Critical Points of Degenerate Metrics on Algebraic Varieties: A Tale of Overparametrization

We study the critical points over an algebraic variety of an optimization problem defined by a quadratic objective that is degenerate. This scenario arises in machine learning when the dataset size is small with respect to the model, and is typically referred to as overparametrization. Our main result relates the degenerate optimization problem to a nondegenerate one via a projection. In the highly-degenerate regime, we find that a central role is played by the ramification locus of the projection. Additionally, we provide tools for counting the number of critical points over projective varieties, and discuss specific cases arising from deep learning. Our work bridges tools from algebraic geometry with ideas from machine learning, and it extends the line of literature around the Euclidean distance degree to the degenerate setting.

Sprecher Networks: A Parameter-Efficient Kolmogorov-Arnold Architecture

We present Sprecher Networks (SNs), a family of trainable neural architectures inspired by the classical Kolmogorov-Arnold-Sprecher (KAS) construction for approximating multivariate continuous functions. Distinct from Multi-Layer Perceptrons (MLPs) with fixed node activations and Kolmogorov-Arnold Networks (KANs) featuring learnable edge activations, SNs utilize shared, learnable splines (monotonic and general) within structured blocks incorporating explicit shift parameters and mixing weights. Our approach directly realizes Sprecher's specific 1965 sum of shifted splines formula in its single-layer variant and extends it to deeper, multi-layer compositions. We further enhance the architecture with optional lateral mixing connections that enable intra-block communication between output dimensions, providing a parameter-efficient alternative to full attention mechanisms. Beyond parameter efficiency with $O(LN + LG)$ scaling (where $G$ is the knot count of the shared splines) versus MLPs' $O(LN^2)$, SNs admit a sequential evaluation strategy that reduces peak forward-intermediate memory from $O(N^2)$ to $O(N)$ (treating batch size as constant), making much wider architectures feasible under memory constraints. We demonstrate empirically that composing these blocks into deep networks leads to highly parameter and memory-efficient models, discuss theoretical motivations, and compare SNs with related architectures (MLPs, KANs, and networks with learnable node activations).

Alternating Gradient Flows: A Theory of Feature Learning in Two-layer Neural Networks

What features neural networks learn, and how, remains an open question. In this paper, we introduce Alternating Gradient Flows (AGF), an algorithmic framework that describes the dynamics of feature learning in two-layer networks trained from small initialization. Prior works have shown that gradient flow in this regime exhibits a staircase-like loss curve, alternating between plateaus where neurons slowly align to useful directions and sharp drops where neurons rapidly grow in norm. AGF approximates this behavior as an alternating two-step process: maximizing a utility function over dormant neurons and minimizing a cost function over active ones. AGF begins with all neurons dormant. At each round, a dormant neuron activates, triggering the acquisition of a feature and a drop in the loss. AGF quantifies the order, timing, and magnitude of these drops, matching experiments across architectures. We show that AGF unifies and extends existing saddle-to-saddle analyses in fully connected linear networks and attention-only linear transformers, where the learned features are singular modes and principal components, respectively. In diagonal linear networks, we prove AGF converges to gradient flow in the limit of vanishing initialization. Applying AGF to quadratic networks trained to perform modular addition, we give the first complete characterization of the training dynamics, revealing that networks learn Fourier features in decreasing order of coefficient magnitude. Altogether, AGF offers a promising step towards understanding feature learning in neural networks.

Learning on a Razor's Edge: the Singularity Bias of Polynomial Neural Networks

Deep neural networks often infer sparse representations, converging to a subnetwork during the learning process. In this work, we theoretically analyze subnetworks and their bias through the lens of algebraic geometry. We consider fully-connected networks with polynomial activation functions, and focus on the geometry of the function space they parametrize, often referred to as neuromanifold. First, we compute the dimension of the subspace of the neuromanifold parametrized by subnetworks. Second, we show that this subspace is singular. Third, we argue that such singularities often correspond to critical points of the training dynamics. Lastly, we discuss convolutional networks, for which subnetworks and singularities are similarly related, but the bias does not arise.

An Invitation to Neuroalgebraic Geometry

In this expository work, 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.

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.

Relative Representations: Topological and Geometric Perspectives

Relative representations are an established approach to zero-shot model stitching, consisting of a non-trainable transformation of the latent space of a deep neural network. Based on insights of topological and geometric nature, we propose two improvements to relative representations. First, we introduce a normalization procedure in the relative transformation, resulting in invariance to non-isotropic rescalings and permutations. The latter coincides with the symmetries in parameter space induced by common activation functions. Second, we propose to deploy topological densification when fine-tuning relative representations, a topological regularization loss encouraging clustering within classes. We provide an empirical investigation on a natural language task, where both the proposed variations yield improved performance on zero-shot model stitching.

Geometry of Lightning Self-Attention: Identifiability and Dimension

We consider function spaces defined by self-attention networks without normalization, and theoretically analyze their geometry. Since these networks are polynomial, we rely on tools from algebraic geometry. In particular, we study the identifiability of deep attention by providing a description of the generic fibers of the parametrization for an arbitrary number of layers and, as a consequence, compute the dimension of the function space. Additionally, for a single-layer model, we characterize the singular and boundary points. Finally, we formulate a conjectural extension of our results to normalized self-attention networks, prove it for a single layer, and numerically verify it in the deep case.

Hyperbolic Delaunay Geometric Alignment

Hyperbolic machine learning is an emerging field aimed at representing data with a hierarchical structure. However, there is a lack of tools for evaluation and analysis of the resulting hyperbolic data representations. To this end, we propose Hyperbolic Delaunay Geometric Alignment (HyperDGA) -- a similarity score for comparing datasets in a hyperbolic space. The core idea is counting the edges of the hyperbolic Delaunay graph connecting datapoints across the given sets. We provide an empirical investigation on synthetic and real-life biological data and demonstrate that HyperDGA outperforms the hyperbolic version of classical distances between sets. Furthermore, we showcase the potential of HyperDGA for evaluating latent representations inferred by a Hyperbolic Variational Auto-Encoder.

Harmonics of Learning: Universal Fourier Features Emerge in Invariant Networks

In this work, we formally prove that, under certain conditions, if a neural network is invariant to a finite group then its weights recover the Fourier transform on that group. This provides a mathematical explanation for the emergence of Fourier features -- a ubiquitous phenomenon in both biological and artificial learning systems. The results hold even for non-commutative groups, in which case the Fourier transform encodes all the irreducible unitary group representations. Our findings have consequences for the problem of symmetry discovery. Specifically, we demonstrate that the algebraic structure of an unknown group can be recovered from the weights of a network that is at least approximately invariant within certain bounds. Overall, this work contributes to a foundation for an algebraic learning theory of invariant neural network representations.