Topological Quantum Field Theories and Operator Algebras

We review "quantum" invariants of closed oriented 3-dimensional manifolds arising from operator algebras.Comment: For proceedings of "International Workshop on Quantum Field Theory and Noncommutative Geometry", Sendai, November 200

Communication protocols and quantum error-correcting codes from the perspective of topological quantum field theory

Topological quantum field theories (TQFTs) provide a general, minimal-assumption language for describing quantum-state preparation and measurement. They therefore provide a general language in which to express multi-agent communication protocols, e.g. local operations, classical communication (LOCC) protocols. Here we construct LOCC protocols using TQFT, and show that LOCC protocols induce quantum error-correcting codes (QECCs) on the agent-environment boundary. Such QECCs can be regarded as implementing, or inducing the emergence of, spacetimes on such boundaries. We investigate this connection between inter-agent communication and spacetime using BF and Chern-Simons theories, and then using topological M-theory.Comment: 52 page

Geometric deep learning: going beyond Euclidean data

Many scientific fields study data with an underlying structure that is a non-Euclidean space. Some examples include social networks in computational social sciences, sensor networks in communications, functional networks in brain imaging, regulatory networks in genetics, and meshed surfaces in computer graphics. In many applications, such geometric data are large and complex (in the case of social networks, on the scale of billions), and are natural targets for machine learning techniques. In particular, we would like to use deep neural networks, which have recently proven to be powerful tools for a broad range of problems from computer vision, natural language processing, and audio analysis. However, these tools have been most successful on data with an underlying Euclidean or grid-like structure, and in cases where the invariances of these structures are built into networks used to model them. Geometric deep learning is an umbrella term for emerging techniques attempting to generalize (structured) deep neural models to non-Euclidean domains such as graphs and manifolds. The purpose of this paper is to overview different examples of geometric deep learning problems and present available solutions, key difficulties, applications, and future research directions in this nascent field

Deep neural networks architectures from the perspective of manifold learning

Despite significant advances in the field of deep learning in ap-plications to various areas, an explanation of the learning pro-cess of neural network models remains an important open ques-tion. The purpose of this paper is a comprehensive comparison and description of neural network architectures in terms of ge-ometry and topology. We focus on the internal representation of neural networks and on the dynamics of changes in the topology and geometry of a data manifold on different layers. In this paper, we use the concepts of topological data analysis (TDA) and persistent homological fractal dimension. We present a wide range of experiments with various datasets and configurations of convolutional neural network (CNNs) architectures and Transformers in CV and NLP tasks. Our work is a contribution to the development of the important field of explainable and interpretable AI within the framework of geometrical deep learning.Comment: 11 pages, 12 figures, PRAI2023. arXiv admin note: substantial text overlap with arXiv:2204.0862

Deep Learning Models to Study Sentence Comprehension in the Human Brain

Recent artificial neural networks that process natural language achieve unprecedented performance in tasks requiring sentence-level understanding. As such, they could be interesting models of the integration of linguistic information in the human brain. We review works that compare these artificial language models with human brain activity and we assess the extent to which this approach has improved our understanding of the neural processes involved in natural language comprehension. Two main results emerge. First, the neural representation of word meaning aligns with the context-dependent, dense word vectors used by the artificial neural networks. Second, the processing hierarchy that emerges within artificial neural networks broadly matches the brain, but is surprisingly inconsistent across studies. We discuss current challenges in establishing artificial neural networks as process models of natural language comprehension. We suggest exploiting the highly structured representational geometry of artificial neural networks when mapping representations to brain data

What do Deep Neural Networks Learn in Medical Images?

Deep learning is increasingly gaining rapid adoption in healthcare to help improve patient outcomes. This is more so in medical image analysis which requires extensive training to gain the requisite expertise to become a trusted practitioner. However, while deep learning techniques have continued to provide state-of-the-art predictive performance, one of the primary challenges that stands to hinder this progress in healthcare is the opaque nature of the inference mechanism of these models. So, attribution has a vital role in building confidence in stakeholders for the predictions made by deep learning models to inform clinical decisions. This work seeks to answer the question: what do deep neural network models learn in medical images? In that light, we present a novel attribution framework using adaptive path-based gradient integration techniques. Results show a promising direction of building trust in domain experts to improve healthcare outcomes by allowing them to understand the input-prediction correlative structures, discover new bio-markers, and reveal potential model biases

Surface Networks

We study data-driven representations for three-dimensional triangle meshes, which are one of the prevalent objects used to represent 3D geometry. Recent works have developed models that exploit the intrinsic geometry of manifolds and graphs, namely the Graph Neural Networks (GNNs) and its spectral variants, which learn from the local metric tensor via the Laplacian operator. Despite offering excellent sample complexity and built-in invariances, intrinsic geometry alone is invariant to isometric deformations, making it unsuitable for many applications. To overcome this limitation, we propose several upgrades to GNNs to leverage extrinsic differential geometry properties of three-dimensional surfaces, increasing its modeling power. In particular, we propose to exploit the Dirac operator, whose spectrum detects principal curvature directions --- this is in stark contrast with the classical Laplace operator, which directly measures mean curvature. We coin the resulting models \emph{Surface Networks (SN)}. We prove that these models define shape representations that are stable to deformation and to discretization, and we demonstrate the efficiency and versatility of SNs on two challenging tasks: temporal prediction of mesh deformations under non-linear dynamics and generative models using a variational autoencoder framework with encoders/decoders given by SNs

On convex conceptual regions in deep network representations

The current study of human-machine alignment aims at understanding the geometry of latent spaces and the correspondence to human representations. G\"ardenfors' conceptual spaces is a prominent framework for understanding human representations. Convexity of object regions in conceptual spaces is argued to promote generalizability, few-shot learning, and intersubject alignment. Based on these insights, we investigate the notion of convexity of concept regions in machine-learned latent spaces. We develop a set of tools for measuring convexity in sampled data and evaluate emergent convexity in layered representations of state-of-the-art deep networks. We show that convexity is robust to basic re-parametrization, hence, meaningful as a quality of machine-learned latent spaces. We find that approximate convexity is pervasive in neural representations in multiple application domains, including models of images, audio, human activity, text, and brain data. We measure convexity separately for labels (i.e., targets for fine-tuning) and other concepts. Generally, we observe that fine-tuning increases the convexity of label regions, while for more general concepts, it depends on the alignment of the concept with the fine-tuning objective. We find evidence that pre-training convexity of class label regions predicts subsequent fine-tuning performance