arXiv subjects

Julio Candanedo

Publications and source records attributed to Julio Candanedo.

6 recordsLinked to original sources

arXiv2026

Diffusion Maps is not Dimensionality Reduction

Diffusion maps (DMAP) are often used as a dimensionality-reduction tool, but more precisely they provide a spectral representation of the intrinsic geometry rather than a complete charting method. To illustrate this distinction, we study a Swiss roll with known isometric coordinates and compare DMAP, Isomap, and UMAP across latent dimensions. For each representation, we fit an oracle affine readout to the ground-truth chart and measure reconstruction error. Isomap most efficiently recovers the low-dimensional chart, UMAP provides an intermediate tradeoff, and DMAP becomes accurate only after combining multiple diffusion modes. Thus the correct chart lies in the span of diffusion coordinates, but standard DMAP do not by themselves identify the appropriate combination.

cs.LG↗

arXiv2026

The Diffusion-Attention Connection

Softmax attention is the row-normalized operator of a diffusion map: both normalize a learned score into a Markov operator, and differ only in what the score is allowed to contain. Decomposing that score reveals three geometric sectors: a metric core with Witten-Laplacian continuum limit, an exact node-potential sector corresponding to a Markov--Witten change of measure, and a circulating sector realized as irreversible Markov--Girsanov transport or as a magnetic $U(1)$ phase. Attention thereby becomes auditable in familiar mathematics: every trained head carries measurable geometry, potential, and flux, while standard mechanisms acquire geometric addresses---Coifman--Lafon normalization as an exact density correction, rotary embeddings as pure gauge, and AdaLN as a Cauchy--Green deformation combined with an Witten deformation. Experiments on pretrained diffusion transformers and language models test this decomposition: enforcing positive-semi-definite geometry is nearly free, consistent with the identification, whereas removing circulation incurs a substantial cost, sharpest on induction.

cs.LG↗

arXiv2026

LISA: Laplacian In-context Spectral Analysis

We propose Laplacian In-context Spectral Analysis (LISA), a method for inference-time adaptation of Laplacian-based time-series models using only an observed prefix. LISA combines delay-coordinate embeddings and Laplacian spectral learning to produce diffusion-coordinate state representations, together with a frozen nonlinear decoder for one-step prediction. We introduce lightweight latent-space residual adapters based on either Gaussian-process regression or an attention-like Markov operator over context windows. Across forecasting and autoregressive rollout experiments, LISA improves over the frozen baseline and is often most beneficial under changing dynamics. This work links in-context adaptation to nonparametric spectral methods for dynamical systems.

cs.LG↗

arXiv2025

Linearized Diffusion Map

We introduce the Linearized Diffusion Map (LDM), a novel linear dimensionality reduction method constructed via a linear approximation of the diffusion-map kernel. LDM integrates the geometric intuition of diffusion-based nonlinear methods with the computational simplicity, efficiency, and interpretability inherent in linear embeddings such as PCA and classical MDS. Through comprehensive experiments on synthetic datasets (Swiss roll and hyperspheres) and real-world benchmarks (MNIST and COIL-20), we illustrate that LDM captures distinct geometric features of datasets compared to PCA, offering complementary advantages. Specifically, LDM embeddings outperform PCA in datasets exhibiting explicit manifold structures, particularly in high-dimensional regimes, whereas PCA remains preferable in scenarios dominated by variance or noise. Furthermore, the complete positivity of LDM's kernel matrix allows direct applicability of Non-negative Matrix Factorization (NMF), suggesting opportunities for interpretable latent-structure discovery. Our analysis positions LDM as a valuable new linear dimensionality reduction technique with promising theoretical and practical extensions.

cs.LG↗

arXiv2024

Diffusion Map Autoencoder

Diffusion-Map-AutoEncoder (DMAE) pairs a diffusion-map encoder (using the Nystr\"om method) with linear or RBF Gaussian-Process latent mean decoders, yielding closed-form inductive mappings and strong reconstructions.

cs.DS↗

arXiv2023

Sparse Partial-Tracing

Matrices and more generally multidimensional arrays, form the backbone of computational studies. In this paper we demonstrate increases in computational efficiency by performing partial-tracing/tensor-contractions on sparse-arrays. It was shown that sparse-arrays are really 3 dense-arrays (dense-shape, index-array, and data-array). Dense-array manipulations of these constituent arrays are used to determine the resulting partial-trace. Because computational arrays are used in a verity of different studies, these methods are broadly applicable.

cs.DS↗