We introduce a novel framework consisting of a class of algebraic structures that generalize one-dimensional monoidal systems into higher dimensions by defining per-axis composition operators subject to non-commutativity and a global interchange law. These structures, defined recursively from a base case of vector-matrix pairs, model directional composition in multiple dimensions while preserving structural coherence through commutative linear operators. We show that the framework that unifies several well-known linear transforms in signal processing and data analysis. In this framework, data indices are embedded into a composite structure that decomposes into simpler components. We show that classic transforms such as the Discrete Fourier Transform (DFT), the Walsh transform, and the Hadamard transform are special cases of our algebraic structure. The framework provides a systematic way to derive these transforms by appropriately choosing vector and matrix pairs. By subsuming classical transforms within a common structure, the framework also enables the development of learnable transformations tailored to specific data modalities and tasks.

Current generative models are able to generate high-quality artefacts but have been shown to struggle with compositional reasoning, which can be defined as the ability to generate complex structures from simpler elements. In this paper, we focus on the problem of compositional representation learning for music data, specifically targeting the fully-unsupervised setting. We propose a simple and extensible framework that leverages an explicit compositional inductive bias, defined by a flexible auto-encoding objective that can leverage any of the current state-of-art generative models. We demonstrate that our framework, used with diffusion models, naturally addresses the task of unsupervised audio source separation, showing that our model is able to perform high-quality separation. Our findings reveal that our proposal achieves comparable or superior performance with respect to other blind source separation methods and, furthermore, it even surpasses current state-of-art supervised baselines on signal-to-interference ratio metrics. Additionally, by learning an a-posteriori masking diffusion model in the space of composable representations, we achieve a system capable of seamlessly performing unsupervised source separation, unconditional generation, and variation generation. Finally, as our proposal works in the latent space of pre-trained neural audio codecs, it also provides a lower computational cost with respect to other neural baselines.

We investigate perturbations of orthonormal bases of $L^2$ via a composition operator $C_h$ induced by a mapping $h$. We provide a comprehensive characterization of the mapping $h$ required for the perturbed sequence to form an orthonormal or Riesz basis. Restricting our analysis to differentiable mappings, we reveal that all Riesz bases of the given form are induced by bi-Lipschitz mappings. In addition, we discuss implications of these results for approximation theory, highlighting the potential of using bijective neural networks to construct complete sequences with favorable approximation properties.

Composition operators have been extensively studied in complex analysis, and recently, they have been utilized in engineering and machine learning. Here, we focus on composition operators associated with maps in Euclidean spaces that are on reproducing kernel Hilbert spaces with respect to analytic positive definite functions, and prove the maps are affine if the composition operators are bounded. Our result covers composition operators on Paley-Wiener spaces and reproducing kernel spaces with respect to the Gaussian kernel on ${\mathbb R}^d$, widely used in the context of engineering.

We study pathwise invariances of centred random fields that can be controlled through the covariance. A result involving composition operators is obtained in second-order settings, and we show that various path properties including additivity boil down to invariances of the covariance kernel. These results are extended to a broader class of operators in the Gaussian case, via the Lo\`eve isometry. Several covariance-driven pathwise invariances are illustrated, including fields with symmetric paths, centred paths, harmonic paths, or sparse paths. The proposed approach delivers a number of promising results and perspectives in Gaussian process regression.