Giovanni Luca Marchetti

Alfredo Reichlin, Giovanni Luca Marchetti, Hang Yin et al. · stanford

Learning from previously collected datasets of expert data offers the promise of acquiring robotic policies without unsafe and costly online explorations. However, a major challenge is a distributional shift between the states in the training dataset and the ones visited by the learned policy at the test time. While prior works mainly studied the distribution shift caused by the policy during the offline training, the problem of recovering from out-of-distribution states at the deployment time is not very well studied yet. We alleviate the distributional shift at the deployment time by introducing a recovery policy that brings the agent back to the training manifold whenever it steps out of the in-distribution states, e.g., due to an external perturbation. The recovery policy relies on an approximation of the training data density and a learned equivariant mapping that maps visual observations into a latent space in which translations correspond to the robot actions. We demonstrate the effectiveness of the proposed method through several manipulation experiments on a real robotic platform. Our results show that the recovery policy enables the agent to complete tasks while the behavioral cloning alone fails because of the distributional shift problem.

Giovanni Luca Marchetti, Gustaf Tegnér, Anastasiia Varava et al.

We introduce a general method for learning representations that are equivariant to symmetries of data. Our central idea is to decompose the latent space into an invariant factor and the symmetry group itself. The components semantically correspond to intrinsic data classes and poses respectively. The learner is trained on a loss encouraging equivariance based on supervision from relative symmetry information. The approach is motivated by theoretical results from group theory and guarantees representations that are lossless, interpretable and disentangled. We provide an empirical investigation via experiments involving datasets with a variety of symmetries. Results show that our representations capture the geometry of data and outperform other equivariant representation learning frameworks.

Alexander Kravberg, Giovanni Luca Marchetti, Vladislav Polianskii et al.

We introduce an algorithm for active function approximation based on nearest neighbor regression. Our Active Nearest Neighbor Regressor (ANNR) relies on the Voronoi-Delaunay framework from computational geometry to subdivide the space into cells with constant estimated function value and select novel query points in a way that takes the geometry of the function graph into account. We consider the recent state-of-the-art active function approximator called DEFER, which is based on incremental rectangular partitioning of the space, as the main baseline. The ANNR addresses a number of limitations that arise from the space subdivision strategy used in DEFER. We provide a computationally efficient implementation of our method, as well as theoretical halting guarantees. Empirical results show that ANNR outperforms the baseline for both closed-form functions and real-world examples, such as gravitational wave parameter inference and exploration of the latent space of a generative model.

Nathan W. Henry, Giovanni Luca Marchetti, Kathlén Kohn

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.

Mohammed Al-Jaff, Giovanni Luca Marchetti, Michael C Welle et al.

Idempotent Generative Networks (IGNs) are deep generative models that also function as local data manifold projectors, mapping arbitrary inputs back onto the manifold. They are trained to act as identity operators on the data and as idempotent operators off the data manifold. However, IGNs suffer from mode collapse, mode dropping, and training instability due to their objectives, which contain adversarial components and can cause the model to cover the data manifold only partially -- an issue shared with generative adversarial networks. We introduce Non-Adversarial Idempotent Generative Networks (NAIGNs) to address these issues. Our loss function combines reconstruction with the non-adversarial generative objective of Implicit Maximum Likelihood Estimation (IMLE). This improves on IGN's ability to restore corrupted data and generate new samples that closely match the data distribution. We moreover demonstrate that NAIGNs implicitly learn the distance field to the data manifold, as well as an energy-based model.

Alejandro García-Castellanos, Giovanni Luca Marchetti, Danica Kragic et al.

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.

Giovanni Luca Marchetti, Erin Connelly, Paul Breiding et al.

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.

Vahid Shahverdi, Giovanni Luca Marchetti, Georg Bökman et al.

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.

Luis Armando Pérez Rey, Giovanni Luca Marchetti, Danica Kragic et al.

We introduce Equivariant Isomorphic Networks (EquIN) -- a method for learning representations that are equivariant with respect to general group actions over data. Differently from existing equivariant representation learners, EquIN is suitable for group actions that are not free, i.e., that stabilize data via nontrivial symmetries. EquIN is theoretically grounded in the orbit-stabilizer theorem from group theory. This guarantees that an ideal learner infers isomorphic representations while trained on equivariance alone and thus fully extracts the geometric structure of data. We provide an empirical investigation on image datasets with rotational symmetries and show that taking stabilizers into account improves the quality of the representations.

Alfredo Reichlin, Giovanni Luca Marchetti, Hang Yin et al.

We address the problem of learning representations from observations of a scene involving an agent and an external object the agent interacts with. To this end, we propose a representation learning framework extracting the location in physical space of both the agent and the object from unstructured observations of arbitrary nature. Our framework relies on the actions performed by the agent as the only source of supervision, while assuming that the object is displaced by the agent via unknown dynamics. We provide a theoretical foundation and formally prove that an ideal learner is guaranteed to infer an isometric representation, disentangling the agent from the object and correctly extracting their locations. We evaluate empirically our framework on a variety of scenarios, showing that it outperforms vision-based approaches such as a state-of-the-art keypoint extractor. We moreover demonstrate how the extracted representations enable the agent to solve downstream tasks via reinforcement learning in an efficient manner.

Giovanni Luca Marchetti, Gabriele Cesa, Pratik Kumar et al.

Lattice reduction is a combinatorial optimization problem aimed at finding the most orthogonal basis in a given lattice. The Lenstra-Lenstra-Lovász (LLL) algorithm is the best algorithm in the literature for solving this problem. In light of recent research on algorithm discovery, in this work, we would like to answer this question: is it possible to parametrize the algorithm space for lattice reduction problem with neural networks and find an algorithm without supervised data? Our strategy is to use equivariant and invariant parametrizations and train in a self-supervised way. We design a deep neural model outputting factorized unimodular matrices and train it in a self-supervised manner by penalizing non-orthogonal lattice bases. We incorporate the symmetries of lattice reduction into the model by making it invariant to isometries and scaling of the ambient space and equivariant with respect to the hyperocrahedral group permuting and flipping the lattice basis elements. We show that this approach yields an algorithm with comparable complexity and performance to the LLL algorithm on a set of benchmarks. Additionally, motivated by certain applications for wireless communication, we extend our method to a convolutional architecture which performs joint reduction of spatially-correlated lattices arranged in a grid, thereby amortizing its cost over multiple lattices.

Giovanni Luca Marchetti, Daniel Kunin, Adele Myers et al.

How do neural networks trained over sequences acquire the ability to perform structured operations, such as arithmetic, geometric, and algorithmic computation? To gain insight into this question, we introduce the sequential group composition task. In this task, networks receive a sequence of elements from a finite group encoded in a real vector space and must predict their cumulative product. The task can be order-sensitive and requires a nonlinear architecture to be learned. Our analysis isolates the roles of the group structure, encoding statistics, and sequence length in shaping learning. We prove that two-layer networks learn this task one irreducible representation of the group at a time in an order determined by the Fourier statistics of the encoding. These networks can perfectly learn the task, but doing so requires a hidden width exponential in the sequence length $k$. In contrast, we show how deeper models exploit the associativity of the task to dramatically improve this scaling: recurrent neural networks compose elements sequentially in $k$ steps, while multilayer networks compose adjacent pairs in parallel in $\log k$ layers. Overall, the sequential group composition task offers a tractable window into the mechanics of deep learning.

Giovanni Luca Marchetti, Christopher Hillar, Danica Kragic et al.

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.

Christian Hägg, Kathlén Kohn, Giovanni Luca Marchetti et al.

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).

Giovanni Luca Marchetti, Vahid Shahverdi, Stefano Mereta et al.

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.

Vahid Shahverdi, Giovanni Luca Marchetti, Kathlén Kohn

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.

Aniss Aiman Medbouhi, Giovanni Luca Marchetti, Vladislav Polianskii et al.

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.

Aniss Aiman Medbouhi, Alejandro García-Castellanos, Giovanni Luca Marchetti et al.

We study the problem of constructing Steiner Minimal Trees (SMTs) in hyperbolic space. Exact SMT computation is NP-hard, and existing hyperbolic heuristics such as HyperSteiner are deterministic and often get trapped in locally suboptimal configurations. We introduce Randomized HyperSteiner (RHS), a stochastic Delaunay triangulation heuristic that incorporates randomness into the expansion process and refines candidate trees via Riemannian gradient descent optimization. Experiments on synthetic data sets and a real-world single-cell transcriptomic data show that RHS outperforms Minimum Spanning Tree (MST), Neighbour Joining, and vanilla HyperSteiner (HS). In near-boundary configurations, RHS can achieve a 32% reduction in total length over HS, demonstrating its effectiveness and robustness in diverse data regimes.