Ali Mohaddes | Searcharxiv

arXiv2025

Regularized Learning for Fractional Brownian Motion via Path Signatures

Fractional Brownian motion (fBm) extends classical Brownian motion by introducing dependence between increments, governed by the Hurst parameter $H\in (0,1)$. Unlike traditional Brownian motion, the increments of an fBm are not independent. Paths generated by fractional Brownian motions can exhibit significant irregularity, particularly when the Hurst parameter is small. As a result, classical regression methods may not perform effectively. Signatures, defined as iterated path integrals of continuous and discrete-time processes, offer a universal nonlinearity property that simplifies the challenge of feature selection in time series data analysis by effectively linearizing it. Consequently, we employ Lasso regression techniques for regularization when handling irregular data. To evaluate the performance of signature Lasso on fractional Brownian motion (fBM), we study its consistency when the Hurst parameter $ H \ne \frac{1}{2} $. This involves deriving bounds on the first and second moments of the signature. For the case $ H > \frac{1}{2} $, we use the signature defined in the Young sense, while for $ H < \frac{1}{2} $, we use the Stratonovich interpretation. Simulation results indicate that signature Lasso can outperform traditional regression methods for synthetic data as well as for real-world datasets.

math.ST↗

arXiv2025

Affine Invariance in Continuous-Domain Convolutional Neural Networks

The notion of group invariance helps neural networks in recognizing patterns and features under geometric transformations. Group convolutional neural networks enhance traditional convolutional neural networks by incorporating group-based geometric structures into their design. This research studies affine invariance on continuous-domain convolutional neural networks. Despite other research considering isometric invariance or similarity invariance, we focus on the full structure of affine transforms generated by the group of all invertible $2 \times 2$ real matrices (generalized linear group $\mathrm{GL}_2(\mathbb{R})$). We introduce a new criterion to assess the invariance of two signals under affine transformations. The input image is embedded into the affine Lie group $G_2 = \mathbb{R}^2 \ltimes \mathrm{GL}_2(\mathbb{R})$ to facilitate group convolution operations that respect affine invariance. Then, we analyze the convolution of embedded signals over $G_2$. In sum, our research could eventually extend the scope of geometrical transformations that usual deep-learning pipelines can handle.

cs.LG↗