We show that deep neural networks trained across diverse tasks exhibit remarkably similar low-dimensional parametric subspaces. We provide the first large-scale empirical evidence that demonstrates that neural networks systematically converge to shared spectral subspaces regardless of initialization, task, or domain. Through mode-wise spectral analysis of over 1100 models - including 500 Mistral-7B LoRAs, 500 Vision Transformers, and 50 LLaMA-8B models - we identify universal subspaces capturing majority variance in just a few principal directions. By applying spectral decomposition techniques to the weight matrices of various architectures trained on a wide range of tasks and datasets, we identify sparse, joint subspaces that are consistently exploited, within shared architectures across diverse tasks and datasets. Our findings offer new insights into the intrinsic organization of information within deep networks and raise important questions about the possibility of discovering these universal subspaces without the need for extensive data and computational resources. Furthermore, this inherent structure has significant implications for model reusability, multi-task learning, model merging, and the development of training and inference-efficient algorithms, potentially reducing the carbon footprint of large-scale neural models.

Certain data are naturally modeled by networks or weighted graphs, be they arterial networks or mobility networks. When there is no canonical labeling of the nodes across the dataset, we talk about unlabeled networks. In this paper, we focus on the question of dimensionality reduction for this type of data. More specifically, we address the issue of interpreting the feature subspace constructed by dimensionality reduction methods. Most existing methods for network-valued data are derived from principal component analysis (PCA) and therefore rely on subspaces generated by a set of vectors, which we identify as a major limitation in terms of interpretability. Instead, we propose to implement the method called barycentric subspace analysis (BSA), which relies on subspaces generated by a set of points. In order to provide a computationally feasible framework for BSA, we introduce a novel embedding for unlabeled networks where we replace their usual representation by equivalence classes of isomorphic networks with that by equivalence classes of cospectral networks. We then illustrate BSA on simulated and real-world datasets, and compare it to tangent PCA.

The increasingly popular independent component analysis (ICA) may only be applied to data following the generative ICA model in order to guarantee algorithmindependent and theoretically valid results. Subspace ICA models generalize the assumption of component independence to independence between groups of components. They are attractive candidates for dimensionality reduction methods, however are currently limited by the assumption of equal group sizes or less general semi-parametric models. By introducing the concept of irreducible independent subspaces or components, we present a generalization to a parameter-free mixture model. Moreover, we relieve the condition of at-most-one-Gaussian by including previous results on non-Gaussian component analysis.

The development of neuromorphic hardware and modeling of biological neural networks requires algorithms with local learning rules. Artificial neural networks using local learning rules to perform principal subspace analysis (PSA) and clustering have recently been derived from principled objective functions. However, no biologically plausible networks exist for minor subspace analysis (MSA), a fundamental signal processing task. MSA extracts the lowest-variance subspace of the input signal covariance matrix. Here, we introduce a novel similarity matching objective for extracting the minor subspace, Minor Subspace Similarity Matching (MSSM). Moreover, we derive an adaptive MSSM algorithm that naturally maps onto a novel neural network with local learning rules and gives numerical results showing that our method converges at a competitive rate.

Given a set $\mathcal{C}=\{C_i\}_{i=1}^m$ of square matrices, the matrix blind joint block diagonalization problem (BJBDP) is to find a full column rank matrix $A$ such that $C_i=A\Sigma_iA^\text{T}$ for all $i$, where $\Sigma_i$'s are all block diagonal matrices with as many diagonal blocks as possible. The BJBDP plays an important role in independent subspace analysis (ISA). This paper considers the identification problem for BJBDP, that is, under what conditions and by what means, we can identify the diagonalizer $A$ and the block diagonal structure of $\Sigma_i$, especially when there is noise in $C_i$'s. In this paper, we propose a ``bi-block diagonalization'' method to solve BJBDP, and establish sufficient conditions under which the method is able to accomplish the task. Numerical simulations validate our theoretical results. To the best of the authors' knowledge, existing numerical methods for BJBDP have no theoretical guarantees for the identification of the exact solution, whereas our method does.

The increasingly popular independent component analysis (ICA) may only be applied to data following the generative ICA model in order to guarantee algorithmindependent and theoretically valid results. Subspace ICA models generalize the assumption of component independence to independence between groups of components. They are attractive candidates for dimensionality reduction methods, however are currently limited by the assumption of equal group sizes or less general semi-parametric models. By introducing the concept of irreducible independent subspaces or components, we present a generalization to a parameter-free mixture model. Moreover, we relieve the condition of at-most-one-Gaussian by including previous results on non-Gaussian component analysis. After introducing this general model, we discuss joint block diagonalization with unknown block sizes, on which we base a simple extension of JADE to algorithmically perform the subspace analysis. Simulations confirm the feasibility of the algorithm.

The increasingly popular independent component analysis (ICA) may only be applied to data following the generative ICA model in order to guarantee algorithmindependent and theoretically valid results. Subspace ICA models generalize the assumption of component independence to independence between groups of components. They are attractive candidates for dimensionality reduction methods, however are currently limited by the assumption of equal group sizes or less general semi-parametric models. By introducing the concept of irreducible independent subspaces or components, we present a generalization to a parameter-free mixture model. Moreover, we relieve the condition of at-most-one-Gaussian by including previous results on non-Gaussian component analysis. After introducing this general model, we discuss joint block diagonalization with unknown block sizes, on which we base a simple extension of JADE to algorithmically perform the subspace analysis. Simulations confirm the feasibility of the algorithm.

The increasingly popular independent component analysis (ICA) may only be applied todata following the generative ICA model in order to guarantee algorithmindependent andtheoretically valid results. Subspace ICA models generalize the assumption of component independence to independence between groups of components. Theyare attractive candidates for dimensionality reduction methods, however are currently limited by the assumption of equal group sizes or less general semi-parametricmodels. By introducing the concept of irreducible independent subspacesor components, we present a generalization to a parameter-free mixture model. Moreover, we relieve the condition of at-most-one-Gaussian by including previous results on non-Gaussian component analysis. After introducing thisgeneral model, we discuss joint block diagonalization with unknown block sizes, on which we base a simple extension of JADE to algorithmically perform the subspace analysis. Simulations confirm the feasibility of the algorithm.