arXiv subjects

Gustavo

Publications and source records attributed to Gustavo.

3 recordsLinked to original sources

arXiv2026

Choosing a parallel heterogeneous ensemble method for tabular classification

Parallel ensemble methods were compared on $56$ small-to-medium tabular classification tasks drawn from OpenML CC18. A set of ``best practice'' recommendations on the use of ensemble methods was derived from these observations. It was later validated on 28 additional tasks using TabArena's precomputed data, where the recommendation set significantly outperformed Single Best and matched or exceeded individual ensemble methods. Two key observations were made. First, Blending and Stacking are inconsistent, but their inconsistencies are independent and happen on different tasks. Second, while Hard Voting's probabilistic classification is rather weak, a consequence of using vote proportions as posterior estimates, Robust Soft Voting's probabilistic classification is particularly successful, especially in the multiclass case.

cs.LG↗

arXiv2026

Morphological Representation Theory in the Fourier Inf-Semilattice: Universal Decomposition of Frequency-Domain Deep Learning Operators

We develop a morphological representation theory for operators acting in the frequency domain of $L^2(\mathbb{R}^n)$. Equipping the space with the \emph{Fourier inf-semilattice} order $\leq_{\!\mathcal{F}}$ (spectral modulus inequality with phase equality), convolution becomes a morphological erosion and the adjoint dilation is the Wiener inverse filter. A key observation is that spatial translation-invariance is vacuous in the spectral order, replaced structurally by \emph{positive homogeneity} of the operator on spectral moduli. Under this hypothesis we prove the \emph{Morphological Representation Theorem} for the Fourier modulus lattice $\hat{L}$: any increasing, USC, positively homogeneous $\Psi:\hat{L}\to\hat{L}$ decomposes \emph{exactly} as a supremum of max-times erosions $\hat{f}/\psi$, indexed by a minimal morphological basis. Three structural corollaries follow: the spectral modulus map $|\cdot|:L^2(\mathbb{R}^n)\to\hat{L}$ is an idempotent cross-lattice projection (the morphological analogue of ReLU) that makes depth non-trivial; wavelet scattering coefficients form the canonical dictionary for the basis via Littlewood--Paley density; and spatial pooling is a Fourier erosion, so that pool-then-unpool is the ideal band-pass opening, U-Net skip connections are Fourier top-hat transforms, and strided pooling preserves the morphological structure if and only if the Nyquist condition holds. The framework carries a natural $\mathbb{C}^*$-group morphology structure, positioning mathematical morphology as the constructive operator-theoretic complement of spectral bias, scattering, and tropical geometry theories.

math.FA↗

arXiv2026

Lattice theory and algebraic models for deep convolutional learning based on mathematical morphology

We develop a rigorous algebraic framework for deep convolutional architectures, CNNs, ResNets, and encoder--decoder networks such as UNet, grounded in lattice theory and mathematical morphology. The central tool is the Matheron--Maragos--Banon--Barrera (MMBB) universal representation theory for translation-invariant operators, which we apply systematically to every layer of a standard deep network. The principal finding is that the standard CNN pipeline (linear convolution~$+$ ReLU~$+$ flat max-pooling) is a cross-lattice operator: the convolution is an erosion in the Fourier inf-semilattice while ReLU is a lattice-join closing and max-pooling is a dilation in the pointwise max-plus lattice, and their composition is a morphological opening in neither. A second finding is that the upper adjoint of ReLU in the pointwise lattice is a global (non-local) operator, the identity on globally non-negative functions and $-\infty$ otherwise, so no local morphological erosion can form an adjunction pair with ReLU. These two results together provide the precise algebraic reason why depth in standard CNNs introduces genuine representational power: the composed layer is not idempotent. Three layer designs that are genuine idempotent openings are identified and fully characterised: the pure max-plus morphological layer (pointwise lattice), the spectral Wiener layer (Fourier lattice), and the self-dual morphological layer. We establish a complete fixed-point and convergence theory. The framework also unifies max-pooling, strided convolution, and the Laplacian pyramid under the Goutsias--Heijmans adjoint pyramid theory, and gives the Activation--Pooling Dilation (APD) factorisation with its correct adjoint.

cs.AI↗