arXiv subjects

Janis Keck

Publications and source records attributed to Janis Keck.

6 recordsLinked to original sources

arXiv2025

Probabilistic Foundations of Fuzzy Simplicial Sets for Nonlinear Dimensionality Reduction

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 \v{C}ech 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.

cs.LG↗

arXiv2025

Frequentist Asymptotics of Variational Laplace

Variational inference is a general framework to obtain approximations to the posterior distribution in a Bayesian context. In essence, variational inference entails an optimization over a given family of probability distributions to choose the member of this family best approximating the posterior. Variational Laplace, an iterative update scheme motivated by this objective, is widely used in different contexts in the cognitive neuroscience community. However, until now, the theoretical properties of this scheme have not been systematically investigated. Here, we study variational Laplace in the light of frequentist asymptotic statistics. Asymptotical frequentist theory enables one to judge the quality of point estimates by their limit behaviour. We apply this framework to find that point estimates generated by variational Laplace enjoy the desirable properties of asymptotic consistency and efficiency in two toy examples. Furthermore, we derive conditions that are sufficient to establish these properties in a general setting. Besides of point estimates, we also study the frequentist convergence of distributions in the sense of total variation distance, which may be useful to relate variational Laplace both to recent findings regarding variational inference as well as to classical frequentist considerations on the Bayesian posterior. Finally, to illustrate the validity of our theoretical considerations, we conduct simulation experiments in our study examples.

math.ST↗

arXiv2025

VIBE: Video-Input Brain Encoder for fMRI Response Modeling

We present VIBE, a two-stage Transformer that fuses multi-modal video, audio, and text features to predict fMRI activity. Representations from open-source models (Qwen2.5, BEATs, Whisper, SlowFast, V-JEPA) are merged by a modality-fusion transformer and temporally decoded by a prediction transformer with rotary embeddings. Trained on 65 hours of movie data from the CNeuroMod dataset and ensembled across 20 seeds, VIBE attains mean parcel-wise Pearson correlations of 0.3225 on in-distribution Friends S07 and 0.2125 on six out-of-distribution films. An earlier iteration of the same architecture obtained 0.3198 and 0.2096, respectively, winning Phase-1 and placing second overall in the Algonauts 2025 Challenge.

cs.LG↗

arXiv2025

Merging Hazy Sets with m-Schemes: A Geometric Approach to Data Visualization

Many machine learning algorithms try to visualize high dimensional metric data in 2D in such a way that the essential geometric and topological features of the data are highlighted. In this paper, we introduce a framework for aggregating dissimilarity functions that arise from locally adjusting a metric through density-aware normalization, as employed in the IsUMap method. We formalize these approaches as m-schemes, a class of methods closely related to t-norms and t-conorms in probabilistic metrics, as well as to composition laws in information theory. These m-schemes provide a flexible and theoretically grounded approach to refining distance-based embeddings.

cs.LG↗

arXiv2024

IsUMap: Manifold Learning and Data Visualization leveraging Vietoris-Rips filtrations

This work introduces IsUMap, a novel manifold learning technique that enhances data representation by integrating aspects of UMAP and Isomap with Vietoris-Rips filtrations. We present a systematic and detailed construction of a metric representation for locally distorted metric spaces that captures complex data structures more accurately than the previous schemes. Our approach addresses limitations in existing methods by accommodating non-uniform data distributions and intricate local geometries. We validate its performance through extensive experiments on examples of various geometric objects and benchmark real-world datasets, demonstrating significant improvements in representation quality.

cs.LG↗

arXiv2024

Fuzzy simplicial sets and their application to geometric data analysis

In this article, we expand upon the concepts introduced by David Spivak about the relationship between the category $\mathbf{UM}$ of uber metric spaces and the category $\mathbf{sFuz}$ of fuzzy simplicial sets. We show that fuzzy simplicial sets can be regarded as natural combinatorial generalizations of metric relations. Furthermore, we take inspiration from UMAP to apply the theory to manifold learning, dimension reduction and data visualization, while refining some of their constructions. We generalize the adjunction between $\mathbf{UM}$ and $\mathbf{sFuz}$, derive an explicit description of colimits in $\mathbf{UM}$, and show that $\mathbf{UM}$ can be embedded into $\mathbf{sFuz}$. Furthermore, we prove analogous results for the category of extended-pseudo metric spaces $\mathbf{EPMet}$. We also provide rigorous definitions of functors that make it possible to recursively merge sets of fuzzy simplicial sets and provide a description of the adjunctions between the category of truncated fuzzy simplicial sets and $\mathbf{sFuz}$, which we relate to persistent homology. Combining those constructions, we can show a surprising connection between the well-known dimension reduction methods UMAP and Isomap and derive an alternative algorithm, which we call IsUMap, that combines some of the strengths of both methods (source code on github). Additionally, we developed a new embedding method that allows to preserve clusters detected in the original metric space that we construct from the data. The visualization of the optimization process gives the user information both about the inner-cluster distributions in the original metric space and their inter-cluster relations. We compare our new method with UMAP, Isomap and t-SNE on a series of low- and high-dimensional datasets, demonstrate how our method improves upon them and provide explanations for observed differences.

math.AT↗