Concept-based models are an emerging paradigm in deep learning that constrains the inference process to operate through human-interpretable variables, facilitating explainability and human interaction. However, these architectures, on par with popular opaque neural models, fail to account for the true causal mechanisms underlying the target phenomena represented in the data. This hampers their ability to support causal reasoning tasks, limits out-of-distribution generalization, and hinders the implementation of fairness constraints. To overcome these issues, we propose Causally reliable Concept Bottleneck Models (C2BMs), a class of concept-based architectures that enforce reasoning through a bottleneck of concepts structured according to a model of the real-world causal mechanisms. We also introduce a pipeline to automatically learn this structure from observational data and unstructured background knowledge (e.g., scientific literature). Experimental evidence suggests that C2BMs are more interpretable, causally reliable, and improve responsiveness to interventions w.r.t.

Mamba extends earlier state space models (SSMs) by introducing input-dependent dynamics, and has demonstrated strong empirical performance across a range of domains, including language modeling, computer vision, and foundation models. However, a surprising weakness remains: despite being built on architectures designed for long-range dependencies, Mamba performs poorly on long-range sequential tasks. Understanding and addressing this gap is important for improving Mamba's universality and versatility. In this work, we analyze Mamba's limitations through three perspectives: expressiveness, inductive bias, and training stability. Our theoretical results show how Mamba falls short in each of these aspects compared to earlier SSMs such as S4D. To address these issues, we propose B2S6, a simple extension of Mamba's S6 unit that combines block-wise selective dynamics with a channel-specific bias. We prove that these changes equip the model with a better-suited inductive bias and improve its expressiveness and stability. Empirically, B2S6 outperforms S4 and S4D on Long-Range Arena (LRA) tasks while maintaining Mamba's performance on language modeling benchmarks.

Graph machine learning architectures are typically tailored to specific tasks on specific datasets, which hinders their broader applicability. This has led to a new quest in graph machine learning: how to build graph foundation models (GFMs) capable of generalizing across arbitrary graphs and features? In this work, we present a recipe for designing GFMs for node-level tasks from first principles. The key ingredient underpinning our study is a systematic investigation of the symmetries that a graph foundation model must respect. In a nutshell, we argue that label permutation-equivariance alongside feature permutation-invariance are necessary in addition to the common node permutation-equivariance on each local neighborhood of the graph. To this end, we first characterize the space of linear transformations that are equivariant to permutations of nodes and labels, and invariant to permutations of features. We then prove that the resulting network is a universal approximator on multisets that respect the aforementioned symmetries. Our recipe uses such layers on the multiset of features induced by the local neighborhood of the graph to obtain a class of graph foundation models for node property prediction.

Relative Positional Encoding (RPE), which encodes the relative distance between any pair of tokens, is one of the most successful modifications to the original Transformer. As far as we know, theoretical understanding of the RPE-based Transformers is largely unexplored. In this work, we mathematically analyze the power of RPE-based Transformers regarding whether the model is capable of approximating any continuous sequence-to-sequence functions. One may naturally assume the answer is in the affirmative--RPE-based Transformers are universal function approximators. However, we present a negative result by showing there exist continuous sequence-to-sequence functions that RPE-based Transformers cannot approximate no matter how deep and wide the neural network is.

Efficient transfer learning algorithms are key to the success of foundation models on diverse downstream tasks even with limited data. Recent works of Basu et al. (2023) and Kaba et al. (2022) propose group averaging (equitune) and optimizationbased methods, respectively, over features from group-transformed inputs to obtain equivariant outputs from non-equivariant neural networks. While Kaba et al. (2022) are only concerned with training from scratch, we find that equitune performs poorly on equivariant zero-shot tasks despite good finetuning results. We hypothesize that this is because pretrained models provide better quality features for certain transformations than others and simply averaging them is deleterious. Hence, we propose λ-equitune that averages the features using importance weights, λs. These weights are learned directly from the data using a small neural network, leading to excellent zero-shot and finetuned results that outperform equitune. Further, we prove that λ-equitune is equivariant and a universal approximator of equivariant functions. Additionally, we show that the method of Kaba et al. (2022) used with appropriate loss functions, which we call equizero, also gives excellent zero-shot and finetuned performance.

The expressive power of neural networks is important for understanding deep learning. Most existing works consider this problem from the view of the depth of a network. In this paper, we study how width affects the expressiveness of neural networks. Classical results state that depth-bounded (e.g.

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We investigate the approximation capabilities of dense neural networks. While universal approximation theorems establish that sufficiently large architectures can approximate arbitrary continuous functions if there are no restrictions on the weight values, we show that dense neural networks do not possess this universality. Our argument is based on a model compression approach, combining the weak regularity lemma with an interpretation of feedforward networks as message passing graph neural networks. We consider ReLU neural networks subject to natural constraints on weights and input and output dimensions, which model a notion of dense connectivity. Within this setting, we demonstrate the existence of Lipschitz continuous functions that cannot be approximated by such networks. This highlights intrinsic limitations of neural networks with dense layers and motivates the use of sparse connectivity as a necessary ingredient for achieving true universality.

Kernel methods are widely utilized in machine learning field to learn, from training data, a latent function in a reproducing kernel Hilbert space. It is well known that the approximator thus obtained usually achieves a linear representation, which brings various computational benefits, while maintaining great representation power (i.e., universal approximation). However, when non-negativity constraints are imposed on the function's outputs, the literature usually takes the kernel method-based approximators as offering linear representations at the expense of limited model flexibility or good representation power by allowing for their nonlinear forms. The main contribution of this paper is to derive a sufficient condition for a positive definite kernel so that it may construct flexible and linear approximators of non-negative functions. We call a kernel function that offers these attributes an inverse M-kernel; it is a generalization of the inverse M-matrix. Furthermore, we show that for a one-dimensional input space, universal exponential/Abel kernels are inverse M-kernels and construct linear universal approxima-tors of non-negative functions. To the best of our knowledge, it is the first time that the existence of linear universal approximators of non-negative functions has been elucidated. We confirm the effectiveness of our results by experiments on the problems of non-negativity-constrained regression, density estimation, and intensity estimation. Finally, we discuss issues and perspectives on multi-dimensional input settings.

Tomakesure functions represented by neural networks are homogeneous, we also prove that HomoGNN and HomoMLP can only represent homogeneous functions, in Section B.2.2.