Fuzzy simplicial sets have become an object of interest in dimensionality reduction and manifold learning, most prominently through their role in UMAP. However, their definition through tools from algebraic topology without a clear probabilistic interpretation detaches them from commonly used theoretical frameworks in those areas. In this work we introduce a framework that explains fuzzy simplicial sets as marginals of probability measures on simplicial sets. In particular, this perspective shows that the fuzzy weights of UMAP arise from a generative model that samples Vietoris-Rips filtrations at random scales, yielding cumulative distribution functions of pairwise distances. More generally, the framework connects fuzzy simplicial sets to probabilistic models on the face poset, clarifies the relation between Kullback-Leibler divergence and fuzzy cross-entropy in this setting, and recovers standard t-norms and t-conorms via Boolean operations on the underlying simplicial sets. We then show how new embedding methods may be derived from this framework and illustrate this on an example where we generalize UMAP using Čech filtrations with triplet sampling. In summary, this probabilistic viewpoint provides a unified probabilistic theoretical foundation for fuzzy simplicial sets, clarifies the role of UMAP within this framework, and enables the systematic derivation of new dimensionality reduction methods.

In this article the notion of the nondecreasing (ND) rank of a matrix or tensor is introduced. A tensor has an ND rank of r if it can be represented as a sum of r outer products of vectors, with each vector satisfying a monotonicity constraint. It is shown that for certain poset orderings finding an ND factorization of rank $r$ is equivalent to finding a nonnegative rank-r factorization of a transformed tensor. However, not every tensor that is monotonic has a finite ND rank. Theory is developed describing the properties of the ND rank, including typical, maximum, and border ND ranks. Highlighted also are the special settings where a matrix or tensor has an ND rank of one or two. As a means of finding low ND rank approximations to a data tensor we introduce a variant of the hierarchical alternating least squares algorithm. Low ND rank factorizations are found and interpreted for two datasets concerning the weight of pigs and a mental health survey during the COVID-19 pandemic.

Natural language is replete with superficially different statements, such as ``Charles Darwin wrote" and ``Charles Darwin is the author of", which carry the same meaning. Large language models (LLMs) should generate the same next-token probabilities in such cases, but usually do not. Empirical workarounds have been explored, such as using k-NN estimates of sentence similarity to produce smoothed estimates. In this paper, we tackle this problem more abstractly, introducing a categorical homotopy framework for LLMs. We introduce an LLM Markov category to represent probability distributions in language generated by an LLM, where the probability of a sentence, such as ``Charles Darwin wrote" is defined by an arrow in a Markov category. However, this approach runs into difficulties as language is full of equivalent rephrases, and each generates a non-isomorphic arrow in the LLM Markov category. To address this fundamental problem, we use categorical homotopy techniques to capture ``weak equivalences" in an LLM Markov category. We present a detailed overview of application of categorical homotopy to LLMs, from higher algebraic K-theory to model categories, building on powerful theoretical results developed over the past half a century.

The absence of intrinsic adjacency relations and orientation systems in hypergraphs creates fundamental challenges for constructing sheaf Laplacians of arbitrary degrees. We resolve these limitations through symmetric simplicial sets derived directly from hypergraphs, called symmetric simplicial lifting, which encode all possible oriented subrelations within each hyperedge as ordered tuples. This construction canonically defines adjacency via facet maps while inherently preserving hyperedge provenance. We establish that the normalized degree zero sheaf Laplacian on our symmetric simplicial lifting reduces exactly to the traditional graph normalized sheaf Laplacian when restricted to graphs, validating its mathematical consistency with prior graph-based sheaf theory. Furthermore, the induced structure preserves all structural information from the original hypergraph, ensuring that every multi-way relational detail is faithfully retained. Leveraging this framework, we introduce Hypergraph Neural Sheaf Diffusion (HNSD), the first principled extension of neural sheaf diffusion to hypergraphs. HNSD operates via normalized degree zero sheaf Laplacian over symmetric simplicial lifting, resolving orientation ambiguity and adjacency sparsity inherent to hypergraph learning. Experimental evaluations demonstrate HNSDs competitive performance across established benchmarks.

Figure 1: We propose a hierarchical Generative AI Architecture (GAIA) using higher-order category theory. Generative AI has become a dominant paradigm for building intelligent systems in the last few years, ranging from large language models developed with the widely used Transformer model Vaswani et al. (2017), or more recently with the structured state space sequence models Gu et al. (2022); Yin et al. (2023), and with the growing use of image diffusion algorithms Song and Ermon (2019); Yin et al. (2023). We can broadly define the problem of generative AI as the construction, maintenance, and deployment of foundation models Bommasani et al. (2022), a storehouse of human knowledge that provides the basic infrastructure for AI across some set of applications. A fundamental question, therefore, to investigate is to study the mathematical basis for foundation models. We propose a mathematical framework for a Generative AI Architecture (GAIA) (see Figure 1) based on the hypothesis that category theory MacLane (1971); Riehl (2017); Lurie (2009) provides a universal mathematical language for foundation models.

We propose a layered hierarchical architecture called UCLA (Universal Causality Layered Architecture), which combines multiple levels of categorical abstraction for causal inference. At the top-most level, causal interventions are modeled combinatorially using a simplicial category of ordinal numbers. At the second layer, causal models are defined by a graph-type category. The non-random ``surgical" operations on causal structures, such as edge deletion, are captured using degeneracy and face operators from the simplicial layer above. The third categorical abstraction layer corresponds to the data layer in causal inference. The fourth homotopy layer comprises of additional structure imposed on the instance layer above, such as a topological space, which enables evaluating causal models on datasets. Functors map between every pair of layers in UCLA. Each functor between layers is characterized by a universal arrow, which defines an isomorphism between every pair of categorical layers. These universal arrows define universal elements and representations through the Yoneda Lemma, and in turn lead to a new category of elements based on a construction introduced by Grothendieck. Causal inference between each pair of layers is defined as a lifting problem, a commutative diagram whose objects are categories, and whose morphisms are functors that are characterized as different types of fibrations. We illustrate the UCLA architecture using a range of examples, including integer-valued multisets that represent a non-graphical framework for conditional independence, and causal models based on graphs and string diagrams using symmetric monoidal categories. We define causal effect in terms of the homotopy colimit of the nerve of the category of elements.

The goal of this paper is to introduce pooling strategies for simplicial convolutional neural networks. Inspired by graph pooling methods, we introduce a general formulation for a simplicial pooling layer that performs: i) local aggregation of simplicial signals; ii) principled selection of sampling sets; iii) downsampling and simplicial topology adaptation. The general layer is then customized to design four different pooling strategies (i.e., max, top-k, self-attention, and separated top-k) grounded in the theory of topological signal processing. Also, we leverage the proposed layers in a hierarchical architecture that reduce complexity while representing data at different resolutions. Numerical results on real data benchmarks (i.e., flow and graph classification) illustrate the advantage of the proposed methods with respect to the state of the art.

Causal inference (Pearl, 2009a; Imbens and Rubin, 2015; Spirtes et al., 2000) and predictive state representations (PSRs) (Singh et al., 2004) in reinforcement learning (Sutton and Barto, 1998), whose roots go back to earlier work on subspace identification in linear systems (Van Overschee and De Moor, 1996) and even earlier work on algebraic theories of context-free languages Chomsky and Schützenberger (1963) and algebraic automata theory (Give'on and Arbib, 1968), both involve structure discovery of a latent variable model through interventions. The use of superficially dissimilar representations - directed acyclic graphs (DAGs) (Pearl, 1989), hybrid undirected and directed graphs (Lauritzen and Richardson, 2002) and hyperedge graphs (Forré and Mooij, 2017; Evans, 2018) in causal inference, versus Hankel matrix and Hilbert space embeddings of dynamical systems - have long obscured their deeper connections. Structure discovery in causal inference and PSRs both involve the determination of a latent structure, which is directional at lower orders, but homotopy equivalences at higher orders induce symmetries. In particular, causal inference involves determining a structure, such as a DAG that encodes direct causal effects between a pair of objects, but multiple DAG models are equivalent because of symmetries induced by conditional independences (Dawid, 2001; Studený et al., 2010a) and correlations induced by latent unobservable confounders that are only revealed over higher-order simplices (e.g., DAGs over n 3 vertices). PSRs represent "hidden state" in dynamical systems by constructing a series of tests,

Causal background knowledge about the existence or the absence of causal edges and paths is frequently encountered in observational studies. The shared directed edges and links of a subclass of Markov equivalent DAGs refined due to background knowledge can be represented by a causal maximally partially directed acyclic graph (MPDAG). In this paper, we first provide a sound and complete graphical characterization of causal MPDAGs and give a minimal representation of a causal MPDAG. Then, we introduce a novel representation called direct causal clause (DCC) to represent all types of causal background knowledge in a unified form. Using DCCs, we study the consistency and equivalency of causal background knowledge and show that any causal background knowledge set can be equivalently decomposed into a causal MPDAG plus a minimal residual set of DCCs. Polynomial-time algorithms are also provided for checking the consistency, equivalency, and finding the decomposed MPDAG and residual DCCs. Finally, with causal background knowledge, we prove a sufficient and necessary condition to identify causal effects and surprisingly find that the identifiability of causal effects only depends on the decomposed MPDAG. We also develop a local IDA-type algorithm to estimate the possible values of an unidentifiable effect. Simulations suggest that causal background knowledge can significantly improve the identifiability of causal effects.

Abstract: Dictionary learning for sparse representations has been successful in many reconstruction tasks. Simplicial learning is an adaptation of dictionary learning, where subspaces become clipped and acquire arbitrary offsets, taking the form of simplices. Such adaptation is achieved through additional constraints on sparse codes. Furthermore, an evolutionary approach can be chosen to determine the number and the dimensionality of simplices composing the simplicial, in which most generative and compact simplicials are favored. This paper proposes an evolutionary simplicial learning method as a generative and compact sparse framework for classification.