There has been much recent interest in designing neural networks (NNs) with relaxed equivariance, which interpolate between exact equivariance and full flexibility for consistent performance gains. In a separate line of work, structured parameter matrices with low displacement rank (LDR)—which permit fast function and gradient evaluation—have been used to create compact NNs, though primarily benefiting classical convolutional neural networks (CNNs). In this work, we propose a framework based on symmetry-based structured matrices to build approximately equivariant NNs with fewer parameters. Our approach unifies the aforementioned areas using Group Matrices (GMs), a forgotten precursor to the modern notion of regular representations of finite groups. GMs allow the design of structured matrices similar to LDR matrices, which can generalize all the elementary operations of a CNN from cyclic groups to arbitrary finite groups. We show GMs can also generalize classical LDR theory to general discrete groups, enabling a natural formalism for approximate equivariance. We test GM-based architectures on various tasks with relaxed symmetry and find that our framework performs competitively with approximately equivariant NNs and other structured matrix-based methods, often with one to two orders of magnitude fewer parameters.

Compositionality in knowledge and language—the ability to represent complex concepts as a combination of simpler ones—is a hallmark of human cognition and communication. Despite recent advances, deep neural networks still struggle to acquire this property reliably. Neural models for emergent communication look to endow artificial agents with compositional language by simulating the pressures that form human language. In this work, we introduce CELEBI (Compressive-Expressive Language Emergence through a discrete Bottleneck and Iterated learning), a novel self-supervised framework for inducing compositional representations through a reconstruction-based communication game between a sender and a receiver. Building on theories of language emergence and the iterated learning framework, we integrate three mechanisms that jointly promote compressibility, expressivity, and efficiency in the emergent language. First, Progressive Decoding incentivizes intermediate reasoning by requiring the receiver to produce partial reconstructions after each symbol. Second, Final-State Imitation trains successive generations of agents to imitate reconstructions rather than messages, enforcing a tighter communication bottleneck. Third, Pairwise Distance Maximization regularizes message diversity by encouraging high distances between messages, with formal links to entropy maximization. Our method significantly improves both the efficiency and compositionality of the learned messages on the Shapes3D and MPI3D datasets, surpassing prior discrete communication frameworks in both reconstruction accuracy and topographic similarity. This work provides new theoretical and empirical evidence for the emergence of structured, generalizable communication protocols from simplicity-based inductive biases.

Persistence diagrams (PD)s play a central role in topological data analysis, and are used in an ever increasing variety of applications. The comparison of PD data requires computing distances among large sets of PDs, with metrics which are accurate, theoretically sound, and fast to compute. Especially for denser multi-dimensional PDs, such comparison metrics are lacking. While on the one hand, Wasserstein-type distances have high accuracy and theoretical guarantees, they incur high computational cost. On the other hand, distances between vectorizations such as Persistence Statistics (PS)s have lower computational cost, but lack the accuracy guarantees and theoretical properties of a true distance over PD space. In this work we introduce a class of pseudodistances called Extended Topological Pseudodistances (ETD)s, which have tunable complexity, and can approximate Sliced and classical Wasserstein distances at the high-complexity extreme, while being computationally lighter and close to Persistence Statistics at the lower complexity extreme, and thus allow users to interpolate between the two metrics. We build theoretical comparisons to show how to fit our new distances at an intermediate level between persistence vectorizations and Wasserstein distances. We also experimentally verify that ETDs outperform PSs in terms of accuracy and outperform Wasserstein and Sliced Wasserstein distances in terms of computational complexity.

Generative modeling over discrete data has recently seen numerous success stories, with applications spanning language modeling, biological sequence design, and graph-structured molecular data. The predominant generative modeling paradigm for discrete data is still autoregressive, with more recent alternatives based on diffusion or flow-matching falling short of their impressive performance in continuous data settings, such as image or video generation. In this work, we introduce Fisher-Flow, a novel flow-matching model for discrete data. Fisher-Flow takes a manifestly geometric perspective by considering categorical distributions over discrete data as points residing on a statistical manifold equipped with its natural Riemannian metric: the \emph{Fisher-Rao metric}. As a result, we demonstrate discrete data itself can be continuously reparameterised to points on the positive orthant of the $d$-hypersphere $\mathbb{S}^d_+$, which allows us to define flows that map any source distribution to target in a principled manner by transporting mass along (closed-form) geodesics of $\mathbb{S}^d_+$. Furthermore, the learned flows in Fisher-Flow can be further bootstrapped by leveraging Riemannian optimal transport leading to improved training dynamics. We prove that the gradient flow induced by Fisher-FLow is optimal in reducing the forward KL divergence. We evaluate Fisher-Flow on an array of synthetic and diverse real-world benchmarks, including designing DNA Promoter, and DNA Enhancer sequences. Empirically, we find that Fisher-Flow improves over prior diffusion and flow-matching models on these benchmarks.

The explicit incorporation of task-specific inductive biases through symmetry has emerged as a general design precept in the development of high-performance machine learning models. For example, group equivariant neural networks have demonstrated impressive performance across various domains and applications such as protein and drug design. A prevalent intuition about such models is that the integration of relevant symmetry results in enhanced generalization. Moreover, it is posited that when the data and/or the model exhibits only approximate or partial symmetry, the optimal or best-performing model is one where the model symmetry aligns with the data symmetry. In this paper, we conduct a formal unified investigation of these intuitions. To begin, we present quantitative bounds that demonstrate how models capturing task-specific symmetries lead to improved generalization. Utilizing this quantification, we examine the more general question of dealing with approximate/partial symmetries. We establish, for a given symmetry group, a quantitative comparison between the approximate equivariance of the model and that of the data distribution, precisely connecting model equivariance error and data equivariance error. Our result delineates the conditions under which the model equivariance error is optimal, thereby yielding the best-performing model for the given task and data.

The Weisfeiler-Lehman (WL) test is a fundamental iterative algorithm for checking the isomorphism of graphs. It has also been observed that it underlies the design of several graph neural network architectures, whose capabilities and performance can be understood in terms of the expressive power of this test. Motivated by recent developments in machine learning applications to datasets involving three-dimensional objects, we study when the WL test is {\em complete} for clouds of Euclidean points represented by complete distance graphs, i.e., when it can distinguish, up to isometry, any arbitrary such cloud. Our main result states that the $(d-1)$-dimensional WL test is complete for point clouds in $d$-dimensional Euclidean space, for any $d\ge 2$, and only three iterations of the test suffice. Our result is tight for $d = 2, 3$. We also observe that the $d$-dimensional WL test only requires one iteration to achieve completeness.