves starting from an arbitrary initial point in the manifold. The obtained results show that, when the system undergoes phase transitions, the geodesic curve obtained by time reversing the computational simulation diverges from the original curve, showing a strange effect that we called the geodesic dispersion phenomenon, which suggests that time irreversibility in random fields is related to the intrinsic geometry of their parametric space.

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arXiv2022

On the Kullback-Leibler divergence between pairwise isotropic Gaussian-Markov random fields

The Kullback-Leibler divergence or relative entropy is an information-theoretic measure between statistical models that play an important role in measuring a distance between random variables. In the study of complex systems, random fields are mathematical structures that models the interaction between these variables by means of an inverse temperature parameter, responsible for controlling the spatial dependence structure along the field. In this paper, we derive closed-form expressions for the Kullback-Leibler divergence between two pairwise isotropic Gaussian-Markov random fields in both univariate and multivariate cases. The proposed equation allows the development of novel similarity measures in image processing and machine learning applications, such as image denoising and unsupervised metric learning.

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arXiv2022

The Curvature Effect in Gaussian Random Fields

Random field models are mathematical structures used in the study of stochastic complex systems. In this paper, we compute the shape operator of Gaussian random field manifolds using the first and second fundamental forms (Fisher information matrices). Using Markov Chain Monte Carlo techniques, we simulate the dynamics of these random fields and compute the Gaussian curvature of the parametric space, analyzing how this quantity changes along phase transitions. During the simulation, we have observed an unexpected phenomenon that we called the \emph{curvature effect}, which indicates that a highly asymmetric geometric deformation happens in the underlying parametric space when there are significant increase/decrease in the system's entropy. This asymmetric pattern relates to the emergence of hysteresis, leading to an intrinsic arrow of time along the dynamics.

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arXiv2021

Geodesic curves in Gaussian random field manifolds

Random fields are mathematical structures used to model the spatial interaction of random variables along time, with applications ranging from statistical physics and thermodynamics to system's biology and the simulation of complex systems. Despite being studied since the 19th century, little is known about how the dynamics of random fields are related to the geometric properties of their parametric spaces. For example, how can we quantify the similarity between two random fields operating in different regimes using an intrinsic measure? In this paper, we propose a numerical method for the computation of geodesic distances in Gaussian random field manifolds. First, we derive the metric tensor of the underlying parametric space (the 3 x 3 first-order Fisher information matrix), then we derive the 27 Christoffel symbols required in the definition of the system of non-linear differential equations whose solution is a geodesic curve starting at the initial conditions. The fourth-order Runge-Kutta method is applied to numerically solve the non-linear system through an iterative approach. The obtained results show that the proposed method can estimate the geodesic distances for several different initial conditions. Besides, the results reveal an interesting pattern: in several cases, the geodesic curve obtained by reversing the system of differential equations in time does not match the original curve, suggesting the existence of irreversible geometric deformations in the trajectory of a moving reference traveling along a geodesic curve.

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arXiv2021

On the curvatures of Gaussian random field manifolds

Information geometry is concerned with the application of differential geometry concepts in the study of the parametric spaces of statistical models. When the random variables are independent and identically distributed, the underlying parametric space exhibit constant curvature, which makes the geometry hyperbolic (negative) or spherical (positive). In this paper, we derive closed-form expressions for the components of the first and second fundamental forms regarding pairwise isotropic Gaussian-Markov random field manifolds, allowing the computation of the Gaussian, mean and principal curvatures. Computational simulations using Markov Chain Monte Carlo dynamics indicate that a change in the sign of the Gaussian curvature is related to the emergence of phase transitions in the field. Moreover, the curvatures are highly asymmetrical for positive and negative displacements in the inverse temperature parameter, suggesting the existence of irreversible geometric properties in the parametric space along the dynamics. Furthermore, these asymmetric changes in the curvature of the space induces an intrinsic notion of time in the evolution of the random field.

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arXiv2017

Information geometry, simulation and complexity in Gaussian random fields

Random fields are useful mathematical objects in the characterization of non-deterministic complex systems. A fundamental issue in the evolution of dynamical systems is how intrinsic properties of such structures change in time. In this paper, we propose to quantify how changes in the spatial dependence structure affect the Riemannian metric tensor that equips the model's parametric space. Defining Fisher curves, we measure the variations in each component of the metric tensor when visiting different entropic states of the system. Simulations show that the geometric deformations induced by the metric tensor in case of a decrease in the inverse temperature are not reversible for an increase of the same amount, provided there is significant variation in the system entropy: the process of taking a system from a lower entropy state A to a higher entropy state B and then bringing it back to A, induces a natural intrinsic one-way direction of evolution. In this context, Fisher curves resemble mathematical models of hysteresis in which the natural orientation is pointed by an arrow of time.

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arXiv2013

Learning from Complex Systems: On the Roles of Entropy and Fisher Information in Pairwise Isotropic Gaussian Markov Random Fields

Markov Random Field models are powerful tools for the study of complex systems. However, little is known about how the interactions between the elements of such systems are encoded, especially from an information-theoretic perspective. In this paper, our goal is to enlight the connection between Fisher information, Shannon entropy, information geometry and the behavior of complex systems modeled by isotropic pairwise Gaussian Markov random fields. We propose analytical expressions to compute local and global versions of these measures using Besag's pseudo-likelihood function, characterizing the system's behavior through its \emph{Fisher curve}, a parametric trajectory accross the information space that provides a geometric representation for the study of complex systems. Computational experiments show how the proposed tools can be useful in extrating relevant information from complex patterns. The obtained results quantify and support our main conclusion, which is: in terms of information, moving towards higher entropy states (A --> B) is different from moving towards lower entropy states (B --> A), since the \emph{Fisher curves} are not the same given a natural orientation (the direction of time).

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