Approximating velocity fields with planted attractors via Neural-ODEs for classification purposes

In this work, Neural ODEs equipped with a curated collection of equilibrium points have been successfully employed for classification tasks. The planted attractors serve as indicators for the target classes, while the velocity field leveraging the universal approximation capabilities of the architecture shapes the dynamical landscape. This process defines the basins of attraction of the trained model, effectively directing each input (provided as an initial condition) toward its corresponding destination target.

Exact Fixed-Point Constraints in Neural-ODEs with Provable Universality

We introduce a technique that enables Neural-ODEs to approximate arbitrary velocity fields with a priori planted fixed-points. Specifically, a recipe is given to explicitly accommodate for a finite collection of points in the reference multi-dimensional space of the Neural-ODE where the velocity field is exactly equal to zero. In this way, the gradient-based training is rigorously constrained inside the prescribed hypothesis class while leaving the expressive power of the Neural-ODE unaltered. We rigorously prove the universality of the Neural-ODE under any local constraints in the velocity field and give a computationally convenient way of imposing the fixed points. Our method is then tested on two paradigmatic physical models.

Spectral Higher-Order Neural Networks

Neural networks are fundamental tools of modern machine learning. The standard paradigm assumes binary interactions (across feedforward linear passes) between inter-tangled units, organized in sequential layers. Generalized architectures have been also designed that move beyond pairwise interactions, so as to account for higher-order couplings among computing neurons. Higher-order networks are however usually deployed as augmented graph neural networks (GNNs), and, as such, prove solely advantageous in contexts where the input exhibits an explicit hypergraph structure. Here, we present Spectral Higher-Order Neural Networks (SHONNs), a new algorithmic strategy to incorporate higher-order interactions in general-purpose, feedforward, network structures. SHONNs leverages a reformulation of the model in terms of spectral attributes. This allows to mitigate the common stability and parameter scaling problems that come along weighted, higher-order, forward propagations.

Spectral Architecture Search for Neural Networks

Architecture design and optimization are challenging problems in the field of artificial neural networks. Working in this context, we here present SPARCS (SPectral ARchiteCture Search), a novel architecture search protocol which exploits the spectral attributes of the inter-layer transfer matrices. SPARCS allows one to explore the space of possible architectures by spanning continuous and differentiable manifolds, thus enabling for gradient-based optimization algorithms to be eventually employed. With reference to simple benchmark models, we show that the newly proposed method yields a self-emerging architecture with a minimal degree of expressivity to handle the task under investigation and with a reduced parameter count as compared to other viable alternatives.

Learning in Wilson-Cowan model for metapopulation

The Wilson-Cowan model for metapopulation, a Neural Mass Network Model, treats different subcortical regions of the brain as connected nodes, with connections representing various types of structural, functional, or effective neuronal connectivity between these regions. Each region comprises interacting populations of excitatory and inhibitory cells, consistent with the standard Wilson-Cowan model. By incorporating stable attractors into such a metapopulation model's dynamics, we transform it into a learning algorithm capable of achieving high image and text classification accuracy. We test it on MNIST and Fashion MNIST, in combination with convolutional neural networks, on CIFAR-10 and TF-FLOWERS, and, in combination with a transformer architecture (BERT), on IMDB, always showing high classification accuracy. These numerical evaluations illustrate that minimal modifications to the Wilson-Cowan model for metapopulation can reveal unique and previously unobserved dynamics.

Automatic Input Feature Relevance via Spectral Neural Networks

Working with high-dimensional data is a common practice, in the field of machine learning. Identifying relevant input features is thus crucial, so as to obtain compact dataset more prone for effective numerical handling. Further, by isolating pivotal elements that form the basis of decision making, one can contribute to elaborate on - ex post - models' interpretability, so far rather elusive. Here, we propose a novel method to estimate the relative importance of the input components for a Deep Neural Network. This is achieved by leveraging on a spectral re-parametrization of the optimization process. Eigenvalues associated to input nodes provide in fact a robust proxy to gauge the relevance of the supplied entry features. Unlike existing techniques, the spectral features ranking is carried out automatically, as a byproduct of the network training. The technique is successfully challenged against both synthetic and real data.

Engineered Ordinary Differential Equations as Classification Algorithm (EODECA): thorough characterization and testing

EODECA (Engineered Ordinary Differential Equations as Classification Algorithm) is a novel approach at the intersection of machine learning and dynamical systems theory, presenting a unique framework for classification tasks [1]. This method stands out with its dynamical system structure, utilizing ordinary differential equations (ODEs) to efficiently handle complex classification challenges. The paper delves into EODECA's dynamical properties, emphasizing its resilience against random perturbations and robust performance across various classification scenarios. Notably, EODECA's design incorporates the ability to embed stable attractors in the phase space, enhancing reliability and allowing for reversible dynamics. In this paper, we carry out a comprehensive analysis by expanding on the work [1], and employing a Euler discretization scheme. In particular, we evaluate EODECA's performance across five distinct classification problems, examining its adaptability and efficiency. Significantly, we demonstrate EODECA's effectiveness on the MNIST and Fashion MNIST datasets, achieving impressive accuracies of $98.06\%$ and $88.21\%$, respectively. These results are comparable to those of a multi-layer perceptron (MLP), underscoring EODECA's potential in complex data processing tasks. We further explore the model's learning journey, assessing its evolution in both pre and post training environments and highlighting its ability to navigate towards stable attractors. The study also investigates the invertibility of EODECA, shedding light on its decision-making processes and internal workings. This paper presents a significant step towards a more transparent and robust machine learning paradigm, bridging the gap between machine learning algorithms and dynamical systems methodologies.

Complex Recurrent Spectral Network

This paper presents a novel approach to advancing artificial intelligence (AI) through the development of the Complex Recurrent Spectral Network ($\mathbb{C}$-RSN), an innovative variant of the Recurrent Spectral Network (RSN) model. The $\mathbb{C}$-RSN is designed to address a critical limitation in existing neural network models: their inability to emulate the complex processes of biological neural networks dynamically and accurately. By integrating key concepts from dynamical systems theory and leveraging principles from statistical mechanics, the $\mathbb{C}$-RSN model introduces localized non-linearity, complex fixed eigenvalues, and a distinct separation of memory and input processing functionalities. These features collectively enable the $\mathbb{C}$-RSN evolving towards a dynamic, oscillating final state that more closely mirrors biological cognition. Central to this work is the exploration of how the $\mathbb{C}$-RSN manages to capture the rhythmic, oscillatory dynamics intrinsic to biological systems, thanks to its complex eigenvalue structure and the innovative segregation of its linear and non-linear components. The model's ability to classify data through a time-dependent function, and the localization of information processing, is demonstrated with an empirical evaluation using the MNIST dataset. Remarkably, distinct items supplied as a sequential input yield patterns in time which bear the indirect imprint of the insertion order (and of the time of separation between contiguous insertions).

A Bridge between Dynamical Systems and Machine Learning: Engineered Ordinary Differential Equations as Classification Algorithm (EODECA)

In a world increasingly reliant on machine learning, the interpretability of these models remains a substantial challenge, with many equating their functionality to an enigmatic black box. This study seeks to bridge machine learning and dynamical systems. Recognizing the deep parallels between dense neural networks and dynamical systems, particularly in the light of non-linearities and successive transformations, this manuscript introduces the Engineered Ordinary Differential Equations as Classification Algorithms (EODECAs). Uniquely designed as neural networks underpinned by continuous ordinary differential equations, EODECAs aim to capitalize on the well-established toolkit of dynamical systems. Unlike traditional deep learning models, which often suffer from opacity, EODECAs promise both high classification performance and intrinsic interpretability. They are naturally invertible, granting them an edge in understanding and transparency over their counterparts. By bridging these domains, we hope to usher in a new era of machine learning models where genuine comprehension of data processes complements predictive prowess.

How a student becomes a teacher: learning and forgetting through Spectral methods

In theoretical ML, the teacher-student paradigm is often employed as an effective metaphor for real-life tuition. The above scheme proves particularly relevant when the student network is overparameterized as compared to the teacher network. Under these operating conditions, it is tempting to speculate that the student ability to handle the given task could be eventually stored in a sub-portion of the whole network. This latter should be to some extent reminiscent of the frozen teacher structure, according to suitable metrics, while being approximately invariant across different architectures of the student candidate network. Unfortunately, state-of-the-art conventional learning techniques could not help in identifying the existence of such an invariant subnetwork, due to the inherent degree of non-convexity that characterizes the examined problem. In this work, we take a leap forward by proposing a radically different optimization scheme which builds on a spectral representation of the linear transfer of information between layers. The gradient is hence calculated with respect to both eigenvalues and eigenvectors with negligible increase in terms of computational and complexity load, as compared to standard training algorithms. Working in this framework, we could isolate a stable student substructure, that mirrors the true complexity of the teacher in terms of computing neurons, path distribution and topological attributes. When pruning unimportant nodes of the trained student, as follows a ranking that reflects the optimized eigenvalues, no degradation in the recorded performance is seen above a threshold that corresponds to the effective teacher size. The observed behavior can be pictured as a genuine second-order phase transition that bears universality traits.