arXiv subjects

Vahid Shahverdi

Publications and source records attributed to Vahid Shahverdi.

10 recordsLinked to original sources

arXiv2026

Singular Learning and Occam's Razor in Deep Monomial Networks

In the optimization of neural networks, gradient dynamics are influenced by critical points that arise from the model's architecture. These critical points occur where the Jacobian of the model's parametrization is rank-deficient, and are the most pronounced singularities studied in Singular Learning Theory. We investigate such points in deep fully-connected networks with monomial activations via tools from polynomial algebra such as Mason's Theorem. We show that, for sufficiently large activation degree, criticality occurs precisely at subnetworks, i.e., at parameter configurations where some neurons are inactive or redundant. This offers a mathematical perspective on the implicit bias in deep neural networks, explaining the tendency of these models to converge toward simpler functions.

cs.LG↗

arXiv2026

On the Geometry and Optimization of Polynomial Convolutional Networks

We study convolutional neural networks with monomial activation functions. Specifically, we prove that their parameterization map is regular and is an isomorphism almost everywhere, up to rescaling the filters. By leveraging on tools from algebraic geometry, we explore the geometric properties of the image in function space of this map - typically referred to as neuromanifold. In particular, we compute the dimension and the degree of the neuromanifold, which measure the expressivity of the model, and describe its singularities. Moreover, for a generic large dataset, we derive an explicit formula that quantifies the number of critical points arising in the optimization of a regression loss.

cs.LG↗

arXiv2026

Learning on a Razor's Edge: Identifiability and Singularity of Polynomial Neural Networks

We study function spaces parametrized by neural networks, referred to as neuromanifolds. Specifically, we focus on deep Multi-Layer Perceptrons (MLPs) and Convolutional Neural Networks (CNNs) with an activation function that is a sufficiently generic polynomial. First, we address the identifiability problem, showing that, for almost all functions in the neuromanifold of an MLP, there exist only finitely many parameter choices yielding that function. For CNNs, the parametrization is generically one-to-one. As a consequence, we compute the dimension of the neuromanifold. Second, we describe singular points of neuromanifolds. We characterize singularities completely for CNNs, and partially for MLPs. In both cases, they arise from sparse subnetworks. For MLPs, we prove that these singularities often correspond to critical points of the mean-squared error loss, which does not hold for CNNs. This provides a geometric explanation of the sparsity bias of MLPs. All of our results leverage tools from algebraic geometry.

cs.LG↗

arXiv2026

Conservation Laws from Data Symmetry in Neural Networks

We explore whether intrinsic symmetries of the training data lead to conserved quantities during gradient-flow training of neural networks. Under the assumption that the loss function is analytic and non-polynomial, we prove that data symmetries generically do not induce any additional integrals of motion. For mean squared error (MSE) loss, on the other hand, there are situations in which data augmentation yields extra conserved quantities. We build a framework, utilizing \emph{tensorizable networks} to describe this phenomenon. Tensorizable networks are a family of architectures whose dependence on parameters and inputs can be separated using an intermediate representation. They include linear and polynomial networks, as well as Lightning Attention.

cs.LG↗

arXiv2026

Identifiable Equivariant Networks are Layerwise Equivariant

We investigate the relation between end-to-end equivariance and layerwise equivariance in deep neural networks. We prove the following: For a network whose end-to-end function is equivariant with respect to group actions on the input and output spaces, there is a parameter choice yielding the same end-to-end function such that its layers are equivariant with respect to some group actions on the latent spaces. Our result assumes that the parameters of the model are identifiable in an appropriate sense. This identifiability property has been established in the literature for a large class of networks, to which our results apply immediately, while it is conjectural for others. The theory we develop is grounded in an abstract formalism, and is therefore architecture-agnostic. Overall, our results provide a mathematical explanation for the emergence of equivariant structures in the weights of neural networks during training -- a phenomenon that is consistently observed in practice.

cs.LG↗

arXiv2025

Moment Constraints and Phase Recovery for Multireference Alignment

Multireference alignment (MRA) refers to the problem of recovering a signal from noisy samples subject to random circular shifts. Expectation--maximization (EM) and variational approaches use statistical modeling to achieve high accuracy at the cost of solving computationally expensive optimization problems. The method of moments, instead, achieves fast reconstructions by utilizing the power spectrum and bispectrum to determine the signal up to shift. Our approach combines the two philosophies by viewing the power spectrum as a manifold on which to constrain the signal. We then maximize the data likelihood function on this manifold with a gradient-based approach to estimate the true signal. Algorithmically, our method involves iterating between template alignment and projections onto the manifold. The method offers increased speed compared to EM and demonstrates improved accuracy over bispectrum-based methods.

cs.IT↗

arXiv2025

Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometry

In this position paper, we promote the study of function spaces parameterized by machine learning models through the lens of algebraic geometry. To this end, we focus on algebraic models, such as neural networks with polynomial activations, whose associated function spaces are semi-algebraic varieties. We outline a dictionary between algebro-geometric invariants of these varieties, such as dimension, degree, and singularities, and fundamental aspects of machine learning, such as sample complexity, expressivity, training dynamics, and implicit bias. Along the way, we review the literature and discuss ideas beyond the algebraic domain. This work lays the foundations of a research direction bridging algebraic geometry and deep learning, that we refer to as neuroalgebraic geometry.

cs.LG↗

arXiv2025

Geometry of Linear Neural Networks: Equivariance and Invariance under Permutation Groups

The set of functions parameterized by a linear fully-connected neural network is a determinantal variety. We investigate the subvariety of functions that are equivariant or invariant under the action of a permutation group. Examples of such group actions are translations or $90^\circ$ rotations on images. We describe such equivariant or invariant subvarieties as direct products of determinantal varieties, from which we deduce their dimension, degree, Euclidean distance degree, and their singularities. We fully characterize invariance for arbitrary permutation groups, and equivariance for cyclic groups. We draw conclusions for the parameterization and the design of equivariant and invariant linear networks in terms of sparsity and weight-sharing properties. We prove that all invariant linear functions can be parameterized by a single linear autoencoder with a weight-sharing property imposed by the cycle decomposition of the considered permutation. The space of rank-bounded equivariant functions has several irreducible components, so it can not be parameterized by a single network-but each irreducible component can. Finally, we show that minimizing the squared-error loss on our invariant or equivariant networks reduces to minimizing the Euclidean distance from determinantal varieties via the Eckart-Young theorem.

cs.LG↗

arXiv2024

Algebraic Complexity and Neurovariety of Linear Convolutional Networks

In this paper, we study linear convolutional networks with one-dimensional filters and arbitrary strides. The neuromanifold of such a network is a semialgebraic set, represented by a space of polynomials admitting specific factorizations. Introducing a recursive algorithm, we generate polynomial equations whose common zero locus corresponds to the Zariski closure of the corresponding neuromanifold. Furthermore, we explore the algebraic complexity of training these networks employing tools from metric algebraic geometry. Our findings reveal that the number of all complex critical points in the optimization of such a network is equal to the generic Euclidean distance degree of a Segre variety. Notably, this count significantly surpasses the number of critical points encountered in the training of a fully connected linear network with the same number of parameters.

math.AG↗

arXiv2024

Function Space and Critical Points of Linear Convolutional Networks

We study the geometry of linear networks with one-dimensional convolutional layers. The function spaces of these networks can be identified with semi-algebraic families of polynomials admitting sparse factorizations. We analyze the impact of the network's architecture on the function space's dimension, boundary, and singular points. We also describe the critical points of the network's parameterization map. Furthermore, we study the optimization problem of training a network with the squared error loss. We prove that for architectures where all strides are larger than one and generic data, the non-zero critical points of that optimization problem are smooth interior points of the function space. This property is known to be false for dense linear networks and linear convolutional networks with stride one.

cs.LG↗