arXiv subjects

Wael Mattar

Publications and source records attributed to Wael Mattar.

5 recordsLinked to original sources

arXiv2026

Multiscaling in Wasserstein Spaces

We present a novel multiscale framework for analyzing sequences of probability measures in Wasserstein spaces over Euclidean domains. Exploiting the intrinsic geometry of optimal transport, we construct a multiscale transform applicable to both absolutely continuous and discrete measures. Central to our approach is a refinement operator based on McCann's interpolants, which preserves the geodesic structure of measure flows and serves as an upsampling mechanism. Building on this, we introduce the optimality number, a scalar that quantifies deviations of a sequence from Wasserstein geodesicity across scales, enabling the detection of irregular dynamics and anomalies. We establish key theoretical guarantees, including stability of the transform and geometric decay of coefficients, ensuring robustness and interpretability of the multiscale representation. Finally, we demonstrate the versatility of our methodology through numerical experiments: denoising and anomaly detection in Gaussian flows, analysis of point cloud dynamics under vector fields, and the multiscale characterization of neural network learning trajectories.

math.NA↗

arXiv2025

Geometric multiscale analysis via nonstationary subdivision schemes

Pyramid transforms are constructive methods for analyzing sequences in a multiscale fashion. Traditionally, these transforms rely on stationary upsampling and downsampling operations. In this paper, we propose employing nonstationary subdivision schemes as upsampling operators that vary with the refinement level. These schemes offer greater flexibility, enabling the development of more expressive multiscale transforms, including geometric multiscale analysis. We establish the fundamental properties of the resulting nonstationary transforms, namely the decay of their detail coefficients and the stability of both the decomposition and the reconstruction, and we demonstrate their effectiveness in capturing and analyzing geometric features. In particular, we apply the framework to assess the consistency of planar samples with a circular shape and to detect localized geometric anomalies, including a study of circles generated by a neural network. The accompanying code is publicly available.

math.NA↗

arXiv2025

Multiscale analysis via pseudo-reversing and applications to manifold-valued sequences

Modeling data using manifold values is a powerful concept with numerous advantages, particularly in addressing nonlinear phenomena. This approach captures the intrinsic geometric structure of the data, leading to more accurate descriptors and more efficient computational processes. However, even fundamental tasks like compression and data enhancement present meaningful challenges in the manifold setting. This paper introduces a multiscale transform that aims to represent manifold-valued sequences at different scales, enabling novel data processing tools for various applications. Similar to traditional methods, our construction is based on a refinement operator that acts as an upsampling operator and a corresponding downsampling operator. Inspired by Wiener's lemma, we term the latter as the reverse of the former. It turns out that some upsampling operators, for example, least-squares-based refinement, do not have a practical reverse. Therefore, we introduce the notion of pseudo-reversing and explore its analytical properties and asymptotic behavior. We derive analytical properties of the induced multiscale transform and conclude the paper with numerical illustrations showcasing different aspects of the pseudo-reversing and two data processing applications involving manifolds.

math.NA↗

arXiv2024

Wavelets Are All You Need for Autoregressive Image Generation

In this paper, we take a new approach to autoregressive image generation that is based on two main ingredients. The first is wavelet image coding, which allows to tokenize the visual details of an image from coarse to fine details by ordering the information starting with the most significant bits of the most significant wavelet coefficients. The second is a variant of a language transformer whose architecture is re-designed and optimized for token sequences in this 'wavelet language'. The transformer learns the significant statistical correlations within a token sequence, which are the manifestations of well-known correlations between the wavelet subbands at various resolutions. We show experimental results with conditioning on the generation process.

cs.LG↗

arXiv2021

Pyramid Transform of Manifold Data via Subdivision Operators

Multiscale transforms have become a key ingredient in many data processing tasks. With technological development, we observe a growing demand for methods to cope with non-linear data structures such as manifold values. In this paper, we propose a multiscale approach for analyzing manifold-valued data using a pyramid transform. The transform uses a unique class of downsampling operators that enable a non-interpolating subdivision schemes as upsampling operators. We describe this construction in detail and present its analytical properties, including stability and coefficient decay. Next, we numerically demonstrate the results and show the application of our method to denoising and anomaly detection.

math.NA↗