One-shot neural network pruning via spectral graph sparsification

Neural network pruning has gained significant attention for its potential to reduce computational resources required for training and inference. A large body of research has shown that networks can be pruned both after training and at initialisation, while maintaining competitive accuracy compared to dense networks. However, current methods rely on iteratively pruning or repairing the network to avoid over-pruning and layer collapse. Recent work has found that by treating neural networks as a sequence of bipartite graphs, pruning can be studied through the lens of spectral graph theory. Therefore, in this work, we propose a novel pruning approach using spectral sparsification, which aims to preserve meaningful properties of a dense graph with a sparse subgraph, by preserving the spectrum of the dense graph's adjacency matrix. We empirically validate and investigate our method, and show that one-shot pruning using spectral sparsification preserves performance at higher levels of sparsity compared to its one-shot counterparts. Additionally, we theoretically analyse our method with respect to local and global connectivity

Sparse random hypergraphs: Non-backtracking spectra and community detection

We consider the community detection problem in a sparse $q$-uniform hypergraph $G$, assuming that $G$ is generated according to the Hypergraph Stochastic Block Model (HSBM). We prove that a spectral method based on the non-backtracking operator for hypergraphs works with high probability down to the generalized Kesten-Stigum detection threshold conjectured by Angelini et al. (2015). We characterize the spectrum of the non-backtracking operator for the sparse HSBM and provide an efficient dimension reduction procedure using the Ihara-Bass formula for hypergraphs. As a result, community detection for the sparse HSBM on $n$ vertices can be reduced to an eigenvector problem of a $2n\times 2n$ non-normal matrix constructed from the adjacency matrix and the degree matrix of the hypergraph. To the best of our knowledge, this is the first provable and efficient spectral algorithm that achieves the conjectured threshold for HSBMs with $r$ blocks generated according to a general symmetric probability tensor.Comment: 61 pages, 8figures. To appear in Information and Inferenc

Spherical and Hyperbolic Toric Topology-Based Codes On Graph Embedding for Ising MRF Models: Classical and Quantum Topology Machine Learning

The paper introduces the application of information geometry to describe the ground states of Ising models by utilizing parity-check matrices of cyclic and quasi-cyclic codes on toric and spherical topologies. The approach establishes a connection between machine learning and error-correcting coding. This proposed approach has implications for the development of new embedding methods based on trapping sets. Statistical physics and number geometry applied for optimize error-correcting codes, leading to these embedding and sparse factorization methods. The paper establishes a direct connection between DNN architecture and error-correcting coding by demonstrating how state-of-the-art architectures (ChordMixer, Mega, Mega-chunk, CDIL, ...) from the long-range arena can be equivalent to of block and convolutional LDPC codes (Cage-graph, Repeat Accumulate). QC codes correspond to certain types of chemical elements, with the carbon element being represented by the mixed automorphism Shu-Lin-Fossorier QC-LDPC code. The connections between Belief Propagation and the Permanent, Bethe-Permanent, Nishimori Temperature, and Bethe-Hessian Matrix are elaborated upon in detail. The Quantum Approximate Optimization Algorithm (QAOA) used in the Sherrington-Kirkpatrick Ising model can be seen as analogous to the back-propagation loss function landscape in training DNNs. This similarity creates a comparable problem with TS pseudo-codeword, resembling the belief propagation method. Additionally, the layer depth in QAOA correlates to the number of decoding belief propagation iterations in the Wiberg decoding tree. Overall, this work has the potential to advance multiple fields, from Information Theory, DNN architecture design (sparse and structured prior graph topology), efficient hardware design for Quantum and Classical DPU/TPU (graph, quantize and shift register architect.) to Materials Science and beyond.Comment: 71 pages, 42 Figures, 1 Table, 1 Appendix. arXiv admin note: text overlap with arXiv:2109.08184 by other author

More than the sum of its parts – pattern mining, neural networks, and how they complement each other

In this thesis we explore pattern mining and deep learning. Often seen as orthogonal, we show that these fields complement each other and propose to combine them to gain from each other’s strengths. We, first, show how to efficiently discover succinct and non-redundant sets of patterns that provide insight into data beyond conjunctive statements. We leverage the interpretability of such patterns to unveil how and which information flows through neural networks, as well as what characterizes their decisions. Conversely, we show how to combine continuous optimization with pattern discovery, proposing a neural network that directly encodes discrete patterns, which allows us to apply pattern mining at a scale orders of magnitude larger than previously possible. Large neural networks are, however, exceedingly expensive to train for which ‘lottery tickets’ – small, well-trainable sub-networks in randomly initialized neural networks – offer a remedy. We identify theoretical limitations of strong tickets and overcome them by equipping these tickets with the property of universal approximation. To analyze whether limitations in ticket sparsity are algorithmic or fundamental, we propose a framework to plant and hide lottery tickets. With novel ticket benchmarks we then conclude that the limitation is likely algorithmic, encouraging further developments for which our framework offers means to measure progress.In dieser Arbeit befassen wir uns mit Mustersuche und Deep Learning. Oft als gegensätzlich betrachtet, verbinden wir diese Felder, um von den Stärken beider zu profitieren. Wir zeigen erst, wie man effizient prägnante Mengen von Mustern entdeckt, die Einsichten über konjunktive Aussagen hinaus geben. Wir nutzen dann die Interpretierbarkeit solcher Muster, um zu verstehen wie und welche Information durch neuronale Netze fließen und was ihre Entscheidungen charakterisiert. Umgekehrt verbinden wir kontinuierliche Optimierung mit Mustererkennung durch ein neuronales Netz welches diskrete Muster direkt abbildet, was Mustersuche in einigen Größenordnungen höher erlaubt als bisher möglich. Das Training großer neuronaler Netze ist jedoch extrem teuer, für das ’Lotterietickets’ – kleine, gut trainierbare Subnetzwerke in zufällig initialisierten neuronalen Netzen – eine Lösung bieten. Wir zeigen theoretische Einschränkungen von starken Tickets und wie man diese überwindet, indem man die Tickets mit der Eigenschaft der universalen Approximierung ausstattet. Um zu beantworten, ob Einschränkungen in Ticketgröße algorithmischer oder fundamentaler Natur sind, entwickeln wir ein Rahmenwerk zum Einbetten und Verstecken von Tickets, die als Modellfälle dienen. Basierend auf unseren Ergebnissen schließen wir, dass die Einschränkungen algorithmische Ursachen haben, was weitere Entwicklungen begünstigt, für die unser Rahmenwerk Fortschrittsevaluierungen ermöglicht