The block diagonal structure of an affinity matrix is a commonly desired property in cluster analysis because it represents clusters of feature vectors by non-zero coefficients that are concentrated in blocks. However, recovering a block diagonal affinity matrix is challenging in real-world applications, in which the data may be subject to outliers and heavy-tailed noise that obscure the hidden cluster structure. To address this issue, we first analyze the effect of different fundamental outlier types in graph-based cluster analysis. A key idea that simplifies the analysis is to introduce a vector that represents a block diagonal matrix as a piece-wise linear function of the similarity coefficients that form the affinity matrix. We reformulate the problem as a robust piece-wise linear fitting problem and propose a Fast and Robust Sparsity-Aware Block Diagonal Representation (FRS-BDR) method, which jointly estimates cluster memberships and the number of blocks. Comprehensive experiments on a variety of real-world applications demonstrate the effectiveness of FRS-BDR in terms of clustering accuracy, robustness against corrupted features, computation time and cluster enumeration performance.

Accurate 3D tracking in highly deformable scenes with occlusions and shadows can facilitate new applications in robotics, augmented reality, and generative AI. However, tracking under these conditions is extremely challenging due to the ambiguity that arises with large deformations, shadows, and occlusions. We introduce MD-Splatting, an approach for simultaneous 3D tracking and novel view synthesis, using video captures of a dynamic scene from various camera poses. MD-Splatting builds on recent advances in Gaussian splatting, a method that learns the properties of a large number of Gaussians for state-of-the-art and fast novel view synthesis. MD-Splatting learns a deformation function to project a set of Gaussians with non-metric, thus canonical, properties into metric space. The deformation function uses a neural-voxel encoding and a multilayer perceptron (MLP) to infer Gaussian position, rotation, and a shadow scalar. We enforce physics-inspired regularization terms based on local rigidity, conservation of momentum, and isometry, which leads to trajectories with smaller trajectory errors. MD-Splatting achieves high-quality 3D tracking on highly deformable scenes with shadows and occlusions. Compared to state-of-the-art, we improve 3D tracking by an average of 23.9 %, while simultaneously achieving high-quality novel view synthesis. With sufficient texture such as in scene 6, MD-Splatting achieves a median tracking error of 3.39 mm on a cloth of 1 x 1 meters in size. Project website: https://md-splatting.github.io/.

Graph auto-encoders are widely used to construct graph representations in Euclidean vector spaces. However, it has already been pointed out empirically that linear models on many tasks can outperform graph auto-encoders. In our work, we prove that the solution space induced by graph auto-encoders is a subset of the solution space of a linear map. This demonstrates that linear embedding models have at least the representational power of graph auto-encoders based on graph convolutional networks. So why are we still using nonlinear graph auto-encoders? One reason could be that actively restricting the linear solution space might introduce an inductive bias that helps improve learning and generalization. While many researchers believe that the nonlinearity of the encoder is the critical ingredient towards this end, we instead identify the node features of the graph as a more powerful inductive bias. We give theoretical insights by introducing a corresponding bias in a linear model and analyzing the change in the solution space. Our experiments are aligned with other empirical work on this question and show that the linear encoder can outperform the nonlinear encoder when using feature information.

DNA sequence alignment involves assigning short DNA reads to the most probable locations on an extensive reference genome. This process is crucial for various genomic analyses, including variant calling, transcriptomics, and epigenomics. Conventional methods, refined over decades, tackle this challenge in two steps: genome indexing followed by efficient search to locate likely positions for given reads. Building on the success of Large Language Models (LLM) in encoding text into embeddings, where the distance metric captures semantic similarity, recent efforts have explored whether the same Transformer architecture can produce numerical representations for DNA sequences. Such models have shown early promise in tasks involving classification of short DNA sequences, such as the detection of coding vs non-coding regions, as well as the identification of enhancer and promoter sequences. Performance at sequence classification tasks does not, however, translate to sequence alignment, where it is necessary to conduct a genome-wide search to successfully align every read. We address this open problem by framing it as an Embed-Search-Align task. In this framework, a novel encoder model DNA-ESA generates representations of reads and fragments of the reference, which are projected into a shared vector space where the read-fragment distance is used as surrogate for alignment. In particular, DNA-ESA introduces: (1) Contrastive loss for self-supervised training of DNA sequence representations, facilitating rich sequence-level embeddings, and (2) a DNA vector store to enable search across fragments on a global scale. DNA-ESA is >97% accurate when aligning 250-length reads onto a human reference genome of 3 gigabases (single-haploid), far exceeds the performance of 6 recent DNA-Transformer model baselines and shows task transfer across chromosomes and species.

Determining suitable low-dimensional representations of complex high-dimensional data is a challenging task in numerous applications. Whether its preprocessing biological datasets prior to their analysis (Nguyen & Holmes, 2019), the visualisation of complex structure in single-cell sequencing data (Lähnemann et al., 2020), or the comparison of different manifold representations (Barannikov et al., 2022): an understanding of structural (dis)similarities is crucial, especially in the context of datasets that are ever-increasing in size and dimensionality. The primary assumption driving such analyses is the manifold hypothesis, which assumes that data is a (noisy) subsample from some unknown manifold. Operating under this assumption, manifold learning methods have made large advances in detecting complex structures in data, but they typically use local measures of the embedding quality, which are ultimately relying on local approximations of manifolds by k-nearest neighbour graphs. However, such approximations--which require specific parameter choices and thresholds--can have a substantial negative impact on both embedding results and the interpretation of evaluation scores. Moreover, countering the increasing popularity of non-linear dimensionality reduction methods that claim to preserve local and global structures, recent work (Chari & Pachter, 2023) sheds some doubt on the assumption that'good' embeddings should also faithfully preserve distances, while raising questions of how to measure the inevitable distortions introduced by representation learning. Thus, there is a need for novel methods in representation learning, which efficiently summarise data across varying levels of similarity, eliminating the need to rely on fixed neighbourhood graphs. Motivated by these considerations, we adopt a more general perspective that does not rely on manifold approximations. To this end, we propose a novel embedding quality measure based on metric space magnitude, a recently-proposed mathematical invariant that encapsulates numerous important geometric characteristics of metric spaces.

In recent years, graph neural networks (GNNs) have become a popular tool for solving various problems over graphs. In these models, the link structure of the graph is typically exploited and nodes' embeddings are iteratively updated based on adjacent nodes. Nodes' contents are used solely in the form of feature vectors, served as nodes' first-layer embeddings. However, the filters or convolutions, applied during iterations/layers to these initial embeddings lead to their impact diminish and contribute insignificantly to the final embeddings. In order to address this issue, in this paper we propose augmenting nodes' embeddings by embeddings generating from their content, at higher GNN layers. More precisely, we propose models wherein a structural embedding using a GNN and a content embedding are computed for each node. These two are combined using a combination layer to form the embedding of a node at a given layer. We suggest methods such as using an auto-encoder or building a content graph, to generate content embeddings. In the end, by conducting experiments over several real-world datasets, we demonstrate the high accuracy and performance of our models.

Problems involving geometric data arise in physics, chemistry, robotics, computer vision, and many other fields. Such data can take numerous forms, for instance points, direction vectors, translations, or rotations, but to date there is no single architecture that can be applied to such a wide variety of geometric types while respecting their symmetries. In this paper we introduce the Geometric Algebra Transformer (GATr), a general-purpose architecture for geometric data. GATr represents inputs, outputs, and hidden states in the projective geometric (or Clifford) algebra, which offers an efficient 16-dimensional vector-space representation of common geometric objects as well as operators acting on them. GATr is equivariant with respect to E(3), the symmetry group of 3D Euclidean space. As a Transformer, GATr is versatile, efficient, and scalable. We demonstrate GATr in problems from n-body modeling to wall-shear-stress estimation on large arterial meshes to robotic motion planning. GATr consistently outperforms both non-geometric and equivariant baselines in terms of error, data efficiency, and scalability.

In this work, we introduce a novel approach based on algebraic topology to enhance graph convolution and attention modules by incorporating local topological properties of the data. To do so, we consider the framework of sheaf neural networks, which has been previously leveraged to incorporate additional structure into graph neural networks' features and construct more expressive, non-isotropic messages. Specifically, given an input simplicial complex (e.g. generated by the cliques of a graph or the neighbors in a point cloud), we construct its local homology sheaf, which assigns to each node the vector space of its local homology. The intermediate features of our networks live in these vector spaces and we leverage the associated sheaf Laplacian to construct more complex linear messages between them. Moreover, we extend this approach by considering the persistent version of local homology associated with a weighted simplicial complex (e.g., built from pairwise distances of nodes embeddings). This i) solves the problem of the lack of a natural choice of basis for the local homology vector spaces and ii) makes the sheaf itself differentiable, which enables our models to directly optimize the topology of their intermediate features.

This paper aims at building the theoretical foundations for manifold learning algorithms in the space of absolutely continuous probability measures on a compact and convex subset of $\mathbb{R}^d$, metrized with the Wasserstein-2 distance $W$. We begin by introducing a natural construction of submanifolds $\Lambda$ of probability measures equipped with metric $W_\Lambda$, the geodesic restriction of $W$ to $\Lambda$. In contrast to other constructions, these submanifolds are not necessarily flat, but still allow for local linearizations in a similar fashion to Riemannian submanifolds of $\mathbb{R}^d$. We then show how the latent manifold structure of $(\Lambda,W_{\Lambda})$ can be learned from samples $\{\lambda_i\}_{i=1}^N$ of $\Lambda$ and pairwise extrinsic Wasserstein distances $W$ only. In particular, we show that the metric space $(\Lambda,W_{\Lambda})$ can be asymptotically recovered in the sense of Gromov--Wasserstein from a graph with nodes $\{\lambda_i\}_{i=1}^N$ and edge weights $W(\lambda_i,\lambda_j)$. In addition, we demonstrate how the tangent space at a sample $\lambda$ can be asymptotically recovered via spectral analysis of a suitable "covariance operator" using optimal transport maps from $\lambda$ to sufficiently close and diverse samples $\{\lambda_i\}_{i=1}^N$. The paper closes with some explicit constructions of submanifolds $\Lambda$ and numerical examples on the recovery of tangent spaces through spectral analysis.

Recent contrastive language image pre-training has led to learning highly transferable and robust image representations. However, adapting these models to video domain with minimal supervision remains an open problem. We explore a simple step in that direction, using language tied self-supervised learning to adapt an image CLIP model to the video domain. A backbone modified for temporal modeling is trained under self-distillation settings with train objectives operating in an action concept space. Feature vectors of various action concepts extracted from a language encoder using relevant textual prompts construct this space. A large language model aware of actions and their attributes generates the relevant textual prompts. We introduce two train objectives, concept distillation and concept alignment, that retain generality of original representations while enforcing relations between actions and their attributes. Our approach improves zero-shot and linear probing performance on three action recognition benchmarks.